The adventure of physics - vol. VI : the strand model - a speculation on unification
Item
Title
The adventure of physics - vol. VI : the strand model - a speculation on unification
Creator
Christoph Schiller
Publisher
Motion Mountain
Description
This book is written for anybody who is intensely curious about nature and motion. Have you ever asked: Why do people, animals, things, images and empty space move? The answer leads to many adventures, and this book presents one of the best of them: the search for a precise, unified and final description of all motion.
The wish to describe all motion is a large endeavour. Fortunately, this large endeavour can be structured in the simple diagram shown in Figure 1. The final and unified descrip- tion of motion, the topic of this book, corresponds to the highest point in the diagram. Searching for this final and unified description is an old quest. In the following, I briefly summarize its history and then present an intriguing, though speculative solution to the riddle.
The search for the final, unified description of motion is a story of many surprises. For example, twentieth-century research has shown that there is a smallest distance in nature. Research has also shown that matter cannot be distinguished from empty space at those small distances. A last surprise dates from this century: particles and space are best described as made of strands, instead of little spheres or points. The present text explains how to reach these unexpected conclusions. In particular, quantum field theory, the standard model of particle physics, general relativity and cosmology are shown to follow from strands. The three gauge interactions, the three particle generations and the three dimensions of space turn out to be due to strands. In fact, all the open questions of twentieth-century physics about the foundations of motion, all the millennium issues, can be solved with the help of strands.
The strand model, as presented in this text, is an unexpected result from a threefold aim that I have pursued since 1990, in the five previous volumes of this series: to present the basics of motion in a way that is up to date, captivating and simple. In retrospect, the aim for maximum simplicity has been central in deducing this speculation. While the previous volumes introduced, in an entertaining way, the established parts of physics, this volume presents, in the same entertaining and playful way, a speculation about uni- fication. Nothing in this volume is established knowledge – yet. The text is the original presentation of the topic.
The search for a final theory is one of the great adventures of life: it leads to the limits of thought. The search overthrows our thinking habits about nature. A change in think- ing habits can produce fear, often hidden by anger. But by overcoming our fears we gain strength and serenity. Changing thinking habits thus requires courage, but it also pro- duces intense and beautiful emotions. Enjoy them!
The wish to describe all motion is a large endeavour. Fortunately, this large endeavour can be structured in the simple diagram shown in Figure 1. The final and unified descrip- tion of motion, the topic of this book, corresponds to the highest point in the diagram. Searching for this final and unified description is an old quest. In the following, I briefly summarize its history and then present an intriguing, though speculative solution to the riddle.
The search for the final, unified description of motion is a story of many surprises. For example, twentieth-century research has shown that there is a smallest distance in nature. Research has also shown that matter cannot be distinguished from empty space at those small distances. A last surprise dates from this century: particles and space are best described as made of strands, instead of little spheres or points. The present text explains how to reach these unexpected conclusions. In particular, quantum field theory, the standard model of particle physics, general relativity and cosmology are shown to follow from strands. The three gauge interactions, the three particle generations and the three dimensions of space turn out to be due to strands. In fact, all the open questions of twentieth-century physics about the foundations of motion, all the millennium issues, can be solved with the help of strands.
The strand model, as presented in this text, is an unexpected result from a threefold aim that I have pursued since 1990, in the five previous volumes of this series: to present the basics of motion in a way that is up to date, captivating and simple. In retrospect, the aim for maximum simplicity has been central in deducing this speculation. While the previous volumes introduced, in an entertaining way, the established parts of physics, this volume presents, in the same entertaining and playful way, a speculation about uni- fication. Nothing in this volume is established knowledge – yet. The text is the original presentation of the topic.
The search for a final theory is one of the great adventures of life: it leads to the limits of thought. The search overthrows our thinking habits about nature. A change in think- ing habits can produce fear, often hidden by anger. But by overcoming our fears we gain strength and serenity. Changing thinking habits thus requires courage, but it also pro- duces intense and beautiful emotions. Enjoy them!
Subject
Physics
Language
English
uri
https://open.umn.edu/opentextbooks/textbooks/the-adventure-of-physics-vol-vi-the-strand-model-a-speculation-on-unification
content
Christoph Schiller
MOTION MOUNTAIN
the adventure of physics – vol.vi
the strand model –
a speculation on unification
www.motionmountain.net
Christoph Schiller
Motion Mountain
The Adventure of Physics
Volume VI
The Strand Model –
A Speculation on Unification
Edition 30, available as free pdf
with films at www.motionmountain.net
Editio trigesima.
Proprietas scriptoris © Chrestophori Schiller
tertio anno Olympiadis trigesimae primae.
Omnia proprietatis iura reservantur et vindicantur.
Imitatio prohibita sine auctoris permissione.
Non licet pecuniam expetere pro aliqua, quae
partem horum verborum continet; liber
pro omnibus semper gratuitus erat et manet.
Thirtieth edition.
Copyright © 1990–2018 by Christoph Schiller,
from the third year of the 24th Olympiad
to the third year of the 31st Olympiad.
This pdf file is licensed under the Creative Commons
Attribution-Noncommercial-No Derivative Works 3.0 Germany
Licence, whose full text can be found on the website
creativecommons.org/licenses/by-nc-nd/3.0/de,
with the additional restriction that reproduction, distribution and use,
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commercial or not, is not allowed without the written consent of
the copyright owner. The pdf file was and remains free for everybody
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electronically, but only in unmodified form and only at no charge.
To Britta, Esther and Justus Aaron
τῷ ἐμοὶ δαὶμονι
Die Menschen stärken, die Sachen klären.
PR EFACE
T
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Christoph Schiller
Motion Mountain – The Adventure of Physics
his book is for anybody who is intensely curious about motion. Why and how do
hings, people, trees, stars, images or empty space move? The answer leads
o many adventures, and this book presents one of the best of them: the search
for a precise, unified and final description of all motion.
The aim to describe all motion – everyday, quantum and relativistic – implies a large
project. This project can be structured using the diagram shown in Figure 1, the so-called
Bronshtein cube. The previous volumes have covered all points in the cube – all domains
of motion – except the highest one. This remaining point contains the final and unified
description of all motion. The present volume briefly summarizes the history of this old
quest and then presents an intriguing, though speculative solution to the riddle.
The search for the final, unified description of motion is a story of many surprises.
First, twentieth-century research has shown that there is a smallest measurable distance
in nature, the Planck length. Then it appeared that matter cannot be distinguished from
empty space at those small distances. A last surprise dates from this century: particles and
space appear to be made of strands, instead of little spheres or points. The present text
explains how to reach these surprising conclusions. In particular, quantum field theory,
the standard model of particle physics, general relativity and cosmology are shown to
follow from strands. The three gauge interactions, the three particle generations and the
three dimensions of space turn out to be due to strands. In fact, all the open questions
of twentieth-century physics about the foundations of motion, including the origin of
colours and of the parameters of the standard model, appear to be answerable.
The strand model, as presented in this text, is an unexpected result from a threefold
aim that the author has pursued since 1990, in the five previous volumes of this series:
to present the basics of motion in a way that is up to date, captivating and simple. While
the previous volumes introduced the established parts of physics, this volume presents,
in the same captivating and playful way, a speculation about unification. Nothing in this
volume is established knowledge – yet. The text is the original presentation of the topic.
The aim for maximum simplicity has been central in deducing this speculation.
The search for a final theory is one of the great adventures of life: it leads to the limits
of thought. The search overthrows several of our thinking habits about nature. This can
produce fear, but by overcoming it we gain strength and serenity. Changing thinking
habits requires courage, but it produces intense and beautiful emotions. Enjoy them.
8
preface
Final, unified description of motion
Adventures: describing precisely all motion, understanding
the origin of colours, space -time and particles, enjoying
extreme thinking, calculating masses and couplings,
catching a further, tiny glimpse of bliss (vol. VI).
PHYSICS:
Describing motion with precision,
i.e., using the least action principle.
Special relativity
Adventures: light,
magnetism, length
c contraction, time
limits dilation and
G
h, e, k
fast E0 = mc2
limits
limit
motion (vol. II).
uniform
tiny
motion
motion
Quantum field theory
(the ‘standard model’)
Adventures: building
accelerators, understanding quarks, stars,
bombs and the basis of
life, matter & radiation
(vol. V).
Quantum theory
Adventures: biology,
birth, love, death,
chemistry, evolution,
enjoying colours, art,
paradoxes, medicine
and high-tech business
(vol. IV and vol. V).
Galilean physics, heat and electricity
The world of everyday motion: human scale, slow and weak.
Adventures: sport, music, sailing, cooking, describing
beauty and understanding its origin (vol. I);
using electricity, light and computers,
understanding the brain and people (vol. III).
F I G U R E 1 A complete map of physics, the science of motion, as first proposed by Matvei Bronshtein
(b. 1907 Vinnytsia, d. 1938 Leningrad). The map is of central importance in the present volume. The
Bronshtein cube starts at the bottom with everyday motion, and shows the connections to the fields of
modern physics. Each connection increases the precision of the description and is due to a limit to
motion that is taken into account. The limits are given for uniform motion by the gravitational constant
G, for fast motion by the speed of light c, and for tiny motion by the Planck constant h, the elementary
charge e and the Boltzmann constant k.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Classical gravity
Adventures:
climbing, skiing,
space travel,
the wonders of
astronomy and
geology (vol. I).
Quantum theory
with classical gravity
Adventures: bouncing
neutrons, understanding tree
growth (vol. V).
Motion Mountain – The Adventure of Physics
General relativity
Adventures: the
night sky, measuring curved and
wobbling space,
exploring black
holes and the
universe, space
and time (vol. II).
An arrow indicates an
increase in precision by
adding a motion limit.
preface
9
Using this bo ok
To get a quick overview, read the first chapter and continue with the summary sections
only. There are summaries at the end of each chapter. In addition, throughout the text,
⊳ Important ideas are marked with a triangle.
Feedback
Supp ort
Your donation to the minuscule, charitable, tax-exempt non-profit organisation that produces, translates and publishes this book series is welcome. For details, see the web page
www.motionmountain.net/donation.html. The German tax office checks the proper use
of your donation. If you want, your name will be included in the sponsor list. Thank you
in advance for your help, on behalf of all readers across the world. And now, enjoy the
reading.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Receiving an email from you at fb@motionmountain.net, either on how to improve the
text or on a solution for one of the prize challenges mentioned on www.motionmountain.
net/prizes.html, would be delightful. All feedback will be used to improve the next edition. For a particularly useful contribution you will be mentioned – if you want – in the
acknowledgements, receive a reward, or both.
Motion Mountain – The Adventure of Physics
Marginal notes refer to bibliographic references, to other pages or to challenge solutions. In the colour edition, such notes and also the pointers to footnotes and to other
websites are typeset in green. In the free pdf edition of this book, available at www.
motionmountain.net, all green pointers and links are clickable. The pdf edition also contains all films; they can be watched directly in Adobe Reader. Over time, links on the internet tend to disappear. Most links can be recovered via www.archive.org, which keeps
a copy of old internet pages.
Challenges are included regularly. Solutions and hints are given in the appendix. Challenges are classified as easy (e), standard student level (s), difficult (d) and research level
(r). Challenges for which no solution has yet been included in the book are marked (ny).
A paper edition of this book is available, either in colour or in black and white, from
www.amazon.com or www.createspace.com. So is a Kindle edition.
CONTENTS
17
1
From m illennium physics to unification
Against a final theory 20 • What went wrong in the past 22 • An encouraging
argument 22
Summary: how to find the final theory of motion
2
P hysics in lim it statements
Simplifying physics as much as possible
Everyday, or Galilean, physics in one statement 26 • Special relativity in one statement 27 • Quantum theory in one statement 28 • Thermodynamics in one statement 30 • General relativity in one statement 30 • Deducing general relativity 32
• Deducing universal gravitation 35 • The size of physical systems in general relativity 35 • A mechanical analogy for the maximum force 35
Planck limits for all physical observables
Physics, mathematics and simplicity 38 • Limits to space, time and size 38 • Mass
and energy limits 39 • Virtual particles – a new definition 40 • Curiosities and
fun challenges about Planck limits 40
Cosmological limits for all physical observables
Size and energy dependence 45 • Angular momentum and action 45 • Speed 46
• Force, power and luminosity 46 • The strange charm of the entropy bound 47
• Curiosities and fun challenges about system-dependent limits to observables 48
• Cosmology in one statement 50 • The cosmological limits to observables 51
• Minimum force 51 • Limits to measurement precision and their challenge to
thought 52 • No real numbers 52 • Vacuum and mass: two sides of the same
coin 52 • Measurement precision and the existence of sets 53
Summary on limits in nature
3
G eneral rel ativity v ersus q ua ntum theory
The contradictions 57 • The origin of the contradictions 58 • The domain of contradictions: Planck scales 59 • Resolving the contradictions 61 • The origin of
points 62
Summary on the clash between the two theories
4
D oes m at ter d iffer from vacuum?
Farewell to instants of time 64 • Farewell to points in space 66 • The generalized indeterminacy relation 68 • Farewell to space-time continuity 68
•
Farewell to dimensionality 71 • Farewell to the space-time manifold 72 • Farewell
to observables, symmetries and measurements 73 • Can space or space-time be a
lattice? 74 • A glimpse of quantum geometry 75 • Farewell to point particles 75
• Farewell to particle properties 77 • A mass limit for elementary particles 78 •
Farewell to massive particles – and to massless vacuum 79 • Matter and vacuum
are indistinguishable 81 • Curiosities and fun challenges on Planck scales 82 •
Common constituents 86
• Experimental predictions 87
Summary on particles and vacuum
5
What is the d ifference b et w een the univ erse a nd nothing?
Cosmological scales 90 • Maximum time 91 • Does the universe have a definite
24
26
26
36
45
54
56
63
64
88
90
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
P reface
Using this book 9 • Feedback 9 • Support 9
Motion Mountain – The Adventure of Physics
7
contents
age? 91 • How precise can age measurements be? 92 • Does time exist? 93 •
What is the error in the measurement of the age of the universe? 94 • Maximum
length 98 • Is the universe really a big place? 98 • The boundary of space – is
the sky a surface? 100 • Does the universe have initial conditions? 101 • Does the
universe contain particles and stars? 101 • Does the universe have mass? 102 • Do
symmetries exist in nature? 104 • Does the universe have a boundary? 104 • Is
the universe a set? 105 • Curiosities and fun challenges about the universe 107
• Hilbert’s sixth problem settled 108 • The perfect physics book 109 • Does the
universe make sense? 110
• Abandoning sets and discreteness eliminates contradictions 111 • Extremal scales and open questions in physics 111 • Is extremal
identity a principle of nature? 112
Summary on the universe
A physical aphorism 114
113
115
6
116
The sha pe of p oints – extension in nature
124
131
133
134
135
137
143
7
The basis of the stra nd m od el
Requirements for a final theory 146
• Introducing strands 150 • Events, processes, interactions and colours 152 • From strands to modern physics 152 • Vacuum 156 • Observable values and limits 157 • Particles and fields 159 • Curiosities and fun challenges about strands 159 • Do strands unify? – The millennium list
of open issues 161 • Are strands final? – On generalizations and modifications 163
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
127
Motion Mountain – The Adventure of Physics
The size and shape of elementary particles
Do boxes exist? 116 • Can the Greeks help? – The limitations of knives 116 • Are
cross sections finite? 117 • Can we take a photograph of a point? 118 • What is
the shape of an electron? 119 • Is the shape of an electron fixed? 120 • Summary
of the first argument for extension 121
The shape of points in vacuum
Measuring the void 123 • What is the maximum number of particles that fit inside
a piece of vacuum? 123 • Summary of the second argument for extension 123
The large, the small and their connection
Is small large? 124 • Unification and total symmetry 125 • Summary of the third
argument for extension 126
Does nature have parts?
Does the universe contain anything? 129 • An amoeba 129 • Summary of the
fourth argument for extension 130
The entropy of black holes
Summary of the fifth argument for extension 132
Exchanging space points or particles at Planck scales
Summary of the sixth argument for extension 134
The meaning of spin
Summary of the seventh argument for extension 135
Curiosities and fun challenges about extension
Gender preferences in physics 137
Checks of extension
Current research based on extended constituents 138 • Superstrings – extension
plus a web of dualities 139 • Why superstrings and supermembranes are so appealing 140 • Why the mathematics of superstrings is difficult 141 • Testing superstrings: couplings and masses 141 • The status of the superstring conjecture 142
Summary on extension in nature
121
146
11
12
contents
• Why strands? – Simplicity 165 • Why strands? – The fundamental circularity
of physics 166 • Funnels – an equivalent alternative to strands 169 • Knots and
the ends of strands 170
Summary on the fundamental principle – and on continuity
170
172
8
244
257
G auge interactions d ed uced from stra nds
Interactions and phase change 222 • Tail deformations versus core deformations 224
Electrodynamics and the first Reidemeister move
Strands and the twist, the first Reidemeister move 226 • Can photons decay, disappear or break up? 228 • Electric charge 228 • Challenge: What topological invariant is electric charge? 229 • Electric and magnetic fields and potentials 230 •
The Lagrangian of the electromagnetic field 231 • U(1) gauge invariance induced
by twists 232 • U(1) gauge interactions induced by twists 234 • The Lagrangian of
QED 234 • Feynman diagrams and renormalization 235 • The anomalous magnetic moment 238 • Maxwell’s equations 240 • Curiosities and fun challenges
about QED 242 • Summary on QED and experimental predictions 242
The weak nuclear interaction and the second Reidemeister move
Strands, pokes and SU(2) 245 • Weak charge and parity violation 246 • Weak bosons 248 • The Lagrangian of the unbroken SU(2) gauge interaction 249 • SU(2)
breaking 250 • Open issue: are the W and Z tangles correct? 251 • The electroweak
Lagrangian 252 • The weak Feynman diagrams 253 • Fun challenges and curiosities about the weak interaction 255 • Summary on the weak interaction and
experimental predictions 255
The strong nuclear interaction and the third Reidemeister move
Strands and the slide, the third Reidemeister move 257 • An introduction to
SU(3) 258 • From slides to SU(3) 261 • The strand model for gluons 266 • The
gluon Lagrangian 268 • Colour charge 269 • Properties of the strong interaction 271 • The Lagrangian of QCD 271 • Renormalization of the strong interaction 272 • Curiosities and fun challenges about SU(3) 272 • Summary on the
strong interaction and experimental predictions 273
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
226
9
Motion Mountain – The Adventure of Physics
Strands, vacuum and particles 172 • Rotation, spin 1/2 and the belt trick 174 • The
belt trick is not unique 178 • An aside: the belt trick saves lives 180 • Fermions
and spin 181 • Bosons and spin 183 • Spin and statistics 184 • Tangle functions:
blurred tangles 184 • Details on fluctuations and averages 186 • Tangle functions
are wave functions 187 • Deducing the Schrödinger equation from tangles 192 •
Mass from tangles 194 • Potentials 196 • Quantum interference from tangles 196
• Deducing the Pauli equation from tangles 198 • Rotating arrows, path integrals
and interference 199 • Measurements and wave function collapse 199 • Hidden
variables and the Kochen–Specker theorem 202 • Many-particle states and entanglement 202 • Mixed states 206 • The dimensionality of space-time 206 •
Operators and the Heisenberg picture 207 • Lagrangians and the principle of least
action 208 • Special relativity: the vacuum 209 • Special relativity: the invariant
limit speed 210 • Dirac’s equation deduced from tangles 212 • Visualizing spinors
and Dirac’s equation using tangles 215 • Quantum mechanics vs. quantum field
theory 217 • A flashback: settling three paradoxes of Galilean physics 218 • Fun
challenges about quantum theory 218
Summary on quantum theory of matter: experimental predictions
220
222
Q ua ntum theory of m at ter d ed uced from stra nd s
contents
13
279
311
10 G eneral rel ativity d ed uced from stra nds
Flat space, special relativity and its limitations 279 • Classical gravitation 280 •
Deducing universal gravitation from black hole properties 281 • Summary on universal gravitation from strands 283 • Curved space 283 • The structure of horizons and black holes 284 • Is there something behind a horizon? 285 • Energy of
black hole horizons 286 • The nature of black holes 286 • Entropy of vacuum and
matter 287 • Entropy of black holes deduced from the strand model 287 • Temperature, radiation and evaporation of black holes 290 • Black hole limits 290 •
Curvature around black holes 292 • The shape of non-rotating black holes 292 •
The field equations of general relativity 293 • Equations from no equation 294 •
The Hilbert action of general relativity 295 • Space-time foam 296 • Gravitons,
gravitational waves and their detection 296 • Open challenge: Improve the argument for the graviton tangle 297 • Other defects in vacuum 297 • The gravity
of superpositions 298 • Torsion, curiosities and challenges about quantum gravity 299 • Predictions of the strand model about gravity 302
Cosmology
The finiteness of the universe 304
• The big bang – without inflation 306 •
The cosmological constant 307 • The value of the matter density 309 • Open challenge: What are the effects of dark matter? 309 • The topology of the universe 310
• Predictions of the strand model about cosmology 310
Summary on millennium issues about relativity and cosmology
313
11 The pa rticle spectrum d ed uced from stra nd s
304
314
319
326
341
346
Particles and quantum numbers from tangles 313
Particles made of one strand
Unknotted curves 316 • Gauge bosons – and Reidemeister moves 317 • Open or
long knots 318 • Closed tangles: knots 318 • Summary on tangles made of one
strand 318
Particles made of two strands
Quarks 320 • Quark generations 323 • The graviton 324 • Glueballs 324 • The
mass gap problem and the Clay Mathematics Institute 325 • A puzzle 325 • Summary on two-stranded tangles 325
Particles made of three strands
Leptons 327 • Open issue: are the lepton tangles correct? 329 • The Higgs boson –
the mistaken section from 2009 329 • The Higgs boson – the corrected section of
2012 331 • 2012 predictions about the Higgs 333 • Quark-antiquark mesons 334
• Meson form factors 338 • Meson masses, excited mesons and quark confinement 338 • CP violation in mesons 339 • Other three-stranded tangles and glueballs 340 • Spin and three-stranded particles 340 • Summary on three-stranded
tangles 341
Tangles of four and more strands
Baryons 341 • Tetraquarks and exotic mesons 344 • Other tangles made of four
or more strands 345 • Summary on tangles made of four or more strands 345
Fun challenges and curiosities about particle tangles
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Summary and predictions about gauge interactions
Predicting the number of interactions in nature 275 • Unification of interactions 275 • No divergences 276 • Grand unification, supersymmetry and other
dimensions 276 • No new observable gravity effects in particle physics 277 • The
status of our quest 277
Motion Mountain – The Adventure of Physics
275
14
352
355
356
372
CPT invariance 351 • Motion through the vacuum – and the speed of light 351
Summary on millennium issues about particles and the vacuum
The omnipresent number 3 353 • Predictions about dark matter, the LHC and the
vacuum 353
395
398
13 E xperimental predictions of the stra nd m od el
Final summary about the millennium issues
400
400
414
416
416
14 The top of Motion Mounta in
Our path to the top
Everyday life: the rule of infinity 401 • Relativity and quantum theory: the absence
of infinity 401
• Unification: the absence of finitude 403
New sights
The beauty of strands 405 • Can the strand model be generalized? 405
• What
is nature? 406 • Quantum theory and the nature of matter and vacuum 407 •
Cosmology 407 • Musings about unification and strands 408 • The elimination
of induction 412
• What is still hidden? 413
A return path: je rêve, donc je suis
What is the origin of colours?
Summary: what is motion?
418
Postface
419
a Knot a nd ta ng le g eom etry
404
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
12 Pa rticle properties d ed uced from stra nds
The masses of the elementary particles
General properties of particle mass values 357 • Boson masses 358 • W/Z boson
mass ratio and mixing angle (in the 2016 tangle model) 359 • The g-factor of the
W boson 361 • The Higgs/Z boson mass ratio 361 • A first approximation for absolute boson mass values 361 • Quark mass ratios 362 • Lepton mass ratios 364
• On the absolute values of particle masses 366 • Analytical estimates for particle
masses 369 • Open issues about mass calculations 370 • On fine-tuning and naturalness 370 • Summary on elementary particle masses and millennium issues 371
Mixing angles
Quark mixing – the experimental data 372 • Quark mixing – explanations 373 •
A challenge 374 • CP violation in quarks 374 • Neutrino mixing 375 • CP violation in neutrinos 376 • Open challenge: calculate mixing angles and phases ab
initio 376 • Summary on mixing angles and the millennium list 377
Coupling constants and unification
Interaction strengths and strands 379 • Strands imply unification 380 • Calculating coupling constants 381 • First hint: the energy dependence of physical quantities 382 • Second hint: the running of the coupling constants at low energy 383
• Third hint: further predictions at low energy 383 • The running of the coupling constants up to Planck energy 384 • Limits for the fine structure constant do
not provide explanations 385 • Charge quantization and topological writhe 385
• Charge quantization and linking number 386 • How to calculate coupling constants 387 • Coupling constants in the strand model 388 • Deducing alpha from
precession 389 • Deducing the weak coupling 390 • Deducing the strong coupling 390 • Open challenge: calculate coupling constants with precision 391 • Electric dipole moments 391 • Five key challenges about coupling strengths 392 •
Summary on coupling constants 394
Motion Mountain – The Adventure of Physics
378
contents
422
C hallenge hints and solu tions
428
B ib liography
452
C redits
Acknowledgments 452 • Film credits 453 • Image credits 453
454
Na m e ind ex
461
Sub ject ind ex
The Strand Model –
A Speculation on
Unification
Where, through the combination of
quantum mechanics and general relativity,
the top of Motion Mountain is reached,
and it is discovered
that vacuum is indistinguishable from matter,
that there is little difference between the large and the small,
that nature can be described by strands,
that particles can be modelled as tangles,
that gauge interactions appear naturally,
that colours are due to strand twisting,
and that a complete description of motion is possible.
Chapter 1
FR OM MILLENNI UM PHYSICS TO
UNIFICAT ION
L
Ref. 2
2. In nature, there is an invariant maximum energy speed, the speed of light ?. This
invariant maximum implies special relativity. Among others, it implies that mass and
energy are equivalent, as is observed.
Page 30
3. In nature, there is an invariant highest momentum flow, the Planck force ?4 /4?. This
invariant maximum implies general relativity, as we will recall below. Among others,
** The photograph on page 16 shows an extremely distant, thus extremely young, part of the universe, with
its large number of galaxies in front of the black night sky (courtesy NASA).
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 1, Ref. 3
1. In nature, motion takes place in three dimensions of space and is described by
the least action principle. Action is a physical quantity that describes how much
change occurs in a process. The least action principle states: motion minimizes change.
Among others, the least change principle implies that motion is predictable, that energy is conserved and that growth and evolution are natural processes, as is observed.
Motion Mountain – The Adventure of Physics
ook at what happens around us. A child that smiles, a nightingale that sings, a
ily that opens: all move. Every shadow, even an immobile one, is due to moving
ight. Every mountain is kept in place by moving electrons. Every star owes its formation and its shine to motion of matter and radiation. Also the darkness of the night
sky** is due to motion: it results from the expansion of space. Finally, human creativity
is due to the motion of molecules, ions and electrons in the brain. Is there a common
language for these and all other observations of nature?
Is there a unified and precise way to describe all motion? How? Is everything that
moves, from people to planets, from light to empty space, made of the same constituents?
What is the origin of motion? Answering these questions is the topic of the present text.
Answering questions about motion with precision defines the subject of physics. Over
the centuries, researchers collected a huge number of precise observations about motion.
We now know how electric signals move in the brain, how insects fly, why colours vary,
how the stars formed, how life evolved, and much more. We use our knowledge about
motion to look into the human body and heal illnesses; we use our knowledge about
motion to build electronics, communicate over large distances, and work for peace; we
use our knowledge about motion to secure life against many of nature’s dangers, including droughts and storms. Physics, the science of motion, has shown time after time that
knowledge about motion is both useful and fascinating.
At the end of the last millennium, humans were able to describe all observed motion
with high precision. This description can be summarized in the following six statements.
18
1 from m illennium physics to unification
Ref. 2
general relativity implies that things fall and that empty space curves and moves, as
is observed.
Ref. 2
4. The evolution of the universe is described by the cosmological constant Λ. It determines the largest distance and the largest age that can presently be observed.
Ref. 4
5. In nature, there is a non-zero, invariant smallest change value, the quantum of action
ℏ. This invariant value implies quantum theory. Among others, it explains what life
and death are, why they exist and how we enjoy the world.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 5
These six statements, the millennium description of physics, describe everything known
about motion in the year 2000. (Actually, 2012 is a more precise, though less striking
date.) These statements describe the motion of people, animals, plants, objects, light,
radiation, stars, empty space and the universe. The six statements describe motion so
precisely that even today there is no difference between calculation and observation,
between theory and practice. This is an almost incredible result, the summary of the
efforts of tens of thousands of researchers during the past centuries.
However, a small set of observations does not yet follow from the six statements. A
famous example is the origin of colours. In nature, colours are consequences of the socalled fine structure constant, a mysterious constant of nature, abbreviated ?, whose value
is measured to be ? = 1/137.035 999 139(31). If ? had another value, all colours would
differ. And why are there three gauge interactions, twelve elementary fermions, thirteen
elementary bosons and three dimensions? What is the origin of particle masses? Why is
the standard model, the sixth statement above, so complicated? How is it related to the
five preceding statements?
A further unexplained observation is the nature of dark matter found around galaxies. We do not know yet what it is. Another unexplained process is the way thinking
forms in our brain. We do not know yet in detail how thinking follows from the above
six statements, though we do know that thinking is not in contrast with them. For this
reason, we will not explore the issue in the following. In the case of dark matter this is
not so clear: dark matter could even be in contrast with the millennium description of
motion.
Finally, why is there motion anyway? In short, even though the millennium description of physics is precise and successful, it is not complete. The list of all those fundamental issues about motion that are unexplained since the year 2000 make up only a
short table. We call them the millennium issues.
Motion Mountain – The Adventure of Physics
Ref. 4
6. In nature, matter and radiation consist of quantum particles. Matter consists of fermions: six quarks, three charged leptons, three neutrinos and their antiparticles. Radiation consists of bosons: the photon, three intermediate weak vector bosons and eight
gluons. In addition, the year 2012 finally brought the discovery of the Higgs boson,
which was already predicted in 1964. Fermions and bosons move and can transform
into each other. The transformations are described by the electromagnetic interaction, the weak nuclear interaction and the strong nuclear interaction. Together with
the masses, quantum numbers, mixing angles and couplings, these transformation
rules form the so-called standard model of particle physics. Among others, the standard model explains how lightning forms, why colours vary, and how the atoms in our
bodies came to be.
from m illennium physics to unification
19
TA B L E 1 The millennium list: everything the standard model and general relativity cannot explain; thus,
also the list of the only experimental data available to test the final, unified description of motion.
O b s e rva b l e P r o p e rt y u n e x p l a i n e d s i n c e t h e y e a r 2 0 0 0
Local quantities unexplained by the standard model: particle properties
Concepts unexplained by the standard model
SU(2)
SU(3)
Renorm. group
?? = 0
? = ∫? SM d?
the origin of the invariant Planck units of quantum field theory
the number of dimensions of physical space and time
the origin of Poincaré symmetry, i.e., of spin, position, energy, momentum
the origin and nature of wave functions
the origin of particle identity, i.e., of permutation symmetry
the origin of the gauge groups, in particular:
the origin of the electromagnetic gauge group, i.e., of the quantization of electric charge, of the vanishing of magnetic charge, and of minimal coupling
the origin of weak interaction gauge group, its breaking and P violation
the origin of strong interaction gauge group and its CP conservation
the origin of renormalization properties
the origin of the least action principle in quantum theory
the origin of the Lagrangian of the standard model of particle physics
Global quantities unexplained by general relativity and cosmology
0
1.2(1) ⋅ 1026 m
?de = Λ?4 /(8π?)
≈ 0.5 nJ/m3
(5 ± 4) ⋅ 1079
?dm
the observed flatness, i.e., vanishing curvature, of the universe
the distance of the horizon, i.e., the ‘size’ of the universe (if it makes sense)
the value and nature of the observed vacuum energy density, dark energy or
cosmological constant
the number of baryons in the universe (if it makes sense), i.e., the average
visible matter density in the universe
the density and nature of dark matter
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
?, ℏ, ?
3+1
SO(3,1)
Ψ
?(?)
Gauge symmetry
U(1)
Motion Mountain – The Adventure of Physics
? = 1/137.036(1) the low energy value of the electromagnetic coupling or fine structure constant
?w or ?w
the low energy value of the weak coupling constant or the value of the weak
mixing angle
?s
the value of the strong coupling constant at one specific energy value
?q
the values of the 6 quark masses
?l
the values of 6 lepton masses
?W
the value of the mass of the ? vector boson
?H
the value of the mass of the scalar Higgs boson
?12 , ?13 , ?23
the value of the three quark mixing angles
?
the value of the CP violating phase for quarks
?
?
?
?12
, ?13
, ?23
the value of the three neutrino mixing angles
?
? , ?1 , ?2
the value of the three CP violating phases for neutrinos
3⋅4
the number of fermion generations and of particles in each generation
J, P, C, etc.
the origin of all quantum numbers of each fermion and each boson
20
1 from m illennium physics to unification
TA B L E 1 (Continued) The millennium list: everything the standard model and general relativity cannot
explain; also the only experimental data available to test the final, unified description of motion.
O b s e rva b l e P r o p e rt y u n e x p l a i n e d s i n c e t h e y e a r 2 0 0 0
?0 (1, ..., c. 1090 )
the initial conditions for c. 1090 particle fields in the universe (if or as long as
they make sense), including the homogeneity and isotropy of matter distribution, and the density fluctuations at the origin of galaxies
Concepts unexplained by general relativity and cosmology
?, ?
R × S3
???
?? = 0
? = ∫? GR d?
the origin of the invariant Planck units of general relativity
the observed topology of the universe
the origin and nature of curvature, the metric and horizons
the origin of the least action principle in general relativity
the origin of the Lagrangian of general relativity
Against a final theory
— It is regularly claimed that a final theory cannot exist because nature is infinite and
mysteries will always remain. But this statement is wrong. First, nature is not infinite.
Second, even if it were infinite, knowing and describing everything would still be
possible. Third, even if knowing and describing everything would be impossible, and
if mysteries would remain, a final theory remains possible. A final theory is not useful
for every issue of everyday life, such as choosing your dish on a menu or your future
profession. A final theory is simply a full description of the foundations of motion:
the final theory just combines and explains particle physics and general relativity.
Page 168
— It is sometimes argued that a final theory cannot exist due to Gödel’s incompleteness
theorem or due to computational irreducibility. However, in such arguments, both
theorems are applied to domains were they are not valid. The reasoning is thus wrong.
— Some state that it is not clear whether a final theory exists at all. We all know from experience that this is wrong. The reason is simple: We are able to talk about everything.
In other words, all of us already have a ‘theory of everything’, or a final theory of
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
We know that a final theory exists: it is the theory that describes how to calculate the
fine structure constant ? = 1/137.036(1). The theory does the same for about two dozen
other constants, but ? is the most famous one. In other terms, the final theory is the
theory that explains all colours found in nature.
A fixed list of arguments are repeated regularly against the search for a final, unified
theory of motion. Reaching the final theory and enjoying the adventure is only possible
if these arguments are known – and then put gently aside.
Motion Mountain – The Adventure of Physics
The millennium list contains everything that particle physics and general relativity
cannot explain. In other words, the list contains every issue that was unexplained in the
domain of fundamental motion in the year 2000. The list is short, but it is not empty.
Every line in the millennium list asks for an explanation. The quest for unification – and
the topic of this text – is the quest for these explanations. We can thus say that a final
theory of motion is a theory that eliminates the millennium list of open issues.
from m illennium physics to unification
—
Ref. 6
Page 160
—
—
Ref. 8
—
Ref. 9
—
Ref. 10
These arguments show us that we can reach the final unified theory – which we symbolically place at the top of Motion Mountain – only if we are not burdened with ideological or emotional baggage. (We get rid of all baggage in the first six chapters of this
volume.) The goal we have set requires extreme thinking, i.e., thinking up to the limits.
After all, unification is the precise description of all motion. Therefore, unification is a
riddle. The search for unification is a pastime. Any riddle is best approached with the
light-heartedness of playing. Life is short: we should play whenever we can.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
—
nature. Also a physical theory is a way to talk about nature, and for the final theory
we only have to search for those concepts that enable us to talk about all of motion
with full precision. Because we are just looking for a way to talk, we know that the
final theory exists. And searching for it is fascinating and exciting, as everybody busy
with this adventure will confirm.
Some claim that the search for a final theory is a reductionist endeavour and cannot
lead to success, because reductionism is flawed. This claim is wrong on three counts.
First, it is not clear whether the search is a reductionist endeavour, as will become
clear later on. Second, there is no evidence that reductionism is flawed. Third, even
if it were, no reason not to pursue the quest would follow. The claim in fact invites
to search with a larger scope than was done in the past decades – an advice that will
turn out to be spot on.
Some argue that searching for a final theory makes no sense as long as the measurement problem of quantum theory is not solved, or consciousness is not understood,
or the origin of life is not understood. Now, the measurement problem is solved by
decoherence, and in order to combine particle physics with general relativity, understanding the details of consciousness or of the origin of life is not required. Neither
is understanding child education required – though this can help.
Some people claim that searching for a final theory is a sign of foolishness or a sin
of pride. Such defeatist or envious comments should simply be ignored. After all, the
quest is just the search for the solution to a riddle.
Some believe that understanding the final theory means to read the mind of god, or
to think like god, or to be like god. This is false, as any expert on god will confirm.
In fact, solving a riddle or reading a physics textbook does not transform people into
gods. This is unfortunate, as such an effect would provide excellent advertising.
Some fear that knowing the final theory yields immense power that harbours huge
dangers of misuse, in short, that knowing the final theory might change people into
devils. However, this fear is purely imaginary; it only describes the fantasies of the
person that is talking. Indeed, the millennium description of physics is already quite
near to the final theory, and nothing to be afraid of has happened. Sadly, another great
advertising opportunity is eliminated.
Some people object that various researchers in the past have thought to have found
the final theory, but were mistaken, and that many great minds tried to find a final
theory, but had no success. That is true. Some failed because they lacked the necessary
tools for a successful search, others because they lost contact with reality, and still
others because they were led astray by prejudices that limited their progress. We just
have to avoid these mistakes.
Motion Mountain – The Adventure of Physics
Ref. 7
Vol. IV, page 143
21
22
1 from m illennium physics to unification
What went wrong in the past
Vol. V, page 268
The twentieth century was the golden age of physics. Scholars searching for the final
theory explored candidates such as grand unified theories, supersymmetry and numerous other options. These candidates will be discussed later on; all were falsified by experiment. In other words, despite a large number of physicists working on the problem,
despite the availability of extensive experimental data, and despite several decades of research, no final theory was found. Why?
During the twentieth century, many successful descriptions of nature were deformed
into dogmatic beliefs about unification. Here are the main examples, with some of their
best known proponents:
An encouraging argument
Page 8
The Bronshtein cube in Figure 1 shows that physics started from the description of motion in everyday life. At the next level of precision, physics introduced the observed limits to motion and added the description of powerful, i.e., as uniform as possible motion (classical gravity), as fast as possible motion (special relativity), and as tiny as possible motion (quantum theory). At the following level of precision, physics achieved all
possible combinations of two of these motion types, by taking care of two motion lim-
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
All these beliefs appeared in the same way: first, some famous scholar – in fact, many
more than those mentioned – explained the idea that guided his discovery; then, he and
most other researchers started to believe the guiding idea more than the discovery itself. The most explored belief were those propagated by Salam and Weinberg: they –
unknowingly – set thousands of researchers on the wrong path for dozens of years. The
most detrimental has been the belief that unification is complicated and difficult: it kept
the smartest physicists from producing progress. In fact, all the mentioned beliefs can
be seen as special cases of the first one. And like the first belief, they are all, as we will
discover in the following, wrong.
Motion Mountain – The Adventure of Physics
— ‘Unification requires generalization of existing theories.’
— ‘Unification requires finding higher symmetries.’ (Werner Heisenberg)
— ‘Unification requires generalizing electroweak mixing to include the strong interaction.’ (Abdus Salam)
— ‘Unification requires extending the standard model of particle physics with supersymmetry.’ (Steven Weinberg)
— ‘Unification requires axiomatization.’ (David Hilbert)
— ‘Unification requires searching for beauty.’ (Paul Dirac)
— ‘Unification requires new quantum evolution equations.’ (Werner Heisenberg)
— ‘Unification requires new field equations of gravitation.’ (Albert Einstein)
— ‘Unification requires more dimensions of space.’ (Theodor Kaluza)
— ‘Unification requires topology change.’ (John Wheeler)
— ‘Unification is independent of Planck’s natural units.’
— ‘Unification requires using complicated mathematics and solving huge conceptual
difficulties.’ (Edward Witten)
— ‘Unification is only for a selected few.’
— ‘Unification is extremely useful, important and valuable.’
from m illennium physics to unification
23
its at the same time: fast and uniform motion (general relativity), fast and tiny motion
(quantum field theory), and tiny and uniform motion (quantum theory with gravity).
The only domain left over is the domain where motion is fast, tiny and as uniform as
possible at the same time. When this last domain is reached, the precise description of
all motion is completed.
But Figure 1 suggests even stronger statements. First of all, no domain of motion is
left: the figure covers all motion. Secondly, the final description appears when general
relativity, quantum field theory and quantum theory with gravity are combined. In other
words, the final theory appears when relativity and quantum theory and interactions are
all described together. But a third conclusion is especially important. Each of these three
fields can be deduced from the unified final theory by eliminating a limitation: either
that of tiny motion, that of straight motion, or that of fast motion. In other words:
⊳ Intermediate steps or theories do not exist before the final theory.
This is a strong statement. In the foundations of motion, apart from the final theory, no
further theory is missing. For example, the figure implies that there is no separate theory
of relativistic quantum gravity or no doubly special relativity.
In particular, Figure 1 implies that, conceptually, we are already close to the final theory. The figure suggests that there is no need for overly elaborate hypotheses or concepts
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Speaking even more bluntly, and against a common conviction of researchers in the field,
the figure suggests: The standard model follows from the final theory by eliminating gravity.
These connections eliminate many candidates for the unified final theory that were proposed in the research literature in the twentieth and twenty-first century. But more importantly, the connections leave open a range of possibilities – and interestingly enough,
this range is very narrow.
The figure allows stronger statements still. Progress towards the final theory is
achieved by taking limitations to motion into account. Whatever path we take from
everyday physics to the final theory, we must take into consideration all limits to motion.
The order can differ, but all limits have to be taken into account. Now, if any intermediate steps – due to additional motion limitations – between quantum field theory and the
final theory existed in the upper part of the figure, corresponding steps would have to
appear also in the lower part of the figure, between everyday physics and classical gravity. In the same way, if any intermediate limits between general relativity and the final
theory really existed, these limits would also have to appear between everyday motion
and quantum theory.
Experiments show that no intermediate steps or limits exist between everyday motion
and the next level of precision. Using the top-down symmetry of Figure 1, this implies:
Motion Mountain – The Adventure of Physics
⊳ General relativity follows from the final theory by eliminating the quantum
of action ℏ, i.e., taking the limit ℏ → 0.
⊳ Quantum field theory, including quantum electrodynamics, follows from
the final theory by eliminating ?, i.e., taking the limit ? → 0.
⊳ Quantum theory with gravity follows from the final theory by eliminating
the speed limit ?, i.e., taking the limit 1/? → 0.
24
1 from m illennium physics to unification
to reach the final theory:
⊳ We just have to add ? to the standard model or ℏ and ? to general relativity.
In short, the final, unified theory of motion cannot be far.
summary: how to find the final theory of motion
Page 147
At this point, after the first half of our adventure, we obtain a detailed requirement list
for the final theory. This list allows us to proceed rapidly towards our goal, without being
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
1. We first simplify modern physics. Twentieth century physics deduced several invariant properties of motion. These invariants, such as the speed of light or the quantum
of action, are called Planck units. The invariant Planck units allow motion to be measured. Above all, these invariants are also found to be limit values, valid for every example of motion.
2. Combining quantum theory and general relativity, we discover that at the Planck limits, the universe, space and particles are not described by points. We find that as long as
we use points to describe particles and space, and as long as we use sets and elements
to describe nature, a unified description of motion is impossible.
3. The combination of quantum theory and general relativity teaches us that space and
particles have common constituents.
4. By exploring black holes, spin, and the limits of quantum theory and gravity, we discover that the common constituents of space and particles are extended, without ends,
one-dimensional and fluctuating: the common constituents of space and particles are
fluctuating strands.
5. We discover that we cannot think or talk without continuity. We need a background
to describe nature. We conclude that to talk about motion, we have to combine continuity and non-continuity in an appropriate way. This is achieved by imagining that
fluctuating strands move in a continuous three-dimensional background.
Motion Mountain – The Adventure of Physics
We have a riddle to solve: we want to describe precisely all motion and discover its origin.
In order to achieve this, we need to find a final theory that solves and explains each open
issue given in the millennium list. This is our starting point.
We proceed in steps. We first simplify quantum theory and gravitation as much as
possible, we explore what happens when the two are combined, and we deduce the requirement list that any final theory must fulfil. Then we deduce the simplest possible
model that fulfils the requirements; we check the properties of the model against every
experiment performed so far and against every open issue from the millennium list. Discovering that there are no disagreements, no points left open and no possible alternatives,
we know that we have found the final theory. We thus end our adventure with a list of
testable predictions for the proposed model.
In short, three lists structure our quest for a final theory: the millennium list of open
issues, the list of requirements for the final theory, and the list of testable predictions. To
get from one list to the next, we proceed along the following legs.
sum m ary: how to find the fina l theory of m otion
25
led astray:
“
Es ist fast unmöglich, die Fackel der Wahrheit
durch ein Gedränge zu tragen, ohne jemandem
den Bart zu sengen.*
Georg Christoph Lichtenberg
”
* ‘It is almost impossible to carry the torch of truth through a crowd without scorching somebody’s beard.’
Georg Christoph Lichtenberg (b. 1742 Ober-Ramstadt, d. 1799 Göttingen) was a famous physicist and essayist.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
At the end of this path, we will thus have unravelled the mystery of motion. It is a truly
special adventure. But be warned: almost all of the story presented here is still speculative, and thus open to question. Everything presented in the following agrees with experiment. Nevertheless, with almost every sentence you will find at least one physicist or
philosopher who disagrees. That makes the adventure even more fascinating.
Motion Mountain – The Adventure of Physics
Page 395
6. We discover a simple fundamental principle that explains how the maximum speed
?, the minimum action ℏ, the maximum force ?4 /4? and the cosmological constant Λ
follow from strands. We also discover how to deduce quantum theory, relativity and
cosmology from strands.
7. We discover that strands naturally yield the existence of three spatial dimensions,
flat and curved space, black holes, the cosmological horizon, fermions and bosons.
We find that all known physical systems are made from strands. Also the process of
measurement and all properties of the background result from strands.
8. We discover that fermions emit and absorb bosons and that they do so with exactly
those properties that are observed for the electromagnetic, the weak and the strong
nuclear interaction. In short, the three known gauge interactions – and their parity
conservation or violation – follow from strands in a unique way. In addition, we discover that other interactions do not exist.
9. We discover that strands naturally yield the known elementary fermions and bosons,
grouped in three generations, and all their observed properties. Other elementary
particles do not exist. We thus recover the standard model of elementary particles.
10. We discover that the fundamental principle allows us to solve all the issues in the
millennium list, and that all properties deduced from strands agree with experiment.
In particular, the strand model allows us to calculate the fine structure constant and
the other gauge coupling strengths. An extensive list of testable predictions can be
given. These predictions will all be tested – by experiment or by calculation – in the
coming years.
11. We discover that motion is due to crossing switches of strands. Motion is an inescapable consequence of observation: motion is an experience that we make because we
are, like every observer, a small, approximate part of a large whole.
Chapter 2
PHYSICS IN LIMIT STAT EMENT S
T
At dinner parties, physicists are regularly asked to summarize physics in a few sentences.
It is useful to have a few simple statements ready to answer such a request. Such statements are not only useful to make other people think; they are also useful in our quest
for the final theory. Here they are.
Everyday, or Galilean, physics in one statement
Everyday motion is described by Galilean physics. It consists of only one statement:
⊳ Motion minimizes change.
In nature, change is measured by physical action ?. More precisely, change is measured
by the time-averaged difference between kinetic energy ? and potential energy ?. In
other words, motion obeys the so-called least action principle, written as
?? = 0 , where
? = ∫(? − ?) d? .
(1)
This statement determines the effort we need to move or throw stones, and explains why
cars need petrol and people need food. In other terms, nature is as lazy as possible. Or:
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
simplifying physics as much as possible
Motion Mountain – The Adventure of Physics
wentieth century physics deduced several invariant properties of motion.
hese invariants, such as the speed of light or the quantum of action, define
he so-called Planck units. The invariant Planck units are important for two reasons: first, they allow motion to be measured; second, the invariants are limit values. In
fact, the Planck units provide bounds for all observables.
The main lesson of modern physics is thus the following: When we simplify physics
as much as possible, we discover that nature limits the possibilities of motion. Such limits
lie at the origin of special relativity, of general relativity and of quantum theory. In fact,
we will see that nature limits every aspect of motion. Exploring the limits of motion will
allow us to deduce several astonishing conclusions. These conclusions contradict all that
we learned about nature so far.
sim plifying physics as m uch as p ossible
27
⊳ Nature is maximally efficient.
Vol. I, page 29
The efficiency or laziness of nature implies that motion is conserved, relative and predictable. In fact, the laziness of motion and nature is valid throughout modern physics,
for all observations, provided a few limit statements are added.
Special rel ativit y in one statement
Ref. 11
The step from everyday, or Galilean, physics to special relativity can be summarized in a
single limit statement on motion. It was popularized by Hendrik Antoon Lorentz:
⊳ There is a maximum energy speed value ? in nature.
Challenge 1 e
Vol. II, page 27
* A physical system is a region of space-time containing mass–energy, the location of which can be followed over time and which interacts incoherently with its environment. The speed of a physical system is
thus an energy speed. The definition of physical system excludes images, geometrical points or incomplete,
entangled situations.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Vol. II, page 100
All results peculiar to special relativity follow from this principle. A few well-known facts
set the framework for the discussion that follows. The speed ? is less than or equal to the
speed of light ? for all physical systems;* in particular, this speed limit is valid both for
composite systems and for elementary particles. No exception has ever been found. (Try
it.)
The energy speed limit is an invariant: the local energy speed limit is valid for all observers. In this context it is essential to note that any observer must be a physical system,
and must be close to the moving energy.
The speed limit ? is realized by massless particles and systems; in particular, it is realized by electromagnetic waves. For matter systems, the speed is always below ?.
Only a maximum energy speed ensures that cause and effect can be distinguished in
nature, or that sequences of observations can be defined. The opposite hypothesis, that
energy speeds greater than ? are possible, which implies the existence of so-called (real)
tachyons, has been explored and tested in great detail; it leads to numerous conflicts with
observations. Tachyons do not exist.
The maximum energy speed forces us to use the concept of space-time to describe
nature, because the existence of a maximum energy speed implies that space and time
mix. It also implies observer-dependent time and space coordinates, length contraction,
time dilation, mass–energy equivalence, horizons for accelerated observers, and all the
other effects that characterize special relativity. Only a maximum speed leads to the principle of maximum ageing that governs special relativity; and only this principle leads to
the principle of least action at low speeds. In addition, only with a finite speed limit is
it possible to define a unit of speed that is valid at all places and at all times. If there
were no global speed limit, there could be no natural measurement standard for speed,
independent of all interactions; speed would not then be a measurable quantity.
Motion Mountain – The Adventure of Physics
For all physical systems and all observers, the local energy speed ? is limited by the speed
of light ?:
? ⩽ ? = 3.0 ⋅ 108 m/s .
(2)
28
2 physics in lim it statem ents
Special relativity also limits the size of systems – whether composite or elementary.
Indeed, the limit speed implies that acceleration ? and size ? cannot be increased independently without bounds, because the two ends of a system must not interpenetrate.
The most important case concerns massive systems, for which we have
?⩽
?2
.
?
(3)
This size limit is induced by the speed of light ?; it is also valid for the displacement ? of
a system, if the acceleration measured by an external observer is used. Finally, the speed
limit implies a relativistic ‘indeterminacy relation’
Δ? Δ? ⩽ ?2
for the length and acceleration indeterminacies. You may wish to take a minute to deduce
this relation from the time–frequency indeterminacy. All this is standard knowledge.
Q uantum theory in one statement
Ref. 12
The difference between Galilean physics and quantum theory can be summarized in a
single statement on motion, due to Niels Bohr:
⊳ There is a minimum action value ℏ in nature.
For all physical systems and all observers, the action ? obeys
? ⩾ ℏ = 1.1 ⋅ 10−34 Js .
Challenge 3 e
Challenge 4 e
(5)
The Planck constant ℏ is the smallest observable action value, and the smallest observable change of angular momentum. The action limit is valid for all systems, thus both
for composite and elementary systems. No exception has ever been found. (Try it.) The
principle contains all of quantum theory. We call it the principle of non-zero action, in
order to avoid confusion with the principle of least action.
The non-zero action limit ℏ is an invariant: it is valid with the same numerical value
for all observers. Again, any such observer must be a physical system.
The action limit is realized by many physical processes, from the absorption of light
to the flip of a spin 1/2 particle. More precisely, the action limit is realized by microscopic
systems where the process involves a single particle.
The non-zero action limit is stated less frequently than the speed limit. It starts from
the usual definition of the action, ? = ∫(? − ?) d?, and states that between two observations performed at times ? and ? + Δ?, even if the evolution of a system is not known, the
measured action is at least ℏ. Since physical action measures the change in the state of a
physical system, there is always a minimum change of state between two different observations of a system.* The non-zero action limit expresses the fundamental fuzziness of
* For systems that seem constant in time, such as a spinning particle or a system showing the quantum Zeno
effect, finding this minimum change is tricky. Enjoy the challenge.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Vol. IV, page 15
Motion Mountain – The Adventure of Physics
Challenge 2 s
(4)
sim plifying physics as m uch as p ossible
Ref. 13
?⩾
Challenge 5 e
Vol. IV, page 24
ℏ
.
??
(6)
In other words, the (reduced) Compton wavelength of quantum theory appears as the
lower limit on the displacement of a system, whenever gravity plays no role. Since this
quantum displacement limit also applies to elementary systems, it also applies to the size
of a composite system. However, for the same reason, this size limit is not valid for the
sizes of elementary particles.
The limit on action also implies Heisenberg’s well-known indeterminacy relation for
the displacement ? and momentum ? of physical systems:
Δ? Δ? ⩾
ℏ
.
2
(7)
This relation is valid for both massless and massive systems. All this is textbook knowledge.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
nature at a microscopic scale.
It can easily be checked that no observation – whether of photons, electrons or macroscopic systems – gives a smaller action than the value ℏ. The non-zero action limit has
been verified for fermions, bosons, laser beams, matter systems, and for any combination
of these. The opposite hypothesis, implying the existence of arbitrary small change, has
been explored in detail: Einstein’s long discussion with Bohr, for example, can be seen as
a repeated attempt by Einstein to find experiments that would make it possible to measure arbitrarily small changes or action values in nature. In every case, Bohr found that
this could not be achieved. All subsequent attempts were equally unsuccessful.
The principle of non-zero action can be used to deduce the indeterminacy relation, the
tunnelling effect, entanglement, permutation symmetry, the appearance of probabilities
in quantum theory, the information-theoretic formulation of quantum theory, and the
existence of elementary particle reactions. Whenever we try to overcome the smallest
action value, the experimental outcome is probabilistic. The minimum action value also
implies that in quantum theory, the three concepts of state, measurement operation, and
measurement result need to be distinguished from each other; this is done by means
of a so-called Hilbert space. Finally, the non-zero action limit is also the foundation of
Einstein–Brillouin–Keller quantization.
The existence of a non-zero action limit has been known from the very beginning
of quantum theory. It is at the basis of – and completely equivalent to – all the usual
formulations of quantum theory, including the many-path and the information-theoretic
formulations.
We also note that only a non-zero action limit makes it possible to define a unit of
action. If there were no action limit, there could be no natural measurement standard
for action: action would not then be a measurable quantity.
The upper bounds for speed and for action for any physical system, ? ⩽ ? and
? ⩽ ?? ⩽ ???, when combined with the quantum of action, imply a limit on the
displacement ? of a system between any two observations:
Motion Mountain – The Adventure of Physics
Ref. 14
29
30
2 physics in lim it statem ents
Thermodynamics in one statement
Thermodynamics can also be summarized in a single statement about motion:
⊳ There is a smallest entropy value ? in nature.
Written symbolically,
Challenge 6 e
Ref. 15
? ⩾ ? = 1.3 ⋅ 10−23 J/K .
1
?
Δ? ⩾ .
?
2
(9)
General rel ativit y in one statement
Page 54
This text can be enjoyed most when a compact and unconventional description of general
relativity is used; it is presented in the following. However, the conclusions do not depend
on this description; the results are also valid if the usual approach to general relativity is
used; this will be shown later on.
The most compact description summarizes the step from universal gravity to general
relativity in a single statement on motion:
⊳ There are maximum force and power values in nature.
For all physical systems and all observers, force ? and power ? are limited by
?⩽
Challenge 7 e
?4
= 3.0 ⋅ 1043 N
4?
and ? ⩽
?5
= 9.1 ⋅ 1051 W .
4?
(10)
No exception has ever been found. (Try it.) These limit statements contain both the speed
of light ? and the gravitational constant ?; they thus qualify as statements about relativistic gravitation. Before we deduce general relativity, let us explore these limits.
The numerical values of the limits are huge. The maximum power corresponds to
converting 50 solar masses into massless radiation within 1 millisecond. And applying
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
This relation was first given by Bohr and then discussed by Heisenberg and many others.
Motion Mountain – The Adventure of Physics
The entropy ? is limited by the Boltzmann constant ?. No exception has ever been found.
(Try it.) This result is almost 100 years old; it was stated most clearly by Leo Szilard. All
of thermodynamics can be deduced from this relation, together with the quantum of
action.
The entropy limit is an invariant: it is valid for all observers. Again, any observer must
be a physical system.
The entropy limit is realized only by physical systems made of a single particle. In
other words, the entropy limit is again realized only by microscopic systems. Therefore
the entropy limit provides the same length limit for physical systems as the action limit.
Like the other limit statements we have examined, the entropy limit can also be
phrased as a indeterminacy relation between temperature ? and energy ?:
Δ
Ref. 16
(8)
sim plifying physics as m uch as p ossible
Ref. 17
Vol. I, page 230
31
the maximum force value along a distance ? costs as much energy as a black hole of
diameter ?.
Force is change of momentum; power is change of energy. Since momentum and energy are conserved, force and power are the flow of momentum and energy through a
surface. Force and power, like electric current, describe the change in time of conserved
quantity. For electric current, the conserved quantity is charge, for force, it is momentum,
for power, it is energy. In other words, like current, also force is a flow across a surface.
This is a simple consequence of the continuity equation. Therefore, every discussion of
maximum force implies a clarification of the underlying surface.
Both the force and the power limits state that the flow of momentum or of energy
through any physical surface – a surface to which an observed can be attached at every
one of its points – of any size, for any observer, in any coordinate system, never exceeds
the limit value. In particular:
Vol. II, page 110
Challenge 8 e
Challenge 9 e
Vol. II, page 107
Vol. II, page 83
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 18
In all other situations, the observed values are strictly smaller than the maximum values.
The force and power limit values are invariants: they are valid for all observers and
for all interactions. Again, any observer must be a physical system and it must be located
on or near the surface used to define the flow of momentum or energy.
The value of the force limit is the energy of a Schwarzschild black hole divided by its
diameter; here the ‘diameter’ is defined as the circumference divided by π. The power
limit is realized when such a black hole is radiated away in the time that light takes to
travel along a length corresponding to the diameter.
An object of mass ? that has the size of its own Schwarzschild radius 2??/?2 is
called a black hole, because according to general relativity, no signals and no light from
inside the Schwarzschild radius can reach the outside world. In this text, black holes
are usually non-rotating and usually uncharged; in this case, the terms ‘black hole’ and
‘Schwarzschild black hole’ are synonymous.
The value of the maximum force, as well as being the mass–energy of a black hole
divided by its diameter, is also the surface gravity of a black hole times its mass. Thus the
force limit means that no physical system of a given mass can be concentrated in a region
of space-time smaller than a (non-rotating) black hole of that mass. (This is the so-called
hoop conjecture.) In fact, the mass–energy concentration limit can easily be transformed
algebraically into the force limit: they are equivalent.
It is easily checked that the maximum force limit is valid for all systems observed
in nature, whether they are microscopic, macroscopic or astrophysical. Neither the
‘gravitational force’ (as long as it is operationally defined) nor the electromagnetic or
nuclear interactions are ever found to exceed this limit.
But is it possible to imagine a system that exceeds the force limit? An extensive discussion shows that this is impossible. For example, the force limit cannot be overcome with
Lorentz boosts. We might think that a boost can be chosen in such a way that a 3-force
value ? in one frame is transformed into any desired value ?? in another, boosted frame.
This thought turns out to be wrong. In relativity, 3-force cannot be increased beyond all
bounds using boosts. In all reference frames, the measured 3-force can never exceed the
Motion Mountain – The Adventure of Physics
⊳ The force limit is only realized at horizons. The power limit is only realized
with horizons.
32
Vol. II, page 107
proper force, i.e., the 3-force value measured in the comoving frame.
Also changing to an accelerated frame does not help to overcome the force limit, because for high accelerations ?, horizons appear at distance ?2 /?, and a mass ? has a
minimum diameter given by ? ⩾ 4??/?2 .
In fact, the force and power limits cannot be exceeded in any thought experiment,
as long as the sizes of observers or of test masses are taken into account. All apparent
exceptions or paradoxes assume the existence of point particles or point-like observers;
these, however, are not physical: they do not exist in general relativity.
Fortunately for us, nearby black holes or horizons are rare. Unfortunately, this means
that neither the force limit nor the power limit are realized in any physical system at
hand, neither at everyday length scales, nor in the microscopic world, nor in astrophysical systems. Even though the force and power limits have never been exceeded, a direct
experimental confirmation of the limits will take some time.
The formulation of general relativity as a consequence of a maximum force is not
common; in fact, it seems that it was only discovered 80 years after the theory of general
relativity had first been proposed.
Deducing general rel ativit y*
Ref. 20
?=
?
.
?
(11)
Since we are at a horizon, we need to insert the maximum possible values. In terms of
* This section can be skipped at first reading.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
In order to elevate the force or power limit to a principle of nature, we have to show that,
just as special relativity follows from the maximum speed, so general relativity follows
from the maximum force.
The maximum force and the maximum power are only realized at horizons. Horizons
are regions of space-time where the curvature is so high that it limits the possibility of
observation. The name ‘horizon’ is due to an analogy with the usual horizon of everyday
life, which also limits the distance to which we can see. However, in general relativity
horizons are surfaces, not lines. In fact, we can define the concept of horizon in general
relativity as a region of maximum force; it is then easy to prove that a horizon is always
a two-dimensional surface, and that it is essentially black (except for quantum effects).
The connection between horizons and the maximum force or power allows us to deduce the field equations in a simple way. First, there is always a flow of energy at a horizon. Horizons cannot be planes, since an infinitely extended plane would imply an infinite energy flow. To characterize the finite extension of a given horizon, we use its radius
? and its total area ?.
The energy flow across a horizon is characterized by an energy ? and a proper length
? of the energy pulse. When such an energy pulse flows perpendicularly across a horizon,
the momentum change d?/d? = ? is given by
Motion Mountain – The Adventure of Physics
Ref. 19
2 physics in lim it statem ents
sim plifying physics as m uch as p ossible
33
the horizon area ? and radius ?, we can rewrite the limit case as
?4
1
?
= 4π?2
4? ?
?
(12)
where we have introduced the maximum force and the maximum possible area 4π?2 of
a horizon of (maximum local) radius ?. The ratio ?/? is the energy per unit area flowing
across the horizon.
Horizons are often characterized by the so-called surface gravity ? instead of the radius
?. In the limit case, two are related by ? = ?2 /2?. This leads to
?=
Ref. 21
1 2
? ?? .
4π?
(13)
?? =
Ref. 22
?2
? ?? .
8π?
(15)
In this form, the relation between energy and area can be applied to general horizons,
including those that are irregularly curved or time-dependent.*
In a well-known paper, Jacobson has given a beautiful proof of a simple connection:
if energy flow is proportional to horizon area for all observers and all horizons, and if
the proportionality constant is the correct one, then general relativity follows. To see
the connection to general relativity, we generalize the horizon relation (15) to general
coordinate systems and general directions of energy flow.
* The horizon relation (15) is well known, though with different names for the observables. Since no communication is possible across a horizon, the detailed fate of energy flowing across a horizon is also unknown.
Energy whose detailed fate is unknown is often called heat, and abbreviated ?. The horizon relation (15)
therefore states that the heat flowing through a horizon is proportional to the horizon area. When quantum
theory is introduced into the discussion, the area of a horizon can be called ‘entropy’ ? and its surface
gravity can be called ‘temperature’ ?; relation (15) can then be rewritten as ?? = ???. However, this translation of relation (15), which requires the quantum of action, is unnecessary here. We only cite it to show
the relation between horizon behaviour and quantum aspects of gravity.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
This horizon relation makes three points. First, the energy flowing across a horizon is limited. Secondly, this energy is proportional to the area of the horizon. Thirdly, the energy
flow is proportional to the surface gravity. These three points are fundamental, and characteristic, statements of general relativity. (We also note that due to the limit property
of horizons, the energy flow towards the horizon just outside it, the energy flow across a
horizon, and the energy inside a horizon are all the same.)
Taking differentials, the horizon relation can be rewritten as
Motion Mountain – The Adventure of Physics
Special relativity shows that at horizons the product ?? of proper length and acceleration is limited by the value ?2 /2. This leads to the central relation for the energy flow at
horizons:
?2
?=
?? .
(14)
8π?
34
2 physics in lim it statem ents
The proof uses tensor notation. We introduce the general surface element dΣ and the
local boost Killing vector field ? that generates the horizon (with suitable norm). We then
rewrite the left-hand side of relation (15) as
?? = ∫ ??? ?? dΣ? ,
(16)
where ??? is the energy–momentum tensor. This is valid in arbitrary coordinate systems and for arbitrary energy flow directions. Jacobson’s main result is that the righthand side of the horizon relation (15) can be rewritten, using the (purely geometric)
Raychaudhuri equation, as
? ?? = ?2 ∫ ??? ?? dΣ? ,
(17)
(19)
Motion Mountain – The Adventure of Physics
where Λ is a constant of integration whose value is not determined by the problem. These
are the full field equations of general relativity, including the cosmological constant Λ.
This value of this constant remains undetermined, though.
The field equations are thus shown to be valid at horizons. Now, it is possible, by
choosing a suitable coordinate transformation, to position a horizon at any desired
space-time event. To achieve this, simply change to the frame of an observer accelerating away from that point at the correct distance, as explained in the volume on relativity.
Therefore, because a horizon can be positioned anywhere at any time, the field equations
must be valid over the whole of space-time.
Since it is possible to have a horizon at every event in space-time, there is the same
maximum possible force (or power) at every event in nature. This maximum force (or
power) is thus a constant of nature.
In other words, the field equations of general relativity are a direct consequence of
the limited energy flow at horizons, which in turn is due to the existence of a maximum
force or power. We can thus speak of the maximum force principle. Conversely, the field
equations imply maximum force and power. Maximum force and general relativity are
thus equivalent.
By the way, modern scholars often state that general relativity and gravity follow from
the existence of a minimum measurable length. The connection was already stated by
Sakharov in 1969. This connection is correct, but unnecessarily restrictive. The maximum
force, which is implicit in the minimal length, is sufficient to imply gravity. Quantum
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
where ??? is the Ricci tensor describing space-time curvature.
Combining these two steps, we find that the energy–area horizon relation (15) can be
rewritten as
?4
∫ ??? ?? dΣ? =
(18)
∫ ??? ?? dΣ? .
8π?
Jacobson shows that this equation, together with local conservation of energy (i.e., vanishing divergence of the energy–momentum tensor), can only be satisfied if
??? =
Vol. II, page 98
Ref. 23
?4
1
(? − ( ? + Λ) ??? ) ,
8π? ??
2
sim plifying physics as m uch as p ossible
35
theory – or ℏ – is (obviously) not necessary to deduce gravity.
Deducing universal gravitation
Challenge 10 e
Challenge 11 e
The size of physical systems in general rel ativit y
General relativity, like the other theories of modern physics, implies a limit on the size ?
of systems. There is a limit to the amount of matter that can be concentrated into a small
volume:
4??
?⩾ 2 .
(20)
?
Ref. 24
Ref. 25
Experimental data are available only for composite systems; all known systems
comply with it. For example, the latest measurements for the Sun give ??⊙ /?3 =
4.925 490 947(1) μs; the error in ? is thus much smaller than the (scaled) error in its
radius, which is known with much smaller precision. The ‘indeterminacy relation’ (21)
is not as well known as that from quantum theory. In fact, tests of it – for example with
binary pulsars – may distinguish general relativity from competing theories. We cannot
yet say whether this inequality also holds for elementary particles.
A mechanical analo gy for the maximum force
The maximum force is central to the theory of general relativity. Indeed, its value (adorned with a factor 2π) appears in the field equations. The importance of the maximum
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 300
The size limit is only realized for black holes, those well-known systems which swallow
everything that is thrown into them. The size limit is fully equivalent to the force limit.
(Also the hoop conjecture is understood to be true.) All composite systems in nature
comply with the lower size limit. Whether elementary particles fulfil or even match this
limit remains open at this point. More about this issue below.
General relativity also implies an ‘indeterminacy relation’ for the measurement errors
of size ? and energy ? of systems:
Δ?
?4
⩽
.
(21)
Δ?
4?
Motion Mountain – The Adventure of Physics
Page 281
Universal gravitation follows from the force limit in the case where both forces and
speeds are much smaller than the maximum values. The first condition implies
√4??? ≪ ?2 , the second ? ≪ ? and ?? ≪ ?2 . Let us apply this to a specific case.
Consider a satellite circling a central mass ? at distance ? with acceleration ?. This
system, with length ? = 2?, has only one characteristic speed. Whenever this speed ? is
much smaller than ?, ?2 must be proportional both to the squared speed calculated by
?? = 2?? and to the squared speed calculated from √4??? . Taken together, these two
conditions imply that ? = ???/?2 , where ? is a numerical factor. A quick check, for
example using the observed escape velocity values, shows that ? = 1.
Forces and speeds much smaller than the limit values thus imply that gravity changes
with the inverse square of distance. In other words, nature’s limit on force implies universal gravitation. Other deductions of universal gravity from limit quantities are given
later.
36
2 physics in lim it statem ents
pl anck limits for all physical observables
The existence of a maximum force in nature is equivalent to general relativity. As a result,
a large part of modern physics can be summarized in four simple and fundamental limit
statements on motion:
? ⩾ℏ
? ⩾?
? ⩽?
General relativity follows from the force limit:
Challenge 12 e
? ⩽
?4
.
4?
(22)
These (corrected) Planck limits are valid for all physical systems, whether composite or
elementary, and for all observers. Note that the limit quantities of quantum theory, thermodynamics, special and general relativity can also be seen as the right-hand sides of the
respective indeterminacy relations. Indeed, the set (4, 7, 9, 21) of indeterminacy relations
is fully equivalent to the four limit statements (22).
We note that the different dimensions of the four fundamental limits (22) in nature
mean that the four limits are independent. For example, quantum effects cannot be used
to overcome the force limit; similarly, the power limit cannot be used to overcome the
speed limit. There are thus four independent limits on motion in nature.
By combining the four fundamental limits, we can obtain limits on a number of physical observables. The following limits are valid generally, for both composite and elementary systems:
time interval:
?⩾√
4?ℏ
?5
=
1.1 ⋅ 10−43 s
(23)
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Quantum theory follows from the action limit:
Thermodynamics follows from the entropy limit:
Special relativity follows from the speed limit:
Motion Mountain – The Adventure of Physics
force becomes clearer when we return to our old image of space-time as a deformable
mattress. Like any material body, a mattress is described by a material constant that
relates the deformation values to the values of applied energy. Similarly, a mattress, like
any material, is described by the maximum stress it can bear before it breaks. These
two values describe all materials, from crystals to mattresses. In fact, for perfect crystals (without dislocations), these two material constants are the same.
Empty space somehow behaves like a perfect crystal, or a perfect mattress: it has a
deformation-energy constant that is equal to the maximum force that can be applied
to it. The maximum force describes the elasticity of space-time. The high value of the
maximum force tells us that it is difficult to bend space.
Now, materials are not homogeneous: crystals are made up of atoms, and mattresses
are made up of foam bubbles. What is the corresponding structure of space-time? This is
a central question in the rest of our adventure. One thing is sure: unlike crystals, vacuum
has no preferred directions. We now take a first step towards answering the question of
the structure of space-time and particles by putting together all the limits found so far.
pl a nck lim its for a ll physical ob servables
time-distance product:
?? ⩾
4?ℏ
?4
?7
4?ℏ
acceleration:
?⩽√
angular frequency:
? ⩽ 2π√
37
=
3.5 ⋅ 10−78 ms
(24)
=
2.8 ⋅ 1051 m/s2
(25)
5.8 ⋅ 1043 /s .
(26)
?5
=
2?ℏ
Adding the knowledge that space and time can mix, we get
distance:
area:
curvature:
mass density:
3.2 ⋅ 10−35 m
(27)
1.0 ⋅ 10−69 m2
(28)
3.4 ⋅ 10−104 m3
(29)
1.0 ⋅ 1069 /m2
(30)
3.2 ⋅ 1095 kg/m3 .
(31)
⊳ Every natural unit or (corrected) Planck unit is the limit value of the corresponding physical observable.
Page 58
Ref. 26
Ref. 27
Most of these limit statements are found scattered throughout the research literature,
though the numerical factors often differ. Each limit has attracted a string of publications.
The existence of a smallest measurable distance and time interval of the order of the
Planck values is discussed in all approaches to quantum gravity. The maximum curvature
has been studied in quantum gravity; it has important consequences for the ‘beginning’
of the universe, where it excludes any infinitely large or small observable. The maximum
mass density appears regularly in discussions on the energy of the vacuum.
In the following, we often call the collection of Planck limits the Planck scales. We will
discover shortly that at Planck scales, nature differs in many ways from what we are used
to at everyday scales.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Of course, speed, action, angular momentum, entropy, power and force are also limited,
as already stated. The limit values are deduced from the commonly used Planck values
simply by substituting 4? for ?. These limit values are the true natural units of nature. In
fact, the ideal case would be to redefine the usual Planck values for all observables to these
extremal values, by absorbing the numerical factor 4 into the respective definitions. In
the following, we call the limit values the corrected Planck units or corrected Planck limits
and assume that the numerical factor 4 has been properly included. In other words:
Motion Mountain – The Adventure of Physics
volume:
4?ℏ
=
?3
4?ℏ
?⩾ 3
=
?
3/2
4?ℏ
?⩾( 3 ) =
?
3
?
?⩽
=
4?ℏ
?5
?⩽
=
16?2 ℏ
?⩾√
38
2 physics in lim it statem ents
“
Die Frage über die Gültigkeit der Voraussetzungen der Geometrie im
Unendlichkleinen hängt zusammen mit der Frage nach dem innern Grunde der
Massverhältnisse des Raumes. Bei dieser Frage, welche wohl noch zur Lehre
vom Raume gerechnet werden darf, kommt die obige Bemerkung zur
Anwendung, dass bei einer discreten Mannigfaltigkeit das Princip der
Massverhältnisse schon in dem Begriffe dieser Mannigfaltigkeit enthalten ist,
bei einer stetigen aber anders woher hinzukommen muss. Es muss also
entweder das dem Raume zu Grunde liegende Wirkliche eine discrete
Mannigfaltigkeit bilden, oder der Grund der Massverhältnisse ausserhalb, in
darauf wirkenden bindenden Kräften, gesucht werden.*
Bernhard Riemann, 1854, Über die Hypothesen, welche der Geometrie zu
Grunde liegen.
Physics, mathematics and simplicit y
”
“
Those are my principles, and if you don’t like
them ... well, I have others.
Groucho Marx***
”
* ‘The question of the validity of the hypotheses of geometry in the infinitely small is connected to the
question of the foundation of the metric relations of space. To this question, which may still be regarded as
belonging to the study of space, applies the remark made above; that in a discrete manifold the principles
of its metric relations are given in the notion of this manifold, while in a continuous manifold, they must
come from outside. Either therefore the reality which underlies space must form a discrete manifold, or the
principles of its metric relations must be sought outside it, in binding forces which act upon it.’
Bernhard Riemann is one of the most important mathematicians. 45 years after this statement, Max
Planck confirmed that natural units are due to gravitation, and thus to ‘binding forces’.
** Interestingly, he also regularly wrote the opposite, as shown on page 86.
*** Groucho Marx (b. 1890 New York City, d. 1977 Los Angeles), well-known comedian.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
L imits to space, time and size
Motion Mountain – The Adventure of Physics
The four limits of nature of equation (22) – on action, entropy, speed and force – are astonishing. Above all, the four limits are simple. For many decades, a silent assumption has
guided many physicists: physics requires difficult mathematics, and unification requires
even more difficult mathematics.
For example, for over thirty years, Albert Einstein searched with his legendary intensity for the final theory by exploring more and more complex equations. He did so even
on his deathbed!** Also most theoretical physicists in the year 2000 held the prejudice
that unification requires difficult mathematics. This prejudice is a consequence of over a
century of flawed teaching of physics. Flawed teaching is thus one of the reasons that the
search for a final theory was not successful for so long.
The summary of physics with limit statements shows that nature and physics are
simple. In fact, the essence of the important physical theories is extremely simple: special relativity, general relativity, thermodynamics and quantum theory are each based on
a simple inequality.
The summary of a large part of physics with inequalities is suggestive. The summary
makes us dream that the description of the remaining parts of physics – gauge fields,
elementary particles and the final theory – might be equally simple. Let us continue to
explore where the dream of simplicity leads us to.
pl a nck lim its for a ll physical ob servables
We have seen that the four fundamental limits of nature (22) result in a minimum distance and a minimum time interval. As the expressions for the limits shows, these minimum intervals arise directly from the unification of quantum theory and relativity: they
do not appear if the theories are kept separate. In other terms, unification implies that
there is a smallest length in nature. This result is important: the formulation of physics as a set of limit statements shows that the continuum model of space and time is not
completely correct. Continuity and manifolds are only approximations, valid for large actions, low speeds and small forces. Formulating general relativity and quantum theory
with limit statements makes this especially clear.
The existence of a force limit in nature implies that no physical system can be smaller
than a Schwarzschild black hole of the same mass. In particular, point particles do not
exist. The density limit makes the same point. In addition, elementary particles are predicted to be larger than the corrected Planck length. So far, this prediction has not been
tested by observations, as the scales in question are so small that they are beyond experimental reach. Detecting the sizes of elementary particles – for example, with electric
dipole measurements – would make it possible to check all limits directly.
Mass and energy limits
for elementary particles: ? ⩽
ℏ
.
??
(32)
Using this limit, we find the well-known mass, energy and momentum limits that are
valid only for elementary particles:
Ref. 28
ℏ?
= 1.1 ⋅ 10−8 kg = 0.60 ⋅ 1019 GeV/c2
4?
for (real) elementary particles:
?⩽√
for (real) elementary particles:
?⩽√
ℏ?5
= 9.8 ⋅ 108 J = 0.60 ⋅ 1019 GeV
4?
for (real) elementary particles:
?⩽√
ℏ?3
= 3.2 kg m/s = 0.60 ⋅ 1019 GeV/c . (33)
4?
These elementary-particle limits are the (corrected) Planck mass, Planck energy and
Planck momentum. They were discussed in 1968 by Andrei Sakharov, though with different numerical factors. They are regularly cited in elementary particle theory. All known
measurements comply with them.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Mass plays a special role in all these arguments. The four limits (22) do not make it possible to extract a limit statement on the mass of physical systems. To find one, we have
to restrict our aim somewhat.
The Planck limits mentioned so far apply to all physical systems, whether composite
or elementary. Other limits apply only to elementary systems. In quantum theory, the
distance limit is a size limit only for composite systems. A particle is elementary if its size
? is smaller than any measurable dimension. In particular, it must be smaller than the
reduced Compton wavelength:
Motion Mountain – The Adventure of Physics
Page 58
39
40
2 physics in lim it statem ents
Virtual particles – a new definition
Vol. II, page 72
Vol. IV, page 193
In fact, there are elementary particles that exceed all three limits that we have encountered so far. Nature does have particles which move faster than light, which show
actions below the quantum of action, and which experience forces larger than the force
limit.
We know from special relativity that the virtual particles exchanged in collisions move
faster than light. We know from quantum theory that the exchange of a virtual particle
implies actions below the minimum action. Virtual particles also imply an instantaneous
change of momentum; they thus exceed the force limit.
In short, virtual particles exceed all the limits that hold for real elementary particles.
Curiosities and fun challenges abou t Pl anck limits*
∗∗
The minimum action may come as a surprise at first, because angular momentum and
spin have the same unit as action; and nature contains particles with spin 0 or with spin
1/2 ℏ. A minimum action indeed implies a minimum angular momentum. However, the
angular momentum in question is total angular momentum, including the orbital part
with respect to the observer. The measured total angular momentum of a particle is never
smaller than ℏ, even if the spin is smaller.
In terms of mass flows, the power limit implies that flow of water through a tube is limited
in throughput. The resulting limit d?/d? ⩽ ?3 /4? for the change of mass with time seems
to be unrecorded in the research literature of the twentieth century.
∗∗
Vol. II, page 107
A further way to deduce the minimum length using the limit statements which structure
this adventure is the following. General relativity is based on a maximum force in nature,
or alternatively, on a maximum mass change per time, whose value is given by d?/d? =
?3 /4?. Quantum theory is based on a minimum action ? in nature, given by ℏ. Since a
distance ? can be expressed as
?
?2 =
,
(34)
d?/d?
we see directly that a minimum action and a maximum rate of change of mass imply
a minimum distance. In other words, quantum theory and general relativity force us to
conclude that in nature there is a minimum distance. In other words, at Planck scales the
term ‘point in space’ has no theoretical or experimental basis.
* Sections called ‘Curiosities’ can be skipped at first reading.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
∗∗
Motion Mountain – The Adventure of Physics
The (corrected) Planck limits are statements about properties of nature. There is no way
to measure values exceeding these limits, with any kind of experiment. Naturally, such a
claim provokes the search for counter-examples and leads to many paradoxes.
pl a nck lim its for a ll physical ob servables
41
∗∗
With the single-particle limits, the entropy limit leads to an upper limit for temperature:
?⩽√
ℏ?5
= 0.71 ⋅ 1032 K .
4??2
(35)
This corresponds to the temperature at which the energy per degree of freedom is given
by the (corrected) Planck energy √ℏ?5 /4? . A more realistic value would have to take
account of the number of degrees of freedom of a particle at Planck energy. This would
change the numerical factor. However, no system that is even near this temperature value
has been studied yet. Only Planck-size horizons are expected to realize the temperature
limit, but nobody has managed to explore them experimentally, so far.
∗∗
∗∗
∗∗
The gravitational attraction between two masses never yields force values high enough
to exceed the force limit. Why? First of all, masses ? and ? cannot come closer together
than the sum of their horizon radii. Using ? = ???/?2 with the distance ? given by the
(naive) sum of the two black hole radii as ? = 2?(? + ?)/?2 , we get
?⩽
?4
??
,
4? (? + ?)2
(36)
which is never larger than the force limit. Thus even two attracting black holes cannot
exceed the force limit – in the inverse-square approximation of universal gravity. In short,
the minimum size of masses means that the maximum force cannot be exceeded.
∗∗
Ref. 29
It is well known that gravity bends space. Therefore, if they are to be fully convincing,
our calculation for two attracting black holes needs to be repeated taking into account
the curvature of space. The simplest way is to study the force generated by a black hole
on a test mass hanging from a wire that is lowered towards a black hole horizon. For an
unrealistic point mass, the force would diverge at the horizon. Indeed, for a point mass
? lowered towards a black hole of mass ? at (conventionally defined radial) distance ?,
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Challenge 13 e
At first sight, it seems that electric charge can be used in such a way that the acceleration
of a charged body towards a charged black hole is increased to a value, when multiplied with the mass, that exceeds the force limit. However, the changes in the horizon for
charged black holes prevent this.
Motion Mountain – The Adventure of Physics
How can the maximum force be determined by gravity alone, which is the weakest interaction? It turns out that in situations near the maximum force, the other interactions
are usually negligible. This is the reason why gravity must be included in a unified description of nature.
42
2 physics in lim it statem ents
the force would be
?=
???
?2 √1 −
2??
??2
.
(37)
∗∗
Challenge 14 e
An absolute power limit implies a limit on the energy that can be transported per unit
time through any imaginable physical surface. At first sight, it may seem that the combined power emitted by two radiation sources that each emit 3/4 of the maximum value
should give 3/2 times the maximum value. However, the combination forms a black hole,
or at least prevents part of the radiation from being emitted by swallowing it between the
two sources.
Challenge 15 e
One possible system that actually achieves the Planck power limit is the final stage of
black hole evaporation. But even in this case, the power limit is not exceeded.
∗∗
Ref. 19
The maximum force limit states that the stress-energy tensor, when integrated over any
physical surface, does not exceed the limit value. No such integral, over any physical
surface, of any tensor component in any coordinate system, can exceed the force limit,
provided that it is measured by a realistic observer, in particular, by an observer with a
realistic proper size. The maximum force limit thus applies to any component of any force
vector, as well as to its magnitude. It applies to gravitational, electromagnetic, and nuclear
forces; and it applies to all realistic observers. It is not important whether the forces are
real or fictitious; nor whether we are discussing the 3-forces of Galilean physics or the
4-forces of special relativity. Indeed, the force limit applied to the zeroth component of
the 4-force is the power limit.
∗∗
The power limit is of interest if applied to the universe as a whole. Indeed, it can be used to
partly explain Olbers’ paradox: the sky is dark at night because the combined luminosity
of all light sources in the universe cannot be brighter than the maximum value.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
∗∗
Motion Mountain – The Adventure of Physics
This diverges at ? = 0, the location of the horizon. However, even a test mass cannot be
smaller than its own gravitational radius. If we want to reach the horizon with a realistic
test mass, we need to choose a small test mass ?: only a small mass can get near the
horizon. For vanishingly small masses, however, the resulting force tends to zero. Indeed,
letting the distance tend to the smallest possible value by letting ? = 2?(? + ?)/?2 →
2??/?2 requires ? → 0, which makes the force ?(?, ?) vanish. If on the other hand,
we remain away from the horizon and look for the maximum force by using a mass as
large as can possibly fit into the available distance (the calculation is straightforward),
then again the force limit is never exceeded. In other words, for realistic test masses,
expression (37) is never larger than ?4 /4?. Taking into account the minimal size of test
masses, we thus see that the maximum force is never exceeded in gravitational systems.
pl a nck lim its for a ll physical ob servables
43
∗∗
Page 35
Challenge 16 s
The force limit and its solid state analogy might be seen to suggest that the appearance of
matter might be nature’s way of preventing space from ripping apart. Does this analogy
make sense?
∗∗
Ref. 23
In fact, the connection between minimum length and gravity is not new. Already in 1967,
Andrei Sakharov pointed out that a minimum length implies gravity. He showed that
regularizing quantum field theory on curved space with a cut-off at small distances will
induce counter-terms that include to lowest order the cosmological constant and then
the Einstein–Hilbert action of general relativity.
∗∗
⊳ No surface is physical if any part of it requires a localization in space-time to
scales below the minimum length.
Ref. 30
For example, a physical surface must not cross any horizon. Only by insisting on physical surfaces can we eliminate unphysical examples that contravene the force and power
limits. For example, this condition was overlooked in Bousso’s early discussion of Bekenstein’s entropy bound – though not in his more recent ones.
Challenge 17 e
The equation ? = ? ? implies that energy and mass are equivalent. What do the equations ? = (4?/?2 )? = (4?/?4 )? for length and ? = ℏ? for action imply?
∗∗
Our discussion of limits can be extended to include electromagnetism. Using the (lowenergy) electromagnetic coupling constant ?, the fine structure constant, we get the following limits for physical systems interacting electromagnetically:
electric charge:
? ⩾ √4π?0 ??ℏ = ? = 0.16 aC
electric field:
?⩽√
?7
?4
=
= 1.9 ⋅ 1062 V/m
64π?o ?ℏ?2
4??
(39)
magnetic field:
?⩽√
?5
?3
=
= 6.3 ⋅ 1053 T
2
64π?0 ?ℏ?
4??
(40)
voltage:
?⩽√
?4
1 ℏ?5
= √
= 6.1 ⋅ 1027 V
16π?0 ??
? 4?
(41)
inductance:
?⩾
1
4?ℏ
1 √ 4?ℏ3
√
=
= 4.4 ⋅ 10−40 H .
4π?o ? ?7
?2
?5
(38)
(42)
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
∗∗
2
Motion Mountain – The Adventure of Physics
We said above that a surface is physical if an observer can be attached to each of its points.
The existence of a smallest length – and a corresponding shortest time interval – implies
44
2 physics in lim it statem ents
With the additional assumption that in nature at most one particle can occupy one Planck
volume, we get
9
π?o ? ?5
√ ?
= 4.7 ⋅ 1084 C/m3
=
?
16?3 ℏ
64?3 ℏ3
charge density:
?e ⩽ √
capacitance:
? ⩾ 4π?0 ?√
4?ℏ
4?
= ?2 √ 5 = 2.6 ⋅ 10−47 F .
3
?
?ℏ
(43)
(44)
For the case of a single conduction channel, we get
electric resistance:
electric current:
Ref. 31
Ref. 32
?⩽√
π?0 ??6
?5
= ?√
= 1.5 ⋅ 1024 A .
?
4ℏ?
(45)
(46)
(47)
∗∗
The general relation that to every limit value in nature there is a corresponding indeterminacy relation is valid also for electricity. Indeed, there is an indeterminacy relation
for capacitors, of the form
Δ? Δ? ⩾ ? ,
(48)
where ? is the positron charge, ? capacity and ? potential difference. There is also an
indeterminacy relation between electric current ? and time ?
Δ? Δ? ⩾ ? .
Ref. 33
Both these relations may be found in the research literature.
(49)
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
The magnetic field limit is significant in the study of extreme stars and black holes. The
maximum electric field plays a role in the theory of gamma-ray bursters. For current,
conductivity and resistance in single channels, the limits and their effects were studied
extensively in the 1980s and 1990s.
The observation of quarks and of collective excitations in semiconductors with charge
?/3 does not necessarily invalidate the charge limit for physical systems. In neither case
is there is a physical system – defined as localized mass–energy interacting incoherently
with the environment – with charge ?/3.
Motion Mountain – The Adventure of Physics
electric conductivity:
1
ℏ
= 2 = 4.1 kΩ
4π?0 ?? ?
?2
? ⩽ 4π?0 ?? =
= 0.24 mS
ℏ
?⩾
cosm olo gical lim its for a ll physical ob serva bles
45
cosmolo gical limits for all physical observables
Vol. II, page 107
Size and energy dependence
Angul ar momentum and action
Challenge 18 e
Ref. 34
Vol. IV, page 151
It only takes a moment to check that the ratio of angular momentum ? to energy ? times
length ? has the dimensions of inverse speed. Since speeds are limited by the speed of
light, we get
1
?system ⩽ ?? .
(50)
?
Indeed, in nature there do not seem to be any exceptions to this limit on angular momentum. In no known system, from atoms to molecules, from ice skaters to galaxies,
does the angular momentum exceed this value. Even the most violently rotating objects,
the so-called extremal black holes, are limited in angular momentum by ? ⩽ ??/?. (Actually, this limit is correct for black holes only if the energy is taken as the irreducible
* Quantum theory refines this definition: a physical system is a part of nature that in addition interacts
incoherently with its environment. In the following discussion we will assume that this condition is satisfied.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
While looking for additional limits in nature, we note a fundamental fact. Any upper
limit for angular momentum, and any lower limit for power, must be system-dependent.
Such limits will not be absolute, but will depend on properties of the system. Now, a
physical system is a part of nature characterized by a boundary and its content.* Thus the
simplest properties shared by all systems are their size (characterized in the following by
the diameter) ? and their energy ?. With these characteristics we can deduce systemdependent limits for every physical observable. The general method is straightforward:
we take the known inequalities for speed, action, power, charge and entropy, and then
extract a limit for any observable, by inserting the length and energy as required. We
then have to select the strictest of the limits we find.
Motion Mountain – The Adventure of Physics
In our quest to understand motion, we have focused our attention on the four fundamental limitations to which motion is subject. Special relativity posits a limit to speed,
namely the speed of light ?. General relativity limits force and power respectively by
?4 /4? and ?5 /4?, and quantum theory introduces a smallest value ℏ for action. Nature
imposes the lower limit ? on entropy. If we include the limit ? on electric charge changes,
these limits induce extremal values for all physical observables, given by the corresponding (corrected) Planck values.
A question arises: does nature also impose limits on physical observables at the opposite end of the measurement scale? For example, there is a highest force and a highest
power in nature. Is there also a lowest force and a lowest power? Is there also a lowest
speed?
We will show that there are indeed such limits, for all observables. We give the general
method to generate such bounds, and explore several examples. This exploration will take
us on an interesting survey of modern physics; we start by deducing system-dependent
limits and then go on to the cosmological limits.
46
2 physics in lim it statem ents
mass times ?2 ; if the usual mass is used, the limit is too large by a factor of 4.) The limit
deduced from general relativity, given by ? ⩽ ?2 ?3 /4?, is not stricter than the one just
given. By the way, no system-dependent lower limit for angular momentum can be deduced.
The maximum value for angular momentum is also interesting when it is seen as an
action limit. Action is the time integral of the difference between kinetic and potential
energy. Though nature always seeks to minimize the action ?, systems, of size ?, that
maximize action are also interesting. You might check for yourself that the action limit
? ⩽ ??/?
Challenge 19 e
(51)
is not exceeded in any physical process.
Speed
?system ⩾ ℏ?2
Challenge 20 e
Challenge 22 s
(52)
This is not a new result; it is just a form of the indeterminacy relation of quantum theory.
It gives a minimum speed for any system of energy ? and diameter ?. Even the extremely
slow radius change of a black hole by evaporation just realizes this minimal speed.
Continuing with the same method, we also find that the limit deduced from general
relativity, ? ⩽ (?2 /4?)(?/?), gives no new information. Therefore, no system-dependent
upper speed limit exists – just the global limit ?.
Incidentally, the limits are not unique. Other limits can be found in a systematic way.
Upper limits can be multiplied, for example, by factors of (?/?)(?4 /4?) or (??)(2/ℏ?),
yielding less strict upper limits. A similar rule can be given for lower limits.
Force, p ower and luminosity
We have seen that force and power are central to general relativity. The force exerted
by a system is the flow of momentum out of the system; emitted power is the flow of
energy out of the system. Thanks to the connection ? = ??? between action ?, force
?, distance ? and time ?, we can deduce
?system ⩾
ℏ 1
.
2? ?2
(53)
Experiments do not reach this limit. The smallest forces measured in nature are those
in atomic force microscopes, where values as small as 1 aN are observed. But even these
values are above the lower force limit.
The power ? emitted by a system of size ? and mass ? is limited by
?3
?
?
⩾ ?system ⩾ 2ℏ? 3 .
?
?
(54)
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Challenge 21 e
1
.
??
Motion Mountain – The Adventure of Physics
Speed times mass times length is an action. Since action values in nature are limited from
below by ℏ, we get a limit for the speed of a system:
cosm olo gical lim its for a ll physical ob serva bles
47
The limit on the left is the upper limit for any engine or lamp, as deduced from relativity;
not even the universe exceeds it. The limit on the right is the minimum power emitted by any system through quantum gravity effects. Indeed, no physical system is completely tight. Even black holes, the systems with the best ability to keep components inside their enclosure, radiate. The power radiated by black holes should just meet this
limit, provided the length ? is taken to be the circumference of the black hole. Thus the
claim of the quantum gravity limit is that the power emitted by a black hole is the smallest power that is emitted by any composite system of the same surface gravity. (However,
the numerical factors in the black hole power appearing in the research literature are not
yet consistent.)
The strange charm of the entropy bound
Ref. 35
?
?c.Planck
⩽
?
? c.Planck
(55)
which gives
?⩽
??3
?,
4?ℏ
(56)
?
?c.Planck
⩽
? c.Planck
?c.Planck ? c.Planck
?
?
?
(57)
we get
?⩽
Ref. 30
π??
2π??
?? =
?? .
ℏ
ℏ
(58)
This is called Bekenstein’s entropy bound. It states that the entropy of any physical system is finite and limited by its mass ? and size ?. No exception has ever been found or
constructed, despite many attempts. Again, the limit value itself is only realized for black
holes.
We need to explain the strange assumption used above. We are investigating the entropy of a horizon. Horizons are not matter, but limits to empty space. The entropy of
horizons is due to the large number of virtual particles found at them. In order to deduce the maximum entropy of expression (57) we therefore have to use the properties of
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
where ? is the surface of the system. Equality is realized only for black holes. The old
question of the origin of the factor 4 in the entropy of black holes is thus answered here:
it is due to the factor 4 in the force or power bound in nature. Time will tell whether this
explanation will be generally accepted.
We can also derive a more general relation by using a mysterious assumption, which
we will discuss afterwards. We assume that the limits for vacuum are opposite to those
for matter. We can then write ?2 /4? ⩽ ?/? for the vacuum. Using
Motion Mountain – The Adventure of Physics
In 1973, Bekenstein discovered a famous limit that connects the entropy ? of a physical
system with its size and mass. No system has a larger entropy than one bounded by a
horizon. The larger the horizon surface, the larger the entropy. We write
48
Ref. 36
2 physics in lim it statem ents
the vacuum. In other words, either we use a mass-to-length ratio for vacuum above the
Planck limit, or we use the Planck entropy as the maximum value for vacuum.
Other, equivalent limits for entropy can be found if other variables are introduced. For
example, since the ratio of the shear viscosity ? to the volume density of entropy (times
?) has the dimensions of action, we can directly write
?⩽
Challenge 23 e
?
?? .
ℏ
(59)
Curiosities and fun challenges abou t system-dependent limits to
observables
∗∗
Challenge 24 r
The content of a system is characterized not only by its mass and charge, but also by
its strangeness, isospin, colour charge, charge and parity. Can you deduce the limits for
these quantities?
∗∗
Challenge 25 s
In our discussion of black hole limits, we silently assumed that they interact, like any
thermal system, in an incoherent way with the environment. Which of the results of this
section change when this condition is dropped, and how? Which limits can be overcome?
∗∗
Challenge 26 e
Can you find a general method to deduce all limits of observables?
∗∗
Bekenstein’s entropy bound leads to some interesting speculations. Let us speculate that
the universe itself, being surrounded by a horizon, meets the Bekenstein bound. The
entropy bound gives a bound to all degrees of freedom inside a system: it tells us that the
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Also the system-dependent limit values for all physical observables, like the Planck values, yield a plethora of interesting questions. We study a few examples.
Motion Mountain – The Adventure of Physics
Again, equality is only attained in the case of black holes. In time, no doubt, the list of
similar bounds will grow longer.
Is there also a smallest, system-dependent entropy? So far, there does not seem to be a
system-dependent minimum value for entropy: the present approach gives no expression
that is larger than ?.
The establishment of the entropy limit is an important step towards making our description of motion consistent. If space-time can move, as general relativity maintains, it
also has an entropy. How could entropy be finite if space-time were continuous? Clearly,
because of the existence of a minimum distance and minimum time in nature, spacetime cannot be continuous, but must have a finite number of degrees of freedom, and
thus a finite entropy.
cosm olo gical lim its for a ll physical ob serva bles
Challenge 27 e
49
number ?d.o.f . of degrees of freedom in the universe is roughly
?d.o.f . ≈ 10132 .
(60)
Compare this with the number ?Pl. vol. of Planck volumes in the universe
?Pl. vol. ≈ 10183
(61)
and with the number ?part. of particles in the universe
?part. ≈ 1091 .
(62)
∗∗
A lower limit for the temperature of a thermal system can be found using the following
idea: the number of degrees of freedom of a system is limited by its surface, or more
precisely, by the ratio between the surface and the Planck surface. We get the limit
?⩾
Vol. II, page 62
Challenge 29 ny
(63)
This is the smallest temperature that a system of mass ? and size ? can have. Alternatively, using the method given above, we can use the limit on the thermal energy
??/2 ⩾ ℏ?/2π? (the thermal wavelength must be smaller than the size of the system)
together with the limit on mass ?2 /4? ⩾ ?/?, and deduce the same result.
We have met the temperature limit already: when the system is a black hole, the limit
yields the temperature of the emitted radiation. In other words, the temperature of black
holes is the lower limit for all physical systems for which a temperature can be defined,
provided they share the same boundary gravity. The latter condition makes sense: boundary gravity is accessible from the outside and describes the full physical system, since it
depends on both its boundary and its content.
So far, no exception to the claim on the minimum system temperature is known. All
systems from everyday life comply with it, as do all stars. Also the coldest known systems in the universe, namely Bose–Einstein condensates and other cold gases produced
in laboratories, are much hotter than the limit, and thus much hotter than black holes
of the same surface gravity. (We saw earlier that a consistent Lorentz transformation for
temperature is not possible; so the minimum temperature limit is only valid for an observer at the same gravitational potential as the system under consideration and stationary relative to it.)
By the way, there seems to be no consistent way to define an upper limit for a sizedependent temperature. Limits for other thermodynamic quantities can be found, but
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Challenge 28 s
4?ℏ ?
.
π?? ?2
Motion Mountain – The Adventure of Physics
We see that particles are only a tiny fraction of what moves around. Most motion must be
movement of space-time. At the same time, space-time moves far less than might be naively expected. To find out how all this happens is the challenge of the unified description
of motion.
50
2 physics in lim it statem ents
we do not discuss them here.
∗∗
When electromagnetism plays a role in a system, the system also needs to be characterized by a charge ?. Our method then gives the following limit for the electric field ?:
? ⩾ 4??
?2
.
?2 ?2
(64)
We write the field limit in terms of the elementary charge ?, though it might be more appropriate to write it using the fine structure constant via ? = √4π?0 ?ℏ? . In observations,
the electric field limit has never been exceeded. For the magnetic field we get
?⩾
(65)
Again, this limit is satisfied by all known systems in nature.
Similar limits can be found for the other electromagnetic observables. In fact, several
of the limits given earlier are modified when electric charge is included. Does the size
limit change when electric charge is taken into account? In fact, an entire research field
is dedicated to deducing and testing the most general limits valid in nature.
∗∗
C osmology in one statement
We now continue our exploration of limits to the largest systems possible. In order to do
that, we have a simple look at cosmology.
Cosmology results from the equations of general relativity when the cosmological
constant is included. Cosmology can thus be summarized by any sufficiently general
statement that includes the cosmological constant Λ. The simplest statement can be deduced from the observation that the present distance ?0 of the night sky horizon is about
?0 ≈ 1/√Λ . From this we can summarize cosmology by stating
⊳ There is a maximum distance value of the order of 1.4/√Λ in nature.
For all systems and all observers, sizes, distances and lengths are limited by the relation
?≲
1.4
= 1.3 × 1026 m = 1.4 × 1010 al .
√Λ
(66)
This expression contains all of cosmology. The details of the numerical factor 1.4 are not
of importance here and we will often neglect it in the following. This statement on length
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Many cosmological limits have not been discussed here nor anywhere else. The following
could all be worth a publication: What is the limit for momentum? Energy? Pressure?
Acceleration? Mass change? Lifetime?
Motion Mountain – The Adventure of Physics
Challenge 30 s
4?? ?2
.
? ?2 ?2
cosm olo gical lim its for a ll physical ob serva bles
Challenge 31 s
51
should be added to the four fundamental Planck limits as a fifth limit statement in nature.
By the way, can you show that the cosmological constant is observer-invariant?
The cosmological limits to observables
Minimum force
The negative energy volume density −Λ?4 /4π? introduced by the positive cosmological
constant Λ corresponds to a negative pressure (both quantities have the same dimensions). When multiplied by the minimum area it yields a force value
?=
Λℏ?
= 4.8 ⋅ 10−79 N .
2π
(68)
Apart from the numerical factor, this is the cosmological force limit, the smallest possible
force in nature. This is also the gravitational force between two corrected Planck masses
located at the cosmological distance √π/4Λ .
As a note, we are led the fascinating conjecture that the full theory of general relativity,
including the cosmological constant, is defined by the combination of a maximum and
a minimum force in nature.
In summary,
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Challenge 32 e
It has never been reached or approached by any observation.
Many cosmological limits are related to black hole limits. The observed average mass
density of the universe is not far from the corresponding black hole limit. The black
hole lifetime limit might thus provide an upper limit for the full lifetime of the universe.
However, the age of the universe is far from that limit by a large factor. In fact, since the
universe’s size and age are increasing, the lifetime limit is pushed further into the future
with every second that passes. The universe evolves so as to escape its own decay.
Motion Mountain – The Adventure of Physics
From the system-dependent limits for speed, action, force and entropy we can deduce
system-dependent limits for all other physical observables. In addition, we note that the
system-dependent limits can (usually) be applied to the universe as a whole; we only need
to insert the size and energy content of the universe. Usually, we can do this through a
limit process, even though the universe itself is not a physical system. In this way, we get
an absolute limit for every physical observable that contains the cosmological constant Λ
and that is on the opposite end of the Planck limit for that observable. We can call these
limits the cosmological limits.
The simplest cosmological limit is the upper limit to length in the universe. Since the
cosmological length limit also implies a maximum possible Compton wavelength, we
get a minimum particle mass and energy. We also get an cosmological lower limit on
luminosity.
For single particles, we find an absolute lower speed limit, the cosmological speed limit,
given by
?
?particle ⩾ c. Planck ? = √4?ℏ/? √Λ ≈ 7 ⋅ 10−53 m/s .
(67)
? universe
52
2 physics in lim it statem ents
⊳ Nature provides two limits for each observable: a Planck limit and a cosmological limit.
Challenge 33 s
Every observable has a lower and an upper limit. You may want to summarize them into
a table. This has important consequences that we will explore now.
L imits to measurement precision and their challenge to thought
No real numbers
Because of the fundamental limits to measurement precision, the measured values of
physical observables do not require the full set of real numbers. In fact, limited precision
implies that observables cannot be described by the real numbers! This staggering result
appears whenever quantum theory and gravity are brought together. But there is more.
Page 58
Ref. 37
There is a limit to the precision of length measurements in nature. This limit is valid both
for length measurements of empty space and for length measurements of matter (or radiation). Now let us recall what we do when we measure the length of a table with a ruler.
To find the ends of the table, we must be able to distinguish the table from the surrounding air. In more precise terms, we must be able to distinguish matter from vacuum.
Whenever we want high measurement precision, we need to approach Planck scales.
But at Planck scales, the measurement values and the measurement errors are of the
same size. In short, at Planck scales, the intrinsic measurement limitations of nature imply that we cannot say whether we are measuring vacuum or matter. We will check this
conclusion in detail later on.
In fact, we can pick any other observable that distinguishes vacuum from matter –
for example, colour, mass, size, charge, speed or angular momentum – and we have the
same problem: at Planck scales, the limits to observables lead to limits to measurement
precision, and therefore, at Planck scales it is impossible to distinguish between matter
and vacuum. At Planck scales, we cannot tell whether a box is full or empty.
To state the conclusion in the sharpest possible terms: vacuum and matter do not differ
at Planck scales. This counter-intuitive result is one of the charms of the search for a
final, unified theory. It has inspired many researchers in the field and some have written
best-sellers about it. Brian Greene was particularly successful in presenting this side of
quantum geometry to the wider public.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Vacuum and mass: t wo sides of the same coin
Motion Mountain – The Adventure of Physics
We now know that in nature, every physical measurement has a lower and an upper
bound. One of the bounds is cosmological, the other is given by the (corrected) Planck
unit. As a consequence, for every observable, the smallest relative measurement error
that is possible in nature is the ratio between the Planck limit and the cosmological limit.
In particular, we have to conclude that all measurements are limited in precision.
All limits, those to observables and those to measurement precision, only appear
when quantum theory and gravity are brought together. But the existence of these limits,
and in particular the existence of limits to measurement precision, forces us to abandon
some cherished assumptions.
cosm olo gical lim its for a ll physical ob serva bles
Page 81
53
Limited measurement precision also implies that at the Planck energy it is impossible
to speak about points, instants, events or dimensionality. Similarly, at the Planck length
it is impossible to distinguish between positive and negative time values: so particles and
antiparticles are not clearly distinguished at Planck scales. All these conclusions are so
far-reaching that we must check them in more detail. We will do this shortly.
Measurement precision and the existence of sets
Page 58
⊳ Nature has no parts.
Page 58
In summary, at Planck scales, perfect separation is impossible in principle. We cannot
distinguish observations. At Planck scales it is impossible to split nature into separate parts
or entities. In nature, elements of sets cannot be defined. Neither discrete nor continuous
sets can be constructed:
⊳ Nature does not contain sets or elements.
Since sets and elements are only approximations, the concept of a ‘set’, which assumes
separable elements, is too specialized to describe nature. Nature cannot be described at
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 58
Motion Mountain – The Adventure of Physics
Page 58
In physics, it is generally assumed that nature is a set of components or parts. These components, called elements by mathematicians, are assumed to be separable from each other.
This tacit assumption is introduced in three main situations: it is assumed that matter
consists of separable particles, that space-time consists of separable events or points, and
that the set of states consists of separable initial conditions. Until the year 2000, physics
has built the whole of its description of nature on the concept of a set.
The existence of a fundamental limit to measurement precision implies that nature is
not a set of such separable elements. Precision limits imply that physical entities can be
distinguished only approximately. The approximate distinction is only possible at energies much lower than the Planck energy √ℏ?5 /4? . As humans, we do live at such small
energies, and we can safely make the approximation. Indeed, the approximation is excellent in practice; we do not notice any error. But at Planck energy, distinction and separation is impossible in principle. In particular, at the cosmic horizon, at the big bang, and
at Planck scales, any precise distinction between two events, two points or two particles
becomes impossible.
Another way to reach this result is the following. Separation of two entities requires
different measurement results – for example, different positions, different masses or different velocities. Whatever observable is chosen, at the Planck energy the distinction
becomes impossible because of the large measurements errors. Only at everyday energies is a distinction possible. In fact, even at everyday energies, any distinction between
two physical systems – for example, between a toothpick and a mountain – is possible
only approximately. At Planck scales, a boundary can never be drawn.
A third argument is the following. In order to count any entities in nature – a set of
particles, a discrete set of points, or any other discrete set of physical observables – the
entities have to be separable. But the inevitable measurement errors contradict separability. Thus at the Planck energy it is impossible to count physical objects with precision:
54
2 physics in lim it statem ents
Planck scales – i.e., with full precision – if any of the concepts used for its description
presupposes sets. However, all concepts used in the past 25 centuries to describe nature
– particles, space, time, observables, phase space, wave functions, Hilbert space, Fock
space, Riemannian space, particle space, loop space or moduli space – are based on sets.
They must all be abandoned at Planck energy.
⊳ No correct mathematical model of nature can be based on sets.
Page 108
In other terms, nature has no parts: nature is one.
None of the approaches to unification pursued in the twentieth century has abandoned sets. This requirement about the final theory is thus powerful and useful. Indeed,
the requirement to abandon sets will be an efficient guide in our search for the unification of relativity and quantum theory. The requirement will even solve Hilbert’s sixth
problem.
If we exclude gauge interactions, we can summarize the rest of physics in a few limit
statements:
Page 19
The speed limit is equivalent to special relativity.
The force limit is equivalent to general relativity.
The action limit is equivalent to quantum theory.
The entropy limit is equivalent to thermodynamics.
The distance limit is equivalent to cosmology.
All these limits are observer-invariant. The invariance of the limits suggests interesting
thought experiments, none of which leads to their violation.
The invariant limits imply that in nature every physical observable is bound on one
end by the corresponding (corrected) Planck unit and on the other end by a cosmological
limit. Every observable in nature has an upper and lower limit value.
The existence of lower and upper limit values to all observables implies that measurement precision is limited. As a consequence, matter and vacuum are indistinguishable, the description of space-time as a continuous manifold of points is not correct, and
nature can be described by sets and parts only approximately. At Planck scales, nature
does not contain sets or elements.
Nature’s limits imply that Planck units are the key to the final theory. Since the most
precise physical theories known, quantum theory and general relativity, can be reduced
to limit statements, there is a good chance that the final, unified theory of physics will
allow an equally simple description. Nature’s limits thus suggest that the mathematics of
the final, unified theory might be simple.
At this point of our adventure, many questions are still open. Answering any of the
open issues of the millennium list still seems out of reach. But this impression is too
pessimistic. Our discussion implies that we only need to find a description of nature that
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
⊳
⊳
⊳
⊳
⊳
Motion Mountain – The Adventure of Physics
summary on limits in nature
sum m ary on lim its in nature
55
is simple and without sets. And a natural way to avoid the use of sets is a description of
empty space, radiation and matter as being made of common constituents. But before we
explore this option, we check the conclusions of this chapter in another way. In particular, as a help to more conservative physicists, we check all conclusions we found so far
without making use of the maximum force principle.
Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Chapter 3
GENER AL R EL AT IVIT Y VER SUS
QUANT UM T HEORY
** ‘One needs to replace habits of thought by necessities of thought.’
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Vol. I, page 247
”
he two accurate descriptions of motion available in the year 2000, namely
hat of general relativity and that of the standard model, are both useful and
horoughly beautiful. This millennium description of motion is useful because its
consequences are confirmed by all experiments, to the full measurement precision. We
are able to describe and understand all examples of motion that have ever been encountered. We can use this understanding to save lives, provide food and enjoy life. We
have thus reached a considerable height in our mountain ascent. Our quest for the full
description of motion is not far from completion.
The results of twentieth century physics are also beautiful. By this, physicists just mean
that they can be phrased in simple terms. This is a poor definition of beauty, but physicists
are rarely experts on beauty. Nevertheless, if a physicist has some other concept of beauty
in physics, avoid him, because in that case he is really talking nonsense.
The simplicity of twentieth-century physics is well-known: all motion observed in
nature minimizes action. Since in physics, action is a measure of change, we can say that
all motion observed in nature minimizes change. In particular, every example of motion
due to general relativity or to the standard model of particle physics minimizes action:
both theories can be described concisely with the help of a Lagrangian.
On the other hand, some important aspects of any type of motion, the masses of the
involved elementary particles and the strength of their coupling, are unexplained by general relativity and by the standard model of particle physics. The same applies to the origin of all the particles in the universe, their initial conditions, and the dimensionality of
space-time. Obviously, the millennium description of physics is not yet complete.
The remaining part of our adventure will be the most demanding. In the ascent of
any high mountain, the head gets dizzy because of the lack of oxygen. The finite amount
of energy at our disposal requires that we leave behind all unnecessary baggage and
everything that slows us down. In order to determine what is unnecessary, we need to
focus on what we want to achieve. Our aim is the precise description of motion. But even
though general relativity and quantum theory are extremely precise, useful and simple,
we do carry a burden: the two theories and their concepts contradict each other.
Motion Mountain – The Adventure of Physics
T
“
Man muß die Denkgewohnheiten durch
Denknotwendigkeiten ersetzen.**
Albert Einstein
g enera l rel ativit y v ersus q ua ntum theory
57
The contradictions
Ref. 38
Vol. II, page 243
In classical physics and in general relativity, the vacuum, or empty space, is a region with
no mass, no energy and no momentum. If particles or gravitational fields are present, the
energy density is not zero, space is curved and there is no complete vacuum.
In everyday life, vacuum has an energy density that cannot be distinguished from
zero. However, general relativity proposes a way to check this with high precision: we
measure the average curvature of the universe. Nowadays, cosmological measurements
performed with dedicated satellites reveal an average energy density ?/? of the intergalactic ‘vacuum’ with the value of
?
≈ 0.5 nJ/m3 .
?
Vol. V, page 122
Vol. V, page 128
Ref. 41
Ref. 40
The approximation is valid for the case in which the cut-off frequency ?max is much larger
than the rest mass ? of the particles corresponding to the field under consideration. The
limit considerations given above imply that the cut-off energy has to be of the order of
the Planck energy √ℏ?5 /4? , about 0.6 ⋅ 1019 GeV= 1.0 GJ. That would give a vacuum
energy density of
?
(71)
≈ 10111 J/m3 ,
?
which is about 10120 times higher than the experimental measurement. In other words,
something is slightly wrong in the calculation due to quantum field theory.*
General relativity and quantum theory contradict each other in other ways. Gravity
is curved space-time. Extensive research has shown that quantum field theory, which
describes electrodynamics and nuclear forces, fails for situations with strongly curved
space-time. In these cases the concept of ‘particle’ is not precisely defined. Quantum
field theory cannot be extended to include gravity consistently, and thus to include general relativity. Without the concept of the particle as a discrete entity, we also lose the
ability to perform perturbation calculations – and these are the only calculations possible
* It is worthwhile to stress that the ‘slight’ mistake lies in the domain of quantum field theory. There is no
mistake and no mystery, despite the many claims to the contrary found in newspapers and in bad research
articles, in general relativity. This well-known point is made especially clear by Bianchi and Rovelli.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 39
In short, cosmological data show that the energy density of intergalactic space is not
exactly zero; nevertheless, the measured value is extremely small and can be neglected
in all laboratory experiments.
On the other hand, quantum field theory tells a different story on vacuum energy
density. A vacuum is a region with zero-point fluctuations. The energy content of a vacuum is the sum of the zero-point energies of all the fields it contains. Indeed, the Casimir
effect ‘proves’ the reality of these zero-point energies. Following quantum field theory,
the most precise theory known, their energy density is given, within one order of magnitude, by
? 4πℎ ?max 3
πℎ 4
? d? = 3 ?max
.
(70)
≈ 3 ∫
?
? 0
?
Motion Mountain – The Adventure of Physics
Ref. 39
(69)
58
Vol. V, page 294
Vol. V, page 44
Vol. II, page 284
Ref. 44, Ref. 45
Ref. 46
The origin of the contradictions
Ref. 47
All contradictions between general relativity and quantum mechanics have the same
origin. In 20th-century physics, motion is described in terms of objects, made up of
* John Archibald Wheeler (b. 1911, Jacksonville, d. 2008, Hightstown), was a physicist and influential teacher
who worked on general relativity.
** As we will see below, the strand model provides a way to incorporate fermions into an extremely accurate approximation of general relativity, without requiring any topology change. This effectively invalidates
Wheeler’s argument.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 42, Ref. 43
in quantum field theory. In short, quantum theory only works because it assumes that
gravity does not exist. Indeed, the gravitational constant ? does not appear in quantum
field theory.
On the other hand, general relativity neglects the commutation rules between physical
quantities discovered in experiments on a microscopic scale. General relativity assumes
that the classical notions of position and momentum of material objects are meaningful.
It thus ignores Planck’s constant ℏ, and only works by neglecting quantum effects.
The concept of measurement also differs. In general relativity, as in classical physics,
it is assumed that arbitrary precision of measurement is possible – for example, by using
finer and finer ruler marks. In quantum mechanics, on the other hand, the precision of
measurement is limited. The indeterminacy relation yields limits that follow from the
mass ? of the measurement apparatus.
The contradictions also concern the concept of time. According to relativity and classical physics, time is what is read from clocks. But quantum theory says that precise
clocks do not exist, especially if gravitation is taken into account. What does ‘waiting 10
minutes’ mean, if the clock goes into a quantum-mechanical superposition as a result of
its coupling to space-time geometry? It means nothing.
Similarly, general relativity implies that space and time cannot be distinguished,
whereas quantum theory implies that matter does make a distinction between them. A
related difference is the following. Quantum theory is a theory of – admittedly weird –
local observables. In general relativity, there are no local observables, as Einstein’s hole
argument shows.
The contradiction between the two theories is shown most clearly by the failure of
general relativity to describe the pair creation of particles with spin 1/2, a typical and
essential quantum process. John Wheeler* and others have argued that, in such a case, the
topology of space necessarily has to change; in general relativity, however, the topology
of space is fixed. Equivalently, quantum theory says that matter is made of fermions, but
fermions cannot be incorporated into general relativity.**
Another striking contradiction was pointed out by Jürgen Ehlers. Quantum theory is
built on point particles, and point particles move on time-like world lines. But following
general relativity, point particles have a singularity inside their black hole horizon; and
singularities always move on space-like world lines. The two theories thus contradict each
other at smallest distances.
No description of nature that contains contradictions can lead to a unified or to a
completely correct description. To eliminate the contradictions, we need to understand
their origin.
Motion Mountain – The Adventure of Physics
Page 64
3 g eneral rel ativity v ersus q uantum theory
g enera l rel ativit y v ersus q ua ntum theory
Ref. 48
59
The d omain of contradictions: Pl anck scales
?S = 2??/?2 .
Ref. 42, Ref. 49
(72)
The gravitational constant ? and the speed of light ? act as conversion constants. Indeed,
as the Schwarzschild radius of an object is approached, the difference between general
relativity and the classical 1/?2 description of gravity becomes larger and larger. For example, the barely measurable gravitational deflection of light by the Sun is due to the
light approaching the Sun to within 2.4 ⋅ 105 times its Schwarzschild radius. Usually, we
are forced to stay away from objects at a distance that is an even larger multiple of the
Schwarzschild radius, as shown in Table 2. Only for this reason is general relativity unne* In the following, we use the terms ‘vacuum’ and ‘empty space’ interchangeably.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Despite their contradictions and the underlying circular definition, both general relativity and quantum theory are successful theories for the description of nature: they agree
with all data. How can this be?
Each theory of modern physics provides a criterion for determining when it is necessary and when classical Galilean physics is no longer applicable. These criteria are the
basis for many arguments in the following chapters.
General relativity shows that it is necessary to take into account the curvature of empty
space* and space-time whenever we approach an object of mass ? to within a distance
of the order of the Schwarzschild radius ?S , given by
Motion Mountain – The Adventure of Physics
particles, and space-time, made up of events. Let us see how these two concepts are
defined.
A particle – and in general any object – is defined as a conserved entity that has a
position and that can move. In fact, the etymology of the word object is connected to the
latter property. In other words, a particle is a small entity with conserved mass, charge,
spin and so on, whose position can vary with time.
An event is a point in space and time. In every physics text, time is defined with the
help of moving objects, usually called ‘clocks’, or moving particles, such as those emitted by light sources. Similarly, length is defined in terms of objects, either with an oldfashioned ruler or in terms of the motion of light, which is itself motion of particles.
Modern physics has sharpened our definitions of particles and space-time. Quantum
mechanics assumes that space-time is given (as a symmetry of the Hamiltonian), and
studies the properties of particles and their motion, both for matter and for radiation.
Quantum theory has deduced the full list of properties that define a particle. General relativity, and especially cosmology, takes the opposite approach: it assumes that the properties of matter and radiation are given (for example, via their equations of state), and
describes in detail the space-time that follows from them, in particular its curvature.
However, one fact remains unchanged throughout all these advances: in the millennium description of nature, the two concepts of particle and of space-time are each defined
with the help of the other. This circular definition is the origin of the contradictions
between quantum mechanics and general relativity. In order to eliminate the contradictions and to formulate a complete theory, we must eliminate this circular definition.
60
3 g eneral rel ativity v ersus q uantum theory
TA B L E 2 The size, Schwarzschild radius and Compton wavelength of some objects appearing in nature.
The lengths in quotation marks make no physical sense, as explained in the text.
Dia- Mass ?
m eter
?
galaxy
neutron star
Sun
Earth
human
molecule
atom (12 C)
proton p
pion π
up-quark u
electron e
neutrino ?e
≈ 1 Zm
10 km
1.4 Gm
13 Mm
1.8 m
10 nm
0.6 nm
2 fm
2 fm
< 0.1 fm
< 4 am
< 4 am
S c h wa r z - R a t i o
schild
?/?S
r a d i u s ?S
≈ 5 ⋅ 1040 kg
2.8 ⋅ 1030 kg
2.0 ⋅ 1030 kg
6.0 ⋅ 1024 kg
75 kg
0.57 zg
20 yg
1.7 yg
0.24 yg
5 ⋅ 10−30 kg
9.1 ⋅ 10−31 kg
< 3 ⋅ 10−36 kg
≈ 70 Tm
4.2 km
3.0 km
8.9 mm
0.11 ym
‘8.5 ⋅ 10−52 m’
‘3.0 ⋅ 10−53 m’
‘2.5 ⋅ 10−54 m’
‘3.6 ⋅ 10−55 m’
‘7 ⋅ 10−57 m’
‘1.4 ⋅ 10−57 m’
‘< 5 ⋅ 10−63 m’
≈ 107
2.4
4.8 ⋅ 105
1.4 ⋅ 109
1.6 ⋅ 1025
1.2 ⋅ 1043
2.0 ⋅ 1043
8.0 ⋅ 1038
5.6 ⋅ 1039
< 1 ⋅ 1040
< 3 ⋅ 1039
n.a.
C om p ton R atio
wav e ?/? C
l e n g t h ?C
(red.)
‘≈ 10−83 m’
‘1.3 ⋅ 10−73 m’
‘1.0 ⋅ 10−73 m’
‘5.8 ⋅ 10−68 m’
‘4.7 ⋅ 10−45 m’
6.2 ⋅ 10−19 m
1.8 ⋅ 10−17 m
2.0 ⋅ 10−16 m
1.5 ⋅ 10−15 m
7 ⋅ 10−14 m
3.9 ⋅ 10−13 m
> 1 ⋅ 10−7 m
≈ 10104
8.0 ⋅ 1076
8.0 ⋅ 1081
2.2 ⋅ 1074
3.8 ⋅ 1044
1.6 ⋅ 1010
3.2 ⋅ 107
9.6
1.4
< 0.001
< 1 ⋅ 10−5
< 3 ⋅ 10−11
?C =
ℏ
.
??
(73)
In this case, Planck’s constant ℏ and the speed of light ? act as conversion factors to
transform the mass ? into a length scale. Of course, this length is only relevant if the
object is smaller than its own Compton wavelength. At these scales we get relativistic
quantum effects, such as particle–antiparticle pair creation or annihilation. Table 2 shows
that the approach distance is near to or smaller than the Compton wavelength only in
the microscopic world, so that such effects are not observed in everyday life. Only for
this reason we do not need quantum field theory to describe common observations.
Combining concepts of quantum field theory and general relativity is required in situations where both conditions are satisfied simultaneously. The necessary approach distance for such situations is calculated by setting ?S = 2? C (the factor 2 is introduced for
simplicity). We find that this is the case when lengths or times are – within a factor of
order 1 – of the order of
?Pl = √ℏ?/?3
= 1.6 ⋅ 10−35 m, the Planck length,
?Pl = √ℏ?/?5
= 5.4 ⋅ 10−44 s, the Planck time.
(74)
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
cessary in everyday life. We recall that objects whose size is given by their Schwarzschild
radius are black holes; smaller objects cannot exist.
Similarly, quantum mechanics shows that Galilean physics must be abandoned and
quantum effects must be taken into account whenever an object is approached to within
distances of the order of the (reduced) Compton wavelength ? C , given by
Motion Mountain – The Adventure of Physics
Challenge 34 e
Object
g enera l rel ativit y v ersus q ua ntum theory
Challenge 35 e
61
R esolving the contradictions
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
The contradictions between general relativity and quantum theory have little practical
consequences. Therefore, for a long time, the contradictions were accommodated by
keeping the two theories separate. It is often said that quantum mechanics is valid at
small scales and general relativity is valid at large scales. This attitude is acceptable as
long as we remain far from the Planck length. However, this accommodating attitude
also prevents us from resolving the circular definition, the contradictions and therefore,
the millennium issues.
The situation resembles the well-known drawing, Figure 2, by Maurits Escher (b. 1898
Leeuwarden, d. 1972 Hilversum) in which two hands, each holding a pencil, seem to
be drawing each other. If one hand is taken as a symbol of vacuum and the other as a
symbol of particles, with the act of drawing taken as the act of defining, the picture gives
a description of twentieth-century physics. The apparent circular definition is solved by
recognizing that the two concepts (the two hands) both originate from a third, hidden
concept. In the picture, this third entity is the hand of the artist. In physics, the third
concept is the common origin of vacuum and particles.
We thus conclude that the contradictions in physics and the circular definition are
solved by common constituents for vacuum and matter. In order to find out what these
common constituents are and what they are not, we must explore the behaviour of nature
at the Planck scales.
Motion Mountain – The Adventure of Physics
Whenever we approach objects at these scales, both general relativity and quantum
mechanics play a role, and effects of quantum gravity appear. Because the values of the
Planck dimensions are extremely small, this level of sophistication is unnecessary in
everyday life, in astronomy and even in particle physics.
In the millennium description of nature, all the contradictions and also the circular
definition just mentioned are effective only at Planck scales. You can check this yourself.
This is the reason that general relativity and quantum theory work so well in practice.
However, to answer the questions posed at the beginning – why do we live in three dimensions, why are there three interactions, and why is the proton 1836.15 times heavier
than the electron? – we require a precise and complete description of nature. To answer
these questions, we must understand physics at Planck scales.
In summary, general relativity and quantum theory do contradict each other. However, the domains where these contradictions play a role, the Planck scales, are not accessible by experiment. As a consequence, the contradictions and our lack of knowledge
of how nature behaves at the Planck scales have only one effect: we do not see the solutions to the millennium issues.
We note that some researchers argue that the Planck scales specify only one of several domains of nature where quantum mechanics and general relativity apply simultaneously. They mention horizons and the big bang as separate domains. However, it is more
appropriate to argue that horizons and the big bang are situations where Planck scales
are essential.
62
3 g eneral rel ativity v ersus q uantum theory
F I G U R E 2 ‘Tekenen’ by Maurits
The origin of p oints
Page 57
⊳ The use of points in space and of separate, point-like particles are the reasons
for the mistaken vacuum energy calculation (71) that is wrong by 120 orders
of magnitude.
In short, only the circular definition of space and matter allows us to define points
and point particles. This puts us in a strange situation. On the one hand, experiment tells
us that describing nature with space points and with point particles works. On the other
hand, reason tells us that this is a fallacy and cannot be correct at Planck scales. We need
a solution.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
General relativity is built on the assumption that space is a continuum of points. Already
at school we learn that lines, surfaces and areas are made of points. We take this as granted, because we imagine that finer and finer measurements are always possible. And all
experiments so far agree with the assumption. Fact is: in this reasoning, we first idealized measurement rulers – which are made of matter – and then ‘deduced’ that points
in space exist.
Quantum theory is built on the assumption that elementary particles are point-like.
We take this as granted, because we imagine that collisions at higher and higher energy
are possible that allow elementary particles to get as close as possible. And all experiments so far agree with the assumption. Fact is: in this reasoning, we first imagined infinite energy and momentum values – which is a statement on time and space properties
– and then ‘deduce’ that point particles exist.
Motion Mountain – The Adventure of Physics
Escher, 1948 – a metaphor for the
way in which ‘particles’ and
‘space-time’ are defined: each with
the help of the other (© M.C. Escher
Heirs).
sum m ary on the cl ash b et ween the t wo theories
63
summary on the cl ash bet ween the t wo theories
General relativity and quantum theory contradict each other. In practice however, this
happens only at Planck scales. The reason for the contradiction is our insistence on a
circular definition of space and particles. Indeed, we need this circularity: Only such a
circular definition allows us to define points and point particles at all.
In order to solve the contradictions between general relativity and quantum theory
and in order to understand nature at Planck scales, we must introduce common constituents for space and particles. But common constituents have an important consequence:
common constituents force us to stop using points to describe nature. We now explore
this connection.
Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Chapter 4
D OES MAT T ER DIFFER FROM
VACUUM?
Vol. II, page 24
T
Farewell to instants of time
Ref. 50, Ref. 51
Ref. 52, Ref. 53
Ref. 54
Page 74
”
Measurement limits appear most clearly when we investigate the properties of clocks and
metre rules. Is it possible to construct a clock that is able to measure time intervals shorter
than the Planck time? Surprisingly, the answer is no, even though the time–energy indeterminacy relation Δ?Δ? ⩾ ℏ seems to indicate that by making Δ? large enough, we can
make Δ? arbitrary small.
Every clock is a device with some moving parts. The moving parts can be mechanical wheels, or particles of matter in motion, or changing electrodynamic fields (i.e.,
photons), or decaying radioactive particles. For each moving component of a clock the
indeterminacy relation applies. As explained most clearly by Michael Raymer, the indeterminacy relation for two non-commuting variables describes two different, but related,
situations: it makes a statement about standard deviations of separate measurements on
many identical systems; and it describes the measurement precision for a joint measurement on a single system. In what follows, we will consider only the second situation.
For a clock to be useful, we need to know both the time and the energy of each hand.
Otherwise it would not be a recording device. More generally, a clock must be a classical system. We need the combined knowledge of the non-commuting variables for each
moving component of the clock. Let us focus on the component with the largest time indeterminacy Δ?. It is evident that the smallest time interval ?? that can be measured by
** Moses Maimonides (b. 1135 Cordoba, d. 1204 Egypt) was a physician, philosopher and influential theologian. However, there is no evidence for ‘time atoms’ in nature, as explained below.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
“
Time is composed of time atoms ... which in
fact are indivisible.
Maimonides**
Motion Mountain – The Adventure of Physics
he appearance of the quantum of action in the description of motion leads
o limitations for all measurements: Heisenberg’s indeterminacy relations.
hese relations, when combined with the effects of gravitation, imply an almost
unbelievable series of consequences for the behaviour of nature at Planck scales. The
most important ones are the necessity to abandon points, instants and events, and the
equivalence of vacuum and matter. Here we show how these surprising and important
conclusions follow from simple arguments based on the indeterminacy relations, the
Compton wavelength and the Schwarzschild radius.
d oes m at ter d iffer from vacuum?
65
a clock is always larger than the quantum limit, i.e., larger than the time indeterminacy
Δ? for the most ‘uncertain’ component. Thus we have
?? ⩾ Δ? ⩾
ℏ
,
Δ?
(75)
where Δ? is the energy indeterminacy of the moving component. Now, Δ? must be smaller than the total energy ? = ?2 ? of the component itself: Δ? < ?2 ?.* Furthermore, a
clock provides information, so signals have to be able to leave it. Therefore the clock must
not be a black hole: its mass ? must be smaller than a black hole of its size, i.e., ? ⩽ ?2 ?/?,
where ? is the size of the clock (neglecting factors of order unity). Finally, for a sensible
measurement of the time interval ??, the size ? of the clock must be smaller than ? ??,
because otherwise different parts of the clock could not work together to produce the
same time display: ? < ???.** If we combine these three conditions, we get
ℏ?
?5 ??
(76)
or
?? ⩾ √
Vol. II, page 163
Ref. 49
Challenge 36 s
Ref. 55, Ref. 56
Ref. 57, Ref. 58
(77)
In summary, from three simple properties of any clock – namely, that it is only a single
clock, that we can read its dial, and that it gives sensible read-outs – we conclude that
clocks cannot measure time intervals shorter than the Planck time. Note that this argument
is independent of the nature of the clock mechanism. Whether the clock operates by
gravitational, electrical, mechanical or even nuclear means, the limit still applies.***
The same conclusion can be reached in other ways. For example, any clock small
enough to measure small time intervals necessarily has a certain energy indeterminacy
due to the indeterminacy relation. Meanwhile, on the basis of general relativity, any energy density induces a deformation of space-time, and signals from the deformed region
arrive with a certain delay due to that deformation. The energy indeterminacy of the
source leads to an indeterminacy in the deformation, and thus in the delay. The expression from general relativity for the deformation of the time part of the line element due
to a mass ? is ?? = ??/??3 . From the mass–energy relation, we see that an energy spread
* Physically, this condition means being sure that there is only one clock: if Δ? > ?, it would be impossible
to distinguish between a single clock and a clock–anticlock pair created from the vacuum, or a component
together with two such pairs, and so on.
** It is amusing to explore how a clock larger than ? ?? would stop working, as a result of the loss of rigidity
in its components.
*** Gravitation is essential here. The present argument differs from the well-known study on the limitations
of clocks due to their mass and their measuring time which was published by Salecker and Wigner and
summarized in pedagogical form by Zimmerman. In our case, both quantum mechanics and gravity are
included, and therefore a different, lower, and more fundamental limit is found. Also the discovery of black
hole radiation does not change the argument: black hole radiation notwithstanding, measurement devices
cannot exist inside black holes.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 59
ℏ?
= ?Pl .
?5
Motion Mountain – The Adventure of Physics
?? ⩾
66
4 d oes m at ter d iffer from vacuum?
Δ? produces an indeterminacy Δ? in the delay:
Δ? =
Δ? ?
.
? ?5
(78)
This determines the precision of the clock. Furthermore, the energy indeterminacy of the
clock is fixed by the indeterminacy relation for time and energy Δ? ⩾ ℏ/Δ?. Combining
all this, we again find the relation ?? ⩾ ?Pl for the minimum measurable time.
We are forced to conclude that in nature, it is impossible to measure time intervals
shorter than the Planck time. Thus
⊳ In nature there is a minimum time interval.
“
Ref. 24
Our greatest pretenses are built up not to hide
the evil and the ugly in us, but our emptiness.
The hardest thing to hide is something that is
not there.
Eric Hoffer,* The Passionate State of Mind
”
In a similar way, we can deduce that it is impossible to make a metre rule, or any other
length-measuring device, that is able to measure lengths shorter than the Planck length.
Obviously, we can already deduce this from ?Pl = ? ?Pl , but an independent proof is also
possible.
For any length measurement, joint measurements of position and momentum are necessary. The most straightforward way to measure the distance between two points is to
put an object at rest at each position. Now, the minimal length ?? that can be measured
* Eric Hoffer (b. 1902 New York City, d. 1983 San Francisco), philosopher.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Farewell to p oints in space
Motion Mountain – The Adventure of Physics
Ref. 60
In other words, at Planck scales the term ‘instant of time’ has no theoretical or experimental
basis. But let us go on. Special relativity, quantum mechanics and general relativity all rely
on the idea that time can be defined for all points of a given reference frame. However,
two clocks a distance ? apart cannot be synchronized with arbitrary precision. Since the
distance between two clocks cannot be measured with an error smaller than the Planck
length ?Pl , and transmission of signals is necessary for synchronization, it is not possible
to synchronize two clocks with a better precision than ?Pl /? = ?Pl , the Planck time. So
use of a single time coordinate for a whole reference frame is only an approximation.
Reference frames do not have a single time coordinate at Planck scales.
Moreover, since the time difference between events can only be measured within a
Planck time, for two events distant in time by this order of magnitude, it is not possible
to say with complete certainty which of the two precedes the other. But if events cannot
be ordered, then the very concept of time, which was introduced into physics to describe
sequences, makes no sense at Planck scales. In other words, after dropping the idea of a
common time coordinate for a complete frame of reference, we are forced to drop the
idea of time at a single ‘point’ as well. The concept of ‘proper time’ loses its meaning at
Planck scales.
d oes m at ter d iffer from vacuum?
67
must be larger than the position indeterminacy of the two objects. From the indeterminacy relation we know that neither object’s position can be determined with a precision
Δ? better than that given by Δ? Δ? = ℏ, where Δ? is the momentum indeterminacy. The
requirement that there be only one object at each end (avoiding pair production from
the vacuum) means that Δ? < ??: together, these requirements give
?? ⩾ Δ? ⩾
ℏ
.
??
(79)
Ref. 42, Ref. 49
Ref. 24
Ref. 61
Ref. 26
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 62, Ref. 63
Length measurements are limited by the Planck length.
Another way to deduce this limit reverses the roles of general relativity and quantum
theory. To measure the distance between two objects, we have to localize the first object
with respect to the other within a certain interval Δ?. The corresponding energy indeterminacy obeys Δ? = ?(?2 ?2 + (Δ?)2 )1/2 ⩾ ?ℏ/Δ?. However, general relativity shows
that a small volume filled with energy changes the curvature of space-time, and thus
changes the metric of the surrounding space. For the resulting distance change Δ?, compared with empty space, we find the expression Δ? ≈ ?Δ?/?4 . In short, if we localize the
first particle in space with a precision Δ?, the distance to a second particle is known
only with precision Δ?. The minimum length ?? that can be measured is obviously larger
than either of these quantities; inserting the expression for Δ?, we find again that the
minimum measurable length ?? is given by the Planck length.
We note that every length measurement requires a joint measurement of position and
momentum. This is particularly obvious if we approach a metre ruler to an object, but it
is equally true for any other length measurement.
We note that, since the Planck length is the shortest possible length, there can be
no observations of quantum-mechanical effects for a situation where the corresponding de Broglie or Compton wavelength is smaller than the Planck length. In proton–
proton collisions we observe both pair production and interference effects. In contrast,
the Planck limit implies that in everyday, macroscopic situations, such as car–car collisions, we cannot observe embryo–antiembryo pair production and quantum interference effects.
Another way to convince oneself that points have no meaning is to observe that a
point is an entity with vanishing volume; however, the minimum volume possible in
3
nature is the Planck volume ?Pl = ?Pl
.
We conclude that the Planck units not only provide natural units; they also provide –
within a factor of order one – the limit values of space and time intervals.
In summary, from two simple properties common to all length-measuring devices,
namely that they are discrete and that they can be read, we arrive at the conclusion that
Motion Mountain – The Adventure of Physics
Furthermore, the measurement cannot be performed if signals cannot leave the objects;
thus, they cannot be black holes. Therefore their masses must be small enough for their
Schwarzschild radius ?S = 2??/?2 to be less than the distance ?? separating them. Again
omitting the factor of 2, we get
ℏ?
?? ⩾ √ 3 = ?Pl .
(80)
?
68
4 d oes m at ter d iffer from vacuum?
⊳ Lengths smaller than the Planck length cannot be measured.
Whatever method is used, be it a metre rule or time-of-flight measurement, we cannot
overcome this fundamental limit. It follows that the concept of a ‘point in space’ has no
experimental or theoretical basis. In other terms,
⊳ In nature there is a minimum length interval.
The limitations on length measurements imply that we cannot speak of continuous space,
except in an approximate sense. As a result of the lack of measurement precision at Planck
scales, the concepts of spatial order, of translation invariance, of isotropy of the vacuum
and of global coordinate systems have no experimental basis.
The generalized indeterminacy rel ation
The limit values for length and time measurements are often expressed by the so-called
generalized indeterminacy relation
Δ?Δ? ⩾ ℏ/2 + ?
?
(Δ?)2
?3
(81)
Δ?Δ? ⩾ ℏ/2 + ?
2
?Pl
(Δ?)2 ,
ℏ
(82)
or
Ref. 24
Ref. 64, Ref. 65
Ref. 66, Ref. 67
Ref. 68
Farewell to space-time continuit y
“
Ich betrachte es als durchaus möglich, dass die Physik nicht auf dem Feldbegriff
begründet werden kann, d.h. auf kontinuierlichen Gebilden. Dann bleibt von
meinem ganzen Luftschloss inklusive Gravitationstheorie nichts bestehen.*
Albert Einstein, 1954, in a letter to Michele Besso.
”
* ‘I consider it as quite possible that physics cannot be based on the field concept, i.e., on continuous structures. In that case, nothing remains of my castle in the air, gravitation theory included.’
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Challenge 37 e
where ? is a numerical factor of order unity. A similar expression holds for the time–
energy indeterminacy relation. The first term on the right-hand side is the usual
quantum-mechanical indeterminacy. The second term is negligible for everyday energies, and is significant only near Planck energies; it is due to the changes in space-time
induced by gravity at these high energies. You should be able to show that the generalized
principle (81) implies that Δ? can never be smaller than ?1/2 ?Pl .
The generalized indeterminacy relation is derived in exactly the same way in which
Heisenberg derived the original indeterminacy relation Δ?Δ? ⩾ ℏ/2, namely by studying
the scattering of light by an object under a microscope. A careful re-evaluation of the
process, this time including gravity, yields equation (81). For this reason, all descriptions
that unify quantum mechanics and gravity must yield this relation, and indeed all known
approaches do so.
Motion Mountain – The Adventure of Physics
Ref. 24
d oes m at ter d iffer from vacuum?
Ref. 69
69
The classical description of nature is based on continuity: it involves and allows differences of time and space that are as small as can be imagined. Between any two points
in time or space, the existence of infinitely many other points is assumed. Measurement
results of arbitrary small values are deemed possible. The same is valid for action values.
However, quantum mechanics begins with the realization that the classical concept of
action makes no sense below the value of ℏ/2; similarly, unified theories begin with the
realization that the classical concepts of time and length make no sense below Planck
scales. Therefore, the continuum description of space-time has to be abandoned in favour
of a more appropriate description.
The minimum length distance, the minimum time interval, and equivalently, the new,
generalized indeterminacy relation appearing at Planck scales show that space, time and
in particular, space-time, are not well described as a continuum. Inserting ?Δ? ⩾ Δ? ⩾
ℏ/Δ? into equation (81), we get
(83)
which of course has no counterpart in standard quantum mechanics. This shows that
also space-time events do not exist. The concept of an ‘event’, being a combination of a
‘point in space’ and an ‘instant of time’, loses its meaning for the description of nature
at Planck scales.
Interestingly, the view that continuity must be abandoned is almost one hundred years
old. Already in 1917, Albert Einstein wrote in a letter to Werner Dällenbach:
Ref. 70
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Wenn die molekulare Auffassung der Materie die richtige (zweckmässige) ist, d.h. wenn ein Teil Welt durch eine endliche Zahl bewegter Punkte
darzustellen ist, so enthält das Kontinuum der heutigen Theorie zu viel Mannigfaltigkeit der Möglichkeiten. Auch ich glaube, dass dieses zu viel daran
schuld ist, dass unsere heutige Mittel der Beschreibung an der Quantentheorie scheitern. Die Frage scheint mir, wie man über ein Diskontinuum
Aussagen formulieren kann, ohne ein Kontinuum (Raum-Zeit) zu Hilfe zu
nehmen; letzteres wäre als eine im Wesen des Problems nicht gerechtfertigte
zusätzliche Konstruktion, der nichts “Reales” entspricht, aus der Theorie zu
verbannen. Dazu fehlt uns aber leider noch die mathematische Form. Wie
viel habe ich mich in diesem Sinne schon geplagt!
Allerdings sehe ich auch hier prinzipielle Schwierigkeiten. Die Elektronen (als Punkte) wären in einem solchen System letzte Gegebenheiten (Bausteine). Gibt es überhaupt letzte Bausteine? Warum sind diese alle von gleicher Grösse? Ist es befriedigend zu sagen: Gott hat sie in seiner Weisheit alle
gleich gross gemacht, jedes wie jedes andere, weil er so wollte; er hätte sie
auch, wenn es ihm gepasst hätte, verschieden machen können. Da ist man
bei der Kontinuum-Auffassung besser dran, weil man nicht von Anfang an
die Elementar-Bausteine angeben muss. Ferner die alte Frage vom Vakuum!
Aber diese Bedenken müssen verblassen hinter der blendenden Tatsache:
Das Kontinuum ist ausführlicher als die zu beschreibenden Dinge...
Lieber Dällenbach! Was hilft alles Argumentieren, wenn man nicht bis zu
einer befriedigenden Auffassung durchdringt; das aber ist verteufelt schwer.
Motion Mountain – The Adventure of Physics
Δ?Δ? ⩾ ℏ?/?4 = ?Pl ?Pl ,
70
4 d oes m at ter d iffer from vacuum?
Es wird einen schweren Kampf kosten, bis man diesen Schritt, der uns da
vorschwebt, wirklich gemacht haben wird. Also strengen Sie Ihr Gehirn an,
vielleicht zwingen Sie es.*
The second half of this text will propose a way to rise to the challenge. At this point
however, we first complete the exploration of the limitations of continuum physics.
In 20th century physics, space-time points are idealizations of events – but this idealization is inadequate. The use of the concept of ‘point’ is similar to the use of the concept
of ‘aether’ a century ago: it is impossible to measure or detect.
⊳ Like the ‘aether’, also ‘points’ lead reason astray.
⊳ Between two points there is not always a third.
Page 86
* ‘If the molecular conception of matter is the right (appropriate) one, i.e., if a part of the world is to be
represented by a finite number of moving points, then the continuum of the present theory contains too
great a manifold of possibilities. I also believe that this ‘too great’ is responsible for our present means of
description failing for quantum theory. The questions seems to me how one can formulate statements about
a discontinuum without using a continuum (space-time) as an aid; the latter should be banned from the
theory as a supplementary construction not justified by the essence of the problem, which corresponds to
nothing “real”. But unfortunately we still lack the mathematical form. How much have I already plagued
myself in this direction!
Yet I also see difficulties of principle. In such a system the electrons (as points) would be the ultimate
entities (building blocks). Do ultimate building blocks really exist? Why are they all of equal size? Is it satisfactory to say: God in his wisdom made them all equally big, each like every other one, because he wanted
it that way; he could also have made them, if he had wanted, all different. With the continuum viewpoint
one is better off, because one doesn’t have to prescribe elementary building blocks from the outset. Furthermore, the old question of the vacuum! But these considerations must pale beside the dazzling fact: The
continuum is more ample than the things to be described...
Dear Dällenbach! All arguing does not help if one does not achieve a satisfying conception; but this is
devilishly difficult. It will cost a difficult fight until the step that we are thinking of will be realized. Thus,
squeeze your brain, maybe you can force it.’
Compare this letter to what Einstein wrote almost twenty and almost forty years later.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
This results again means that space and time are not continuous. Of course, at large scales
they are – approximately – continuous, in the same way that a piece of rubber or a liquid
seems continuous at everyday scales, even though it is not at a small scale. But in nature,
space, time and space-time are not continuous entities.
Motion Mountain – The Adventure of Physics
All paradoxes resulting from the infinite divisibility of space and time, such as Zeno’s
argument on the impossibility of distinguishing motion from rest, or the Banach–Tarski
paradox, are now avoided. We can dismiss them straight away because of their incorrect
premises concerning the nature of space and time.
The consequences of the Planck limits for measurements of time and space can be
expressed in other ways. It is often said that given any two points in space or any two
instants of time, there is always a third in between. Physicists sloppily call this property
continuity, while mathematicians call it denseness. However, at Planck scales this property cannot hold, since there are no intervals smaller than the Planck time. Thus points
and instants are not dense, and
d oes m at ter d iffer from vacuum?
71
But there is more to come. The very existence of a minimum length contradicts the
theory of special relativity, in which it is shown that lengths undergo Lorentz contraction
when the frame of reference is changed. There is only one conclusion: special relativity
(and general relativity) cannot be correct at very small distances. Thus,
⊳ Space-time is not Lorentz-invariant (nor diffeomorphism-invariant) at
Planck scales.
Even the number of spatial dimensions makes no sense at Planck scales. Let us remind
ourselves how to determine this number experimentally. One possible way is to determine how many points we can choose in space such that all the distances between them
are equal. If we can find at most ? such points, the space has ? − 1 dimensions. But if
reliable length measurement at Planck scales is not possible, there is no way to determine
reliably the number of dimensions of space with this method.
Another way to check for three spatial dimensions is to make a knot in a shoe string
and glue the ends together: since it stays knotted, we know that space has three dimensions, because there is a mathematical theorem that in spaces with greater or fewer than
three dimensions, knots do not exist. Again, at Planck scales, we cannot say whether a
string is knotted or not, because measurement limits at crossings make it impossible to
say which strand lies above the other.
There are many other methods for determining the dimensionality of space.* In all
cases, the definition of dimensionality is based on a precise definition of the concept of
* For example, we can determine the dimension using only the topological properties of space. If we draw a
so-called covering of a topological space with open sets, there are always points that are elements of several
sets of the covering. Let ? be the maximal number of sets of which a point can be an element in a given
covering. The minimum value of ? over all possible coverings, minus one, gives the dimension of the space.
In fact, if physical space is not a manifold, the various methods for determining the dimensionality may
give different answers. Indeed, for linear spaces without norm, the dimensionality cannot be defined in a
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Farewell to dimensionalit y
Motion Mountain – The Adventure of Physics
Ref. 24, Ref. 71
All the symmetries that are at the basis of special and general relativity are only approximately valid at Planck scales.
The imprecision of measurement implies that most familiar concepts used to describe
spatial relations become useless. For example, the concept of a metric loses its usefulness
at Planck scales, since distances cannot be measured with precision. So it is impossible
to say whether space is flat or curved. The impossibility of measuring lengths exactly is
equivalent to fluctuations of the curvature, and thus of gravity.
In short, space and space-time are not smooth at Planck scales. This conclusion has
important implications. For example, the conclusion implies that certain mathematical
solutions found in books on general relativity, such as the Eddington–Finkelstein coordinates and the Kruskal–Szekeres coordinates do not describe nature! Indeed, these
coordinate systems, which claim to show that space-time goes on behind the horizon of
a black hole, are based on the idea that space-time is smooth everywhere. However,
quantum physics shows that space-time is not smooth at the horizon, but fluctuates
wildly there. In short, quantum physics confirms what common sense already knew:
Behind a horizon, nothing can be observed, and thus there is nothing there.
72
4 d oes m at ter d iffer from vacuum?
neighbourhood. At Planck scales, however, length measurements do not allow us to say
whether a given point is inside or outside a given region. In short, whatever method we
use, the lack of precise length measurements means that
⊳ At Planck scales, the dimensionality of physical space is not defined.
Farewell to the space-time manifold
Ref. 72
Ref. 73
But there are more surprises. At Planck scales, since both temporal and spatial order
break down, there is no way to say if the distance between two nearby space-time regions
unique way. Different definitions (fractal dimension, Lyapunov dimension, etc.) are possible.
* Where does the incorrect idea of continuous space-time have its roots? In everyday life, as well as in physics, space-time is a book-keeping device introduced to describe observations. Its properties are extracted
from the properties of observables. Since observables can be added and multiplied, like numbers, we infer that they can take continuous values, and, in particular, arbitrarily small values. It is then possible to
define points and sets of points. A special field of mathematics, topology, shows how to start from a set of
points and construct, with the help of neighbourhood relations and separation properties, first a topological
space, then, with the help of a metric, a metric space. With the appropriate compactness and connectedness
relations, a manifold, characterized by its dimension, metric and topology, can be constructed.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
⊳ Physical space-time cannot be a set of mathematical points.
Motion Mountain – The Adventure of Physics
Vol. I, page 57
The reasons for the problems with space-time become most evident when we remember
Euclid’s well-known definition: ‘A point is that which has no part.’ As Euclid clearly understood, a physical point, as an idealization of position, cannot be defined without some
measurement method. Mathematical points, however, can be defined without reference
to a metric. They are just elements of a set, usually called a ‘space’. (A ‘measurable’ or
‘metric’ space is a set of points equipped with a measure or a metric.)
In the case of physical space-time, the concepts of measure and of metric are more
fundamental than that of a point. Confusion between physical and mathematical space
and points arises from the failure to distinguish a mathematical metric from a physical
length measurement.*
Perhaps the most beautiful way to make this point is the Banach–Tarski theorem,
which clearly shows the limits of the concept of volume. The theorem states that a sphere
made up of mathematical points can be cut into five pieces in such a way that the pieces
can be put together to form two spheres, each of the same volume as the original one.
However, the necessary ‘cuts’ are infinitely curved and detailed: the pieces are wildly disconnected. For physical matter such as gold, unfortunately – or fortunately – the existence of a minimum length, namely the atomic distance, makes it impossible to perform
such a cut. For vacuum, the puzzle reappears. For example, the energy of zero-point
fluctuations is given by the density times the volume; following the Banach–Tarski theorem, the zero-point energy content of a single sphere should be equal to the zero-point
energy of two similar spheres each of the same volume as the original one. The paradox
is resolved by the Planck length, which provides a fundamental length scale even for vacuum, thus making infinitely complex cuts impossible. Therefore, the concept of volume
is only well defined at Planck scales if a minimum length is introduced.
To sum up:
d oes m at ter d iffer from vacuum?
73
is space-like or time-like.
⊳ At Planck scales, time and space cannot be distinguished from each other.
In addition, we cannot state that the topology of space-time is fixed, as general relativity
implies. The topology changes, mentioned above, that are required for particle reactions
do become possible. In this way another of the contradictions between general relativity
and quantum theory is resolved.
In summary, space-time at Planck scales is not continuous, not ordered, not endowed
with a metric, not four-dimensional, and not made up of points. It satisfies none of the
defining properties of a manifold.* We conclude that the concept of a space-time manifold
has no justification at Planck scales. This is a strong result. Even though both general
relativity and quantum mechanics use continuous space-time, the combined theory does
not.
Vol. V, page 358
* A manifold is what looks locally like a Euclidean space. The exact definition can be found in the previous
volume.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
If space and time are not continuous, no quantities defined as derivatives with respect
to space or time are precisely defined. Velocity, acceleration, momentum, energy and
so on are only well defined under the assumption of continuity. That important tool,
the evolution equation, is based on derivatives and can thus no longer be used. Therefore the Schrödinger and Dirac equations lose their basis. Concepts such as ‘derivative’,
‘divergence-free’ and ‘source free’ lose their meaning at Planck scales.
All physical observables are defined using length and time measurements. Each physical unit is a product of powers of length and time (and mass) units. (In the SI system,
electrical quantities have a separate base quantity, the ampere, but the argument still
holds: the ampere is itself defined in terms of a force, which is measured using the three
base units of length, time and mass.) Since time and length are not continuous, at Planck
scales, observables cannot be described by real numbers.
In addition, if time and space are not continuous, the usual expression for an observable field, ?(?, ?), does not make sense: we have to find a more appropriate description.
Physical fields cannot exist at Planck scales. Quantum mechanics also relies on the possibility to add wave functions; this is sometimes called the superposition principle. Without
fields and superpositions, all of quantum mechanics comes crumbling down.
The lack of real numbers has severe consequences. It makes no sense to define multiplication of observables by real numbers, but only by a discrete set of numbers. Among
other implications, this means that observables do not form a linear algebra. Observables are not described by operators at Planck scales. In particular, the most important
observables are the gauge potentials. Since they do not form an algebra, gauge symmetry
is not valid at Planck scales. Even innocuous-looking expressions such as [??, ??] = 0 for
?? ≠ ??, which are at the root of quantum field theory, become meaningless at Planck
scales. Since at those scales superpositions cannot be backed up by experiment, even
the famous Wheeler–DeWitt equation, sometimes assumed to describe quantum gravity, cannot be valid.
Motion Mountain – The Adventure of Physics
Farewell to observables, symmetries and measurements
74
4 d oes m at ter d iffer from vacuum?
Similarly, permutation symmetry is based on the premise that we can distinguish two
points by their coordinates, and then exchange particles between those locations. As we
have just seen, this is not possible if the distance between the two particles is very small.
We conclude that permutation symmetry has no experimental basis at Planck scales.
Even discrete symmetries, like charge conjugation, space inversion and time reversal,
cannot be correct in this domain, because there is no way to verify them exactly by measurement. CPT symmetry is not valid at Planck scales.
Finally we note that all types of scaling relations break down at small scales, because
of the existence of a smallest length. As a result, the renormalization group breaks down
at Planck scales.
In summary, due to the impossibility of accurate measurements,
⊳ All symmetries break down at Planck scales.
⊳ The concept of measurement has no significance at Planck scales.
This results from the limitations on time and length measurements.
Can space or space-time be a l at tice?
Ref. 75
Ref. 76
Ref. 77
Ref. 78
⊳ Space is not discrete. Neither is space-time.
We will discover more evidence for this negative conclusion later on.
But in fact, many discrete models of space and time have a much bigger limitation.
Any such model has to answer a simple question: Where is a particle during the jump
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 74
Let us take a breath. Can a space or even a space-time lattice be an alternative to continuity?
Discrete models of space-time have been studied since the 1940s. Recently, the idea
that space or space-time could be described as a lattice – like a crystal – has been explored
most notably by David Finkelstein and by Gerard ’t Hooft. The idea of space as a lattice
is based on the idea that, if there is a minimum distance, then all distances are multiples
of this minimum.
In order to get an isotropic and homogeneous situation for large, everyday scales,
the structure of space cannot be periodic, but must be random. But not only must it be
random in space, it must also be fluctuating in time. In fact, any fixed structure for spacetime would violate the result that there are no lengths smaller than the Planck length: as
a result of the Lorentz contraction, any moving observer would find lattice distances
smaller than the Planck value. Worse still, the fixed lattice idea conflicts with general
relativity, in particular with the diffeomorphism-invariance of the vacuum.
Thus, neither space nor space-time can be a lattice. A minimum distance does exist in
nature; however, we cannot hope that all other distances are simple multiples of it.
Motion Mountain – The Adventure of Physics
For example, supersymmetry cannot be valid at Planck scale. All mentioned conclusions
are consistent: if there are no symmetries at Planck scales, there are also no observables,
since physical observables are representations of symmetry groups. And thus,
d oes m at ter d iffer from vacuum?
75
from one lattice point to the next? This simple question eliminates most naive space-time
models.
A glimpse of quantum geometry
Vol. V, page 261
In every example of motion, some object is involved. One of the important discoveries
of the natural sciences was that all objects are composed of small constituents, called
elementary particles. Quantum theory shows that all composite, non-elementary objects
have a finite, non-vanishing size. The naive statement is: a particle is elementary if it behaves like a point particle. At present, only the leptons (electron, muon, tau and the neutrinos), the quarks, the radiation quanta of the electromagnetic, weak and strong nuclear
interactions (the photon, the W and Z bosons, and the gluons) and the Higgs boson have
been found to be elementary. Protons, atoms, molecules, cheese, people, galaxies and so
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Farewell to p oint particles
Motion Mountain – The Adventure of Physics
Ref. 27
Given that space-time is not a set of points or events, it must be something else. We have
three hints at this stage. The first is that in order to improve our description of motion
we must abandon ‘points’, and with them, abandon the local description of nature. Both
quantum mechanics and general relativity assume that the phrase ‘observable at a point’
has a precise meaning. Because it is impossible to describe space as a manifold, this
expression is no longer useful. The unification of general relativity and quantum physics
forces the adoption of a non-local description of nature at Planck scales. This is the first
hint.
The existence of a minimum length implies that there is no way to physically distinguish between locations that are even closer together. We are tempted to conclude that
no pair of locations can be distinguished, even if they are one metre apart, since on any
path joining two points, no two locations that are close together can be distinguished.
The problem is similar to the question about the size of a cloud or of an atom. If we
measure water density or electron density, we find non-vanishing values at any distance
from the centre of the cloud or the atom; however, an effective size can still be defined,
because it is very unlikely that the effects of the presence of a cloud or of an atom can
be seen at distances much larger than this effective size. Similarly, we can guess that two
points in space-time at a macroscopic distance from each other can be distinguished because the probability that they will be confused drops rapidly with increasing distance.
In short, we are thus led to a probabilistic description of space-time. This is the second
hint. Space-time becomes a macroscopic observable, a statistical or thermodynamic limit
of some microscopic entities. This is our second hint.
We note that a fluctuating structure for space-time also avoids the problems of fixed
structures with Lorentz invariance. In summary, the experimental observations of special relativity – Lorentz invariance, isotropy and homogeneity – together with the notion
of a minimum distance, point towards a description of space-time as fluctuating. This is
the third hint.
Several research approaches in quantum gravity have independently confirmed that
a non-local and fluctuating description of space-time at Planck scales resolves the contradictions between general relativity and quantum theory. These are our first results on
quantum geometry. To clarify the issue, we turn to the concept of the particle.
76
Page 60
Vol. IV, page 107
Ref. 79
4 d oes m at ter d iffer from vacuum?
on are all composite, as shown in Table 2.
Although the naive definition of ‘elementary particle’ as point particle is all we need in
the following argument, the definition is not precise. It seems to leave open the possibility
that future experiments could show that electrons or quarks are not elementary. This is
not so! In fact, the precise definition is the following:
⊳ Any particle is elementary if it is smaller than its own Compton wavelength.
ℏ?
= ?Pl .
?3
(84)
In other words, there is no way to observe that an object is smaller than the Planck length.
Thus,
⊳ There is no way to deduce from observations that a particle is point-like.
The term ‘point particle’ makes no sense at all.
Of course, there is a relation between the existence of a minimum length for empty
space and the existence of a minimum length for objects. If the term ‘point of space’
is meaningless, then the term ‘point particle’ is also meaningless. And again, the lower
limit on particle size results from the combination of quantum theory and general relativity.*
The minimum size for particles can be tested. A property connected with the size is
the electric dipole moment. This describes the deviation of its charge distribution from
* We note that the existence of a minimum size for a particle has nothing to do with the impossibility, in
quantum theory, of localizing a particle to within less than its Compton wavelength.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
?>√
Motion Mountain – The Adventure of Physics
If such a small particle were composite, there would be a lighter particle inside it, which
would have a larger Compton wavelength than the composite particle. This is impossible,
since the size of a composite particle must be larger than the Compton wavelength of its
components. (The alternative possibility that all components are heavier than the composite does not lead to satisfying physical properties: for example, it leads to intrinsically
unstable components.)
The size of an object, such as those given in Table 2, is defined as the length at which
differences from point-like behaviour are observed. The size ? of an object is determined
by measuring how it scatters a beam of probe particles. For example, the radius of the
atomic nucleus was determined for the first time in Rutherford’s experiment using alpha
particle scattering. In daily life as well, when we look at objects, we make use of scattered
photons. In general, in order for scattering to be useful, the effective wavelength ? =
ℏ/?? of the probe must be smaller than the object size ? to be determined. We thus
need ? > ? = ℏ/?? ⩾ ℏ/??. In addition, in order for a scattering experiment to be
possible, the object must not be a black hole, since, if it were, it would simply swallow
the approaching particle. This means that its mass ? must be smaller than that of a
black hole of the same size; in other words, from equation (72) we must have ? < ??2 /?.
Combining this with the previous condition we get, for the size ? of an object, the relation
d oes m at ter d iffer from vacuum?
Ref. 80
spherical. Some predictions from the standard model of elementary particles give as an
upper limit for the electron dipole moment ?? a value of
|?? |
< 10−39 m ,
?
Ref. 81
(85)
where ? is the charge of the electron. This predicted value is ten thousand times smaller
than the Planck length ?Pl . Since the Planck length is the smallest possible length, we
seem to have a contradiction here. However, a more careful and recent prediction from
the standard model only states
|?? |
< 3 ⋅ 10−23 m ,
?
Ref. 82
77
(86)
which is not in contradiction with the minimal length. The experimental limit in 2013 is
(87)
In the coming years, the experimental limit value will approach the Planck length. In
summary, no point particle is known. In fact, not even a particle smaller than the Planck
length is known.
Farewell to particle properties
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Planck scales have other strange consequences. In quantum field theory, the difference
between a virtual particle and a real particle is that a real particle is ‘on shell’, obeying
?2 = ?2 ?4 + ?2 ?2 , whereas a virtual particle is ‘off shell’. Because of the fundamental
limits of measurement precision, at Planck scales we cannot determine whether a particle
is real or virtual.
That is not all. Antimatter can be described as matter moving backwards in time.
Since the difference between backwards and forwards cannot be determined at Planck
scales, matter and antimatter cannot be distinguished at Planck scales.
Every particle is characterized by its spin. Spin describes two properties of a particle:
its behaviour under rotations (and thus, if the particle is charged, its behaviour in magnetic fields) and its behaviour under particle exchange. The wave function of a particle
with spin 1 remains invariant under a rotation of 2π, whereas that of a particle with spin
1/2 changes sign. Similarly, the combined wave function of two particles with spin 1 does
not change sign under exchange of particles, whereas for two particles with spin 1/2 it
does.
We see directly that both transformations are impossible to study at Planck scales.
Given the limit on position measurements, the position of a rotation axis cannot be well
defined, and rotations become impossible to distinguish from translations. Similarly,
positional imprecision makes it impossible to determine precise separate positions for
exchange experiments; at Planck scales it is impossible to say whether particle exchange
has taken place or not, and whether the wave function has changed sign or not. In short,
Motion Mountain – The Adventure of Physics
|?? |
< 8.7 ⋅ 10−31 m .
?
78
4 d oes m at ter d iffer from vacuum?
F I G U R E 3 Andrei Sakharov (1921–1989).
⊳ At Planck scales, spin cannot be defined or measured, and neither fermion
nor boson behaviour can be defined or measured.
A mass limit for elementary particles
?<
Ref. 28
ℏ
ℏ?
=√
= ?Pl = 2.2 ⋅ 10−8 kg = 1.2 ⋅ 1019 GeV/c2 .
? ?Pl
?
(88)
The limit ?Pl , the so-called Planck mass, corresponds roughly to the mass of a human
embryo that is ten days old, or equivalently, to that of a small flea. In short, the mass
of any elementary particle must be smaller than the Planck mass. This fact was already
noted as ‘well known’ by Andrei Sakharov* in 1968; he explains that these hypothetical
particles are sometimes called ‘maximons’. And indeed, the known elementary particles
all have masses well below the Planck mass. (In fact, the question why their masses are
so very much smaller than the Planck mass is one of the most important questions of
high-energy physics. We will come back to it.)
* Andrei Dmitrievich Sakharov, Soviet nuclear physicist (b. 1921 Moscow, d. 1989 Moscow). One of the
keenest thinkers in physics, Sakharov, among others, invented the Tokamak, directed the construction of
nuclear bombs, and explained the matter–antimatter asymmetry of nature. Like many others, he later campaigned against nuclear weapons, a cause for which he was put into jail and exile, together with his wife,
Yelena Bonner. He received the Nobel Peace Prize in 1975.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
The size ? of any elementary particle must by definition be smaller than its own (reduced)
Compton wavelength ℏ/??. Moreover, the size of a particle is always larger than the
Planck length: ? > ?Pl . Combining these two requirements and eliminating the size ?,
we get a constraint on the mass ? of any elementary particle, namely
Motion Mountain – The Adventure of Physics
Challenge 38 e
In particular, this implies that supersymmetry cannot be valid at Planck scales.
And we can continue. Due to measurement limitations, also spatial parity cannot be
defined or measured at Planck scales.
We have thus shown that at Planck scales, particles do not interact locally, are not
point-like, cannot be distinguished from antiparticles, cannot be distinguished from virtual particles, have no definite spin and have no definite spatial parity. We deduce that
particles do not exist at Planck scales. Let us explore the remaining concept: particle mass.
d oes m at ter d iffer from vacuum?
Ref. 83
79
There are many other ways to arrive at the mass limit for particles. For example, in
order to measure mass by scattering – and that is the only way for very small objects –
the Compton wavelength of the scatterer must be larger than the Schwarzschild radius;
otherwise the probe will be swallowed. Inserting the definitions of the two quantities and
neglecting the factor 2, we again get the limit ? < ?Pl . In fact it is a general property
of descriptions of nature that a minimum space-time interval leads to an upper limit for
masses of elementary particles.
Farewell to massive particles – and to massless vacuum
The Planck mass divided by the Planck volume, i.e., the Planck density, is given by
?Pl =
(89)
and is a useful concept in the following. One way to measure the (gravitational) mass ?
enclosed in a sphere of size ?, and thus (roughly) of volume ?3 , is to put a test particle
in orbit around it at that same distance ?. Universal gravitation then gives for the mass
? the expression ? = ??2 /?, where ? is the speed of the orbiting test particle. From
? < ?, we deduce that ? < ?2 ?/?; since the minimum value for ? is the Planck distance,
we get (again neglecting factors of order unity) a limit for the mass density ?, namely
? < ?Pl .
(90)
Δ? ⩾
ℏ
.
??
(91)
Note that for everyday situations, this error is extremely small, and other errors, such as
the technical limits of the balance, are much larger.
To check this result, we can explore another situation. We even use relativistic expressions, in order to show that the result does not depend on the details of the situation or
the approximations. Imagine having a mass ? in a box of size ?, and weighing the box
with a scale. (It is assumed that either the box is massless or that its mass is subtracted by
the scale.) The mass error is given by Δ? = Δ?/?2 , where Δ? is due to the indeterminacy
in the kinetic energy of the mass inside the box. Using the expression ?2 = ?2 ?4 + ?2 ?2 ,
we get that Δ? ⩾ Δ?/?, which again reduces to equation (91). Now that we are sure of
the result, let us continue.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
In other words, the Planck density is the maximum possible value for mass density.
Interesting things happen when we try to determine the error Δ? of a mass measurement in a Planck volume. Let us return to the mass measurement by an orbiting probe.
From the relation ?? = ??2 we deduce by differentiation that ?Δ? = ?2 Δ? + 2??Δ? >
2??Δ? = 2??Δ?/?. For the error Δ? in the velocity measurement we have the indeterminacy relation Δ? ⩾ ℏ/?Δ? + ℏ/?? ⩾ ℏ/??. Inserting this in the previous inequality,
and again forgetting the factor of 2, we find that the mass measurement error Δ? of a
mass ? enclosed in a volume of size ? is subject to the condition
Motion Mountain – The Adventure of Physics
Challenge 39 e
?5
= 5.2 ⋅ 1096 kg/m3
?2 ℏ
80
4 d oes m at ter d iffer from vacuum?
M
R
R
F I G U R E 4 A thought experiment
showing that matter and vacuum
cannot be distinguished when the
size of the enclosing box is of the
order of a Planck length.
⩾ ? ⩾
ℏ
.
??
(empty box)
(92)
We see directly that for sizes ? of the order of the Planck scales, the two limits coincide;
in other words, we cannot distinguish a full box from an empty box in that case.
To be sure of this strange result, we check whether it also occurs if, instead of measuring the gravitational mass, as we have just done, we measure the inertial mass. The
inertial mass for a small object is determined by touching it: physically speaking, by
performing a scattering experiment. To determine the inertial mass inside a region of
size ?, a probe must have a wavelength smaller than ?, and a correspondingly high energy. A high energy means that the probe also attracts the particle through gravity. (We
thus find the intermediate result that at Planck scales, inertial and gravitational mass can-
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?2 ?
?
(full box)
Motion Mountain – The Adventure of Physics
From equation (91) we deduce that for a box of Planck dimensions, the mass measurement error is given by the Planck mass. But from above we also know that the mass
that can be put inside such a box must not be larger than the Planck mass. Therefore,
for a box of Planck dimensions, the mass measurement error is larger than (or at best
equal to) the mass contained in it: Δ? ⩾ ?Pl . In other words, if we build a balance
with two boxes of Planck size, one empty and the other full, as shown in Figure 4, nature
cannot decide which way the balance should hang! Note that even a repeated or a continuous measurement will not resolve the situation: the balance will change inclination
at random, staying horizontal on average.
The argument can be rephrased as follows. The largest mass that we can put in a box
of size ? is a black hole with a Schwarzschild radius of the same value; the smallest mass
present in such a box – corresponding to what we call a vacuum – is due to the indeterminacy relation and is given by the mass with a Compton wavelength that matches
the size of the box. In other words, inside any box of size ? we have a mass ?, the limits
of which are given by:
d oes m at ter d iffer from vacuum?
81
not be distinguished. Even the balance experiment shown in Figure 4 illustrates this: at
Planck scales, the two types of mass are always inextricably linked.) Now, in any scattering experiment, for example in a Compton-type experiment, the mass measurement
is performed by measuring the wavelength change ?? of the probe before and after the
scattering. The mass indeterminacy is given by
Δ? Δ??
=
.
?
??
(93)
Mat ter and vacuum are indistinguishable
Vol. V, page 128
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 84
We can put these results in another way. On the one hand, if we measure the mass of
a piece of vacuum of size ?, the result is always at least ℏ/??: there is no possible way
to find a perfect vacuum in an experiment. On the other hand, if we measure the mass
of a particle, we find that the result is size-dependent: at Planck scales it approaches the
Planck mass for every type of particle, be it matter or radiation.
To use another image, when two particles approach each other to a separation of the
order of the Planck length, the indeterminacy in the length measurements makes it impossible to say whether there is something or nothing between the two objects. In short,
matter and vacuum are interchangeable at Planck scales. This is an important result: since
mass and empty space cannot be differentiated, we have confirmed that they are made
of the same ‘fabric’, of the same constituents. This idea, already suggested above, is now
common to all attempts to find a unified description of nature.
This approach is corroborated by attempts to apply quantum mechanics in highly
curved space-time, where a clear distinction between vacuum and particles is impossible,
as shown by the Fulling–Davies–Unruh effect. Any accelerated observer, and any observer in a gravitational field, detects particles hitting him, even if he is in a vacuum. The
effect shows that for curved space-time the idea of vacuum as particle-free space does
not work. Since at Planck scales it is impossible to say whether or not space is flat, it is
impossible to say whether it contains particles or not.
In short, all arguments lead to the same conclusion: vacuum, i.e., empty space-time,
cannot be distinguished from matter at Planck scales. Another common way to express
this state of affairs is to say that when a particle of Planck energy travels through space
it will be scattered by the fluctuations of space-time itself, as well as by matter, and the
Motion Mountain – The Adventure of Physics
In order to determine the mass in a Planck volume, the probe has to have a wavelength of
the Planck length. But we know from above that there is always a minimum wavelength
indeterminacy, given by the Planck length ?Pl . In other words, for a Planck volume the
wavelength error – and thus the mass error – is always as large as the Planck mass itself:
Δ? ⩾ ?Pl . Again, this limit is a direct consequence of the limit on length and space
measurements.
This result has an astonishing consequence. In these examples, the measurement error
is independent of the mass of the scatterer: it is the same whether or not we start with a
situation in which there is a particle in the original volume. We thus find that in a volume
of Planck size, it is impossible to say whether or not there is something there when we
probe it with a beam!
82
4 d oes m at ter d iffer from vacuum?
Nature's energy scale
Eeveryday
EPlanck
F I G U R E 5 Planck effects make the
energy axis an approximation.
Curiosities and fun challenges on Pl anck scales
“
There is nothing in the world but matter in
motion, and matter in motion cannot move
otherwise than in space and time.
Lenin, Materialism and empirio-criticism.
∗∗
Observers are made of matter. Observers are not made of radiation. Observers are not
made of vacuum. Observers are thus biased, because they take a specific standpoint.
But at Planck scales, vacuum, radiation and matter cannot be distinguished. Two consequences follow: first, only at Planck scales would a description be free of any bias in
favour of matter. Secondly, on the other hand, observers do not exist at all at Planck
energy. Physics is thus only possible below Planck energy.
∗∗
If measurements become impossible near Planck energy, we cannot even draw a diagram
with an energy axis reaching that value. A way out is shown Figure 5. The energy of
elementary particles cannot reach the Planck energy.
∗∗
Challenge 40 s
By the standards of particle physics, the Planck energy is rather large. Suppose we wanted
to impart this amount of energy to protons using a particle accelerator. How large would
a Planck accelerator have to be?
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
”
Lenin’s statement is wrong. And this is not so much because the world contains moving
matter, moving radiation, moving vacuum and moving horizons, which is not exactly
what Lenin claimed. Above all, his statement is wrong because at Planck scales, there is
no matter, no radiation, no horizon, no space and no time. These concepts only appear
at low energy. The rest of our adventure clarifies how.
Motion Mountain – The Adventure of Physics
two cases are indistinguishable. These surprising results rely on a simple fact: whatever
definition of mass we use, it is always measured via combined length and time measurements. (This is even the case for normal weighing scales: mass is measured by the
displacement of some part of the machine.) Mass measurement is impossible at Planck
scales. The error in such mass measurements makes it impossible to distinguish vacuum
from matter. In particular, the concept of particle is not applicable at Planck scale.
d oes m at ter d iffer from vacuum?
83
∗∗
Challenge 41 s
By the standards of everyday life, the Planck energy is rather small. Measured in litres of
gasoline, how much fuel does it correspond to?
∗∗
The usual concepts of matter and of radiation are not applicable at Planck scales. Usually,
it is assumed that matter and radiation are made up of interacting elementary particles.
The concept of an elementary particle implies an entity that is discrete, point-like, real
and not virtual, has a definite mass and a definite spin, is distinct from its antiparticle,
and, most of all, is distinct from vacuum, which is assumed to have zero mass. All these
properties are lost at Planck scales. At Planck scales, the concepts of ‘mass’, ‘vacuum’,
‘elementary particle’, ‘radiation’ and ‘matter’ do not make sense.
∗∗
∗∗
We now have a new answer to the old question: why is there something rather than
nothing? At Planck scales, there is no difference between something and nothing. We
can now honestly say about ourselves that we are made of nothing.
∗∗
Ref. 85
Page 279
∗∗
Ref. 86
Ref. 67
Challenge 44 s
Quantum mechanics alone gives, via the Heisenberg indeterminacy relation, a lower
limit to the spread of measurements, but, strangely enough, not on their precision, i.e.,
not on the number of significant digits. Wolfgang Jauch gives an example: atomic lattice
constants are known to a much higher precision than the positional indeterminacy of
single atoms inside the crystal.
It is sometimes claimed that measurement indeterminacies smaller than the Planck
values are possible for large enough numbers of particles. Can you show why this is
incorrect, at least for space and time?
∗∗
The idea that vacuum is not empty is not new. More than two thousand years ago,
Aristotle argued for a filled vacuum, although his arguments were incorrect as seen
from today’s perspective. Also in the fourteenth century there was much discussion
on whether empty space was composed of indivisible entities, but the debate died down
again.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Challenge 43 r
Special relativity implies that no length or energy can be invariant. Since we have come
to the conclusion that the Planck energy and the Planck length are invariant, it appears
that there must be deviations from Lorentz invariance at high energy. What effects would
follow? What kind of experiment could measure them? If you have a suggestion, publish it! Several attempts are being explored. We will settle the issue later on, with some
interesting insights.
Motion Mountain – The Adventure of Physics
Challenge 42 s
Do the large errors in mass measurements imply that mass can be negative at Planck
energy?
84
4 d oes m at ter d iffer from vacuum?
∗∗
Challenge 45 s
A Planck-energy particle falling in a gravitational field would gain energy. But the Planck
energy is the highest energy in nature. What does this apparent contradiction imply?
∗∗
Ref. 59
One way to generalize the results presented here is to assume that, at Planck energy,
nature is event-symmetric, i.e., symmetric under exchange of any two events. This idea,
developed by Phil Gibbs, provides an additional formulation of the strange behaviour of
nature at extreme scales.
∗∗
Vol. II, page 274
Because there is a minimum length in nature, so-called singularities do not exist. The
issue, hotly debated for decades in the twentieth century, thus becomes uninteresting.
∗∗
Ref. 87
∗∗
Vol. I, page 26
If vacuum and matter cannot be distinguished, we cannot distinguish between objects
and their environment. However, this was one of the starting points of our journey. Some
interesting adventures still await us!
∗∗
Vol. III, page 322
We have seen earlier that characterizing nature as made up of particles and vacuum creates problems when interactions are included. On the one hand interactions are the
difference between the parts and the whole, while on the other hand interactions are
exchanges of quantum particles. This apparent contradiction can be used to show that
something is counted twice in the usual characterization of nature. Noting that matter
and space-time are both made of the same constituents resolves the issue.
∗∗
Challenge 46 d
Is there a smallest possible momentum? And a smallest momentum error?
∗∗
Given that time becomes an approximation at Planck scales, can we still ask whether
nature is deterministic?
Let us go back to the basics. We can define time, because in nature change is not ran-
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Because mass and energy density are limited, any object of finite volume has only a finite number of degrees of freedom. The calculation of the entropy of black holes has
confirmed that entropy values are always finite. This implies that perfect baths do not
exist. Baths play an important role in thermodynamics (which must therefore be viewed
as only an approximation), and also in recording and measuring devices: when a device
measures, it switches from a neutral state to a state in which it shows the result of the
measurement. In order not to return to the neutral state, the device must be coupled to
a bath. Without a bath, a reliable measuring device cannot exist. In short, perfect clocks
and length-measuring devices do not exist, because nature puts a limit on their storage
ability.
Motion Mountain – The Adventure of Physics
Vol. V, page 145
d oes m at ter d iffer from vacuum?
Challenge 47 s
Page 410
85
dom, but gradual. What is the situation now that we know that time is only approximate?
Is non-gradual change possible? Is energy conserved? In other words, are surprises possible in nature?
It is correct to say that time is not defined at Planck scales, and that therefore that
determinism is an undefinable concept, but it is not a satisfying answer. What happens at
‘everyday’ scales? One answer is that at our everyday scales, the probability of surprises
is so small that the world indeed is effectively deterministic. In other words, nature is not
really deterministic, but the departure from determinism is not measurable, since every
measurement and observation, by definition, implies a deterministic world. The lack of
surprises would be due to the limitations of our human nature – more precisely, of our
senses and brain.
Can you imagine any other possibility? In truth, it is not possible to prove these answers at this point, even though the rest of our adventure will do so. We need to keep
any possible alternative in mind, so that we remain able to check the answers.
Page 115
If matter and vacuum cannot be distinguished, then each has the properties of the other.
For example, since space-time is an extended entity, matter and radiation are also extended entities. Furthermore, as space-time is an entity that reaches the borders of the
system under scrutiny, particles must also do so. This is our first hint at the extension of
matter; we will examine this argument in more detail shortly.
∗∗
∗∗
Challenge 48 s
When can matter and vacuum be distinguished? At what energy? This issue might be
compared to the following question: Can we distinguish between a liquid and a gas by
looking at a single atom? No, only by looking at many. Similarly, we cannot distinguish
between matter and vacuum by looking at one point, but only by looking at many. We
must always average. However, even averaging is not completely successful. Distinguishing matter from vacuum is like distinguishing clouds from the clear sky: like clouds,
matter has no precise boundary.
∗∗
Challenge 49 e
If the dimensionality of space is undefined at Planck scales, what does this mean for
superstrings?
∗∗
Vol. I, page 27
Since vacuum, particles and fields are indistinguishable at Planck scales, we also lose
the distinction between states and permanent, intrinsic properties of physical systems
at those scales. This is a strong statement: the distinction was the starting point of our
exploration of motion; the distinction allowed us to distinguish systems from their en-
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 287
The impossibility of distinguishing matter and vacuum implies a lack of information at
Planck scales. In turn, this implies an intrinsic basic entropy associated with any part of
the universe at Planck scales. We will come back to this topic shortly, when we discuss
the entropy of black holes.
Motion Mountain – The Adventure of Physics
∗∗
86
4 d oes m at ter d iffer from vacuum?
vironment. In other words, at Planck scales we cannot talk about motion! This is a strong
statement – but it is not unexpected. We are searching for the origin of motion, and we
are prepared to encounter such difficulties.
C ommon constituents
“
One can give good reasons why reality cannot at all be represented by a
continuous field. From the quantum phenomena it appears to follow with
certainty that a finite system of finite energy can be completely described by a
finite set of numbers (quantum numbers). This does not seem to be in
accordance with a continuum theory, and must lead to an attempt to find a
purely algebraic theory for the description of reality. But nobody knows how to
obtain the basis of such a theory.
Albert Einstein, 1955, the last sentences of The Meaning of Relativity – Including
the Relativistic Theory of the Non-Symmetric Field, fifth edition. These were also
his last published words.
”
”
Page 69
* ‘Yet it has been suggested that the introduction of a space-time continuum, in view of the molecular
structure of all events in the small, may possibly be considered as contrary to nature. Perhaps the success of
Heisenberg’s method may point to a purely algebraic method of description of nature, to the elimination of
continuous functions from physics. Then, however, one must also give up, in principle, the use of the spacetime continuum. It is not inconceivable that human ingenuity will some day find methods that will make it
possible to proceed along this path. Meanwhile, however, this project resembles the attempt to breathe in
an airless space.’
See also what Einstein thought twenty years before. The new point is that he believes that an algebraic
description is necessary. He repeats the point in the next quote.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 69
In this rapid journey, we have destroyed all the experimental pillars of quantum theory:
the superposition of wave functions, space-time symmetry, gauge symmetry, renormalization symmetry and permutation symmetry. We also have destroyed the foundations
of special and general relativity, namely the concepts of the space-time manifold, fields,
particles and mass. We have even seen that matter and vacuum cannot be distinguished.
It seems that we have lost every concept used for the description of motion, and thus
made its description impossible. It seems that we have completely destroyed our two
‘castles in the air’, general relativity and quantum theory. And it seems that we are trying
to breathe in airless space. Is this pessimistic view correct, or can we save the situation?
First of all, since matter and radiation are not distinguishable from vacuum, the quest
for unification in the description of elementary particles is correct and necessary. There
is no alternative to tearing down the castles and to continuing to breathe.
Motion Mountain – The Adventure of Physics
“
Es ist allerdings darauf hingewiesen worden, dass bereits die Einführung eines
raum-zeitlichen Kontinuums angesischts der molekularen Struktur allen
Geschehens im Kleinen möglicherweise als naturwidrig anzusehen sei.
Vielleicht weise der Erfolg von Heisenbergs Methode auf eine rein algebraische
Methode der Naturbeschreibung, auf die Ausschaltung kontinuierlicher
Funktionen aus der Physik hin. Dann aber muss auch auf die Verwendung des
Raum-Zeit-Kontinuums prinzipiell verzichtet werden. Es ist nicht undenkbar,
dass der menschliche Scharfsinn einst Methoden finden wird, welche die
Beschreitung dieses Weges möglich machen. Einstweilen aber erscheint dieses
Projekt ähnlich dem Versuch, in einem luftleeren Raum zu atmen.*
Albert Einstein, 1936, in Physik und Realität.
d oes m at ter d iffer from vacuum?
87
Secondly, after tearing down the castles, the invariant Planck limits ?, ℏ and ?4 /4? still
remain as a foundation.
Thirdly, after tearing down the castles, one important result appears. Since the concepts of ‘mass’, ‘time’ and ‘space’ cannot be distinguished from each other, a new, single
entity or concept is necessary to define both particles and space-time. In short, vacuum
and particles must be made of common constituents. In other words, we are not in airless space, and we uncovered the foundation that remains after we tore down the castles.
Before we go on exploring these common constituents, we check what we have deduced
so far against experiment.
Experimental predictions
Challenge 50 r
Vol. V, page 146
Ref. 89, Ref. 90
Ref. 90
Ref. 92, Ref. 93
Ref. 94
This energy value is between 1.4 ⋅ 1019 GeV and over 1022 GeV for the best measurement
to date. This is between just above the Planck energy and over one thousand times the
Planck energy. However, despite this high characteristic energy, no dispersion has been
found: even after a trip of ten thousand million years, all light arrives within one or two
seconds.
Another candidate experiment is the direct detection of distance fluctuations between
bodies. Gravitational wave detectors are sensitive to extremely small noise signals in
length measurements. There should be a noise signal due to the distance fluctuations
induced near Planck energy. The indeterminacy in measurement of a length ? is predicted to be
? 2/3
??
(95)
⩾ ( Pl ) .
?
?
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 92, Ref. 91
Motion Mountain – The Adventure of Physics
Ref. 88
A race is going on both in experimental and in theoretical physics: to be the first to
suggest and to be the first to perform an experiment that detects a quantum gravity effect
– apart possibly from (a part of) the Sokolov–Ternov effect. Here are some proposals.
At Planck scales, space fluctuates. We might think that the fluctuations of space could
blur the images of faraway galaxies, or destroy the phase relation between the photons.
However, no blurring is observed, and the first tests show that light from extremely distant galaxies still interferes. The precise prediction of the phase washing effect is still
being worked out; whatever the exact outcome, the effect is too small to be measured.
Another idea is to measure the speed of light at different frequencies from faraway
light flashes. There are natural flashes, called gamma-ray bursts, which have an extremely
broad spectrum, from 100 GeV down to visible light at about 1 eV. These flashes often
originate at cosmological distances ?. Using short gamma-ray bursts, it is thus possible
to test precisely whether the quantum nature of space-time influences the dispersion of
light signals when they travel across the universe. Planck-scale quantum gravity effects
might produce a dispersion. Detecting a dispersion would confirm that Lorentz symmetry breaks down at Planck scales.
The difference in arrival time Δ? between two photon energies ?1 and ?2 defines a
characteristic energy by
(? − ?2 ) ?
?char = 1
.
(94)
? Δ?
88
Page 66
Ref. 95
Ref. 92
Ref. 96
summary on particles and vacuum
Combining quantum theory and general relativity leads us to several important results
on the description of nature:
Ref. 98
— Vacuum and particles mix at Planck scales, because there is no conceivable way to distinguish whether a Planck-sized region is part of a particle or of empty space. Matter,
radiation and vacuum cannot be distinguished at Planck scales. Equivalently, empty
space and particles are made of fluctuating common constituents.
— We note that all arguments of this chapter equally imply that vacuum and particles
mix near Planck scales. For example, matter, radiation and vacuum cannot be distinguished near Planck scales.
— The constituents of vacuum and particles cannot be points. There is no conceivable
way to prove that points exist, because the smallest measurable distance in nature is
the Planck length.
— Particles, vacuum and continuous space do not exist at Planck scales. They disappear
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 97
This expression is deduced simply by combining the measurement limit of a ruler, from
quantum theory, with the requirement that the ruler not be a black hole. The sensitivity
of the detectors to noise might reach the required level in the twenty-first century. The
noise induced by quantum gravity effects has also been predicted to lead to detectable
quantum decoherence and vacuum fluctuations. So far, no such effect has been found.
A further candidate experiment for measuring quantum gravity effects is the detection of the loss of CPT symmetry at high energies. Especially in the case of the decay of
certain elementary particles, such as neutral kaons, the precision of experimental measurement is approaching the detection of Planck-scale effects. However, no such effect
has been found yet.
Another possibility is that quantum gravity effects may change the threshold energy at
which certain particle reactions become possible. It may be that extremely high-energy
photons or cosmic rays will make it possible to prove that Lorentz invariance is indeed
broken at Planck scales. However, no such effect has been found yet.
In the domain of atomic physics, it has also been predicted that quantum gravity
effects will induce a gravitational Stark effect and a gravitational Lamb shift in atomic
transitions. However, no such effect has been found yet.
Other proposals start from the recognition that the bound on the measurability of
observables also puts a bound on the measurement precision for each observable. This
bound is of no importance in everyday life, but it is important at Planck energy. One
proposal is to search for a minimal noise in length measurements, e.g., in gravitational
wave detectors. But no such noise has been found yet.
In summary, the experimental detection of quantum gravity effects might be possible,
despite their weakness, at some time during the twenty-first century. The successful prediction and detection of such an effect would be one of the highlights of physics, as it
would challenge the usual description of space and time even more than general relativity did. On the other hand, most unified models of physics predict the absence of any
measurable quantum gravity effect.
Motion Mountain – The Adventure of Physics
Ref. 95
4 d oes m at ter d iffer from vacuum?
sum m ary on pa rticles a nd vacuum
89
in a yet unclear Planck scale mixture.
— The three independent Planck limits ?, ℏ and ?4 /4? remain valid also in domains
where quantum theory and general relativity are combined.
Page 54
All these results must be part of the final theory that we are looking for. Generally speaking, we found the same conclusions that we found already in the chapter on limit statements. We thus continue along the same path that we took back then: we explore the
universe as a whole.
Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Chapter 5
WHAT IS T HE DIFFER ENCE BET WEEN
T HE UNIVERSE AND NOT HING?
C osmological scales
”
Antiquity
The description of motion requires the application of general relativity whenever the
scale ? of the situation is of the order of the Schwarzschild radius, i.e., whenever
? ≈ ?S = 2??/?2 .
Challenge 51 s
(96)
It is straightforward to confirm that, with the usually quoted mass ? and size ? of
everything visible in the universe, this condition is indeed fulfilled. We do need general
relativity, and thus curved space-time, when talking about the whole of nature.
Similarly, quantum theory is required for the description of the motion of an object
whenever we approach it within a distance ? of the order of the (reduced) Compton
wavelength ?C , i.e., whenever
ℏ
? ≈ ?C =
.
(97)
??
Obviously, for the total mass of the universe this condition is not fulfilled. However, we
are not interested in the motion of the universe itself; we are interested in the motion
of its components. In the description of these components, quantum theory is required
whenever pair production and annihilation play a role. This is the case in the early his** ‘The frontier is the really productive place of understanding.’ Paul Tillich (b. 1886 Starzeddel,
d. 1965 Chicago), theologian, socialist and philosopher.
*** ‘Here are lions.’ This was written across unknown and dangerous regions on ancient maps.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
“
Hic sunt leones.***
”
his strange question is the topic of the current leg of our mountain ascent. In
he last section we explored nature in the vicinity of Planck scales. In fact,
he other limit, namely the description of motion at large, cosmological scales, is
equally fascinating. As we proceed, many incredible results will appear, and at the end
we will discover a surprising answer to the question in the title.
Motion Mountain – The Adventure of Physics
T
“
Die Grenze ist der eigentlich fruchtbare Ort der
Erkenntnis.**
Paul Tillich, Auf der Grenze.
w hat is the d ifference b et w een the univ erse a nd nothing?
91
tory of the universe and near the horizon, i.e., for the most distant events that we can
observe in space and time. We are thus obliged to include quantum theory in any precise description of the universe.
Since at cosmological scales we need both quantum theory and general relativity, we
start our investigation with the study of time, space and mass, by asking at large scales
the same questions that we asked above at Planck scales.
Maximum time
Is it possible to measure time intervals of any imaginable size? General relativity shows
that in nature there is a maximum time interval, with a value of about
13 800 million years, or 435 Ps,
“
Vol. II, page 308
One should never trust a woman who tells one
her real age. A woman who would tell one that,
would tell one anything.
Oscar Wilde**
”
In light of all measurements, it may seem silly to question the age of the universe. The
age value is found in many books and tables and its precise determination is one of the
most important quests in modern astrophysics. But is this quest reasonable?
In order to measure the duration of a movement or the age of a system, we need a
clock that is independent of that movement or system, and thus outside the system. How* This implies that so-called ‘oscillating universe’ models, in which it is claimed that ‘before’ the big bang
there were other phenomena, cannot be justified on the basis of nature or observations. They are based on
beliefs.
** Oscar Wilde, (b. 1854 Dublin, d. 1900 Paris), poet and playwright, equally famous for his wit.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
D oes the universe have a definite age?
Motion Mountain – The Adventure of Physics
Challenge 52 s
providing an upper limit to the measurement of time. It is called the age of the universe,
and has been deduced from two sets of measurements: the expansion of space-time and
the age of matter.
We are all familiar with clocks that have been ticking for a long time: the hydrogen
atoms in our body. All hydrogen atoms were formed just after the big bang. We can
almost say that the electrons in these atoms have been orbiting their nuclei since the
dawn of time. In fact, the quarks inside the protons in these atoms have been moving a
few hundred thousand years longer than the electrons.
We thus have an upper time limit for any clock made of atoms. Even ‘clocks’ made of
radiation (can you describe one?) yield a similar maximum time. Now, the study of the
spatial expansion of the universe leads to the same maximum age. No clock or measurement device was ticking longer ago than this maximum time, and no clock could provide
a record of having done so.
In summary, it is not possible to measure time intervals greater than the maximum
time, either by using the history of space-time or by using the history of matter or radiation.* The maximum time is thus rightly called the age of the universe. Of course, this
is not a new idea; but looking at the age issue in more detail does reveal some surprises.
92
5 w hat is the d ifference b et ween the univ erse a nd nothing?
Challenge 53 s
?3
= 0.39 ⋅ 1070 m−2
?ℏ
(98)
as a limit for the surface curvature ? in nature. In other words, the universe has never
been as small as a point, never had zero age, never had infinite density, and never had
infinite curvature. It is not difficult to get a similar limit for temperature or any other
physical quantity near the big bang. In short, since events do not exist,
⊳ The big bang cannot have been an event.
There never was an initial singularity or a beginning of the universe.
In short, the situation is consistently muddled. Neither the age of the universe nor
its origin makes sense. What is going wrong? Or rather, how are things going wrong?
What happens if instead of jumping directly to the big bang, we approach it as closely as
possible? To clarify the issue, we ask about the measurement error in our statement that
the universe is fourteen thousand million years old. This turns out to be a fascinating
topic.
How precise can age measurements be?
“
No woman should ever be quite accurate about
her age. It looks so calculating.
Oscar Wilde
”
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
?<
Motion Mountain – The Adventure of Physics
ever, there are no clocks outside the universe, and no clock inside it can be independent.
In fact, we have just seen that no clock inside the universe can run throughout its full
history. In particular, no clock can run through its earliest history.
Time can be defined only if it is possible to distinguish between matter and space.
Given this distinction, we can talk either about the age of space, by assuming that matter
provides suitable and independent clocks – as is done in general relativity – or about the
age of matter, such as stars or galaxies, by assuming that the extension of space-time, or
possibly some other matter, provides a good clock. Both possibilities are being explored
experimentally in modern astrophysics – and both give the same result, of about fourteen
thousand million years, which was mentioned above. Despite this correspondence, for
the universe as a whole, an age cannot be defined, because there is no clock outside it!
The issue of the starting point of time makes this difficulty even more apparent. We
may imagine that going back in time leads to only two possibilities: either the starting
instant ? = 0 is part of time or it is not. (Mathematically, this means that the segment
representing time is either closed or open.) Both these possibilities imply that it is possible to measure arbitrarily small times; but we know from the combination of general
relativity and quantum theory that this is not the case. In other words, neither possibility
is correct: the beginning cannot be part of time, nor can it not be part of it. There is only
one solution to this contradiction: there was no beginning at all.
The lack of a beginning is consistent with a minimum length or a minimum action.
Indeed, both imply that there is a maximum curvature for space-time. Curvature can be
measured in several ways: for example, surface curvature is an inverse area. Within a
factor of order one, we find
w hat is the d ifference b et w een the univ erse a nd nothing?
Ref. 55, Ref. 56
The first way to measure the age of the universe* is to look at clocks in the usual sense of
the word, namely at clocks made of matter. As explained in the part on quantum theory,
Salecker and Wigner showed that a clock built to measure a total time ? with a precision
Δ? has a minimum mass ? given by
?>
Ref. 94
93
ℏ ?
.
?2 (Δ?)2
(99)
A simple way to incorporate general relativity into this result was suggested by Ng and
van Dam. Any clock of mass ? has a minimum resolution Δ? due to the curvature of
space that it introduces, given by
??
Δ? > 3 .
(100)
?
?<
(101)
where ?Pl = √ℏ?/?5 = 5.4 ⋅ 10−44 s is the Planck time. (As usual, we have omitted factors
of order one in this and in all the following results of this chapter.) In other words, the
higher the accuracy of a clock, the shorter the time during which it works dependably.
The precision of a clock is limited not only by the expense of building it, but also by
nature itself. Nevertheless, it is easy to check that for clocks used in daily life, this limit
is not even remotely approached. For example, you may wish to calculate how precisely
your own age can be specified.
As a consequence of the inequality (101), a clock trying to achieve an accuracy of one
Planck time can do so for at most one Planck time! A real clock cannot achieve Plancktime accuracy. If we try to go beyond the limit (101), fluctuations of space-time hinder the
working of the clock and prevent higher precision. With every Planck time that passes,
the clock accumulates a measurement error of at least one Planck time. Thus, the total
measurement error is at least as large as the measurement itself. This conclusion is also
valid for clocks based on radiation.
In short, measuring age with a clock always involves errors. Whenever we try to reduce these errors to the smallest possible level, the Planck level, the clock becomes so
imprecise over large times that age measurements become impossible.
D oes time exist?
“
Ref. 99
Time is waste of money.
Oscar Wilde
”
* The age ?0 is not the same as the Hubble time ? = 1/?0 . The Hubble time is only a computed quantity and
(almost) always larger than the age; the relation between the two depends on the values of the cosmological
constant, the density and other properties of the universe. For example, for the standard ‘hot big bang’
scenario, i.e., for the matter-dominated Einstein–de Sitter model, we have the simple relation ? = (3/2) ?0.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Challenge 54 e
(Δ?)3
,
?2Pl
Motion Mountain – The Adventure of Physics
If ? is eliminated, these two results imply that a clock with a precision Δ? can only measure times ? up to a certain maximum value, namely
94
Vol. I, page 40
Challenge 55 e
What is the error in the measurement of the age of the universe?
Challenge 57 e
It is now straightforward to apply our discussion about the measurement of time to the
age of the universe. The inequality (101) implies that the highest precision possible for a
clock is about 10−23 s, or about the time light takes to move across a proton. The finite
age of the universe also yields a maximum relative measurement precision. Inequality
(101) can be written as
Δ?
? 2/3
(102)
> ( Pl ) .
?
?
Inserting the age of the universe for ?, we find that no time interval can be measured
with a precision of more than about 40 decimals.
To clarify the issue, we can calculate the error in measurement as a function of the
observation energy ?meas , the energy of the measurement probe. There are two limit
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 68
Challenge 56 e
Ever since people began to study physics, the concept of ‘time’ has designated what is
measured by a clock. But the inequality (101) for a maximum clock time implies that
perfect clocks do not exist, and thus that time is only an approximate concept: perfect
time does not exist. Thus, in nature there is no ‘idea’ of time, in the Platonic sense.
In fact, the discussion so far can be seen as proof that combining quantum theory and
general relativity, because of the resulting measurement errors, prevents the existence of
perfect or ‘ideal’ examples of any classical observable or any everyday concept.
Time does not exist. Yet it is obviously a useful concept in everyday life. The key to
understanding this is measurement energy. Any clock – in fact, any system of nature –
is characterized by a simple number, namely the highest ratio of its kinetic energy to the
rest energy of its components. In daily life, this ratio is about 1 eV/10 GeV = 10−10 . Such
low-energy systems are well suited for building clocks. The more precisely the motion
of the main moving part – the pointer of the clock – can be kept constant and monitored, the higher the precision of the clock. To achieve very high precision, the pointer
must have very high mass. Indeed, in any clock, both the position and the speed of the
pointer must be measured, and the two measurement errors are related by the quantummechanical indeterminacy relation Δ? Δ? > ℏ/?. High mass implies low intrinsic fluctuation. Furthermore, in order to screen the pointer from outside influences, even more
mass is needed. This connection between mass and accuracy explains why more accurate
clocks are usually more expensive.
The standard indeterminacy relation ?Δ? Δ? > ℏ is valid only at everyday energies.
However, we cannot achieve ever higher precision simply by increasing the mass without
limit, because general relativity changes the indeterminacy relation to Δ? Δ? > ℏ/? +
?(Δ?)2 ?/?3 . The additional term on the right-hand side, negligible at everyday scales, is
proportional to energy. Increasing it by a large amount limits the achievable precision of
the clock. The smallest measurable time interval turns out to be the Planck time.
In summary, time exists, as a good approximation, only for low-energy systems. Any
increase in precision beyond a certain limit requires an increase in the energy of the
components; at Planck energy, this increase will prevent an increase in precision.
Motion Mountain – The Adventure of Physics
Page 58
5 w hat is the d ifference b et ween the univ erse a nd nothing?
w hat is the d ifference b et w een the univ erse a nd nothing?
95
Relative
measurement error
1
quantum
error
total
error
quantum
gravity
error
ΔEmin
E
Energy
Eopt
EPl
FIGURE 6
cases. For low energies, the error is due to quantum effects and is given by
Δ?
1
∼
?
?meas
(103)
Δ? ?meas
∼
?
?Pl
(104)
so that the total error varies as shown in Figure 6. In particular, very high energies do not
reduce measurement errors: any attempt to reduce the measurement error for the age of
the universe below 10−23 s would require energies so high that the limits of space-time
would be reached, making the measurement itself impossible. We reached this conclusion through an argument based on clocks made of particles. We will see below that
trying to determine the age of the universe from its expansion leads to the same limitation.
Imagine observing a tree which, as a result of some storm or strong wind, has fallen
towards second tree, touching it at the very top, as shown in Figure 7. It is possible to
determine the heights of both trees by measuring their separation and the angles at the
base. The error in the heights will depend on the errors in measurement of the separation
and angles.
Similarly, the age of the universe can be calculated from the present distance and
speed of objects – such as galaxies – observed in the night sky. The present distance
? corresponds to separation of the trees at ground level, and the speed ? to the angle
between the two trees. The Hubble time ? of the universe (which is usually assumed to
be larger than the age of the universe) then corresponds to the height at which the two
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
which decreases with increasing measurement energy. For high energies, however, the
error is due to gravitational effects and is given by
Motion Mountain – The Adventure of Physics
Measurement
errors as a
function of
measurement
energy.
96
5 w hat is the d ifference b et ween the univ erse a nd nothing?
v
T
d
trees meet. This age – in a naive sense, the time since the galaxies ‘separated’ – is given,
within a factor of order one, by
?
?= .
(105)
?
Let us explore this in more detail. For any measurement of ?, we have to choose the
object, i.e., a distance ?, as well as an observation time Δ?, or, equivalently, an observation
energy Δ? = 2πℏ/Δ?. We will now investigate the consequences of these choices for
equation (106), always taking into account both quantum theory and general relativity.
At everyday energies, the result of the determination of the age of the universe ?0 is
about (13.8 ± 0.1) ⋅ 109 Ga. This value is deduced by measuring red-shifts, i.e., velocities,
and distances, using stars and galaxies in distance ranges, from some hundred thousand
light years up to a red-shift of about 1. Measuring red-shifts does not produce large velocity errors. The main source of experimental error is the difficulty in determining the
distances of galaxies.
What is the smallest possible error in distance? Obviously, inequality (102) implies
Δ?
? 2/3
> ( Pl )
?
?
(107)
* At higher red-shifts, the speed of light, as well as the details of the expansion, come into play. To continue
with the analogy of the trees, we find that the trees are not straight all the way up to the top and that they
grow on a slope, as suggested by Figure 8.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
In simple terms, this is the method used to determine the age of the universe from the
expansion of space-time, for galaxies with red-shifts below unity.* The (positive) measurement error Δ? becomes
Δ? Δ? Δ?
=
+
.
(106)
?
?
?
Motion Mountain – The Adventure of Physics
F I G U R E 7 Trees and galaxies.
w hat is the d ifference b et w een the univ erse a nd nothing?
big bang
97
space
4ct0/9
light cone:
what we
can see
time
other galaxies
in the night sky
t0
our galaxy
Challenge 58 e
Challenge 59 e
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 99
thus giving the same indeterminacy in the age of the universe as the one we found above
in the case of material clocks.
We can try to reduce the age error in two ways: by choosing objects at either small or
large distances. Let us start with small distances. In order to get high precision at small
distances, we need high observation energies. It is fairly obvious that at observation
energies near the Planck value, Δ?/? approaches unity. In fact, both terms on the righthand side of equation (106) become of order one. At these energies, Δ? approaches ?
and the maximum value for ? approaches the Planck length, for the same reason that
at Planck energy the maximum measurable time is the Planck time. In short, at Planck
scales it is impossible to say whether the universe is old or young.
Let us consider the other extreme, namely objects extremely far away, say with a redshift of ? ≫ 1. Relativistic cosmology requires the diagram of Figure 7 to be replaced by
the more realistic diagram of Figure 8. The ‘light onion’ replaces the familiar light cone
of special relativity: light converges near the big bang. In this case the measurement
error for the age of the universe also depends on the distance and velocity errors. At
the largest possible distances, the signals an object sends out must be of high energy,
because the emitted wavelength must be smaller than the universe itself. Thus, inevitably,
we reach Planck energy. However, we have seen that in such high-energy situations,
both the emitted radiation and the object itself are indistinguishable from the space-time
background. In other words, the red-shifted signal we would observe today would have
a wavelength as large as the size of the universe, with a correspondingly small frequency.
There is another way to describe the situation. At Planck energy or near the horizon,
the original signal has an error of the same size as the signal itself. When measured at
the present time, the red-shifted signal still has an error of the same size as the signal. As
a result, the error in the horizon distance becomes as large as the value to be measured.
In short, even if space-time expansion and large scales are used, the instant of the socalled beginning of the universe cannot be determined with an error smaller than the
Motion Mountain – The Adventure of Physics
F I G U R E 8 The speed and distance of remote
galaxies.
98
Challenge 60 ny
5 w hat is the d ifference b et ween the univ erse a nd nothing?
age of the universe itself: a result we also found at Planck distances. If we aim for perfect
precision, we just find that the universe is 13.8 ± 13.8 thousand million years old! In
other words, in both extremal situations, it is impossible to say whether the universe has a
non-vanishing age.
We have to conclude that the anthropocentric concept of ‘age’ does not make any
sense for the universe as a whole. The usual textbook value is useful only for ranges of
time, space and energy in which matter and space-time are clearly distinguished, namely
at everyday, human-scale energies; the value has no more general meaning.
You may like to examine the issue of the fate of the universe using the same arguments.
But we will now continue on the path outlined at the start of this chapter; the next topic
on this path is the measurement of length.
Maximum length
Ref. 99
Is the universe really a big pl ace?
Ref. 100
Vol. II, page 308
Ref. 99
Astronomers and Hollywood films answer this question in the affirmative. Indeed, the
distance to the horizon of the universe is often included in tables. Cosmological models specify that the scale factor ?, which fixes the distance to the horizon, grows with
* In cosmology, we need to distinguish between the scale factor ?, the Hubble radius ?/? = ??/?,̇ the
horizon distance ℎ and the size ? of the universe. The Hubble radius is a computed quantity giving the
distance at which objects move away with the speed of light. The Hubble radius is always smaller than the
horizon distance, at which in the standard Einstein–de Sitter model, for example, objects move away with
twice the speed of light. However, the horizon itself moves away with three times the speed of light.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 99
Motion Mountain – The Adventure of Physics
Vol. II, page 243
General relativity shows that the horizon distance, i.e., the distance of objects with infinite
red-shift, is finite. In the usual cosmological model, for hyperbolic (open) and parabolic
(marginal) evolutions of the universe, the size of the universe is assumed infinite.* For
elliptical evolution, the total size is finite and depends on the curvature. However, in this
case also the present measurement limit yields a minimum size for the universe many
times larger than the horizon distance.
Quantum field theory, on the other hand, is based on flat and infinite space-time. Let
us see what happens when the two theories are combined. What can we say about measurements of length in this case? For example, would it be possible to construct and use
a metre rule to measure lengths larger than the distance to the horizon?
Admittedly, we would have no time to push the metre rule out up to the horizon,
because in the standard big bang model the horizon moves away from us faster than the
speed of light. (We should have started using the metre rule right at the big bang.) But
just for fun, let us assume that we have actually managed to do this. How far away can
we read off distances? In fact, since the universe was smaller in the past, and since every
observation of the sky is an observation of the past, Figure 8 shows that the maximum
spatial distance away from us at which an object can be seen is only 4??0/9. Obviously,
for space-time intervals, the maximum remains ??0 .
Thus, in all cases it turns out to be impossible to measure lengths larger than the horizon distance, even though general relativity sometimes predicts such larger distances.
This result is unsurprising, and in obvious agreement with the existence of a limit for
measurements of time intervals. The real surprises come next.
w hat is the d ifference b et w een the univ erse a nd nothing?
99
time ?; for the case of the standard mass-dominated Einstein–de Sitter model, i.e., for a
vanishing cosmological constant and flat space, we have
?(?) = ? ?2/3 ,
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Challenge 62 e
where the numerical constant ? relates the commonly accepted horizon distance to the
commonly accepted age. Indeed, observation shows that the universe is large, and is
getting larger. But let us investigate what happens if we add some quantum theory to this
result from general relativity. Is it really possible to measure the distance to the horizon?
We look first at the situation at high (probe) energies. We saw above that space-time
and matter are not distinguishable at Planck scales. Therefore, at Planck energy we cannot state whether or not objects are localized. At Planck scales, the distinction between
matter and vacuum – so basic to our thinking – disappears.
Another way to say this is that we cannot claim that space-time is extended at Planck
scales. Our concept of extension derives from the possibility of measuring distances and
time intervals, and from observations such as the ability to align several objects behind
one another. Such observations are not possible at Planck scales and energies, because
of the inability of probes to yield useful results. In fact, all of the everyday observations
from which we deduce that space is extended are impossible at Planck scales, where the
basic distinction between vacuum and matter, namely between extension and localization,
disappears. As a consequence, at Planck energy the size of the universe cannot be measured. It cannot even be called larger than a matchbox.
The problems encountered with probes of high probe energies have drastic consequences for the size measurement of the universe. All the arguments given above for
the errors in measurement of the age can be repeated for the distance to the horizon. To
reduce size measurement errors, a measurement probe needs to have high energy. But
at high energy, measurement errors approach the value of the measurement results. At
the largest distances and at Planck energy, the measurement errors are of the same magnitude as the measured values. If we try to determine the size of the universe with high
precision, we get no precision at all.
The inability to get precise values for the size of the universe should not come unexpected. For a reliable measurement, the standard must be different, independent, and
outside the system to be measured. For the universe this is impossible.
Studying the size of the big bang also produces strange results. The universe is said to
have been much smaller near the big bang because, on average, all matter is moving away
from all other matter. But if we try to follow the path of matter into the past with high
precision, using Planck energy probes, we get into trouble: since measurement errors are
as large as measurement data, we cannot claim that the universe was smaller near the big
bang than it is today: there is no way to reliably distinguish size values.
There are other confirmations too. If we had a metre rule spanning the whole universe,
even beyond the horizon, with zero at the place where we live, what measurement error
would it produce for the horizon? It does not take long to work out that the expansion
of space-time, from Planck scales to the present size, implies an expansion in the error
from Planck size to a length of the order of the present distance to the horizon. Again,
the error is as large as the measurement result. And again, the size of the universe turns
Motion Mountain – The Adventure of Physics
Challenge 61 e
(108)
100
5 w hat is the d ifference b et ween the univ erse a nd nothing?
out not to be a meaningful property.
Since this reasoning also applies if we try to measure the diameter of the universe
instead of its radius, it is impossible to say whether the antipodes in the sky really are
distant from each other!
We can summarize the situation by noting that anything said about the size of the
universe is as limited as anything said about its age. The height of the sky depends on
the observation energy. If we start measuring the sky at standard observation energies,
and try to increase the precision of measurement of the distance to the horizon, the
measurement error increases beyond all bounds. At Planck energy, the volume of the
universe is indistinguishable from the Planck volume – and vice versa.
The boundary of space – is the sky a surface?
Challenge 63 ny
Page 85
* The measurement errors also imply that we cannot say anything about translational symmetry at cosmological scales. Can you confirm this? In addition, at the horizon it is impossible to distinguish between
space-like and time-like distances. Even worse, concepts such as ‘mass’ or ‘momentum’ become muddled
at the horizon. This means that, as at Planck energy, we are unable to distinguish between object and background, and between state and intrinsic properties. We thus confirm the point made above.
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Page 58
Motion Mountain – The Adventure of Physics
Challenge 64 s
The horizon of the universe – essentially, the black part of the night sky – is a fascinating entity. Everybody interested in cosmology wants to know what happens there. In
newspapers the horizon is sometimes called the boundary of space. Some surprising insights – which have not yet made it to the newspapers – appear when we combine general
relativity and quantum mechanics.
We have seen that the errors in measuring the distance of the horizon are substantial.
They imply that we cannot pretend that all points of the sky are equally far away from us.
Thus we cannot say that the sky is a surface; it could be a volume. In fact, there is no way
to determine the dimensionality of the horizon, nor the dimensionality of space-time
near the horizon.*
Thus measurements do not allow us to determine whether the boundary is a point, a
surface, or a line. It may be a very complex shape, even knotted. In fact, quantum theory
tells us that it must be all of these from time to time: that the sky fluctuates in height and
shape.
In short, measurement errors prevent the determination of the topology of the sky.
In fact, this is not a new result. As is well known, general relativity is unable to describe
particle–antiparticle pair creation particles with spin 1/2. The reason for this inability is
the change in space-time topology required by such processes. The universe is full of
these and many other quantum processes; they imply that it is impossible to determine
or define the microscopic topology for the universe and, in particular, for the horizon.
Can you find at least two other arguments to confirm this conclusion?
Worse still, quantum theory shows that space-time is not continuous at a horizon:
this can easily be deduced using the Planck-scale arguments from the previous section.
Time and space are not defined at horizons.
Finally, there is no way to decide by measurement whether the various points on the
horizon are different from each other. On the horizon, measurement errors are of the
same order as the size of the horizon. The distance between two points in the night sky
is thus undefined. Therefore it is unclear what the diameter of the horizon is.
w hat is the d ifference b et w een the univ erse a nd nothing?
101
In summary, the horizon has no specific distance or shape. The horizon, and thus the
universe, cannot be shown to be manifolds. This unexpected result leads us to a further
question.
D oes the universe have initial conditions?
Ref. 101
Page 19
D oes the universe contain particles and stars?
Vol. II, page 308
Vol. IV, page 115
The total number of stars in the universe, about 1023±1, is included in every book on
cosmology. A smaller number can be counted on clear nights. But how dependable is the
statement? We can ask the same question about particles instead of stars. The commonly
quoted numbers are 1080±1 baryons and 1089±1 photons. However, the issue is not simple.
Neither quantum theory nor general relativity alone make predictions about the number
of particles. What happens if we combine the two theories?
In order to define the number of particles in a region, quantum theory first of all requires a vacuum state to be defined. The number of particles is defined by comparing
the system with the vacuum. If we neglect or omit general relativity by assuming flat
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Page 306
Motion Mountain – The Adventure of Physics
Page 58
One often reads about the quest for the initial conditions of the universe. But before
joining this search, we should ask whether and when such initial conditions make any
sense.
Obviously, our everyday description of motion requires knowledge of initial conditions, which describe the state of a system, i.e., all those aspects that differentiate it from
a system with the same intrinsic properties. Initial conditions – like the state – are attributed to a system by an outside observer.
Quantum theory tells us that initial conditions, or the state of a system, can only be
defined by an outside observer with respect to an environment. It is already difficult
to be outside the universe – but even inside the universe, a state can only be defined if
matter can be distinguished from vacuum. This is impossible at Planck energy, near the
big bang, or at the horizon. Thus the universe has no state. This means also that it has
no wave function.
The limits imposed by the Planck values confirm this conclusion in other ways. First
of all, they show that the big bang was not a singularity with infinite curvature, density
or temperature, because infinitely large values do not exist in nature. Secondly, since
instants of time do not exist, it is impossible to define the state of any system at a given
time. Thirdly, as instants of time do not exist, neither do events, and so the big bang was
not an event, and neither an initial state nor an initial wave function can be ascribed to
the universe. (Note that this also means that the universe cannot have been created.)
In short, there are no initial conditions for the universe. Initial conditions make sense
only for subsystems, and only far from Planck scales. Thus, for initial conditions to exist,
the system must be far from the horizon and it must have evolved for some time ‘after’
the big bang. Only when these two requirements are fulfilled can objects move in space.
Of course, this is always the case in everyday life. The lack of initial conditions means
that we have solved the first issue from the millennium list.
At this point in our mountain ascent, where neither time nor length is clearly defined
at cosmological scales, it should come as no surprise that there are similar difficulties
concerning the concept of mass.
102
Challenge 65 e
Challenge 66 s
5 w hat is the d ifference b et ween the univ erse a nd nothing?
Vol. II, page 308
Mass distinguishes objects from the vacuum. The average mass density of the universe,
about 10−26 kg/m3 , is often cited in texts. Is it different from a vacuum? Quantum theory
shows that, as a result of the indeterminacy relation, even an empty volume of size ? has
a mass. For a zero-energy photon inside such a vacuum, we have ?/? = Δ? > ℏ/Δ?, so
that in a volume of size ?, we have a minimum mass of at least ?min(?) = ℎ/??. For a
spherical volume of radius ? there is thus a minimal mass density given approximately
by
? (?)
ℏ
?min ≈ min3
= 4 .
(109)
?
??
For the universe, if the standard horizon distance ?0 of 13 800 million light years is inserted, the value becomes about 10−142 kg/m3 . This describes the density of the vacuum.
In other words, the universe, with a textbook density of about 10−26 kg/m3 , seems to be
clearly different from vacuum. But are we sure?
We have just deduced that the radius of the horizon is undefined: depending on the
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
D oes the universe have mass?
Motion Mountain – The Adventure of Physics
space-time, this procedure poses no problem. However, if we include general relativity,
and thus a curved space-time, especially one with a strangely behaved horizon, the answer is simple: there is no vacuum state with which we can compare the universe, for two
reasons. First, nobody can explain what an empty universe would look like. Secondly,
and more importantly, there is no way to define a state of the universe. The number of
particles in the universe thus becomes undefinable. Only at everyday energies and for
finite dimensions are we able to speak of an approximate number of particles.
In the case of the universe, a comparison with the vacuum is also impossible for purely
practical reasons. The particle counter would have to be outside the system. (Can you
confirm this?) In addition, it is impossible to remove particles from the universe.
The impossibility of defining a vacuum state, and thus the number of particles in the
universe, is not surprising. It is an interesting exercise to investigate the measurement
errors that appear when we try to determine the number of particles despite this fundamental impossibility.
Can we count the stars? In principle, the same conclusion applies as for particles.
However, at everyday energies the stars can be counted classically, i.e., without taking
them out of the volume in which they are enclosed. For example, this is possible if
the stars are differentiated by mass, colour or any other individual property. Only near
Planck energy or near the horizon are these methods inapplicable. In short, the number
of stars is only defined as long as the observation energy is low, i.e., as long as we stay
away from Planck energy and from the horizon.
So, despite appearances on human scales, there is no definite number of particles in
the universe. The universe cannot be distinguished from vacuum by counting particles.
Even though particles are necessary for our own existence and functioning, a complete
count of them cannot be made.
This conclusion is so strange that we should try to resist it. Let us try another method
of determining the content of matter in the universe: instead of counting particles in the
universe, let us weigh the universe.
w hat is the d ifference b et w een the univ erse a nd nothing?
Vol. I, page 100
Vol. II, page 65
Challenge 68 e
observation energy, it can be as small as the Planck length. This implies that the density
of the universe lies somewhere between the lowest possible value, given by the density of
vacuum ?min just mentioned, and the highest possible one, namely the Planck density.*
In short, the total mass of the universe depends on the energy of the observer.
Another way to measure the mass of the universe would be to apply the original definition of mass, as given in classical physics and as modified by special relativity. Thus, let
us try to collide a standard kilogram with the universe. It is not hard to see that whatever
we do, using either low or high energies for the standard kilogram, the mass of the universe cannot be constrained by this method. We would need to produce or to measure a
velocity change Δ? for the rest of the universe after the collision. To hit all the mass in
the universe at the same time, we need high energy; but then we are hindered by Planck
energy effects. In addition, a properly performed collision measurement would require
a mass outside the universe, which is rather difficult to achieve.
Yet another way to measure the mass would be to determine the gravitational mass
of the universe through straightforward weighing. But the lack of balances outside the
universe makes this an impractical solution, to say the least.
Another way out might be to use the most precise definition of mass provided by general relativity, the so-called ADM mass. However, the definition of this requires a specified
behaviour at infinity, i.e., a background, which the universe lacks.
We are then left with the other general-relativistic method: determining the mass of
the universe by measuring its average curvature. Let us take the defining expressions for
average curvature ? for a region of size ?, namely
Challenge 69 e
Ref. 102
1
2
?curvature
=
3 4π?2 − ? 15 4π?3 /3 − ?
=
.
4π
?4
4π
?5
(111)
We have to insert the horizon radius ?0 and either its surface area ?0 or its volume ?0 .
However, given the error margins on the radius and the volume, especially at Planck
energy, we again find no reliable result for the radius of curvature.
An equivalent method starts with the usual expression provided by Rosenfeld for the
indeterminacy Δ? in the scalar curvature for a region of size ?, namely
Δ? >
2
16π?Pl
.
?4
(112)
However, this expression also shows that the error in the radius of curvature behaves like
the error in the distance to the horizon.
We find that at Planck energy, the average radius of curvature of nature lies between
infinity and the Planck length. This implies that the mass density of the universe lies
Challenge 67 e
Vol. V, page 144
* In fact, at everyday energies the density of the universe lies midway between the two values, yielding the
strange relation
?20 /?20 ≈ ?2Pl /?2Pl = ?4 /?2 .
(110)
But this fascinating relation is not new. The approximate equality can be deduced from equation 16.4.3
(p. 620) of Steven Weinberg, Gravitation and Cosmology, Wiley, 1972, namely ????? = 1/?20 . The
relation is required by several cosmological models.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
?=
Motion Mountain – The Adventure of Physics
Vol. II, page 189
103
104
Challenge 70 s
5 w hat is the d ifference b et ween the univ erse a nd nothing?
between the minimum value and the Planck value. There is thus no method to determine
the mass of the universe at Planck energy. (Can you find one?)
In summary, mass measurements of the universe vary with the energy scale. Both at
the lowest and at the highest energies, a precise mass value cannot be determined. The
concept of mass cannot be applied to the universe as a whole: the universe has no mass.
D o symmetries exist in nature?
It is common to take ‘boundary’ and ‘horizon’ as synonyms in the case of the universe,
because they are the same for all practical purposes. Knowledge of mathematics does
not help us here: the properties of mathematical boundaries – for example, that they
themselves have no boundary – are not applicable to the universe, since space-time is
not continuous. We need other, physical arguments.
The boundary of the universe is supposed to represent the boundary between something and nothing. There are three possible interpretations of ‘nothing’:
— ‘Nothing’ could mean ‘no matter’. But we have just seen that this distinction cannot be made at Planck scales. So either the boundary will not exist at all or it will
encompass the horizon as well as the whole universe.
— ‘Nothing’ could mean ‘no space-time’. We then have to look for those domains
where space and time cease to exist. These occur at Planck scales and at the horizon.
Again, either the boundary will not exist or it will encompass the whole universe.
— ‘Nothing’ could mean ‘neither space-time nor matter’. The only possibility is a
boundary that encloses domains beyond the Planck scales and beyond the horizon;
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D oes the universe have a boundary?
Motion Mountain – The Adventure of Physics
Challenge 71 e
We have already seen that at the horizon, space-time translation symmetry breaks down.
Let us have a quick look at the other symmetries.
What happens to permutation symmetry? Permutation is an operation on objects in
space-time. It thus necessarily requires a distinction between matter, space and time. If
we cannot distinguish positions, we cannot talk about exchange of particles. Therefore, at
the horizon, general relativity and quantum theory together make it impossible to define
permutation symmetry.
Let us explore CPT symmetry. As a result of measurement errors or of limiting maximum or minimum values, it is impossible to distinguish between the original and the
transformed situations. Therefore we cannot claim that CPT is a symmetry of nature at
horizon scales. In other words, matter and antimatter cannot be distinguished at the
horizon.
Also gauge symmetry is not valid at horizon scale, as you may wish to check in detail
yourself. For its definition, the concept of gauge field requires a distinction between time,
space and mass; at the horizon this is impossible. We therefore also deduce that at the
horizon, concepts such as algebras of observables cannot be used to describe nature.
Renormalization breaks down too.
All symmetries of nature break down at the horizon. None of the vocabulary we use
to talk about observations – including terms such as such as ‘magnetic field’, ‘electric
field’, ‘potential’, ‘spin’, ‘charge’, or ‘speed’ – can be used at the horizon.
w hat is the d ifference b et w een the univ erse a nd nothing?
105
but again, such a boundary would also encompass all of nature.
Challenge 72 s
Ref. 103
Is the universe a set?
“
”
Cicero
We are used to thinking of the universe the sum of all matter and all space-time. In doing so, we imply that the universe is a set of mutually distinct components. This idea
has been assumed in three situations: in claiming that matter consists of particles; that
space-time consists of events (or points); and that different states consist of different initial conditions. However, our discussion shows that the universe is not a set of such
distinguishable elements. We have encountered several proofs: at the horizon, at the big
bang and at Planck scales, it becomes impossible to distinguish between events, between
particles, between observables, and between space-time and matter. In those domains,
distinctions of any kind become impossible. We have found that distinguishing between
two entities – for example, between a toothpick and a mountain – is only approximately possible. It is approximately possible because we live at energies well below the
Planck energy. The approximation is so good that we do not notice the error when we
distinguish cars from people and from toothpicks. Nevertheless, our discussion of the
situation at Planck energy shows that a perfect distinction is impossible in principle. It
is impossible to split the universe into separate parts.
Another way to reach this result is the following. Distinguishing between two entities requires different measurement results: for example, different positions, masses
or sizes. Whatever quantity we choose, at Planck energy the distinction becomes impossible. Only at everyday energies is it approximately possible.
In short, since the universe contains no distinguishable parts, there are no (mathematical) elements in nature. Simply put: the universe is not a set. We envisaged this possibility earlier on; now it is confirmed. The concepts of ‘element’ and ‘set’ are already too
* ‘The mistress and queen of all things is reason.’ Tusculanae Disputationes, 2.21.47. Marcus Tullius Cicero
(106–43 bce), was an influential lawyer, orator and politician at the end of the Roman republic.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Vol. III, page 327
Domina omnium et regina ratio.*
Motion Mountain – The Adventure of Physics
This is puzzling. When combining quantum theory and relativity, we do not seem to
be able to find a conceptual definition of the horizon that distinguishes it from what
it includes. A distinction is possible in general relativity alone, and in quantum theory
alone; but as soon as we combine the two, the boundary becomes indistinguishable from
its content. The interior of the universe cannot be distinguished from its horizon. There is
no boundary of the universe.
The difficulty in distinguishing the horizon from its contents suggests that nature may
be symmetric under transformations that exchange interiors and boundaries. This idea
is called holography, because it vaguely recalls the working of credit-card holograms. It
is a busy research field in high-energy physics.
We note that if the interior and the boundary of the universe cannot be distinguished,
the constituents of nature can neither be points nor tiny objects of any kind. The constituents of nature must be extended. But before we explore this topic, we continue with
our search for differences between the universe and nothing. The search leads us to our
next question.
106
Challenge 74 s
* Some people knew this long before physicists. For example, the belief that the universe is or contains
information was ridiculed most thoroughly in the popular science-fiction parody by Douglas Adams,
The Hitchhiker’s Guide to the Galaxy, 1979, and its sequels.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Challenge 73 e
specialized to describe the universe. The universe must be described by a mathematical
concept that does not contain any set. The new concept must be more general than that
of a set.
This is a powerful result: a precise description of the universe cannot use any concept
that presupposes the existence of sets. But all the concepts we have used so far to describe
nature, such as space-time, metric, phase space, Hilbert space and its generalizations, are
based on elements and sets. They must all be abandoned at Planck energies, and in any
precise description.
Elements and sets must be abandoned. Note that this radical conclusion is deduced
from only two statements: the necessity of using quantum theory whenever the dimensions are of the order of the Compton wavelength, and of using general relativity
whenever the dimensions are of the order of the Schwarzschild radius. Together, they
mean that no precise description of nature can contain elements and sets. The difficulties in complying with this result explain why the unification of the two theories has
not so far been successful. Not only does unification require that we stop using space,
time and mass for the description of nature; it also requires that all distinctions, of any
kind, should be only approximate. But all physicists have been educated on the basis of
exactly the opposite creed!
Many past speculations about the final unified description of nature depend on sets.
In particular, all studies of quantum fluctuations, mathematical categories, posets, involved mathematical spaces, computer programs, Turing machines, Gödel’s incompleteness theorem, creation of any sort, space-time lattices, quantum lattices and Bohm’s unbroken wholeness presuppose sets. In addition, all speculations by cosmologists about the
origin of the universe presuppose sets. But since these speculations presuppose sets, they
are wrong. You may also wish to check the religious explanations you know against this
criterion. In fact, no approach used by theoretical physicists up to the year 2000 satisfied
the requirement that elements and sets must be abandoned.
The task of abandoning sets is not easy. This is shown with a simple test: do you know
of a single concept not based on elements or sets?
In summary, the universe is not a set. In particular, the universe is not a physical
system. Specifically, it has no state, no intrinsic properties, no wave function, no initial
conditions, no energy, no mass, no entropy and no cosmological constant. The universe
is thus neither thermodynamically closed nor open; and it contains no information. All
thermodynamic quantities, such as entropy, temperature and free energy, are defined
using ensembles. Ensembles are limits of systems which are thermodynamically either
open or closed. As the universe is neither open nor closed, no thermodynamic quantity
can be defined for it.* All physical properties are defined only for parts of nature. Only
parts of nature are approximated or idealized as sets, and thus only parts of nature are
physical systems.
Motion Mountain – The Adventure of Physics
Ref. 104
5 w hat is the d ifference b et ween the univ erse a nd nothing?
w hat is the d ifference b et w een the univ erse a nd nothing?
107
Curiosities and fun challenges abou t the universe
“
“
Insofern sich die Sätze der Mathematik auf die
Wirklichkeit beziehen, sind sie nicht sicher, und
sofern sie sicher sind, beziehen sie sich nicht auf
die Wirklichkeit.*
Albert Einstein
”
”
Die ganzen Zahlen hat der liebe Gott gemacht,
alles andere ist Menschenwerk.**
Leopold Kronecker
∗∗
∗∗
Ref. 105
In 2002, Seth Lloyd estimated how much information the universe can contain, and how
many calculations it has performed since the big bang. This estimate is based on two
ideas: that the number of particles in the universe is a well-defined quantity, and that the
universe is a computer, i.e., a physical system. We now know that neither assumption
is correct. The universe contains no information. Conclusions such as this one show the
power of the criteria that we have deduced for any precise or complete description of
motion.
∗∗
Challenge 75 s
Astronomers regularly take pictures of the cosmic background radiation and its variations. Is it possible that these photographs will show that the spots in one direction of
the sky are exactly the same as those in the diametrically opposite direction?
∗∗
* ‘In so far as mathematical statements describe reality, they are not certain, and as far as they are certain,
they are not a description of reality.’
** ‘Gracious god made the integers, all else is the work of man.’ Leopold Kronecker (b. 1823 Liegnitz,
d. 1891 Berlin) was a well-known mathematician. Among others, the Kronecker delta and the Kronecker
product are named for him.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
If vacuum cannot be distinguished from matter or radiation, and if the universe cannot
be distinguished from nothing, then it is incorrect to claim that “the universe appeared
from nothing.” The naive idea of creation is a logical impossibility. “Creation” results
from a lack of imagination.
Motion Mountain – The Adventure of Physics
In mathematics, 2 + 2 = 4. This statement is an idealization of statements such as ‘two
apples plus two apples makes four apples.’ However, we now know that at Planck energy,
the statement about apples is not a correct statement about nature. At Planck energy,
objects cannot be counted or even defined, because separation of objects is not possible
at that scale. We can count objects only because we live at energies much lower than the
Planck energy.
The statement by Kronecker must thus be amended. Since all integers are low-energy
approximations, and since we always use low-energy approximations when talking or
thinking, we provokingly conclude: man also makes the integers.
108
Ref. 106
Challenge 76 s
5 w hat is the d ifference b et ween the univ erse a nd nothing?
In 1714, the famous scientist and thinker Leibniz (b. 1646 Leipzig, d. 1716 Hannover)
published his Monadologie. In it he explores what he calls a ‘simple substance’, which
he defined to be a substance that has no parts. He called it a monad and describes some
of its properties. However, mainly thanks to his incorrect deductions, the term has not
been generally adopted. What is the physical concept most closely related to that of a
monad?
∗∗
Challenge 77 s
We usually speak of the universe, implying that there is only one of them. Yet there is
a simple case to be made that ‘universe’ is an observer-dependent concept, since the
idea of ‘all’ is observer-dependent. Does this mean that there are many universes, or a
‘multiverse’?
∗∗
Is the ‘radius’ of the universe observer-invariant?
∗∗
Challenge 79 e
Is the cosmological constant Λ observer-invariant?
∗∗
Challenge 80 s
If all particles were removed (assuming one knew where to put them), there wouldn’t be
much of a universe left. True?
∗∗
Can you show that the distinction between matter and antimatter is not possible at the
cosmic horizon? And the distinction between real and virtual particles?
∗∗
Challenge 82 s
At Planck energy, interactions cannot be defined. Therefore, ‘existence’ cannot be
defined. In short, at Planck energy we cannot say whether particles exist. True?
Hilbert ’ s sixth problem set tled
Vol. III, page 281
Ref. 107
In the year 1900, David Hilbert* gave a famous lecture in which he listed 23 of the great
challenges facing mathematics in the twentieth century. Most of these provided challenges to many mathematicians for decades afterwards. A few are still unsolved, among
them the sixth, which challenged mathematicians and physicists to find an axiomatic
treatment of physics. The problem has remained in the minds of many physicists since
that time. Scholars have developed axiomatic treatments if classical mechanics, electrodynamics and special relativity. Then they did this for quantum theory, quantum field
theory and general relativity.
Whenever we combine quantum theory and general relativity, we must abandon the
concept of point particle, of space point and of event. Mathematically speaking, when
we combine quantum theory and general relativity, we find that nature does not contain
* David Hilbert (b. 1862 Königsberg, d. 1943 Göttingen) was the greatest mathematician of his time. His
textbooks are still in print.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Challenge 81 e
Motion Mountain – The Adventure of Physics
Challenge 78 e
w hat is the d ifference b et w een the univ erse a nd nothing?
109
sets, and that the universe is not a set. However, all mathematical systems – be they
algebraic systems, order systems, topological systems or a mixture of these – are based
on elements and sets. Mathematics does not have axiomatic systems without elements
or sets. The reason for this is simple: every (mathematical) concept contains at least one
element and one set. However, nature is different. And since nature does not contain
sets, an axiomatic description of nature is impossible.
All concepts used in physics before the year 2000 depend on elements and sets. For
humans, it is difficult even to think without first defining a set of possibilities. Yet nature
does not contain sets.
⊳ There is no axiomatic description of nature.
The perfect physics bo ok
Vol. I, page 435
A perfect physics book describes all of nature with full precision. In particular, a perfect
physics book describes itself, its own production, its own author, its own readers and its
own contents. Can such a book exist?
Since the universe is not a set, a perfect physics book can exist, as it does not contradict
any property of the universe. Since the universe is not a set and since it contains no
information, the paradox of the perfect physics book disappears. Indeed, any existing
physics book attempts to be perfect. But now a further question arises.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 218
Motion Mountain – The Adventure of Physics
Vol. I, page 436
And since an axiomatic formulation of physics is impossible, we conclude that the final,
unified theory cannot be based on axioms. This is surprising at first, because separate
axiomatic treatments of quantum theory and general relativity are possible. However,
axiomatic systems in physics are always approximate. The need to abandon axioms is one
of the reasons why reaching a unified description of nature is a challenge.
The impossibility of an axiomatic system for physics is also confirmed in another way.
Physics starts with a circular definition: space-time and vacuum are defined with the help
of objects and objects are defined with the help of space-time and vacuum. In fact, physics has never been axiomatic! Physicists have always had to live with circular definitions.
The situation is similar to a child’s description of the sky as ‘made of air and clouds’.
Looking closely, we discover that clouds are made up of water droplets. We find that
there is air inside clouds, and that there is also water vapour away from clouds. When
clouds and air are viewed through a microscope, there is no clear boundary between the
two. We cannot define either of the terms ‘cloud’ and ‘air’ without the other.
Like clouds and air, also objects and vacuum are indistinguishable. Virtual particles
are found in vacuum, and vacuum is found inside objects. At Planck scales there is no
clear boundary between the two; we cannot define either of the terms ‘particle’ and
‘vacuum’ without the other. But despite the lack of a clean definition, and despite the
logical problems that can ensue, in both cases the description works well at large, everyday scales.
In summary, an axiomatic description of nature is impossible. In particular, the final,
unified theory must contain circular definitions. We will find out how to realize the
requirement later on.
110
5 w hat is the d ifference b et ween the univ erse a nd nothing?
TA B L E 3 Statements about the universe when explored at highest precision, i.e., at Planck scales
The universe has no beginning.
The universe has no volume.
The universe’s particle number is undefined.
The universe has no energy.
The universe contains no matter.
The universe has no initial conditions.
The universe has no wave function.
The universe contains no information.
The universe is not open.
The universe does not interact.
The universe cannot be said to exist.
The universe cannot be distinguished from a
single event.
The universe is not composite.
The universe is not a concept.
There is no plural for ‘universe’.
The universe was not created.
D oes the universe make sense?
Vol. III, page 252
Challenge 83 r
Challenge 84 s
”
Is the universe really the sum of matter–energy and space-time? Or of particles and
vacuum? We have heard these statements so often that we may forget to check them.
We do not need magic, as Faust thought: we only need to list what we have found so far,
especially in this section, in the section on Planck scales, and in the chapter on brain and
language. Table 3 shows the result.
Not only are we unable to state that the universe is made of space-time and matter;
we are unable to say anything about the universe at all! It is not even possible to say that
it exists, since it is impossible to interact with it. The term ‘universe’ does not allow us to
make a single sensible statement. (Can you find one?) We are only able to list properties
it does not have. We are unable to find any property that the universe does have. Thus,
the universe has no properties! We cannot even say whether the universe is something
or nothing. The universe isn’t anything in particular. The term universe has no content.
By the way, there is another well-known, non-physical concept about which nothing
can be said. Many scholars have explored it in detail. What is it?
* ‘Thus I have devoted myself to magic, [ ... ] that I understand how the innermost world is held together.’
Goethe was a German poet.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
“
Drum hab ich mich der Magie ergeben,
[ ... ]
Daß ich erkenne, was die Welt
Im Innersten zusammenhält.*
Goethe, Faust.
Motion Mountain – The Adventure of Physics
The universe has no age.
The universe has no size.
The universe has no shape.
The universe has no mass.
The universe has no density.
The universe has no cosmological constant.
The universe has no state.
The universe is not a physical system.
The universe is not isolated.
The universe has no boundaries.
The universe cannot be measured.
The universe cannot be distinguished from
nothing.
The universe contains no moments.
The universe is not a set.
The universe cannot be described.
The universe cannot be distinguished from
vacuum.
w hat is the d ifference b et w een the univ erse a nd nothing?
Vol. III, page 283
111
In short, the term ‘universe’ is not at all useful for the description of motion. We
can obtain a confirmation of this strange conclusion from an earlier chapter. There we
found that any concept needs defined content, defined limits and a defined domain of
application. In this section, we have found that the term ‘universe’ has none of these;
there is thus no such concept. If somebody asks why the universe exists, the answer is:
not only does the use of the word ‘why’ wrongly suggest that something may exist outside
the universe, providing a reason for it and thus contradicting the definition of the term
‘universe’ itself; but more importantly, the universe does not exist, because there is no
such concept as a ‘universe’.
In summary, any sentence containing the word ‘universe’ is meaningless. The word only
seems to express something, but it doesn’t.* This conclusion may be interesting, even
strangely beautiful, but does it help us to understand motion more precisely? Yes, it
does.
Extremal scales and open questions in physics
Page 19
At the beginning of this volume, we listed all the fundamental properties of nature that
are unexplained either by general relativity or by quantum theory. We called it the millennium list. The results of this chapter provide us with surprising statements on many
of the items. In fact, many of the statements are not new at all, but are surprisingly familiar. Let us compare systematically the statements from this chapter, on the universe,
with those of the previous chapter, on Planck scales. The comparison is given in Table 4.
* Of course, the term ‘universe’ still makes sense if it is defined more restrictively: for example, as everything
interacting with a particular human or animal observer in everyday life. But such a definition, equating
‘universe’ and ‘environment’, is not useful for our quest, as it lacks the precision required for a description
of motion.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Challenge 85 e
Our discussion of the term ‘universe’ shows that the term cannot include any element or
set. And the same applies to the term ‘nature’. Nature cannot be made of atoms. Nature
cannot be made of space-time points. Nature cannot be made of separate, distinct and
discrete entities.
The difficulties in giving a sharp definition of ‘universe’ also show that the fashionable
term ‘multiverse’ makes no sense. There is no way to define such a term, since there is
no empirical way and also no logical way to distinguish ‘one’ universe from ‘another’:
the universe has no boundary. In short, since the term ‘universe’ has no content, the
term ‘multiverse’ has even less. The latter term has been created only to trick the media
and various funding agencies. In fact, the same might be said of the former term...
So far, by taking into account the limits on length, time, mass and all the other quantities we have encountered, we have reached a number of almost painful conclusions about
nature. However, we have also received something in exchange: all the contradictions
between general relativity and quantum theory that we mentioned at the beginning of
this chapter are now resolved. We changed the contradictions to circular definitions. Although we have had to leave many cherished habits behind us, in exchange we have the
promise of a description of nature without contradictions. But we get even more.
Motion Mountain – The Adventure of Physics
Aband oning sets and discreteness eliminates contradictions
112
5 w hat is the d ifference b et ween the univ erse a nd nothing?
TA B L E 4 Properties of nature at maximal, everyday and minimal scales
At
every d ay
scale
At P l anck
scales
requires quantum theory and relativity
intervals can be measured precisely
length and time intervals appear
space-time is not continuous
points and events cannot be distinguished
space-time is not a manifold
space is 3-dimensional
space and time are indistinguishable
initial conditions make sense
space-time fluctuates
Lorentz and Poincaré symmetry
CPT symmetry
renormalization
permutation symmetry
interactions and gauge symmetries
number of particles
algebras of observables
matter indistinguishable from vacuum
boundaries exist
nature is a set
true
false
limited
true
true
true
false
true
false
true
do not apply
does not apply
does not apply
does not apply
do not exist
undefined
undefined
true
false
false
false
true
unlimited
false
false
false
true
false
true
false
apply
applies
applies
applies
exist
defined
defined
false
true
true
true
false
limited
true
true
true
false
true
false
true
do not apply
does not apply
does not apply
does not apply
do not exist
undefined
undefined
true
false
false
First, Table 4 shows that each unexplained property listed there is unexplained at both
limits of nature, the small and the large limit. Worse, many of these unexplained general
properties do not even make sense at the two limits of nature!
Secondly, and more importantly, nature behaves in the same way at the cosmological
horizon scale and at the Planck scale. In fact, we have not found any difference between
the two cases. (Can you discover one?) We are thus led to the hypothesis that nature
does not distinguish between the large and the small. Nature seems to be characterized
by extremal identity.
Is extremal identit y a principle of nature?
The idea of extremal identity incorporates some rather general points:
—
—
—
—
—
All open questions about nature appear at both size extremes.
Any description of nature requires both general relativity and quantum theory.
Nature, or the universe, is not a set.
Initial conditions and evolution equations make no sense at nature’s limits.
There is a relation between local and global issues in nature.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
At hori zon scale
Motion Mountain – The Adventure of Physics
Challenge 86 r
P hysical p ropert y of nature
sum m ary on the univ erse
113
— The concept of ‘universe’ has no content.
Challenge 87 e
Ref. 108
Extremal identity thus looks like a useful hypothesis in the search for a unified description of nature. To be a bit more provocative, it seems that extremal identity may be the
only hypothesis incorporating the idea that the universe is not a set. Therefore, extremal
identity seems to be essential in the quest for unification.
Extremal identity is beautiful in its simplicity, in its unexpectedness and in the richness of its consequences. You might enjoy exploring it by yourself. In fact, the exploration of extremal identity is currently the subject of much activity in theoretical physics,
although often under different names.
The simplest approach to extremal identity – in fact, one that is too simple to be correct – is inversion. Indeed, extremal identity seems to imply a connection such as
?↔
Ref. 103
Page 85
?? ↔
2
??
?Pl
?? ??
(113)
relating distances ? or coordinates ?? with their inverse values using the Planck length
?Pl . Can this mapping be a symmetry of nature? At every point of space? For example,
if the horizon distance is inserted, the relation (113) implies that lengths smaller than
?Pl /1061 ≈ 10−96 m never appear in physics. Is this the case? What would inversion imply
for the big bang?
More involved approaches to extremal identity come under the name of space-time
duality and holography. They are subject of intense research. Numerous fascinating questions are contained in extremal identity; there is a lot of fun ahead of us.
Above all, we need to find the correct version of the inversion relation (113). Inversion
is neither sufficient nor correct. It is not sufficient because it does not explain any of the
millennium issues left open by general relativity and quantum theory. It only relates some
of them, but it does not solve any of them. (You may wish to check this for yourself.)
In other words, we need to find the precise description of quantum geometry and of
elementary particles.
However, inversion is also simply wrong. Inversion is not the correct description of
extremal identity because it does not realize a central result discovered above: it does
not connect states and intrinsic properties, but keeps them distinct. In particular, inversion does not take interactions into account. And most open issues at this point of our
mountain ascent concern the properties and the appearance of interactions.
summary on the universe
The exploration of the universe allows us to formulate several additional requirements
for the final theory that we are looking for:
— Whenever we combine general relativity and quantum theory, the universe teaches
us that it is not a set of parts. For this reason, any sentence or expression containing
the term ‘universe’ is meaningless whenever full precision is required.
— We learned that a description of nature without sets solves the contradictions between
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Challenge 89 e
or
Motion Mountain – The Adventure of Physics
Challenge 88 s
2
?Pl
?
114
5 w hat is the d ifference b et ween the univ erse a nd nothing?
general relativity and quantum theory.
— We found, again, that despite the contradictions between quantum theory and general relativity, the Planck limits ?, ℏ and ?4 /4? remain valid.
— We then found an intriguing relation between Planck scales and cosmological scales:
they seem to pose the same challenges to their description. There is a close relationship between large and small scales in nature.
We can now answer the question in the chapter title: there seems to be little difference
– if any at all – between the universe and nothing. We can express this result in the
following catchy statement:
⊳ The universe cannot be observed.
A physical aphorism
Ref. 55, Ref. 56
Inserting numbers, we find rather precisely that the time ? is the present age of the universe.
With the right dose of humour we can see this result as a sign that time is now ripe,
after so much waiting, for us to understand the universe down to the Planck scales. We
are thus getting nearer to the top of Motion Mountain. Be prepared for a lot of fun.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Challenge 90 s
Here is a humorous ‘proof’ that we really are near the top of Motion Mountain. Salecker
and Wigner, and then Zimmerman, formulated the fundamental limit for the measurement precision ? attainable by a clock of mass ?. It is given by ? = √ℏ?/?2 ? , where
? is the time to be measured. We can then ask what maximum time ? can be measured
with a precision of a Planck time ?Pl , given a clock of the mass of the whole universe. We
get a maximum time of
?2 ?2
? = Pl ? .
(114)
ℏ
Motion Mountain – The Adventure of Physics
In our journey, the confusion and tension are increasing. But in fact we are getting close
to our goal, and it is worth continuing.
Chapter 6
T HE SHAPE OF POINT S – EX T ENSION
IN NAT UR E
”
he usual expressions for the reduced Compton wavelength ? = ℏ/?? and for
he Schwarzschild radius ?s = 2??/?2 , taken together, imply the conclusion
hat at Planck energies, what we call ‘space points’ and ‘point particles’ must actually be described by extended constituents that are infinite and fluctuating in size. We
will show this result with the following arguments:
We conclude the chapter with some experimental and theoretical checks of extension
** ‘Nothing is so difficult that it could not be investigated.’ Terence is Publius Terentius Afer (b. c. 190
Carthago, d. 159 bce Greece), important Roman poet. He writes this in his play Heauton Timorumenos,
verse 675.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
1. Any experiment trying to measure the size or the shape of an elementary particle
with high precision inevitably leads to the result that at least one dimension of the
particle is of macroscopic size.
2. There is no evidence that empty space consists of points, as they cannot be measured
or detected. In addition, in order to build up a measurable entity, such as the vacuum,
that is extended in three dimensions, its constituents must also be extended.
3. The existence of minimum measurable distances and time intervals implies the existence of space-time duality: a symmetry between very large and very small distances.
Space-time duality in turn implies that the fundamental constituents that make up
vacuum and matter are extended.
4. The constituents of the universe, and thus of vacuum, matter and radiation, cannot
form a (mathematical) set. And any precise description of nature without sets must
use extended constituents.
5. The Bekenstein–Hawking expression for the entropy of black holes – in particular
its surface dependence – confirms that both vacuum and particles are composed of
extended constituents.
6. The attempt to extend statistical properties to Planck scales shows that both particles
and space points behave as extended constituents at high energies, namely as braids
or tangles.
7. The belt trick provides a model for fermions that matches observations and again
suggests extended constituents in matter.
Motion Mountain – The Adventure of Physics
T
“
Nil tam difficile est, quin quaerendo investigari
possiet.**
Terence
116
6 the sha pe of p oints
and an overview of present research efforts.
“
Also, die Aufgabe ist nicht zu sehen, was noch
nie jemand gesehen hat, sondern über dasjenige
was jeder schon gesehen hat zu denken was
noch nie jemand gedacht hat.*
Erwin Schrödinger
”
the size and shape of elementary particles
D o boxes exist?
Can the Greeks help? – The limitations of knives
The Greeks deduced the existence of atoms by noting that matter cannot be divided indefinitely. There must be uncuttable particles, which they called atoms. Twenty-five centuries later, experiments in the field of quantum physics confirmed the conclusion, but
* ‘Our task is not to see what nobody has ever seen, but to think what nobody has ever thought about that
which everybody has seen already.’ Erwin Schrödinger (b. 1887 Vienna, d. 1961 Vienna) discovered the
equation that brought him international fame and the Nobel Prize in Physics.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
The first and simplest way to determine the size of a compact particle such as a sphere,
or to find at least an upper limit, is to measure the size of a box it fits in. To be sure that
the particle is inside, we must first be sure that the box is tight: that is, whether anything
(such as matter or radiation) can leave the box.
But there is no way to ensure that a box has no holes! We know from quantum physics
that any wall is a finite potential hill, and that tunnelling is always possible. In short, there
is no way to make a completely tight box.
Let us cross-check this result. In everyday life, we call particles ‘small’ when they
can be enclosed. Enclosure is possible in daily life because walls are impenetrable. But
walls are only impenetrable for matter particles up to about 10 MeV and for photons up
to about 10 keV. In fact, boxes do not even exist at medium energies. So we certainly
cannot extend the idea of ‘box’ to Planck energy.
Since we cannot conclude that particles are of compact size by using boxes, we need
to try other methods.
Motion Mountain – The Adventure of Physics
Page 88
Size is the length of vacuum taken by an object. This definition comes naturally in everyday life, quantum theory and relativity. To measure the size of an object as small as an
elementary particle, we need high energy. The higher the energy, the higher the precision
with which we can measure the size.
However, near the Planck energy, vacuum and matter cannot be distinguished: it is
impossible to define the boundary between the two, and thus it is impossible to define
the size of an object. As a consequence, every object, and in particular every elementary
particle, becomes as extended as the vacuum! There is no measurement precision at all
at Planck scales. Can we save the situation? Let us take a step back. Do measurements at
least allow us to say whether particles can be contained inside small spheres?
the siz e a nd sha pe of elem entary pa rticles
117
modified it: nowadays, the elementary particles are the ‘atoms’ of matter and radiation.
Despite the huge success of the concept of elementary particle, at Planck energy, we
have a different situation. The use of a knife, like any other cutting process, is the insertion of a wall. Walls and knives are potential hills. All potential hills are of finite height,
and allow tunnelling. Therefore a wall is never perfect, and thus neither is a knife. In
short, any attempt to divide matter fails to work when we approach Planck scales. At
Planck energy, any subdivision is impossible.
The limitations of knives and walls imply that at Planck energy, an attempted cut does
not necessarily lead to two separate parts. At Planck energy, we can never state that the
two parts have been really, completely separated: the possibility of a thin connection
between the two parts to the right and left of the blade can never be excluded. In short,
at Planck scales we cannot prove compactness by cutting objects.
Are cross sections finite?
Vol. I, page 25
That is quite a statement. Are particles really not of finite, bounded size? Right at the
start of our mountain ascent, we distinguished objects from their environment. Objects
are by definition localized, bounded and compact. All objects have a boundary, i.e., a
surface which does not itself have a boundary. Objects are also bounded in abstract
ways: also the set of symmetries of an object, such as its geometric symmetry group or
its gauge group, is bounded. In contrast, the environment is not localized, but extended
and unbounded. But all these basic assumptions fail us at Planck scales. At Planck energy,
it is impossible to determine whether something is bounded or compact. Compactness
and locality are only approximate properties; they are not applicable at high energies.
In particular, the idea of a point particle is an approximate concept, valid only at low
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
⊳ Quantum particles are extended.
Motion Mountain – The Adventure of Physics
Ref. 109
To sum up: despite all attempts, we cannot show that elementary particles are point-like.
Are they, at least, of finite size?
To determine the size of a particle, we can try to determine its departure from pointlikeness. Detecting this departure requires scattering. For example, we can suspend the
particle in a trap and then shoot a probe at it. What happens in a scattering experiment at
highest energies? This question has been studied by Leonard Susskind and his colleagues.
When shooting at the particle with a high-energy probe, the scattering process is characterized by an interaction time. Extremely short interaction times imply sensitivity to
size and shape fluctuations, due to the quantum of action. An extremely short interaction time also provides a cut-off for high-energy shape and size fluctuations, and thus
determines the measured size. As a result, the size measured for any microscopic, but
extended, object increases when the probe energy is increased towards the Planck value.
In summary, even though at experimentally achievable energies the size of an elementary particle is always smaller than the measurement limit, when we approach the
Planck energy, the particle size increases beyond all bounds. So at high energies we cannot give an upper limit to the size of a particle – except the universe itself. In other
words, since particles are not point-like at everyday energies, at Planck energy they are
enormous:
118
6 the sha pe of p oints
TA B L E 5 Effects of various camera shutter times on photographs
O b s e rvat i o n p o s s i b i l i t i e s a n d e f f e c t s
1h
1s
20 ms
10 ms
high
high
lower
lower
0.25 ms
lower
1 μs
c. 10 ps
very low
lowest
10 fs
100 zs
shorter times
higher
high
very high
10−43 s
highest
Ability to see faint quasars at night if motion is compensated
Everyday motion completely blurred
Interruption by eyelids; small changes impossible to see
Effective eye/brain shutter time; tennis ball impossible to see while
hitting it
Shortest commercial photographic camera shutter time; ability to
photograph fast cars
Ability to photograph flying bullets; strong flashlight required
Study of molecular processes; ability to photograph flying light
pulses; laser light required to get sufficient illumination
Light photography impossible because of wave effects
X-ray photography impossible; only ?-ray imaging left over
Photographs get darker as illumination decreases; gravitational effects significant
Imaging impossible
energies.
We conclude that particles at Planck scales are as extended as the vacuum. Let us
perform another check.
“
Ref. 110
Ref. 47, Ref. 24
Καιρὸν γνῶθι.*
”
Pittacus
Humans – or any other types of observers – can only observe the world with finite resolution in time and in space. In this respect, humans resemble a film camera. Every
camera has a resolution limit: it can only distinguish two events if they are a certain
minimum distance apart and separated by a certain minimum time. What is the best
resolution possible? The value was (almost) discovered in 1899: the Planck time and the
Planck length. No human, no film camera and no apparatus can measure space or time
intervals smaller than the Planck values. But what would happen if we took photographs
with shutter times that approach the Planck time?
Imagine that you have the world’s best shutter and that you are taking photographs
at shorter and shorter times. Table 5 gives a rough overview of the possibilities. When
shutter times are shortened, photographs get darker and sharper. When the shutter time
reaches the oscillation time of light, strange things happen: light has no chance to pass
undisturbed; signal and noise become indistinguishable; and the moving shutter will
produce colour shifts. In contrast to our everyday experience, the photograph would get
* ‘Recognize the right moment.’ Also rendered as: ‘Recognize thy opportunity.’ Pittacus (Πιττακος) of
Mytilene (c. 650–570 BCE ), was a Lesbian tyrant and lawmaker; he was also one of the ‘Seven Sages’ of
ancient Greece.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Can we take a photo graph of a p oint?
Motion Mountain – The Adventure of Physics
D uration Blur
the siz e a nd sha pe of elem entary pa rticles
119
Since particles are not point-like, they have a shape. How can we determine it? We determine the shape of an everyday object by touching it from all sides. This works with
plants, people or machines. It even works with molecules, such as water molecules. We
can put them (almost) at rest, for example in ice, and then scatter small particles off them.
Scattering is just a higher-energy version of touching. However, scattering cannot determine shapes of objects smaller than the wavelength of the probes used. To determine
the shape of an object as small as an electron, we need the highest energies available. But
we already know what happens when approaching Planck scales: the shape of a particle
becomes the shape of all the space surrounding it. In short, the shape of an electron
cannot be determined in this way.
Another way to determine the shape is to build a tight box around the system under
investigation and fill it with molten wax. We then let the wax cool and observe the hollow
part. However, near Planck energy, boxes do not exist. We are unable to determine the
shape in this way.
A third way to measure the shape of an object is to cut it into pieces and then study
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What is the shape of an electron?
Motion Mountain – The Adventure of Physics
more blurred and incorrect at extremely short shutter times. Photography is impossible
not only at long but also at short shutter times.
The difficulty of taking photographs is independent of the wavelength used. The limits
move, but do not disappear. With a shutter time of ?, photons of energy lower than ℏ/?
cannot pass the shutter undisturbed.
In short, the blur decreases when shutter times usual in everyday life are shortened,
but increases when shutter times are shortened further towards Planck times. As a result,
there is no way to detect or confirm the existence of point objects by taking pictures.
Points in space, as well as instants of time, are imagined concepts: they do not belong in
a precise description of nature.
At Planck shutter times, only signals with Planck energy can pass through the shutter.
Since at these energies matter cannot be distinguished from radiation or from empty
space, all objects, light and vacuum look the same. It is impossible to say what nature
looks like at very short times.
But the situation is worse than this: a Planck shutter cannot exist at all, as it would
need to be as small as a Planck length. A camera using it could not be built, as lenses do
not work at this energy. Not even a camera obscura – without any lens – would work,
as diffraction effects would make image production impossible. In other words, the idea
that at short shutter times a photograph of nature shows a frozen image of everyday life,
like a stopped film, is completely wrong. In fact, a shutter does not exist even at medium
energy: shutters, like walls, stop existing at around 10 MeV. At a single instant of time,
nature is not frozen at all. Zeno criticized this idea in his discussions of motion, though
not as clearly as we can do now. At short times, nature is blurred. In particular, point
particles do not exist.
In summary, whatever the intrinsic shape of what we call a ‘point’ might be, we know
that, being always blurred, it is first of all a cloud. Whatever method is used to photograph an elementary particle, the picture is always extended. Therefore we need to study
its shape in more detail.
120
Ref. 47
Is the shape of an electron fixed?
Page 85
Only an object composed of localized constituents, such as a house or a molecule, can
have a fixed shape. The smaller the system, the more quantum fluctuations play a role.
No small entity of finite size – in particular, no elementary particle – can have a fixed
shape. In every thought experiment involving a finite shape, the shape itself fluctuates.
But we can say more.
The distinction between particles and environment rests on the idea that particles
have intrinsic properties. In fact, all intrinsic properties, such as spin, mass, charge, and
parity, are localized. But we have seen that no intrinsic property is measurable or definable at Planck scales. Thus it is impossible to distinguish particles from the environment.
In addition, at Planck energy particles have all the properties that the environment has.
In particular, particles are extended.
In short, we cannot prove by experiments that at Planck energy elementary particles
are finite in size in all directions. In fact, all experiments we can think of are compatible
with extended particles, with ‘infinite’ size. More precisely, a particle always reaches the
borders of the region of space-time under exploration. In simple words, we can also say
that particles have tails.
Not only are particles extended, but their shape cannot be determined by the methods
just explored. The only remaining possibility is that suggested by quantum theory: the
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Challenge 91 e
the pieces. As is well known, the term ‘atom’ just means ‘uncuttable’ or ‘indivisible’.
However, neither atoms nor indivisible particles can exist. Indeed, cutting is just a lowenergy version of a scattering process. And the process does not work at high energies.
Therefore, there is no way to prove that an object is indivisible at Planck scales. Our
everyday intuition leads us completely astray at Planck energy.
We could try to distinguish transverse and longitudinal shape, with respect to the direction of motion. However, for transverse shape we get the same issues as for scattering;
transverse shape diverges for high energy. And to determine longitudinal shape, we need
at least two infinitely high potential walls. We already know that this is impossible.
A further, indirect way of measuring shapes is to measure the moment of inertia.
A finite moment of inertia means a compact, finite shape. But when the measurement
energy is increased towards Planck scales, rotation, linear motion and exchange become
mixed up. We do not get meaningful results.
Yet another way to determine shapes is to measure the entropy of a collection of
particles we want to study. This allows us to determine the dimensionality and the number of internal degrees of freedom. But at high energies, a collection of electrons would
become a black hole. We will study this issue separately below, but again we find no new
information.
Are these arguments watertight? We assumed three dimensions at all scales, and that
the shape of the particle itself is fixed. Maybe these assumptions are not valid at Planck
scales? Let us check the alternatives. We have already shown that because of the fundamental measurement limits, the dimensionality of space-time cannot be determined at
Planck scales. Even if we could build perfect three-dimensional boxes, holes could remain in other dimensions. It does not take long to see that all the arguments against
compactness work even if space-time has additional dimensions.
Motion Mountain – The Adventure of Physics
Ref. 47
6 the sha pe of p oints
the sha pe of p oints in vacuum
Vol. IV, page 108
Ref. 28
121
Summary of the first argument for extension
Ref. 47
the shape of points in vacuum
Ref. 112
“
Thus, since there is an impossibility that [finite]
quantities are built from contacts and points, it
is necessary that there be indivisible material
elements and [finite] quantities.
Aristotle,*** Of Generation and Corruption.
”
We are used to the idea that empty space is made of spatial points. However, at Planck
scales, no measurement can give zero length, zero mass, zero area or zero volume. There
Ref. 111
* Examples are the neutron, positronium, or the atoms. Note that the argument does not change when the
elementary particle itself is unstable, like the muon. The possibility that all components are heavier than
the composite, which would avoid this argument, does not seem to lead to satisfying physical properties:
for example, it leads to intrinsically unstable composites.
** Thus at Planck scales there is no quantum Zeno effect.
*** Aristotle (b. 384/3 Stageira, d. 322 bce Chalkis), Greek philosopher and scientist.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Challenge 92 e
Point particles do not exist at Planck scales. At Planck scales, all thought experiments
with partciles suggest that matter and radiation are made of extended and fluctuating
constituents of infinite size.
For extended constituents, the requirement of a non-local description is satisfied. The
argument forbidding composition of elementary particles is circumvented, as extended
constituents have no mass. Thus the concept of Compton wavelength cannot be defined
or applied to extended constituents, and elementary particles can have constituents if
these constituents are extended and massless. However, if the constituents are infinitely
extended, how can compact, point-like particles be formed from them? We will look at
a few options shortly.
Motion Mountain – The Adventure of Physics
shape of a particle fluctuates.
We reach the same conclusions for radiation particles. The box argument shows that
radiation particles are also extended and fluctuating.
Incidentally, we have also settled an important question about elementary particles.
We have already seen that any particle that is smaller than its own Compton wavelength
must be elementary. If it were composite, there would be a lighter component inside
it; this lighter particle would have a larger Compton wavelength than the composite
particle. This is impossible, since the size of a composite particle must be larger than
the Compton wavelength of its components.*
However, an elementary particle can have constituents, provided that they are not
compact. The difficulties of compact constituents were described by Andrei Sakharov
in the 1960s. If the constituents are extended, the previous argument does not apply, as
extended constituents have no localized mass. As a result, if a flying arrow – Zeno’s
famous example – is made of extended constituents, it cannot be said to be at a given
position at a given time. Shortening the observation time towards the Planck time makes
an arrow disappear in the cloud that makes up space-time.**
122
* Imagining the vacuum as a collection of compact constituents, such as spheres, with Planck size in all
directions would avoid the Banach–Tarski paradox, but would not allow us to deduce the number of dimensions of space and time. It would also contradict all the other results of this section. Therefore we do
not explore it further.
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Ref. 113
Challenge 93 s
is no way to state that something in nature is a point without contradicting experimental
results.
Furthermore, the idea of a point is an extrapolation of what is found in small empty
boxes getting smaller and smaller. But we have just seen that at high energies small boxes
cannot be said to be empty. In fact, boxes do not exist at all, as they can never have
impenetrable walls at high energies.
Also, the idea of a point as a limiting subdivision of empty space is untenable. At
small distances, space cannot be subdivided, as division requires some sort of dividing
wall, which is impossible.
Even the idea of repeatedly putting a point between two others cannot be applied. At
high energy, it is impossible to say whether a point is exactly on the line connecting the
outer two points; and near Planck energy, there is no way to find a point between them
at all. In fact, the term ‘in between’ makes no sense at Planck scales.
We thus find that space points do not exist, just as point particles do not exist. But
there are other reasons why space cannot be made of points. In order to form space.
points need to be kept apart somehow. Indeed, mathematicians have a strong argument
for why physical space cannot be made of mathematical points: the properties of mathematical spaces described by the Banach–Tarski paradox are quite different from those
of the physical vacuum. The Banach–Tarski paradox states that a sphere made of mathematical points can be cut into five pieces which can be reassembled into two spheres
each of the same volume as the original sphere. Mathematically, there are sets of points
for which the concept of volume makes no sense. Physically speaking, we conclude that
the concept of volume does not exist for continuous space; it is only definable if an intrinsic length exists. And in nature, an intrinsic length exists for matter and for vacuum:
the Planck length. And any concept with an intrinsic length must be described by one
or several extended constituents.* In summary, in order to build up space, we need extended constituents.
Also the number of space dimensions is problematic. Mathematically, it is impossible
to define the dimension of a set of points on the basis of the set structure alone. Any
compact one-dimensional set has as many points as any compact three-dimensional set
– indeed, as any compact set of any dimensionality greater than zero. To build up the
physical three-dimensional vacuum, we need constituents that organize their neighbourhood. The fundamental constituents must possess some sort of ability to form bonds,
which will construct or fill precisely three dimensions. Bonds require extended constituents. A collection of tangled constituents extending to the maximum scale of the region
under consideration would work perfectly. Of course, the precise shape of the fundamental constituents is not yet known. In any case, we again find that any constituents of
physical three-dimensional space must be extended.
In summary, we need extension to define dimensionality and to define volume. This
result is not surprising. We deduced above that the constituents of particles are extended.
Since vacuum is not distinguishable from matter, we would expect the constituents of
Motion Mountain – The Adventure of Physics
Ref. 47
6 the sha pe of p oints
the sha pe of p oints in vacuum
123
vacuum to be extended as well. Stated simply, if elementary particles are not point-like,
then points in the vacuum cannot be either.
Measuring the void
What is the maximum number of particles that fit inside a piece
of vacuum?
Summary of the second argument for extension
Vol. I, page 338
Ref. 114
Planck scales imply that space is made of fluctuating extended constituents of huge size.
Like particles, also space and vacuum are not made of points, but of a web. Vacuum
requires a statistical description.
More than two thousand years ago, the Greeks argued that matter must be made of
particles because salt can be dissolved in water and because fish can swim through water.
Now that we know more about Planck scales, we have to reconsider this argument. Like
fish swimming through water, particles can move through vacuum; but since vacuum
has no bounds and cannot be distinguished from matter, vacuum cannot be made of
localised particles. However, another possibility allows for motion of particles through
a vacuum: both vacuum and particles might be made of a web of extended constituents.
Let us study this possibility in more detail.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Another approach to counting the number of points in a volume is to fill a piece of vacuum with point particles.
The maximum mass that fits into a piece of vacuum is a black hole. But in this case
too, the maximum mass depends only on the surface of the given region of vacuum. The
maximum mass increases less rapidly than the volume. In other words, the number of
physical points inside a region of space is only proportional to the surface area of the
region. We are forced to conclude that vacuum must be made of extended constituents
crossing the whole region, independently of its shape.
Motion Mountain – The Adventure of Physics
To check whether the constituents of the vacuum are extended, let us perform a few
additional thought experiments. First, let us measure the size of a point in space. The
clearest definition of size is in terms of the cross section. How can we determine the cross
section of a point? We can determine the cross section of a piece of vacuum and then
determine the number of points inside it. However, at Planck energy, we get a simple
result: the cross section of a volume of empty space is independent of depth. At Planck
energy, vacuum has a surface, but no depth. In other words, at Planck energy we can only
state that a Planck layer covers the surface of a region. We cannot say anything about its
interior. One way to picture this result is to say that what we call ‘space points’ are in
fact long tubes.
Another way to determine the size of a point is to count the points found in a given
volume of space-time. One approach is to count the possible positions of a point particle
in a volume. However, at Planck energy point particles are extended and indistinguishable from vacuum. At Planck energy, the number of points is given by the surface area
of the volume divided by the Planck area. Again, the surface dependence suggests that
particles and the constituents of space are long tubes.
124
6 the sha pe of p oints
the l arge, the small and their connection
“
”
Is small l arge?
Ref. 116
“
[Zeno of Elea maintained:] If the existing are
many, it is necessary that they are at the same
time small and large, so small to have no size,
and so large to be without limits.
Simplicius***
”
* William Shakespeare (1564 Stratford upon Avon–1616 Stratford upon Avon) wrote theatre plays that are
treasures of world literature.
** There is also an S-duality, which connects large and small coupling constants, and a U-duality, which is
the combination of S- and T-duality.
*** Simplicius of Cilicia (c. 499 – 560), neoplatonist philosopher.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
If two observables cannot be distinguished, there is a symmetry transformation connecting them. For example, by a change of observation frame, an electric field may (partially)
change into a magnetic one. A symmetry transformation means that we can change the
viewpoint (i.e., the frame of observation) in such a way that the same observation is described by one quantity from one viewpoint and by the corresponding quantity from the
other viewpoint.
When measuring a length at Planck scales it is impossible to say whether we are
measuring the length of a piece of vacuum, the Compton wavelength of a body, or the
Schwarzschild diameter of a body. For example, the maximum size for an elementary
object is its Compton wavelength. The minimum size for an elementary object is its
Schwarzschild radius. The actual size of an elementary object is somewhere in between.
If we want to measure the size precisely, we have to go to Planck energy; but then all these
quantities are the same. In other words, at Planck scales, there is a symmetry transformation between Compton wavelength and Schwarzschild radius. In short, at Planck scales
there is a symmetry between mass and inverse mass.
As a further consequence, at Planck scales there is a symmetry between size and inverse size. Matter–vacuum indistinguishability means that there is a symmetry between
length and inverse length at Planck energy. This symmetry is called space-time duality
or T-duality in the research literature of superstrings.** Space-time duality is a symmetry between situations at scale ? ?Pl and at scale ??Pl /?, or, in other words, between
? and (??Pl )2 /?, where the number ? is usually conjectured to have a value somewhere
between 1 and 1000.
Duality is a genuine non-perturbative effect. It does not exist at low energy, since
duality automatically also relates energies ? and ?2Pl /? = ℏ?3 /??, i.e., it relates energies
below and above Planck scale. Duality is not evident in everyday life. It is a quantum
symmetry, as it includes Planck’s constant in its definition. It is also a general-relativistic
effect, as it includes the gravitational constant and the speed of light. Let us study duality
in more detail.
Motion Mountain – The Adventure of Physics
Ref. 115
I could be bounded in a nutshell and count
myself a king of infinite space, were it not that I
have bad dreams.
William Shakespeare,* Hamlet.
the l a rge, the sm a ll a nd their connection
125
Vol. III, page 92
Ref. 117
Ref. 115
So far, we have shown that at Planck energy, time and length cannot be distinguished,
and that vacuum and matter cannot be distinguished. Duality shows that mass and inverse mass cannot be distinguished. As a consequence, we deduce that length, time, and
mass cannot be distinguished from each other at all energies and scales! And since every
observable is a combination of length, mass and time, space-time duality means that there
is a symmetry between all observables. We call it the total symmetry.*
Total symmetry implies that there are many specific types of duality, one for each
pair of quantities under investigation. Indeed, the number of duality types discovered is
increasing every year. It includes, among others, the famous electric–magnetic duality we
first encountered in electrodynamics, coupling constant duality, surface–volume duality,
space-time duality, and many more. All this confirms that there is an enormous amount
of symmetry at Planck scales. In fact, similar symmetries have been known right from
the beginning of research in quantum gravity.
Most importantly, total symmetry implies that gravity can be seen as equivalent to all
other forces. Space-time duality thus shows that unification is possible. Physicists have
* A symmetry between size and Schwarzschild radius, i.e., a symmetry between length and mass, leads
to general relativity. Additionally, at Planck energy there is a symmetry between size and Compton
wavelength. In other words, there is a symmetry between length and inverse mass. This implies a symmetry between coordinates and wave functions, i.e., a symmetry between states and observables. It leads to
quantum theory.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Unification and total symmetry
Motion Mountain – The Adventure of Physics
To explore the consequences of duality, we can compare it to rotational symmetry in
everyday life. Every object in daily life is symmetrical under a full rotation of 2π. For the
rotation of an observer, angles make sense only as long as they are smaller than 2π. If a
rotating observer were to insist on distinguishing angles of 0, 2π, 4π etc., he would get a
new copy of the universe at each full turn.
2
Similarly, in nature, scales ? and ?Pl
/? cannot be distinguished. Lengths make no
sense when they are smaller than ?Pl . If, however, we insist on using even smaller values
and on distinguishing them from large ones, we get a new copy of the universe at those
small scales. Such an insistence is part of the standard continuum description of motion, where it is assumed that space and time are described by the real numbers, which
are defined over arbitrarily small intervals. Whenever the (approximate) continuum de2
scription with infinite extension is used, the ? ↔ ?Pl
/? symmetry pops up.
Duality implies that diffeomorphism invariance is only valid at medium scales, not at
extremal ones. At extremal scales, quantum theory has to be taken into account in the
proper manner. We do not yet know how to do this.
Space-time duality means that introducing lengths smaller than the Planck length (as
when one defines space points, which have size zero) means at the same time introducing
things with very large (‘infinite’) value. Space-time duality means that for every small
enough sphere the inside equals the outside.
Duality means that if a system has a small dimension, it also has a large one, and vice
versa. There are thus no small objects in nature. So space-time duality is consistent with
the idea that the basic constituents are extended.
126
Summary of the third argument for extension
Challenge 95 e
Unification implies thinking in terms of duality and the concepts that follow from it.
The large and the small are connected. Duality points to one single type of extended
constituents that defines all physical observables.
We still need to understand exactly what happens to duality when we restrict ourselves
to low energies, as we do in everyday life. We explore this now.
* Renormalization energy does connect different energies, but not in the correct way; in particular, it does
not include duality.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
always dreamt about unification. Duality tells us that this dream can indeed be realized.
It may seem that total symmetry completely contradicts what was said in the previous
section, where we argued that all symmetries are lost at Planck scales. Which result is
correct? Obviously, both of them are.
At Planck scales, all low-energy symmetries are indeed lost. In fact, all symmetries
that imply a fixed energy are lost. However, duality and its generalizations combine both
small and large dimensions, or large and small energies. Most of the standard symmetries of physics, such as gauge, permutation and space-time symmetries, are valid at each
fixed energy separately. But nature is not made this way. The precise description of nature
requires us to take into consideration large and small energies at the same time. In everyday life, we do not do that. The physics of everyday life is an approximation to nature
valid at low and fixed energies. For most of the twentieth century, physicists tried to
reach higher and higher energies. We believed that precision increases with increasing
energy. But when we combine quantum theory and gravity we are forced to change this
approach. To achieve high precision, we must take high and low energy into account at
the same time.*
The great differences between the phenomena that occur at low and high energies are
the main reason why unification is so difficult. We are used to dividing nature along
a scale of energies: high-energy physics, atomic physics, chemistry, biology, and so on.
But we are not allowed to think in this way any more. We have to take all energies into
account at the same time. That is not easy, but we do not have to despair. Important
conceptual progress was made in the last decade of the twentieth century. In particular,
we now know that we need only one constituent for all things that can be measured.
Since there is only one constituent, total symmetry is automatically satisfied. And
since there is only one constituent, there are many ways to study it. We can start from any
(low-energy) concept in physics and explore how it looks and behaves when we approach
Planck scales. In the present section, we are looking at the concept of ‘point’. Obviously,
the conclusions must be the same whatever concept we start with, be it electric field, spin,
or any other. Such studies thus provide a check for the results in this section.
Motion Mountain – The Adventure of Physics
Challenge 94 d
6 the sha pe of p oints
d oes nature hav e pa rts?
127
does nature have parts?
“
Ref. 118
”
Pluralitas non est ponenda sine necessitate.*
William of Occam
Another argument, independent of those given so far, points towards a model of nature
based on extended constituents. We know that any concept for which we can distinguish parts is described by a set. We usually describe nature as a set of objects, positions,
instants and so on. The most famous set-theoretic description of nature is the oldest
known, given by Democritus:
The world is made of indivisible particles and void.
Vol. III, page 327
The conclusion does not come as a surprise. We have already encountered another reason
to doubt that nature is a set. Whatever definition we use for the term ‘particle’, Democritus cannot be correct for a purely logical reason. The description he provided is not
complete. Every description of nature that defines nature as a set of parts fails to explain
the number of these parts. In particular, the number of particles and the number of dimensions of space-time must be specified if we describe nature as made from particles
and vacuum. For example, we saw that it is rather dangerous to make fun of the famous
statement by Arthur Eddington
* ‘Multitude should not be introduced without necessity.’ This famous principle is commonly called Occam’s razor. William of Ockham (b. 1285/1295 Ockham, d. 1349/50 Munich), or Occam in the common
Latin spelling, was one of the great thinkers of his time. In his famous statement he expresses that only
those concepts which are strictly necessary should be introduced to explain observations. It can be seen as
the requirement to abandon beliefs when talking about nature. But at this stage of our mountain ascent it
has an even more direct interpretation: the existence of any multitude in nature is questionable.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
⊳ Nature cannot be a set.
Motion Mountain – The Adventure of Physics
This description was extremely successful in the past: there are no discrepancies with
observations. However, after 2500 years, the conceptual difficulties of this approach are
obvious.
We know that Democritus was wrong, first of all, because vacuum and matter cannot be distinguished at Planck scales. Thus the word ‘and’ in his sentence is already a
mistake. Secondly, because of the existence of minimal scales, the void cannot be made
of ‘points’, as we usually assume. Thirdly, the description fails because particles are not
compact objects. Finally, total symmetry implies that we cannot distinguish parts in
nature. Nothing can be distinguished from anything else with complete precision, and
thus the particles or points in space that make up the naive model of the world cannot
exist.
In summary, quantum theory and general relativity together show that in nature, all
partitions and all differences are only approximate. Nothing can really be distinguished
from anything else with complete precision. In other words, there is no way to define a
‘part’ of nature, whether for matter, space, time, or radiation.
128
Ref. 119
Ref. 120
In fact, practically all physicists share this belief, although they usually either pretend to
favour some other number, or worse, keep the number unspecified.
In modern physics, many specialized sets are used to describe nature. We have used
vector spaces, linear spaces, topological spaces and Hilbert spaces. But so far, we consistently refrained, like all physicists, from asking about the origin of their sizes (mathematically speaking, of their dimensionality or cardinality). In fact, it is just as unsatisfying
to say that the universe contains some specific number of atoms as it is to say that spacetime is made of point-like events arranged in 3+1 dimensions. Both are statements about
set sizes, in the widest sense. In a complete, unified description of nature the number of
smallest particles and the number of space-time points must not be fixed beforehand,
but must result from the description.
Any part of nature is by definition smaller than the whole of nature, and different from
other parts. As a result, no description of nature by a set can possibly yield the number
of particles or the dimensionality of space-time. As long as we insist on using spacetime or Hilbert spaces for the description of nature, we cannot understand the number
of dimensions or the number of particles.
That is not too bad, as we know already that nature is not made of parts. We know
that parts are only approximate concepts. In short, if nature were made of parts, it could
not be a unity, or a ‘one.’ On the other hand, if nature is a unity, it cannot have parts.*
Nature cannot be separable exactly. It cannot be made of particles.
To sum up, nature cannot be a set. Sets are lists of distinguishable elements. When
general relativity and quantum theory are unified, nature shows no elements: nature
stops being a set at Planck scales. This result clarifies a discussion we started earlier in
relation to classical physics. There we discovered that matter objects were defined using
space and time, and that space and time were defined using objects. Along with the
results of quantum theory, this implies that in modern physics particles are defined in
terms of the vacuum and the vacuum in terms of particles. Circularity is not a good
idea, but we can live with it – at low energy. But at Planck energy, vacuum and particles
are indistinguishable from each other. Particles and vacuum – thus everything – are
the same. We have to abandon the circular definition. This is a satisfactory outcome;
however, it also implies that nature is not a set.
Also space-time duality implies that space is not a set. Space-time duality implies
that events cannot be distinguished from each other, and thus do not form elements of
some space. Phil Gibbs has given the name event symmetry to this property of nature.
This thought-provoking term, although still containing the term ‘event’, emphasizes the
impossibility to use a set to describe space-time.
In short,
* As a curiosity, practically the same discussion can already be found in Plato’s Parmenides, written in the
fourth century bce. There, Plato musically ponders different arguments on whether nature is or can be a
unity or a multiplicity, i.e., a set. It seems that the text is based on the real visit to Athens by Parmenides
and Zeno. (Their home city, Elea, was near Naples.) Plato does not reach a conclusion. Modern physics,
however, does.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 121
I believe there are 15,747,724,136,275,002,577,605,653,961,181,555,468,044,
717,914,527,116,709,366,231,425,076,185,631,031,296 protons in the universe
and the same number of electrons.
Motion Mountain – The Adventure of Physics
Page 101
6 the sha pe of p oints
d oes nature hav e pa rts?
129
⊳ Nature cannot be made of vacuum and particles.
This is a bizarre result. Atomists, from Democritus to Galileo, have been persecuted
throughout history. Were their battles all in vain? Let us continue to clarify our thoughts.
D oes the universe contain anything?
An amoeba
“
”
We have found that parts are approximate concepts. The parts of nature are not strictly
smaller than nature itself. As a result, any ‘part’ must be extended. Let us try to extract
some more information about the constituents of nature.
In any unified theory, all the concepts that appear must be only approximately parts
of the whole. Thus we need an entity Ω, describing nature, which is not a set but which
can be approximated by one. This is strange. We are all convinced very early in our lives
that we are a part of nature. Our senses provide us with this information. We are not
used to thinking otherwise. But now we have to.
Let us straight away eliminate a few options for Ω. One concept without parts is the
empty set. Perhaps we need to construct a description of nature from the empty set? We
could be inspired by the usual construction of the natural numbers from the empty set.
However, the empty set makes only sense as the opposite of some full set. So the empty
set is not a candidate for Ω.
Another possible way to define approximate parts is to construct them from multiple
copies of Ω. But in this way we would introduce a new set through the back door. Furthermore, new concepts defined in this way would not be approximate.
We need to be more imaginative. How can we describe a whole which has no parts,
but which has parts approximately? Let us recapitulate. The world must be described by
a single entity, sharing all properties of the world, but which can be approximated as a set
of parts. For example, the approximation should yield a set of space points and a set of
particles. But also, whenever we look at any ‘part’ of nature, without any approximation,
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Vol. III, page 290
A theory of everything describing nothing is
not better than a theory of nothing describing
everything.
Anonymous
Motion Mountain – The Adventure of Physics
To state that the universe contains something implies that we are able to distinguish the
universe from its contents. However, we now know that precise distinctions are impossible. If nature is not made of parts, it is wrong to say that the universe contains
something.
Let us go further. We need a description of nature that allows us to state that at Planck
energy nothing can be distinguished from anything else. For example, it must be impossible to distinguish particles from each other or from the vacuum. There is only one
solution: everything – or at least, what we call ‘everything’ in everyday life – must be
made of the same single constituent. All particles are made of one ‘piece’. Every point
in space, every event, every particle and every instant of time must be made of the same
single constituent.
130
6 the sha pe of p oints
Summary of the fourth argument for extension
Challenge 96 r
* This is the simplest model; but is it the only way to describe nature?
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 146
The lack of particles and of sets in nature leads to describing nature by a single constituent. Nature is thus modelled by an entity which is one single ‘object’ (to eliminate distinguishability), which is extended (to eliminate localizability) and which is fluctuating (to
ensure approximate continuity). Nature is a far-reaching, fluctuating fold. Nature is similar to an amoeba. The tangled branches of the amoeba allow a definition of length via
counting of the folds. In this way, discreteness of space, time, and particles could also be
realized; the quantization of space-time, matter and radiation thus follows. Any flexible
and deformable entity is also a perfect candidate for the realization of diffeomorphism
invariance, as required by general relativity.
A simple candidate for the extended fold is the image of a fluctuating, flexible tube
of Planck diameter. Counting tubes implies determining distances or areas. The minimum possible count (one) gives the minimum distance, from which quantum theory
is derived. In fact, at this point we can use as a model any flexible object with a small
dimension, such as a tube, a thin sheet, a ball chain or a woven collection of rings. We
will explore these options below.
Motion Mountain – The Adventure of Physics
we should not be able to distinguish it from the whole world. Composite objects are
not always larger than their constituents. On the other hand, composed objects must
usually appear to be larger than their constituents. For example, space ‘points’ or ‘point’
particles are tiny, even though they are only approximations. Which concept without
boundaries can be at their origin? Using usual concepts, the world is everywhere at
the same time; if nature is to be described by a single constituent, this entity must be
extended.
The entity has to be a single one, but it must seem to be multiple: it has to be multiple
approximately, as nature shows multiple aspects. The entity must be something folded.
It must be possible to count the folds, but only approximately. (An analogy is the question of how many grooves there are on an LP or a CD: depending on the point of view,
local or global, one gets different answers.) Counting folds would correspond to a length
measurement.
The simplest model would be a single entity which is extended and fluctuating,
reaches spatial infinity, allows approximate localization, and thus allows approximate
definition of parts and points.* In more vivid imagery, nature could be described by
some deformable, folded and tangled entity: a giant, knotted amoeba. An amoeba slides
between the fingers whenever we try to grab a part of it. A perfect amoeba flows around
any knife trying to cut it. The only way to hold it would be to grab it in its entirety. However, for someone himself made of amoeba strands, this is impossible. He can only grab
it approximately, by catching part of it and approximately blocking it, for example using
a small hole, so that the escape takes a long time.
the entropy of b l ack holes
131
the entropy of bl ack holes
We are still collecting arguments to determining the shape of fundamental constituents.
Another approach is to study situations where particles appear in large numbers. Systems
composed of many particles behave differently depending on whether the particles are
point-like or extended. In particular, their entropy is different. Studying large-number
entropy thus allows us to determine the shape of components. The most revealing situations are those in which large numbers of particles are crammed in a small volume.
Therefore we are led to study the entropy of black holes. Indeed, black holes tell us a lot
about the fundamental constituents of nature.
A black hole is a body whose gravity is so strong that even light cannot escape. It is
easily deduced from general relativity that any body whose mass ? fits inside the socalled Schwarzschild radius
?S = 2??/?2
(115)
?=
Ref. 35
Ref. 123
Ref. 125
Ref. 124
?=?
4π??2
ℏ?
(116)
where ? is the Boltzmann constant and ? = 4π?S2 is the surface of the black hole horizon.
This important result has been derived in many different ways. The various derivations
confirm that space-time and matter are equivalent: they show that the entropy value can
be interpreted as an entropy either of matter or of space-time. In the present context, the
two main points of interest are that the entropy is finite, and that it is proportional to the
area of the black hole horizon.
In view of the existence of minimum lengths and times, the finiteness of the entropy
is not surprising: it confirms the idea that matter is made of a finite number of discrete
constituents per given volume (or area). It also shows that these constituents behave statistically: they fluctuate. In fact, quantum gravity implies a finite entropy for any object,
not only for black holes. Jacob Bekenstein has shown that the entropy of an object is
always smaller than the entropy of a (certain type of) black hole of the same mass.
The entropy of a black hole is proportional to its horizon area. Why? This question
has been the topic of a stream of publications.* A simple way to understand the entropy–
surface proportionality is to look for other systems in nature whose entropy is proportional to system surface instead of system volume. In general, the entropy of a collection
of flexible one-dimensional objects, such as polymer chains, shares this property. Indeed,
the entropy of a polymer chain made of ? monomers, each of length ?, whose ends are
* The result can be derived from quantum statistics alone. However, this derivation does not yield the
proportionality coefficient.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 122
?
??3
?
=
? or
2
4?Pl
4ℏ?
Motion Mountain – The Adventure of Physics
Ref. 57, Ref. 58
is a black hole. A black hole can be formed when a whole star collapses under its own
weight. Such a black hole is a macroscopic body, with a large number of constituents.
Therefore it has an entropy. The entropy ? of a macroscopic black hole was determined
by Bekenstein and Hawking, and is given by
132
Ref. 126
6 the sha pe of p oints
kept a distance ? apart, is given by
?(?) = ?
for ?? ≫ √?? ≫ ? .
(117)
Summary of the fifth argument for extension
Page 287
Black hole entropy is best understood as resulting from extended constituents that tangle
and fluctuate. And black hole entropy confirms that vacuum and particles are made of
common constituents.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
This formula can be derived in a few lines from the properties of a random walk on a lattice, using only two assumptions: the chains are extended; and they have a characteristic
internal length ? given by the smallest straight segment. Expression (117) is only valid
if the polymers are effectively infinite: in other words, if the length ?? of the chain and
the elongation ?√? , are much larger than the radius ? of the region of interest. If the
chain length is comparable to or smaller than the region of interest, we get the usual extensive entropy, satisfying ? ∼ ?3 . Thus only flexible extended constituents yield an ? ∼ ?2
dependence.
However, there is a difficulty. From the expression for the entropy of a black hole we
deduce that the elongation ?√? is given by ?√? ≈ ?Pl ; thus it is much smaller than
the radius of a general macroscopic black hole, which can have a diameter of several
kilometres. On the other hand, the formula for long constituents is only valid when the
chains are longer than the distance ? between the end points.
This difficulty disappears when we remember that space near a black hole is strongly
curved. All lengths have to be measured in the same coordinate system. It is well known
that for an outside observer, any object of finite size falling into a black hole seems to
cover the complete horizon for long times (whereas for an observer attached to the object
it falls into the hole in its original size). In short, an extended constituent can have a
proper length of Planck size but still, when seen by an outside observer, be as long as the
horizon of the black hole.
We thus find that black holes are made of extended constituents. Another viewpoint
can confirm this result. Entropy is (proportional to) the number of yes-or-no questions
needed to know the exact state of the system. But if a system is defined by its surface, as
a black hole is, its components must be extended.
Finally, imagining black holes as made of extended constituents is also consistent with
the so-called no-hair theorem: black holes’ properties do not depend on what falls into
them – as long as all matter and radiation particles are made of the same extended components. The final state of a black hole only depends on the number of extended constituents.
Motion Mountain – The Adventure of Physics
Ref. 109
3?2
2??2
excha nging space p oints or pa rticles at pl a nck sca les
133
exchanging space points or particles at pl anck
scales
Let us now focus on the exchange behaviour of fundamental constituents in nature. We
saw above that ‘points’ in space have to be abandoned in favour of continuous, fluctuating constituents common to space, time and matter. Is such a constituent a boson or a
fermion? If we exchange two points of empty space, in everyday life, nothing happens.
Indeed, at the basis of quantum field theory is the relation
[?, ?] = ?? − ?? = 0
(118)
Ref. 47
Ref. 127
At Planck energy this cannot be correct. Quantum gravity effects modify the right-hand
side: they add an energy-dependent term that is negligible at experimentally accessible
energies but becomes important at Planck energy. We know from our experience with
* The same reasoning applies to the so-called fermionic or Grassmann coordinates used in supersymmetry.
They cannot exist at Planck energy.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
This means that ‘points’ are neither bosons nor fermions.* ‘Points’ have more complex
exchange properties. In fact, the term on the right-hand side will be energy-dependent,
to an increasing extent as we approach Planck scales. In particular, as we have seen,
gravity implies that a double exchange does not lead back to the original situation at
Planck scales.
Constituents obeying this or similar relations have been studied in mathematics for
many decades: they are called braids. Thus space is not made of points at Planck scales,
but of braids or their generalizations, namely tangles. We find again that quantum theory
and general relativity taken together imply that the vacuum must be made of extended
constituents.
We now turn to particles. All particles in nature behave in a similar way: we know that
at low, everyday energies, particles of the same type are identical. Experiments sensitive
to quantum effects show that there is no way to distinguish them: any system of several identical particles has permutation symmetry. On the other hand, we know that at
Planck energy all low-energy symmetries disappear. We also know that at Planck energy
permutation cannot be carried out, as it implies exchanging positions of two particles.
At Planck energy, nothing can be distinguished from vacuum; thus no two entities can
be shown to have identical properties. Indeed, no two particles can be shown to be indistinguishable, as they cannot even be shown to be separate.
What happens when we slowly approach Planck energy? At everyday energies, permutation symmetry is defined by commutation or anticommutation relations between
particle creation operators
?† ?† ± ?† ?† = 0 .
(120)
Motion Mountain – The Adventure of Physics
between any two points with coordinates ? and ?, making them bosons. But at Planck
scales, because of the existence of minimal distances and areas, this relation must at least
be changed to
2
[?, ?] = ?Pl
+ ... .
(119)
134
Ref. 47
Ref. 127
6 the sha pe of p oints
Planck scales that, in contrast to everyday life, exchanging particles twice cannot lead
back to the original situation. A double exchange at Planck energy cannot have no effect, because at Planck energy such statements are impossible. The simplest extension of
the commutation relation (120) for which the right-hand side does not vanish is braid
symmetry. This again suggests that particles are made of extended constituents.
Summary of the sixth argument for extension
Extrapolating both point and particle indistinguishability to Planck scales suggests extended, braided or tangled constituents.
the meaning of spin
* With a flat (or other) background, it is possible to define a local energy–momentum tensor. Thus particles
can be defined. Without a background, this is not possible, and only global quantities can be defined.
Without a background, even particles cannot be defined. Therefore, in this section we assume that we have
a slowly varying space-time background.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 172
Motion Mountain – The Adventure of Physics
Vol. IV, page 112
As last argument we will now show that the extension of particles makes sense even
at everyday energy. Any particle is a part of the universe. A part is something that is
different from anything else. Being ‘different’ means that exchange has some effect. Distinction means detection of exchange. In other words, any part of the universe is also
described by its exchange behaviour.
In nature, particle exchange is composed of rotations. In other words, parts of nature
are described by their rotation behaviour. This is why, for microscopic particles, exchange
behaviour is specified by spin. Spin distinguishes particles from vacuum.*
We note that volume does not distinguish vacuum from particles; neither does rest
mass or charge: nature provides particles without measurable volume, rest mass or
charge, such as photons. The only observables that distinguish particles from vacuum
are spin and momentum. In fact, linear momentum is only a limiting case of angular
momentum. We thus find again that rotation behaviour is the basic aspect distinguishing particles from vacuum.
If spin is the central property that distinguishes particles from vacuum, finding a
model for spin is of central importance. But we do not have to search for long. A model
for spin 1/2 is part of physics folklore since almost a century. Any belt provides an example, as we discussed in detail when exploring permutation symmetry. Any localized
structure with any number of tails attached to it – tails that reach the border of the region
of space under consideration – has the same properties as a spin 1/2 particle. The only
condition is that the tails themselves are unobservable. It is a famous exercise to show that
such a model, like one of those shown in Figure 9, is indeed invariant under 4π rotations
but not under 2π rotations, and that two such particles get entangled when exchanged,
but get untangled when exchanged twice. Such a tail model has all the properties of spin
1/2 particles, independently of the precise structure of the central region, which is not
important at this point. The tail model even has the same problems with highly curved
space as real spin 1/2 particles have. We will explore the issues in more detail shortly.
curiosities a nd fun cha llenges a b ou t extension
position
of spin 1/2
particle
135
flexible bands
in unspecified
number
reaching the
border
of space
particle.
Ref. 128
Summary of the seventh argument for extension
Exploring the properties of particle spin suggests the existence of extended constituents
in elementary fermions. We note that gravitation is not used explicitly in the argument.
It is used implicitly, however, in the definition of the locally flat space-time and of the
asymptotic region to where the tails are reaching.
curiosities and fun challenges abou t extension
“
No problem is so big or complicated that it
can’t be run away from.
Charles Schulz
”
In case that this section has not provided enough food for thought, here is some more.
∗∗
Challenge 98 s
Quantum theory implies that even if tight walls exist, the lid of a box made of them could
never be tightly shut. Can you provide the argument?
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Challenge 97 e
The tail model thus confirms that rotation is partial exchange. More interestingly, it
shows that rotation implies connection with the border of space. Extended particles can
be rotating. Particles can have spin 1/2 provided that they have tails going to the border
of space. If the tails do not reach the border, the model does not work. Spin 1/2 thus
even seems to require extension.
It is not hard to extend this idea to include spin 1 particles. In short, both bosons and
fermions can be modelled with extended constituents.
Motion Mountain – The Adventure of Physics
F I G U R E 9 Possible models for a spin 1/2
136
6 the sha pe of p oints
∗∗
Challenge 99 e
Can you provide an argument against the idea of extended constituents in nature? If so,
publish it!
∗∗
Challenge 100 s
Does duality imply that the cosmic background fluctuations (at the origin of galaxies and
clusters) are the same as vacuum fluctuations?
∗∗
Challenge 101 s
Does duality imply that a system with two small masses colliding is equivalent to a system
with two large masses gravitating?
∗∗
It seems that in all arguments so far we have assumed that time is continuous, even
though we know it is not. Does this change the conclusions?
∗∗
Duality also implies that in some sense large and small masses are equivalent. A mass
2
? in a radius ? is equivalent to a mass ?2Pl /? in a radius ?Pl
/?. In other words, duality
2
transforms mass density from ? to ?Pl /?. Vacuum and maximum density are equivalent!
Vacuum is thus dual to black holes.
∗∗
Total symmetry and space-time duality together imply that there is a symmetry between
all values an observable can take. Do you agree?
∗∗
Challenge 104 s
Any description is a mapping from nature to mathematics, i.e., from observed differences
(and relations) to thought differences (and relations). How can we do this accurately, if
differences are only approximate? Is this the end of physics?
∗∗
Challenge 105 d
Duality implies that the notion of initial conditions for the big bang makes no sense,
as we saw earlier by considering the minimal distance. As duality implies a symmetry
between large and small energies, the big bang itself becomes a vague concept. What else
do extended constituents imply for the big bang?
∗∗
Challenge 106 d
Can you show that going to high energies or selecting a Planck-size region of space-time
is equivalent to visiting the big bang?
∗∗
Ref. 129
Challenge 107 s
In 2002, Andrea Gregori made a startling prediction for any model using extended constituents that reach the border of the universe: if particles are extended in this way, their
mass should depend on the size of the universe. Thus particle masses should change with
time, especially around the big bang. Is this conclusion unavoidable?
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Challenge 103 s
Motion Mountain – The Adventure of Physics
Challenge 102 d
checks of extension
137
∗∗
Challenge 108 s
What is wrong with the following argument? We need lines to determine areas, and we
need areas to determine lines. This implies that at Planck scales, we cannot distinguish
areas from lengths at Planck scales.
∗∗
Ref. 129, Ref. 130
Challenge 109 s
We need a description for the expansion of the universe in terms of extended constituents. Various approaches are being explored. Can you speculate about the solution?
Gender preferences in physics
Vol. I, page 338
The idea that nature is described by extended constituents is taken for granted in all
current research approaches to unification. How can we be sure that extension is correct?
The arguments presented so far provide several possible checks. We start with some
options for theoretical falsification.
Challenge 110 e
Page 111
— Any explanation of black hole entropy without extended constituents would invalidate the need for extended constituents.
— A single thought experiment invalidating extended constituents would prove extension wrong.
— Extended constituents must appear if we start from any physical (low-energy)
concept – not only from length measurements – and study how the concept behaves
at Planck scales.
— Invalidating the requirement of extremal identity, or duality, would invalidate the
need for extended constituents. As Edward Witten likes to say, any unified model of
nature must include duality.
— If the measurement of length could be shown to be unrelated to the counting of folds
of extended constituents, extension would become unnecessary.
— Finding any property of nature that contradicts extended constituents would spell the
end of extension.
Any of these options would signal the end for almost all current unification attempts.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
checks of extension
Motion Mountain – The Adventure of Physics
Why has extension appeared so late in the history of physics? Here is a not too serious
answer. When we discussed the description of nature as made of tiny balls moving in a
void, we called this as a typically male idea. This implies that the female part is missing.
Which part would that be?
From a general point of view, the female part of physics might be the quantum description of the vacuum, the container of all things. We can speculate that if women
had developed physics, the order of its discoveries might have been different. Instead
of studying matter first, as men did, women might have studied the vacuum first. And
women might not have needed 2500 years to understand that nature is not made of a
void and little balls, but that everything in nature is made of extended constituents. It is
curious that (male) physics took so long for this discovery.
138
6 the sha pe of p oints
Fortunately, theoretical falsification has not yet occurred. But physics is an experimental
science. What kind of data could falsify the idea of extended constituents?
Ref. 131
Ref. 132
Page 19
— Observing a single particle in cosmic rays with energy above the corrected Planck
energy would invalidate the invariant limits and thus also extension. However, the
present particle energy record, about 0.35 ZeV, is a million times lower than the
Planck energy.
— Paul Mende has proposed a number of checks on the motion of extended objects in
space-time. He argues that an extended object and a mass point move differently; the
differences could be noticeable in scattering or dispersion of light near masses.
Current research based on extended constituents
Ref. 114
”
The Greeks deduced the existence of atoms from the observation that fish can swim
through water. They argued that only if water is made of atoms could a fish make its
way through it, by pushing the atoms aside. We can ask a similar question of a particle
flying through a vacuum: why is it able to do so? A vacuum cannot be a fluid or a solid
composed of small constituents, as its dimensionality would not then be fixed. Only one
possibility remains: both vacuum and particles are made of extended constituents.
The idea of describing matter as composed of extended constituents dates from the
1960s. That of describing nature as composed of ‘infinitely’ extended constituents dates
from the 1980s. In addition to the arguments presented so far, current research provides
several other approaches that arrive at the same conclusion.
∗∗
Ref. 133
Bosonization, the construction of fermions using an infinite number of bosons, is a central aspect of modern unification attempts. It also implies coupling duality, and thus the
extension of fundamental constituents.
∗∗
Ref. 134, Ref. 135
Research into quantum gravity – in particular the study of spin networks, spin foams
and loop quantum gravity – has shown that the vacuum can be thought of as a collection
of extended constituents.
* Isaiah Berlin (b. 1909 Riga, d. 1997 Oxford) was an influential political philosopher and historian of ideas.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
“
To understand is to perceive patterns.
Isaiah Berlin*
Motion Mountain – The Adventure of Physics
Experimental falsification of extension has not yet occurred. In fact, experimental falsification is rather difficult. It seems easier and more productive to confirm extension.
Confirmation is a well-defined project: it implies to deduce all those aspects of nature
that are given in the millennium list of unexplained properties. Among others, confirmation requires to find a concrete model, based on extended constituents, for the electron,
the muon, the tau, the neutrinos, the quarks and all bosons. Confirmation also requires
using extended constituents to realize an old dream of particle physics: to deduce the
values of the coupling constants and particle masses. Before we attempt this deduction,
we have a look at some other attempts.
checks of extension
139
∗∗
Ref. 136
In the 1990s, Dirk Kreimer showed that high-order QED Feynman diagrams are related
to knot theory. He thus proved that extension arrives by the back door even when electromagnetism is described in terms of point particles.
∗∗
Ref. 137
A popular topic in particle physics, ‘holography’, relates the surface and the volume of
physical systems at high energy. It implies extended constituents of nature.
∗∗
Vol. IV, page 157
It is long known that wave function collapse can be seen as the result of extended constituents. We will explore the details below.
∗∗
Ref. 140, Ref. 141
Ref. 142, Ref. 143
Page 347
At the start of the twenty-first century, a number of new approaches to describe elementary particles appeared, such as models based on string nets, models based on bands,
models based on ribbons, and models based on knots. All these attempts make use of
extended constituents. Several of them are discussed in more detail below.
Despite the use of extension, none of these attempts solved a single problem from the
millennium list. One approach – especially popular between the years 1984 and 2010 –
merits a closer look.
Superstrings – extension plus a web of dualities
Ref. 144
Ref. 145
”
Superstrings and supermembranes – often simply called strings and membranes – are extended constituents in the most investigated physics conjecture ever. The approach contains a maximum speed, a minimum action and a maximum force (or tension). The approach thus incorporates special relativity, quantum theory and general relativity. This
attempt to achieve the final description of nature uses four ideas that go beyond standard
general relativity and quantum theory:
1. Particles are conjectured to be extended. Originally, particles were conjectured to be
one-dimensional oscillating superstrings. In a subsequent generalization, particles are
conjectured to be fluctuating higher-dimensional supermembranes.
2. The conjecture uses higher dimensions to unify interactions. A number of space-time
dimensions much higher than 3+1, typically 10 or 11, is necessary for a mathematically
consistent description of superstrings and membranes.
3. The conjecture is based on supersymmetry. Supersymmetry is a symmetry that relates
matter to radiation, or equivalently, fermions to bosons. Supersymmetry is the most
general local interaction symmetry that can be constructed mathematically. Supersymmetry is the reason for the terms ‘superstring’ and ‘supermembrane’.
4. The conjecture makes heavy use of dualities. In the context of high-energy physics,
dualities are symmetries between large and small values of physical observables. Important examples are space-time duality and coupling constant duality. Dualities are
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
“
Throw physic to the dogs; I’ll none of it.
William Shakespeare, Macbeth.
Motion Mountain – The Adventure of Physics
Ref. 138, Ref. 139
140
6 the sha pe of p oints
global interaction and space-time symmetries. They are essential for the inclusion of
gauge interaction and gravitation in the quantum description of nature. Dualities also
express a fundamental equivalence between space-time and matter–radiation. Dualities also imply and contain holography, the idea that physical systems are completely
fixed by the states on their bounding surface.
Ref. 146
By incorporating these four ideas, the superstring conjecture – named so by Brian Greene,
one of its most important researchers – acquires a number of appealing characteristics.
Why superstrings and supermembranes are so appealing
* This argument is questionable, because general relativity already cures that divergence.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 122
Motion Mountain – The Adventure of Physics
Ref. 147
First of all, the superstring conjecture is unique: the Lagrangian is claimed to be unique
and to have no adjustable parameters. Furthermore, as we would expect from a description involving extended constituents, the conjecture includes gravity. In addition, the
conjecture describes interactions: it describes gauge fields. The conjecture thus expands
quantum field theory, while retaining all its essential points. In this way, the conjecture
fulfils most of the requirements for a unified description of motion that we have deduced
so far. For example, particles are not point-like, there are minimal length and time intervals, and all other limit quantities appear. (However, sets are still used.)
The superstring conjecture has many large symmetries, which arise from the many
dualities it contains. These symmetries connect many situations that seem intuitively to
be radically different: this makes the conjecture extremely fascinating, but also difficult
to picture.
The conjecture shows special cancellations of anomalies and of other inconsistencies. Historically, the first example was the Green–Schwarz anomaly cancellation; superstrings also solve other anomalies and certain inconsistencies of quantum field theory.
Edward Witten, the central figure of the field, liked to say that quantum theory cures
the infinities that appear in ?2 /? when the distance ? goes to zero; in the same way, superstrings cure the infinities that appear in ?2 /? when the distance ? goes to zero.*
Also following Witten, in the superstring conjecture, the interactions follow from the
particle definitions: interactions do not have to be added. That is why the superstring
conjecture predicts gravity, gauge theory, supersymmetry and supergravity.
About gravity, one of the pretty results of the superstring conjecture is that superstrings and black holes are complementary to each other. This was argued by Polchinsky,
Horowitz and Susskind. As expected, superstrings explain the entropy of black holes.
Strominger and Vafa showed this in 1996.
The superstring conjecture naturally includes holography, the idea that the degrees of
freedom of a physical system are determined by its boundary. In particular, holography
provides for a deep duality between gauge theory and gravity. More precisely, there is a
correspondence between quantum field theory in flat space and the superstring conjecture in certain higher-dimensional spaces that contain anti-de Sitter space.
In short, the superstring conjecture implies fascinating mathematics. Conformal invariance enters the Lagrangian. Concepts such as the Virasoro algebra, conformal field
theory, topological field theory and many related ideas provide vast and fascinating generalizations of quantum field theory.
checks of extension
141
Why the mathematics of superstrings is difficult
Ref. 148
One of the main results of quantum chromodynamics or QCD, the theory of strong interactions, is the explanation of mass relations such as
?proton ∼ e−?/?Pl ?Pl
Page 378
Ref. 149
and
? = 11/2π , ?Pl ≈ 1/25 .
(121)
Here, the value of the strong coupling constant ?Pl is taken at the Planck energy. In
other words, a general understanding of masses of bound states of the strong interaction,
such as the proton, requires little more than a knowledge of the Planck energy and the
coupling constant at that energy. The approximate value ?Pl ≈ 1/25 is an empirical value
based on experimental data.
Any unified theory must allow us to calculate the three gauge coupling constants as a
function of energy, thus also ?Pl . At present, most researchers regard the search for the
vacuum state – the precise embedding of four dimensions in the total ten – as the main
difficulty facing the superstring conjecture. Without knowledge of the vacuum state, no
calculations of coupling constants or masses are possible.
The vacuum state of the superstring conjecture is expected to be one of an rather
involved set of topologically distinct manifolds. It was first estimated that there are only
10500 possible vacuum states; recent estimates raised the number to 10272 000 candidate
vacuum states. Since the universe contains 1080 atoms, it seems easier to find a particular
atom somewhere in the universe than to find the correct vacuum state. The advantages
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Testing superstrings: couplings and masses
Motion Mountain – The Adventure of Physics
The superstring conjecture, like all modern descriptions of physics, is claimed to be described by a Lagrangian. The Lagrangian is constructed starting from the Lagrangian for
the motion of a classical superstring of matter. Then the Lagrangian for the corresponding quantum superstring fields is constructed, and then higher dimensions, supersymmetry, dualities and membranes are incorporated. This formulation of the superstring
conjecture takes for granted the existence of a space-time background.
The Lagrangian of the superstring conjecture is extremely complex, much too complex to write it down here. It is not as simple as the Lagrangian of the standard model
of particle physics or the Lagrangian of general relativity. But the complexity of the Lagrangian is not the only reason why the studying the superstring conjecture is difficult.
It turns out that exploring how the known 4 dimensions of space-time are embedded
in the 10 or 11 dimensions of the superstring conjecture is extremely involved. The topology and the size of the additional dimensions is unclear. There are only few people who
are able to study these options.
Indeed, a few years ago a physicist and a mathematician listened to a talk on superstrings, describing nature in eleven dimensions. The mathematician listened intensely
and obviously enjoyed the talk. The physicist did not understand anything and got more
and more annoyed. At the end, the physicist had a terrible headache, whereas the mathematician was full of praise. ‘But how can you even understand this stuff?’, asked the
physicist. ‘I simply picture it in my head!’ ‘But how do you imagine things in eleven
dimensions?’ ‘Easy! I first imagine them in ? dimensions and then let ? go to 11.’
142
6 the sha pe of p oints
due to a unique Lagrangian are thus lost.
We can also describe the problems with the calculation of particle masses in the following way. The superstring conjecture predicts states with Planck mass and with zero
mass. The zero-mass particles are then thought to get their actual mass, which is tiny
compared with the Planck mass, from the Higgs mechanism. However, the Higgs mechanism and its measured properties – or any other parameter of the standard model –
have not yet been deduced from superstrings.
The status of the superstring conjecture
“
Ref. 150
* ‘Nothing great has been achieved without passion, nor can it be achieved without it.’ Hegel, an influential
philospher, writes this towards the end of the third and last part of his Enzyklopädie der philosophischen
Wissenschaften im Grundrisse, §474, 296.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 151
”
It is fair to say that nowadays, superstring researchers are stuck. Despite the huge collective effort, not a single calculation of an experimentally measurable value has been
performed. For example, the superstring conjecture has not predicted the masses of any
elementary particle, nor the value of any coupling constant, nor the number of gauge
interactions, nor the number of particle generations. In fact, none of the open issues in
physics that are listed the millennium list has been solved by the superstring conjecture.
This disappointing situation is the reason that many scholars, including several Nobel
Prize winners, dismiss the superstring conjecture altogether.
What are the reasons that the superstring conjecture, like several other approaches
based on extended constituents, was unsuccessful? First of all, superstrings and supermembranes are complex: superstrings and supermembranes move in many dimensions,
carry mass, have tension and carry (supersymmetric) fields. In fact, the precise mathematical definition of a superstring or a supermembrane and their features is so complex
that already understanding the definition is beyond the capabilities of most physicists.
But a high complexity always nourishes the doubt that some of the underlying assumptions do not apply to nature.
Superstrings are complex entities. And no researcher tried to make them simple. Put
in different terms, the superstring conjecture was not successful because its basic principles have never been clarified. It is estimated that, from 1984 to 2010, over 10 000 manyears have been invested in the exploration of the superstring conjecture. Compare this
with about a dozen man-years for the foundations and principles of electrodynamics,
a dozen man-years for the foundations and principles of general relativity, and a dozen
man-years for the foundation and principles of quantum theory. The lack of clear foundations of the superstring conjecture is regularly underlined even by its supporters, such
as Murray Gell-Mann. And despite this gap, no research papers on the basic principles
exist – to this day.
Apart from the complexity of the conjecture, a further aspect about superstrings and
supermembranes has been getting growing attention: the original claim that there is a
unique well-defined Lagrangian has been retracted; it not even made by the most outgo-
Motion Mountain – The Adventure of Physics
Page 19
Es ist nichts Großes ohne Leidenschaft vollbracht worden, noch kann es ohne
solche vollbracht werden.*
Friedrich Hegel, Enzyklopädie.
sum m ary on extension in nature
143
ing proponents any more. In other words, it is not clear which specific supermembrane
conjecture should be tested against experiment in the first place.
These developments effectively dried out the research field. At the latest in 2014, during the Strings conference, it became clear that the research community has quietly given
up its quest to achieve a unified theory with the help of superstrings or supermembranes.
Researchers are now looking for other microscopic models of nature.
summary on extension in nature
“
”
We have explored nature at her limits: we have studied the Planck limits, explored threedimensionality, curvature, particle shape, renormalization, spin and bosonization; we
have investigated the cosmological constant problem and searched for a ‘backgroundfree’ description of nature. As a result, we have found that at Planck scales, all these
explorations lead to the same conclusions:
— Points and sets do not describe nature correctly.
— Matter and vacuum are two sides of the same medal.
— What we usually call space-time points and point particles are in fact made up of
common and, above all, extended constituents.
* ‘We must know, we will know.’ This was Hilbert’s famous personal credo.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
We can reach the conclusions in an even simpler way. What do quantum theory and
black holes have in common? They both suggest that nature is made of extended entities.
We will confirm below that both the Dirac equation and black hole entropy imply that
particles, space and horizons are built from extended constituents.
Despite using extension as fundamental aspect, and despite many interesting results,
all the attempts from the twentieth century – including the superstring conjecture and all
quantum gravity models, but also supersymmetry and supergravity – have not been successful in understanding or in describing nature at the Planck scale. The reasons for this
lack of success were the unclear relation to the Planck scale, the lack of clear principles,
the use of incorrect assumptions, the use of sets and, above all, the unclear connection
to experiment.
To be successful, we need a different approach to calculations with extended constituents. We need an approach that is built on Planck units, is based on clear principles, has
few but correct assumptions, and provides predictions that stand up against experimental
tests.
In our quest for a final theory of physics, one way to advance is by raising the following issue. The basis for the superstring conjecture is formed by four assumptions:
extension, duality, higher dimensions and supersymmetry. Can we dispense with any of
them? Now, duality is closely related to extension, for which enough theoretical and experimental evidence exists, as we have argued above. On the other hand, the expressions
for the Schwarzschild radius and for the Compton wavelength imply, as we found out
Motion Mountain – The Adventure of Physics
Ref. 152
Wir müssen wissen, wir werden wissen.*
David Hilbert
144
Page 71, page 77
Page 74, page 77
6 the sha pe of p oints
earlier on, that the dimensionality of space and the statistics of particles are undefined at
Planck scales. In other words, nature does not have higher dimensions nor supersymmetry at Planck scales. Indeed, all known experiments confirm this conclusion. In our
quest for a final theory of motion, we therefore drop the two incorrect assumptions and
continue our adventure.
In summary, extension is the central property of the fundamental entities of nature
that make up space, horizons, particles and interactions at Planck scales. We can thus
phrase our remaining quest in the following specific way:
⊳ How do extended entities relate the Planck constants ?, ℏ, ? and ? to the
electromagnetic, the weak and the strong interactions?
Challenge 111 e
Motion Mountain – The Adventure of Physics
This question is rarely asked so specifically. Attempts to answer it are even rarer. (Can
you find one?) Up to this point, we discovered: Finding the Planck origin of the gauge
interactions using extension means finding the final theory. To be successful in this quest,
we need three resources: simplicity, playfulness and intrepidity.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Chapter 7
T HE BA SIS OF T HE ST R AND MODEL
”
he two extremely precise descriptions of motion that were discovered in
he twentieth century – quantum field theory and general relativity – are
he low-energy approximations of how nature behaves at Planck scales. In order
to understand nature at Planck scales, and thus to find the unified and final description
of motion, we follow the method that has been the most effective during the history of
physics: we search for the simplest possible description. Simplicity was used successfully,
for example, in the discovery of special relativity, in the discovery of quantum theory,
and in the discovery of general relativity. We therefore use the guidance provided by
simplicity to deduce a promising speculation for the unified and final theory of motion.
Page 161
Page 163
The central requirement for any unified description is that it leads from Planck scales,
and thus from Planck units, to quantum field theory, to the standard model of elementary particles and to general relativity. In simple terms, as detailed below, the unified
description must be valid for all observations and provide complete precision.
From the preceding chapters, we know already quite a bit about the unified description. In particular, any unified description of general relativity and quantum theory must
use extended constituents. We discovered a number of reasons that are central for this
conclusion. All these reasons appear only when quantum theory and general relativity
are combined. First of all, only constituents that are extended allow us to deduce black
hole entropy. Secondly, only extended constituents allow us to model that elementary
particles are not point-like or that physical space is not made of points. Thirdly, only
extended constituents allow us to model a smallest measurable space and time interval.
Fourthly, only extended constituents allow us to model spin 1/2 in locally flat space-time.
But we are not only looking for a unified theory; we are also looking for the final
theory. This implies a second requirement: the final theory must be unmodifiable. As
we will show below, if a candidate for a final theory can be modified, or generalized, or
reduced to special cases, or varied in any other way, it is not final.
In the preceding chapters we have deduced many additional, requirements that a final
** Ernest Rutherford (b. 1871 Brightwater, d. 1937 Cambridge) was an important physicist and researcher;
he won the Nobel Prize in Chemistry for his work on atoms and radioactivity.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
R equirements for a final theory
Motion Mountain – The Adventure of Physics
T
“
We haven’t got the money, so we have to think.
Ernest Rutherford**
the basis of the stra nd m odel
147
The fundamental principle of the strand model
Strand model :
?1
Observation :
Some
deformation,
but no
passing
through
? = ℏ/2
Δ? = ?Pl
Δ? = ?Pl
? = ?/2
?2
F I G U R E 10 The fundamental principle of the strand model: the simplest observation in nature, a
‘‘point-like’’ fundamental event, is defined by a crossing switch in three spatial dimensions. The crossing
switch defines the action ℏ/2, the Planck length, the Planck time and half the Boltzmann constant ?/2.
TA B L E 6 General requirements for a final and unified description of nature and of motion.
R equirements for the final and unified
description
Precision
must be complete; the unified description must precisely describe
all motion – everyday, quantum and relativistic – and explain all
open issues from the millennium list, given (again) in Table 8 on
page 162, including the fine structure constant.
must be impossible, as explained on page 163.
must be clear. (Otherwise the unified description is not
falsifiable.)
must not differ at Planck scales because of limits of measurement
precision; vacuum and particles therefore must be described by
common fundamental constituents.
must determine all observables.
must be as simple as possible, to satisfy Occam’s razor.
must be extended and fluctuating, to explain black hole entropy,
spin, minimum measurement intervals, space-time homogeneity
and isotropy of space.
must be the only unobservable entities. (If they were observable,
the theory would not be final, because the properties of the
entities would need explanation; if additional unobservable
entities would exist, the theory would be fiction, not science.)
Modification
Fundamental principles
Vacuum and particles
Fundamental constituents
Fundamental constituents
Fundamental constituents
Fundamental constituents
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Aspect
Motion Mountain – The Adventure of Physics
Challenge 112 e
theory must realize. The full list of requirements is given in Table 6. Certain requirements
follow from the property that the description must be final, others from the property
that it must be unified, and still others from the property that it must describe nature
with quantum theory and general relativity. More specifically, every requirement appears
when the expressions for the Compton wavelength and for the Schwarzschild radius are
combined. So far, the table is not found elsewhere in the research literature.
148
7 the basis of the stra nd m od el
TA B L E 6 (Continued) General requirements for a final and unified description of nature and of motion.
R equirements for the final and unified
description
Non-locality
must be part of the description; non-locality must be negligible at
everyday scales, but important at Planck scales.
must not exist, due to limits of measurement precision; points and
sets only exist approximately, at everyday scales.
must not exist at Planck scales, due to the lack of points and sets.
must not exist at Planck scales, due to limits of measurement
precision; systems only exist approximately at everyday scales.
must not be a system, due to limits of measurement precision.
must not be an event, and thus not be a beginning, as this would
contradict the non-existence of points and sets in nature.
must not exist, due to the limits of measurements.
must be limit values for each observable (within a factor of order
one); infinitely large or small measurement values must not exist.
must imply quantum field theory, the standard model of particle
physics, general relativity and cosmology.
must follow from the final unified theory by eliminating ?.
Physical points and sets
Evolution equations
Physical systems
Universe
Big bang
Singularities
Planck’s natural units
Planck scale description
Circularity of definitions
Axiomatic description
Dimensionality of space
Symmetries
Large and small scales
Challenge 113 e
must follow from the final unified theory by eliminating ℏ.
must define all observables, including coupling constants.
must be as simple as possible, to satisfy Occam’s razor.
is required, as background independence is logically impossible.
must be equal to physical space-time at everyday scale, but must
differ globally and at Planck scales.
of physical concepts must be part of the final, unified description,
as a consequence of being ‘precise talk about nature’.
must be impossible, as nature is not described by sets; Hilbert’s
sixth problem must have no solution.
must be undefined at Planck scales, as space is undefined there.
must be undefined at Planck scales, due to the limits to
measurement precision.
must be similar, due to the limits to measurement precision.
The requirement list given in Table 6 can be considerably shortened. Shortening the
list of requirements is possible because the various requirements are consistent with each
other. In fact, shortening is possible because a detailed check confirms a suspicion that
arose during the last chapters: extension alone is sufficient to explain all those requirements that seem particularly surprising or unusual, such as the lack of points or the lack
of axioms. Such a shortened list also satisfies our drive for simplicity. After shortening,
two requirements for a final theory remain:
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Quantum field theory,
including QED, QAD,
QCD
General relativity
Planck’s natural units
Relation to experiment
Background dependence
Background space-time
Motion Mountain – The Adventure of Physics
Aspect
the basis of the stra nd m odel
149
⊳ The final theory must describe nature at and below the Planck scale* as made
of extended constituents fluctuating in a background. Extended constituents
must explain particles, space, interactions and horizons.
⊳ In the final theory, the fluctuations of the extended constituents must explain all motion. The Planck-scale fluctuations must describe all observed
examples of everyday, quantum and relativistic motion with complete precision, imply all interactions, all concepts of physics and explain all fundamental constants.
Challenge 114 s
“
Nur die ergangenen Gedanken haben Wert.***
Friedrich Nietzsche
”
* The final theory must not describe nature beyond the Planck scales. A more relaxed requirement is that the
predictions of the theory must be independent of any fantasies of what might occur beyond Planck scales.
** In Dutch: draden, in French: fils, in German: Fäden, in Italian: fili.
*** ‘Only thoughts conceived while walking have value.’ Friedrich Nietzsche (b. 1844 Röcken,
d. 1900 Weimar) was philologist, philosopher and sick.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 165
Motion Mountain – The Adventure of Physics
Ref. 153
This requirement summary is the result of our journey up to this point. The summary
forms the starting point for the final leg of our adventure. If you do not agree with these
two requirements, take a rest and explore your disagreement in all its details.
Looking at the table of requirements for the final theory – both the full one and the
shortened one – we note something astonishing. Even though all requirements appear
when quantum physics and general relativity are combined, each of these requirements
contradicts both quantum physics and general relativity. The final theory thus differs from
both pillars of modern physics. A final theory cannot be found if we remain prisoners
of either quantum theory or general relativity. To put it bluntly, each requirement for
the final theory contradicts every result of twentieth century physics! This unexpected
conclusion is the main reason that past attempts failed to discover the final theory. In fact,
most attempts do not fulfil the requirements because various scholars explicitly disagree
with one or several of them.
The requirement of the extension of the fundamental constituents is the central result. A final theory must make a statement about these constituents. The fundamental
constituents, also called fundamental degrees of freedom, must explain everything we observe and know about nature. In particular, the constituents must explain the curvature
of space, the entropy of black holes, the origin of gauge interactions and the spectrum,
mass and other properties of all elementary particles. The fundamental constituents must
be extended. Extension is the reason that the final theory contradicts both general relativity and quantum theory; but extension must allows these theories as excellent approximations. In short, extension is the key to finding the final theory.
The requirement for fluctuating extended constituents resulted from our drive for extreme simplicity. With this requirement, the search for a candidate final theory does not
take long. Of the few candidates that satisfy the requirement, it appears that the simplest
is the one based on fluctuating featureless strands. In this approach, strands,** not points,
are assumed to be the fundamental constituents of vacuum, horizons, matter and radiation.
150
7 the basis of the stra nd m od el
Introducing strands
The strand model starts with a simple idea:
⊳ Nature is made of unobservable, fluctuating, featureless strands.
We will discover that everything observed in nature – vacuum, fermions, bosons and
horizons – is made of strands. Strands are the common and extended constituents of
everything. Even though strands are unobservable and featureless, all observations are
due to strands.
⊳ All observations, all change and all events are composed of the fundamental
event, the crossing switch.
⊳ Planck units are defined through crossing switches of strands.
Page 147
The definition of the Planck units with the crossing switch is illustrated in Figure 10. All
measurements are consequence of this definition. All observations and everything that
happens are composed of fundamental events. The fundamental principle thus specifies
why and how Planck units are the natural units of nature. In particular, the four basic
Planck units are associated in the following way:
⊳ The (corrected) Planck length ?Pl = √4?ℏ/?3 appears as the effective diameter of strands. Since the Planck length is a limit that cannot be achieved
by measurements, strands with such a diameter remain unobservable.*
⊳ The Planck entropy, i.e., the Boltzmann constant ?, is the natural unit associated to the counting and statistics of crossings.*
⊳ The (corrected) Planck time ?Pl = √4?ℏ/?5 appears as the shortest possible
duration of a crossing switch.*
Crossing switches that are faster than the Planck time do not play a role, as they are
unobservable and unmeasurable. Let us see why.
How can we imagine a minimum time interval in nature? A crossing switch could be
arbitrarily fast, couldn’t it? So how does the Planck time arise? To answer, we must recall
the role of the observer. The observer is a physical system, also made of strands. The
* In other words, the strand model sets ℏ = ?Pl = ?Pl = ? = 1. The strange numerical values that these
constants have in the SI, the international system of units, follow from the strange definitions of the metre,
second, kilogram and kelvin.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
⊳ Planck’s quantum of action ℏ/2 appears as the action value associated to a
crossing switch. The action ℏ corresponds to a double crossing switch, or
full turn of one strand segment around another.*
Motion Mountain – The Adventure of Physics
To describe all observations with precision, the strand model uses only one fundamental
principle:
the basis of the stra nd m odel
151
A twist :
t2
t1
A twirl :
t2
F I G U R E 11 An example of strand deformation leading to a crossing switch (above) and one that does
not lead to a crossing switch (below).
Page 36
Challenge 115 r
** The issue of time remains subtle also in the strand model. The requirement of consistency with macroscopic experience, realized with shivering space or space-time, allows us to side-step the issue. An alternative approach might be to picture a crossing switch and its fluctuations in 4 space-time dimensions, thus
visualizing how the minimum time interval is related to minimum distance. This might be worth exploring.
But also in this approach, the fuzziness due to shivering is at the basis of minimum time.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Challenge 116 s
Challenge 117 e
observer cannot define a really continuous background space-time; careful consideration
tells us that the space-time defined by the observer is somewhat fuzzy: it is effectively
shivering. The average shivering amplitude is, in the best possible case, of the order of a
Planck time and length. Therefore, crossing switches faster than the Planck time are not
observable by an observer made of strands.**
Strands are impenetrable; the switch of a crossing thus always requires the motion of
strand segments around each other. The simplest example of a deformation leading to a
crossing switch is shown in Figure 11.
Can you deduce the strand processes for the Planck momentum, the Planck force and
the Planck energy?
Exploring strand processes we find: the fundamental principle implies that every
Planck unit is an observer-invariant limit value. Therefore, the fundamental principle
naturally contains special and general relativity, quantum theory and thermodynamics
(though not elementary particle physics!). In theory, this argument is sufficient to show
that the fundamental principle contains all these parts of twentieth century physics. In
practice, however, physicists do not change their thinking habits that quickly; thus we
need to show this result in more detail.
Motion Mountain – The Adventure of Physics
t1
152
7 the basis of the stra nd m od el
A strand crossing
phase
shortest distance
position
orientation
F I G U R E 12 The definition of a crossing, its position, its orientation and its phase. The shortest distance
defines a local density.
Events, pro cesses, interactions and colours
⊳ Any event, any observation, any measurement and any interaction is composed of switches of crossings between two strand segments.
⊳ Particle masses, the elementary electric charge ? and the fine structure constant ? = 137.036(1) are due to crossing switches.
The value of the fine structure constant and the standard model are not evident consequences of the fundamental principle; nevertheless, they are natural consequences –
as we will find out.
From strands to modern physics
Every observation and every process is a sequence of crossing switches of unobservable
strands. In turn, crossing switches are automatic consequences of the shape fluctuations
of strands. We will show below that all the continuous quantities we are used to – physical space, physical time, gauge fields and wave functions – result from averaging crossing
switches over the background space. The main conceptual tools necessary in the following are:
⊳ A crossing of strands is a local minimum of strand distance. The position,
orientation and phase of a crossing are defined by the space vector corres-
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
The crossing switch is the fundamental process in nature. We will show that describing
events and interactions with the help of crossing switches leads, without alternative, to
the complete standard model of particle physics, with all its known gauge interactions
and all its known particle spectrum.
In particular, we will show:
Motion Mountain – The Adventure of Physics
In the strand model, every physical process is described as a sequence of crossing
switches. But every physical process is also a sequence of events. We thus deduce that
events are processes:
the basis of the stra nd m odel
153
ponding to the local minimum of distance, as shown in Figure 12.
Page 172
The position, orientation and phase of crossings will lead, as shown later on, to the position, orientation and phase of wave functions. The sign of the orientation is defined by
arbitrarily selecting one strand as the starting strand. The even larger arbitrariness in the
definition of the phase will be of great importance later on: it implies the existence of the
three known gauge groups.
⊳ A crossing switch is the rotation of the crossing orientation by an angle π at
a specific position. More precisely, a crossing switch is the inversion of the
orientation at a specific position.
⊳ Events are (one or several) observable crossing switches of unobservable
strands.
Page 165
Since all observations are events, all experimental observations should follow from the
strand definition of an event. We will confirm this in the rest of this text. The strands
are featureless: they have no mass, no tension, no stiffness, no branches, no fixed length,
no ends, and they cannot be pulled, cut or pushed through each other. Strands have
no measurable property at all: strands are unobservable. Only crossing switches are observable. Featureless strands are thus among the simplest possible extended constituents.
How simple are they? We will discuss this issue shortly.
⊳ Strands are one-dimensional curves in three-dimensional space that reach
the border of space.
In practice, the border of space has one of two possible meanings. Whenever space is
assumed to be flat, the border of space is spatial infinity. Whenever we take into account
the properties of the universe as a whole, the border of space is the cosmic horizon.
Imagining the strands as having Planck diameter does not make them observable,
as this measurement result cannot be realized. (We recall that the Planck length is the
lower bound on any length measurement.) In low energy situations, a vanishing strand
diameter is an excellent approximation.
⊳ In a purist definition, featureless strands have no diameter – neither the
Planck length nor zero. They are better thought as long thin clouds.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 353
Motion Mountain – The Adventure of Physics
We note that the definitions make use of all three dimensions of space; therefore the number of crossings and of crossing switches is independent of the direction of observation.
This contrasts with the definition of crossing used in two-dimensional knot diagrams; in
such two-dimensional projections, the number of crossings does depend on the direction of the projection.
We note that strand fluctuations do not conserve the number of crossings; due to
fluctuations, crossings disappear and appear and disappear over time. This appearance
and disappearance will turn out to be related to virtual particles.
The fundamental principle declares that events are not points on manifolds; instead,
154
7 the basis of the stra nd m od el
Vacuum :
Elementary
spin 1/2
fermion :
spin
Elementary
spin 1 boson :
spin
Horizon :
detail below.
Page 166
⊳ Physical space, or vacuum, is a physical system made of tangles that has size,
curvature and other measurable properties.
⊳ Continuous background space is introduced by the observer only to be able
to describe observations. Every observer introduces his own background.
It does not need to coincide with physical space, and it does not do so at
the location of matter or black holes. But every observer’s background is
continuous and has three spatial and one temporal dimension.
At this point of the discussion, we simply assume background space. Later on we will
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 169
Strands are unobservable and featureless, and thus have no diameter. Due to shape fluctuations, or equivalently, due to the shivering of space-time, the strands can be thought
as having an effective diameter, akin to the diameter of a long thin cloud; this effective
diameter is just a guide to our thinking. Since it is due to the shivering of the background space-time, the strand diameter is invariant under boosts. Funnels, mentioned
below, might be a better visualization of the purist definition of strand. To keep this introduction as intuitive as possible, however, we stick with the idea of strands having an
effective, invariant Planck diameter.
The strand model distinguishes physical space from background space. We will show
shortly why both concepts are required. With this distinction, the strand model asserts
that matter and radiation, vacuum and horizons, are all built from fluctuating strands in
a continuous background. We first clarify the two basic space concepts.
Motion Mountain – The Adventure of Physics
F I G U R E 13 A first illustration of the basic physical systems found in nature; they will be explored in
the basis of the stra nd m odel
155
TA B L E 7 Correspondences between all known physical systems and mathematical tangles.
Physical system
Strand content
Ta n g l e t y p e
Vacuum and dark energy
many unknotted and untangled
infinite strands
two infinite twisted strands
many infinite twisted strands
many woven infinite strands
one infinite curved strand
unlinked, trivial tangle
Graviton
Gravity wave
Horizon
Elementary vector boson
(radiation)
Classical electromagnetic
wave (radiation)
Elementary quark (matter)
Elementary lepton (matter)
two infinite linked strands
three infinite linked strands
see why background space appears and why it needs to be three-dimensional. The size
of the background space is assumed to be large; larger than any physical scale under
discussion. In most situations of everyday life, when space is flat, background space and
physical space coincide. However, they differ in situations with curvature and at Planck
energy.
All strand fluctuations are possible, as long as strands do not interpenetrate. For example, there is no speed limit for strands. Whenever strand fluctuations lead to a crossing
switch, they lead to an observable effect – be it a vacuum fluctuation, a particle reaction
or a horizon fluctuation.
⊳ Fluctuations are a consequence of the embedding of strands in a continuous
background.
In the strand model, even isolated physical systems are surrounded by a bath of fluctuating vacuum strands. The properties of fluctuations, such as their spectrum, their
density etc., are fixed once and for all by the embedding. Fluctuations are necessary for
the self-consistency of the strand model.
Due to the impenetrability of strands – which itself is a consequence of the embedding in a continuous background – any disturbance of the vacuum strands at one location propagates. We will see below what disturbances exist and how they differ from
fluctuations.
Fluctuating strands that lead to crossing switches explain everything that does happen,
and explain everything that does not happen. Our main aim in the following is to classify
all possible strand fluctuations and all possible strand configurations, in particular, all
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
⊳ Fluctuations change the position, shape and length of strands; fluctuations
thus change position, orientation and phase of strand crossings. However,
fluctuations never allow one strand to pass through another.
Motion Mountain – The Adventure of Physics
Page 206
many infinite curved strands
rational tangle
many rational tangles
woven, web-like tangle
any one from a family of
tangled curves
many helically
deformed/tangled curves
rational tangle
braided tangle
156
7 the basis of the stra nd m od el
Vacuum
The strand model :
Observation :
time average
of crossing
switches
Nothing
(only for long
observation
times)
F I G U R E 14 An illustration of the strand model for the vacuum.
Page 154
Page 313
Page 150
Some examples of important tangles are given in Figure 13. They will be discussed in
detail in the following. Among others, we will discover that knots and knotted tangles
do not play a role in the strand model; only linked, but unknotted tangles do.
We observe that vacuum, matter and radiation are all made of the same fundamental
constituents, as required for a final theory. We will discover below that classifying localized tangles naturally leads to the elementary particles that make up the standard model
of particle physics – and to no other elementary particle.
We will also discover that strand fluctuations and the induced crossing switches in
every physical system lead to the evolution equations and the Lagrangians of quantum
field theory and of general relativity. In this way, strands describe every physical process
observed in nature, including all known interactions and every type of motion.
The fundamental principle relates crossing switches and observations. The fundamental principle was discovered because it appears to be the only simple definition
of Planck units that on the one hand yields space-time, with its continuity, local isotropy and curvature, and on the other hand realizes the known connection between the
quantum of action, spin and rotation.
Vacuum
We now construct, step by step, all important physical systems, concepts and processes
from tangles. We start with the most important.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
⊳ A tangle is a configuration of one or more strands that are linked or knotted.
Tangles are characterized by their topology, i.e., by the precise way that they
are linked or knotted.
Motion Mountain – The Adventure of Physics
states that differ from flat vacuum states. By doing so, we will be able to classify every
process and every system that we observe in nature.
We will discover that all physical systems can be constructed from strands. Table 7
gives a first overview of how vacuum, particles and horizons result from strand tangles.
the basis of the stra nd m odel
157
⊳ Vacuum, or physical space, is formed by the time average of many unknotted
fluctuating strands.
Page 236
With the definition of the vacuum as a time average, the strand model yields a minimum
length and a continuous vacuum at the same time. In this way, many issues about the
alleged contradiction between continuity and minimum length are put to rest. In particular, physical space is not fundamentally discrete: a minimum length appears, though
only in domains where physical space is undefined. On the other hand, the continuity of physical space results from an averaging process. Therefore, physical space is not
fundamentally continuous: the strand model describes physical space as a homogeneous
distribution of crossing switches. This is the strand version of Wheeler’s idea space-time
foam.
Observable values and limits
The fundamental principle implies the following definitions of the basic observables:
* We recall that since over a century, the concept of aether is superfluous, because it is indistinguishable
from the concept of vacuum.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
⊳ We do not make any statement on the numerical density of strands in vacuum, or, equivalently, on their average spacing. Since strands are not observable, such a statement is not sensible. In particular, strands in vacuum
are not tightly packed.
Motion Mountain – The Adventure of Physics
Page 279
Figure 14 visualizes the definition. In the following, vacuum and physical space are always taken to be synonyms; the exploration will show that this is the most sensible use
of the two concepts.* However, as mentioned, the strand model distinguishes physical
space from background space. In particular, since matter and vacuum are made of the
same constituents, it is impossible to speak of physical space at the location of matter. At
the location of matter, it is only possible to use the concept of background space.
When the strand fluctuations in flat vacuum are averaged over time, there are no
crossing switches. Equivalently, if we use concepts to be introduced shortly, flat vacuum
shows, averaged over time, no knots and no tangles, so that it is observed to be empty of
matter and radiation. Temporary tangles that appear for a short time through vacuum
fluctuations will be shown later to represent virtual particles.
We note that the (flat) physical vacuum state, which appears after averaging the strand
crossings, is continuous, Lorentz invariant and unique. These are important points for the
consistency of the model. Later we will also discover that curvature and horizons have
a natural description in terms of strands; exploring them will yield the field equations
of general relativity. The strand model thus replaces what used to be called ‘space-time
foam’ or ‘quantum foam’.
We also note that Figure 14 implies, despite appearances, that vacuum is isotropic. To
see this, we need to recall that the observables are the crossing switches, not the strands,
and that the observed vacuum isotropy results from the isotropy of the time-averaged
strand fluctuations.
158
7 the basis of the stra nd m od el
A fermion
Strand model :
time average
of crossing
switches
tails
Observed
probability
density :
crossing
orientations
core
Page 172
positions
phases
spin
orientation
position
phase
F I G U R E 15 The tangle model of a spin 1/2 particle. More details will be given below.
⊳ The time interval between two events is the maximum number of crossing
switches that can be measured between them. Time measurement is thus
defined as counting Planck times.
⊳ The entropy of any physical system is related to the logarithm of the number of possible measurable crossing switches. Entropy measurement is thus
defined through the counting of crossing switches. The strand model thus
states that any large physical system – be it made of matter, radiation, empty
space or horizons – has entropy.
Page 147
It is well-known that all other physical observables are defined using these four basic
ones. In other words, all physical observables are defined with crossing switches. We
also note that even though counting always yields an integer, the result of a physical
measurement is often an average of many counting processes. As a result of averaging
and fluctuations, measured values can be non-integer multiples of Planck units. Therefore, space, time, action, entropy and all other observables are effectively real numbers,
and thus continuous. Continuity is thus reconciled with the existence of a minimum
measurable length and time interval.
Finally, we note that defining observables with the help of crossing switches automatically makes the Planck units ?, ℏ, ?4 /4?, ? and all their combinations both observerinvariant and limit values. All these conclusions agree with the corresponding requirements for a final theory of nature.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
⊳ The physical action of a physical system evolving from an initial to a final
state is the number of crossing switches that can be measured. Action measurement is thus defined as counting crossing switches. Physical action is
thus a measure for the change that a system undergoes.
Motion Mountain – The Adventure of Physics
⊳ The distance between two particles is the maximum number of crossing
switches that can be measured between them. Length measurement is thus
defined as counting Planck lengths.
the basis of the stra nd m odel
159
Particles and fields
Strands also define particles, as illustrated in Figure 15:
⊳ A quantum particle is a tangle of fluctuating strands. The tangle core, the
region where the strands are linked, defines position, speed, phase and spin
of the particle. The tangle tails reach up to the border of space.
Page 174
Page 222
Page 212
Page 226
Why do crossing switches have such a central role in the strand model? An intuitive explanation follows from their role in the definition of observables. All measurements – be
they measurements of position, speed, mass or any other observable – are electromagnetic. In other words, all measurements in nature are, in the end, detection of photons.
And the strand model shows that photon absorption and detection are intimately related
to the crossing switch, as we will find out below.
∗∗
Is there a limit to the fluctuations of strands? Yes and no. On the one hand, the ‘speed’
of fluctuations is unlimited. On the other hand, fluctuations with a ‘curvature radius’
smaller than a Planck length do not lead to observable effects. Note that the terms ‘speed’
and ‘radius’ are between quotation marks because they are unobservable. Care is needed
when talking about strands and their fluctuations.
∗∗
What are strands made of? This question tests whether we are really able to maintain
the fundamental circularity of the unified description. Strands are featureless. They have
no measurable properties: they have no branches, carry no fields and, in particular, they
cannot be divided into parts. The ‘substance’ that strands are made of has no properties.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Curiosities and fun challenges abou t strands
Motion Mountain – The Adventure of Physics
Page 314
As shown in more detail soon, this definition of quantum particles yields, depending on
the tangle details, either fermion or boson behaviour, and reproduces the spin–statistics
theorem.
Boson tangles will allow us to model field intensities. In particular, boson tangles
allow us to deduce the electromagnetic and the two nuclear fields, as well as the corresponding gauge symmetries of the standard model of particle physics.
Modelling fermions as tangles will allow us to deduce Dirac’s equation for relativistic quantum particles (and the Schrödinger equation for non-relativistic particles). Still
later, by classifying all possible tangles, we will discover that only a finite number of possible elementary particles exist, and that the topological type of tangle determines the
mass, mixings, quantum numbers, charges and couplings of each elementary particle.
We could also speak of a tangle model.
In the 1960s, John Wheeler stated that a unified description of nature must explain
‘mass without mass, charge without charge, field without field’. The strand model realizes
this aim, as we will find out.
Before we deduce modern physics, we first take a break and explore some general
issues of the strand model.
160
Challenge 118 e
Page 304
7 the basis of the stra nd m od el
Thus strands are not made of anything. This may seem surprising at first. Strands are
extended, and we naturally imagine them as sequence of points. But this is a fallacy.
Given the way that observations and events are defined, there is no way to observe, to
label or to distinguish points on strands. Crossing switches do not allow doing so, as is
easily checked: the mathematical points we imagine on a strand are not physical points.
‘Points’ on strands are unobservable: they simply do not exist.
But strands must be made of something, we might insist. Later we will find out that
in the strand model, the universe is made of a single strand folded in a complicated way.
Nature is one strand. Therefore, strands are not made of something, they are made of
everything. The substance of strands is nature itself.
∗∗
Since there is only one strand in nature, strands are not a reductionist approach. At
Planck scale, nature is one and indivisible.
∗∗
Challenge 119 e
Can macroscopic determinism arise at all from randomly fluctuating strands?
∗∗
Challenge 120 s
Do parallel strands form a crossing? Do two distant strands form a crossing?
∗∗
Challenge 121 s
Is a crossing switch defined in more than three dimensions?
∗∗
Challenge 122 s
Can you find a way to generalize or to modify the strand model?
∗∗
In hindsight, the fundamental principle resembles John Wheeler’s vision ‘it from bit’.
He formulated it, among others, in 1989 in his often-cited essay Information, physics,
quantum: the search for links.
∗∗
Looking back, we might equally note a relation between the strand model and the ex-
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 77
What are particles? In the strand model, elementary particles are (families of) tangles of
strands. In other words, elementary particles are not the basic building blocks of matter –
strands are. If particles could really be elementary, it would be impossible to understand
their properties.
In the strand model, particles are not really elementary, but neither are they, in the
usual sense, composed. Particles are tangles of unobservable strands. In this way, the
strand model retains the useful aspects of the idea of elementary particle but gets rid of
its limitations. In a sense, the strand model can be seen as eliminating the concepts of
elementariness and of particle. This confirms and realizes another requirement that we
had deduced earlier on.
Motion Mountain – The Adventure of Physics
∗∗
the basis of the stra nd m odel
Challenge 123 e
161
pression ‘it from qubit’ that is propagated by David Deutsch. A qubit is a quantummechanical two-level system. What is the difference between the fundamental principle
and a qubit?
∗∗
Ref. 154
Page 299
D o strands unify? – The millennium list of open issues
Page 64
Page 73
Page 127
Challenge 124 e
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 90
Does the strand model reproduce all the paradoxical results we found in the first
chapters? Yes, it does. The strand model implies that vacuum cannot be distinguished
from matter at Planck scales: both are made of strands. The strand model implies that
observables are not real numbers at Planck scales. The strand model implies that the
universe and the vacuum are the same, when explored at high precision: both are made
of one strand. The strand model also implies that the number of particles in the universe
is not clearly defined and that nature is not a set. You can check by yourself that all other
paradoxes appear automatically. Furthermore, almost all requirements for a final theory
listed in Table 6 are fulfilled. Only two requirements of the table must be discussed in
more detail: the requirements of complete precision and of unmodifiability. We start
with complete precision.
If strands really describe all of nature, they must explain the inverse square dependence with distance of the electrostatic and of the gravitational interaction. But that is
not sufficient. If the strand model is a final, unified description, it must provide complete
precision. This requires, first of all, that the model describes all experiments. As will be
shown below, this is indeed the case, because the strand model contains both general relativity and the standard model of particle physics. But secondly and most importantly,
the model must also settle all those questions that were left unanswered by twentiethcentury fundamental physics. Because the questions, the millennium list of open issues,
are so important, they are given, again, in Table 8.
Motion Mountain – The Adventure of Physics
Page 395
Is the strand model confirmed by other, independent research? Yes, a few years after the
strand model appeared, this started to happen. For example, in a long article exploring
the small scale structure of space-time from various different research perspectives in
general relativity, Steven Carlip comes to the conclusion that all these perspectives suggest the common result that ‘space at a fixed time is thus threaded by rapidly fluctuating
lines’. This is exactly what the strand model states.
Other theoretical approaches that confirm the strand model are mentioned in various
places later in the text. Despite such developments, the essential point remains to check
how the strand model compares with experiment. Given that the strand model turns out
to be unmodifiable, there are no ways to amend predictions that turn out to be wrong. If
a single prediction of the strand model turns out to be incorrect, the model is doomed.
But so far, no experimental prediction of the strand model contradicts experiments.
162
7 the basis of the stra nd m od el
TA B L E 8 The millennium list: everything the standard model and general relativity cannot explain; thus,
also the list of the only experimental data available to test the final, unified description of motion.
O b s e rva b l e P r o p e rt y u n e x p l a i n e d s i n c e t h e y e a r 2 0 0 0
Local quantities unexplained by the standard model: particle properties
Concepts unexplained by the standard model
SU(2)
SU(3)
Renorm. group
?? = 0
? = ∫? SM d?
the origin of the invariant Planck units of quantum field theory
the number of dimensions of physical space and time
the origin of Poincaré symmetry, i.e., of spin, position, energy, momentum
the origin and nature of wave functions
the origin of particle identity, i.e., of permutation symmetry
the origin of the gauge groups, in particular:
the origin of the electromagnetic gauge group, i.e., of the quantization of
electric charge, of the vanishing of magnetic charge, and of minimal coupling
the origin of weak interaction gauge group, its breaking and P violation
the origin of strong interaction gauge group and its CP conservation
the origin of renormalization properties
the origin of the least action principle in quantum theory
the origin of the Lagrangian of the standard model of particle physics
Global quantities unexplained by general relativity and cosmology
0
1.2(1) ⋅ 1026 m
?de = Λ?4 /(8π?)
≈ 0.5 nJ/m3
(5 ± 4) ⋅ 1079
?dm
the observed flatness, i.e., vanishing curvature, of the universe
the distance of the horizon, i.e., the ‘size’ of the universe (if it makes sense)
the value and nature of the observed vacuum energy density, dark energy or
cosmological constant
the number of baryons in the universe (if it makes sense), i.e., the average
visible matter density in the universe
the density and nature of dark matter
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
?, ℏ, ?
3+1
SO(3,1)
Ψ
?(?)
Gauge symmetry
U(1)
Motion Mountain – The Adventure of Physics
? = 1/137.036(1) the low energy value of the electromagnetic coupling or fine structure constant
?w or ?w
the low energy value of the weak coupling constant or the value of the weak
mixing angle
?s
the value of the strong coupling constant at one specific energy value
?q
the values of the 6 quark masses
?l
the values of 6 lepton masses
?W
the value of the mass of the ? vector boson
?H
the value of the mass of the scalar Higgs boson
?12 , ?13 , ?23
the value of the three quark mixing angles
?
the value of the CP violating phase for quarks
?
?
?
?12
, ?13
, ?23
the value of the three neutrino mixing angles
?
? , ?1 , ?2
the value of the three CP violating phases for neutrinos
3⋅4
the number of fermion generations and of particles in each generation
J, P, C, etc.
the origin of all quantum numbers of each fermion and each boson
the basis of the stra nd m odel
163
TA B L E 8 (Continued) The millennium list: everything the standard model and general relativity cannot
explain; also the only experimental data available to test the final, unified description of motion.
O b s e rva b l e P r o p e rt y u n e x p l a i n e d s i n c e t h e y e a r 2 0 0 0
?0 (1, ..., c. 1090 )
the initial conditions for c. 1090 particle fields in the universe (if or as long as
they make sense), including the homogeneity and isotropy of matter distribution, and the density fluctuations at the origin of galaxies
Concepts unexplained by general relativity and cosmology
?, ?
R × S3
???
?? = 0
? = ∫? GR d?
the origin of the invariant Planck units of general relativity
the observed topology of the universe
the origin and nature of curvature, the metric and horizons
the origin of the least action principle in general relativity
the origin of the Lagrangian of general relativity
— Reproduce quantum theory, the standard model, general relativity and cosmology.
— Explain masses, mixing angles and coupling constants.
Are strands final? – On generalizations and modifications
“
The chief attraction of the theory lies in its
logical completeness. If a single one of the
conclusions drawn from it proves wrong, it
must be given up; to modify it without
destroying the whole structure seems
impossible.
Albert Einstein, The Times, 28. 11. 1919.
”
If a description of motion claims to be final, it must explain all aspects of motion. To
be a full explanation, such a description must not only be logically and experimentally
complete, it must also be unmodifiable. Even though Einstein made the point for general relativity, this important aspect is rarely discussed with clarity. In particular, any
unmodifiable explanation has two main properties: first, it cannot be generalized, and
second, it is not itself a generalization.
Generalizing models is a sport among theoretical and mathematical physicists. If you
have a description of a part of nature, they will try to find more general cases. For any
candidate unified description, they will try to explore the model in more than three dimensions, with more than three generations of quarks, with more complicated gauge
symmetries, with different types of supersymmetry, with more Higgs bosons, or with
additional heavy neutrinos. In the case of the strand model, researchers will also ex-
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Of course, only the second point is the definite test for a final, unified description. But
we need the first point as well. The following chapters deal with both points.
Motion Mountain – The Adventure of Physics
The open issues in the millennium list must be resolved by any final, unified model
of nature, and thus also by the strand model. All the open issues can be summarized in
two general points:
164
7 the basis of the stra nd m od el
plore models with more complicated entities than strands, such as bands or bifurcating
entities, and any other generalization they can imagine.
⊳ Can a final description of nature have generalizations? No.
⊳ Can the unified theory be a generalization of existing theories? No.
Ref. 155
* Independently, David Deutsch made a similar point with his criterion that an explanation is only correct
if it is hard to vary. Used in the case of a final theory, we can say that the final theory must be an explanation
of general relativity and of the standard model. This implies that the final theory must be hard to vary. This
matches the above conclusion that the final theory must be unmodifiable.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Challenge 125 e
Because neither general relativity nor the standard model of particle physics are able to
explain the millennium issues, any generalization of them would also be unable to do so.
Generalizations have no explanatory power. If the unified theory were a generalization
of the two existing theories, it could not explain any of the millennium issues of Table 8!
Therefore, general relativity and the standard model of particle physics must be approximations, but not special cases, of the final theory. In particular, if the strand model is a
final description, approximations of the strand model must exist, but special cases must
not. This is indeed the case, as we will find out.
To summarize, a final theory must be an explanation of all observations. An explanation of an observation is the recognition that it follows unambiguously, without alternative, from a general property of nature. We conclude that the final, unified description
of motion must neither allow generalization nor must it be a generalization of either
the standard model or general relativity. The unified theory cannot be generalized and
cannot be ‘specialized’; the unified theory must be unmodifiable.* This requirement is
extremely strong; you may check that it eliminates most past attempts at unification. For
example, this requirement eliminates grand unification, supersymmetry and higher dimensions as aspects of the final theory: indeed, these ideas generalize the standard model
Motion Mountain – The Adventure of Physics
Indeed, if it were possible to generalize the final description, it would lose the ability to
explain any of the millennium issues! If a candidate unified theory could be generalized,
it would not be final. In short, if the strand model is a final description, the efforts of
theoretical and mathematical physicists just described must all be impossible. So far,
investigations confirm this prediction: no generalization of the strand model has been
found yet.
Where does this fondness for generalization come from? In the history of physics,
generalizations often led to advances and discoveries. In the past, generalizations often
led to descriptions that had a wider range of validity. As a result, generalizing became
the way to search for new discoveries. Indeed, in the history of physics, the old theory
often was a special case of the new theory. This relation was so common that usually,
approximation and special case were taken to be synonyms. This connection leads to a
second point.
General relativity and the standard model of particle physics must indeed be approximations of the final theory. But can either general relativity or the standard model be
special cases of the final, unified theory? Or, equivalently:
the basis of the stra nd m odel
Challenge 126 e
165
of elementary particles and they are modifiable. Therefore, all these ideas lack explanatory power.
A final and unified theory must be an unmodifiable explanation of general relativity
and the standard model. Because neither supersymmetry, nor the superstring conjecture,
nor loop quantum gravity explain the standard model of particle physics, they are not
unified theories. Because these models are modifiable, they are not final theories. In
fact, at least one of these two aspects is lacking in every candidate final theory proposed
in the twentieth century.
We will discover below that the strand model is unmodifiable. Its fundamental principle cannot be varied in any way without destroying the whole description. Indeed, no
modification of the strand model or of the fundamental principle has been found so
far. We will also discover that the strand model explains the standard model of particle
physics and explains general relativity. The strand model is thus a candidate for the final
theory.
“
Page 115
Ref. 140
Ref. 156
Ref. 157
Ref. 158
”
Antiquity
Let us assume that we do not know yet whether the strand model can be modified
or not. Two other reasons still induce us to explore featureless strands as basis for a
unified description. First, featureless strands are the simplest known model that unifies quantum field theory and general relativity. Second, featureless strands are the only
known model that realizes an important requirement: a unified description must not be
based on points, sets or any axiomatic system. Let us explore the issue of simplicity first.
In order to reproduce three-dimensional space, Planck units, spin, and black-hole entropy, the fundamental constituents must be extended and fluctuating. We have deduced
this result in detail in the previous chapter. The extension must be one-dimensional, because this is the simplest option, and it is also the only option compatible with threedimensional space. In fact, one-dimensional strands explain the three-dimensionality of
space, because tangles of one-dimensional strands exist only in three spatial dimensions.
In four or more dimensions, any tangle or knot can be undone; this is impossible in three
spatial dimensions.
No simpler model than featureless strands is possible. All other extended constituents
that have been explored – ribbons, bands, strings, membranes, posets, branched lines,
networks, crystals and quantum knots – increase the complexity of the model. In fact
these constituents increase the complexity in two ways: they increase the number of features of the fundamental constituents and they complicate the mapping from the model
to observation.
First, no other model based on extension uses featureless constituents. In all other
models, the fundamental constituents have properties such as tension, field values, coordinates, quantum numbers, shape, twists, orientation, non-trivial topological information, etc. In some models, space-time is non-commutative or fermionic. All these features are assumed; they are added to the model by fiat. As such, they allow alternatives
* ‘Simplicity is the seal of truth.’
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 105
Simplex sigillum veri.*
Motion Mountain – The Adventure of Physics
Why strands? – Simplicit y
166
Ref. 150
7 the basis of the stra nd m od el
“
Without the concepts place, void and time,
change cannot be. [...] It is therefore clear [...]
that their investigation has to be carried out, by
studying each of them separately.
Aristotle Physics, Book III, part 1.
”
The strand model describes strands as fluctuating in a background space-time of three
plus one space-time dimensions. The background space-time is introduced by the observer. The background is thus different for every observer; however, all such backgrounds have three dimensions of space and one of time. The observer – be it a machine,
an animal or a human – is itself made of strands, so that in fact, the background space is
itself the product of strands.
We therefore have a fundamental circular definition: we describe strands with a back-
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Why strands? – The fundamental circul arit y of physics
Motion Mountain – The Adventure of Physics
and are difficult if not impossible to justify. In addition, these features increase the complexity of the possible processes. In contrast, the strand model has no justification issue
and no complexity issue.
Secondly, the link between more complicated models and experiment is often intricate and sometimes not unique. As an example, the difficulties to relate superstrings to
experiments are well-known. In contrast, the strand model argues that the experimentally accessible Dirac equation of quantum field theory and the experimentally accessible
field equations of general relativity arise directly, from an averaging procedure of crossing switches. Indeed, the strand model proposes to unify these two halves of physics with
only one fundamental principle: strand crossing switches define Planck units. In fact, we
will find out that the strand model describes not only vacuum and matter, but also gauge
interactions and particle properties as natural consequences of the structure of nature at
Planck scales. The comparable ideas in other models are much more elaborate.
We remark that building three-dimensional physical space from strands is even simpler than building it from points! In order to build three-dimensional space from points,
we need concepts such as sets, neighbourhoods, topological structures and metric structures. And despite all these intricate concepts, the concept of space defined in this way
still has no defined physical length scale; in short, it is not the same as physical space.
In contrast, in order to build three-dimensional physical space from strands, we need
no fundamental points, sets, or metric structures; we only need long-time averages of
strands and their crossings. And the length scale is built in.
All this suggests that the strand model, based on featureless, one-dimensional and
fluctuating constituents, might be the model for unification with the smallest number
of concepts, thus satisfying Occam’s razor. In fact, we will discover that strands indeed
are the simplest way to model particles, interactions and the vacuum, while fulfilling the
requirements of a final theory.
The simplicity of a model helps in two ways. First, the simpler a model is, the freer it
is of ideology, preconceptions and beliefs. Secondly, the simpler a model is, the easier it
can be checked against observation. In particular, a simple model allows simple checking
of its solution of paradoxes. Above all, we can resolve the most important paradox of
physics.
the basis of the stra nd m odel
167
TA B L E 9 The differences between nature and any description.
Description
Nature is not a set.
Descriptions need sets to allow talking and
thinking.
Descriptions need events, points and
continuous 3 + 1-dimensional space-time to
allow formulating them.
Descriptions need point particles to allow
talking and thinking.
Descriptions need locality to allow talking and
thinking.
Descriptions need a background to allow
talking and thinking.
Descriptions need to break duality to allow
talking and thinking.
Axiomatic descriptions are needed for precise
talking and thinking.
Nature has no events, no points and no
continuity.
Nature has no point particles.
Nature is not local.
Nature has no background.
Nature shows something akin to ? ↔ 1/?
duality.
Nature is not axiomatic but contains circular
definitions.
Page 108
⊳ We use both continuous background space-time and discrete strands to describe nature.
In a few words: A unified model of physics allows talking about motion with highest
precision; this requirement forces us to use, at the same time, both continuous spacetime and discrete strands. This double use is not a contradiction but, as just explained,
the result of a circular definition. Since we, the talkers, are part of nature, a unified model
means that we, the talkers, talk about ourselves.
We stress that despite the circularity of physics, Gödel’s incompleteness theorem does
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
ground, and the background with strands. Strands thus do not provide an axiomatic
system in the mathematical sense. This fulfils one of the requirements for the unified
description.
Why does the fundamental circular definition arise? Physics is talking (and thinking) about nature and motion. A unified model of physics is talking about motion with
highest precision. This implies that on the one hand, as talkers, we must use concepts that
allow us to talk. Talking and thinking requires that we use continuous space and time:
on short, we must use a background. The background must be continuous, without minimum length. On the other hand, to talk with precision, we must have a minimum length,
and use strands. There is no way to get rid of this double and apparently contradictory
requirement. More such contradictory requirements are listed in Table 9. We know that
nature is not a set, has no points, no point particles and no locality, but that it is dual.
But in order to talk about nature, we need a background that lacks all these properties.
Because there is no way to get rid of these apparently contradictory requirements, we
don’t:
Motion Mountain – The Adventure of Physics
Nature
168
7 the basis of the stra nd m od el
Universe’s
horizon or
‘border
of space’
(pink)
Universe’s
tangle
(blue
lines)
Background
space
(grey)
Background
space
(grey)
Physical
space or
vacuum
(white)
Particle
tangle
(tangled
blue
lines)
Physical
space or
vacuum
(white)
Vol. III, page 308
Page 108
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 159
not apply to the situation. In fact, the theorem does not apply to any unified theory
of physics for two reasons. First, the incompleteness theorem applies to self-referential
statements, not to circular definitions. Self-referential statements do not appear in physics, not in sensible mathematics and not in the strand model. Secondly, Gödel’s theorem
applies to mathematical structures based on sets, and the final theory is not based on
sets.
We do not state that background space and time exist a priori, as Immanuel Kant
states, but only that background space and time are necessary for thinking and talking, as
Aristotle states. In fact, physical space and time result from strands, and thus do not exist
a priori; however, background space and time are required concepts for any description
of observations, and thus necessary for thinking and talking. Figure 16 illustrates the
solution proposed by the strand model.
We have always to be careful to keep the fundamental circular definition of strands
and backgrounds in our mind. Any temptation to resolve it leads astray. For example, if
we attempt to define sets or elements (or points) with the help of measurements, we are
hiding or forgetting the fundamental circularity. Indeed, many physicists constructed
and still construct axiomatic systems for their field. The fundamental circularity implies
that axiomatic systems are possible for parts of physics, but not for physics as a whole.
Indeed, there are axiomatic descriptions of classical mechanics, of electrodynamics, of
quantum theory, of quantum field theory, and even of general relativity. But there is no
axiomatic system for all of physics – i.e., for the description of all motion – and there
cannot be one.
A further issue must be discussed in this context. As mentioned, strands fluctuate
in a background space, and only crossing switches can be observed. In particular, this
implies that the mathematical points of the background space cannot be observed. In
other words, despite using mathematical points to describe the background space (and
strands themselves), none of them have physical significance. Physical points do not exist
Motion Mountain – The Adventure of Physics
F I G U R E 16 In the strand model, physical space – or vacuum – and background space are distinct, both
near the horizon and near particles.
the basis of the stra nd m odel
A strand :
169
A funnel :
F I G U R E 17 Two equivalent depictions of the fundamental constituents of nature: strands and funnels.
Another type of constituent also fulfils all the conditions for a unified description. As
shown in Figure 17, as an alternative to fluctuating strands, we can use fluctuating funnels
as fundamental constituents. In the description with funnels, nature resembles a complicated tangle of a three-dimensional space that is projected back into three dimensions.
Funnels show that the strand model only requires that the effective minimal effective
diameter of a strand is the Planck length; it could have other diameters as well. Funnels
also show that due to varying diameters, strands can, through their fluctuations, literally
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Funnels – an equivalent alternative to strands
Motion Mountain – The Adventure of Physics
Ref. 160
in the strand model. Physical locations of events are due to crossing switches, and can
at best be localized to within a Planck length. The same limitation applies to physical
events and to physical locations in time. A natural Planck-scale non-locality is built into
the model. This realizes a further requirement that any unified description has to fulfil.
The situation for physicists working on unification is thus harder – and more fascinating – than that for biologists, for example. Biology is talking about living systems.
Biologists are themselves living systems. But in the case of biologists, this does not lead
to a circular definition. Biology does not use concepts that contain circular definitions: a
living being has no problems describing other living beings. Even neurobiologists, who
aim to explore the functioning of the brain, face no fundamental limit doing so, even
though they explore the human brain using their own brain: a brain has no problem
describing other brains. In contrast, physicists working on unification need to live with
circularity: a fundamental, precise description of motion requires to be conscious of our
own limitations as describing beings. And our main limitation is that we cannot think
without continuous space and time, even though these concepts do not apply to nature.
We conclude: A unified description cannot be axiomatic, cannot be based on observable physical points, must distinguish physical space from background space, and cannot be
background-independent. Many models based on extended constituents also use backgrounds. However, most models also allow the definition of sets and axiomatic descriptions. Such models thus cannot be candidates for a unified description of nature. In
contrast, the strand model keeps the fundamental circularity of physics intact; it does
not allow an axiomatic formulation of fundamental physics, and only allows points or
sets as approximate concepts.
170
Page 129
Challenge 127 e
7 the basis of the stra nd m od el
Knots and the ends of strands
summary on the fundamental principle – and on
continuit y
Page 147
We have introduced featureless, fluctuating strands as common constituents of space,
matter, radiation and horizons. We defined fundamental events as crossing switches of
strands. All physical processes are composed of fundamental events. Events and the
values of all physical observables are defined with the help of Planck units, which in turn
are due to crossing switches of strands. The definition of all physical observables through
Planck units with the help of crossing switches of strands is the fundamental principle.
Using the fundamental principle, continuity of any kind – of space, fields, wave functions or time – results from the time averaging of crossing switches. This issue will be
explored in detail below.
The strand model fulfils the general requirements for the final and unified description
listed in Table 6, provided that it describes all motion with full precision and that it is
* Two issues that put this equivalence into question are ending funnels and diameter behaviour under
boosts. The first issue is subject of research, but it is expected that it poses no problem. The second issue is mitigated by the shivering of the background space.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
In the original strand model, developed in the year 2008, strands that contain knots were
part of the allowed configurations. This has the disadvantage that the formation of a knot
requires at least one loose end that is pulled through a strand configuration. Such loose
ends, however, produce a number of issues that are difficult to explain, especially during
the emission and absorption of knotted tangles.
Later, in 2015, it became clear that strands without knots are sufficient to recover the
standard model of particle physics. This approach is shown in the following. Knots and
the ends of strands play no role in the model any more; the aim of highest possible simplicity is now realized.
Motion Mountain – The Adventure of Physics
be everywhere in space and thus effectively fill space, even if their actual density is low.
Funnels resemble many other research topics. Funnels are similar to wormholes; however, both their ends lead, at the border of space, ‘into’ usual three-dimensional space.
Funnels are also similar to D-branes, except that they are embedded in three spatial dimensions, not ten. Funnels also resemble a part of an exotic manifold projected into
three-dimensional space. Fluctuating funnels also remind us of the amoeba mentioned
above. However, the similarities with wormholes, D-branes or exotic manifolds are of
little help: so far, none of these approaches has led to viable models of unification.
A first check shows that the funnel alternative seems completely equivalent to strands.*
You might enjoy testing that all the conclusions deduced in the following pages appear
unchanged if strands are replaced by funnels. In particular, also funnels allow us to deduce quantum field theory, the standard model and general relativity. Due to the strict
equivalence between strands and funnels, the choice between the two alternatives is a
matter of taste or of visualization, but not a matter of physics. We use strands in the
following, as they are simpler to draw.
sum m ary on the fundamental principle – a nd on continuity
Page 162
171
unmodifiable.
At this point, therefore, we must start the comparison with experiment. We need
to check whether strands describe all motion with complete precision. Fortunately, the
task is limited: we only need to check whether strands solve each of the millennium
issues listed in Table 8. If the strand model can solve those issues, then it reproduces all
observations about motion and provides a final and unified description of nature. If the
issues are not solved, the strand model is worthless.
Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Chapter 8
QUANT UM T HEORY OF MAT T ER
DEDUCED FR OM ST R ANDS
W
In nature, particles move in the vacuum. The vacuum is free of matter and energy. In the
strand model,
⊳ Vacuum is a collection of fluctuating, unknotted and untangled strands.
Page 156
Page 236
The vacuum is illustrated in Figure 14. The time average of unknotted and untangled
strands has no energy and no matter content, because there are – averaged over time –
no crossing switches and no tangles. The temporary crossing switches that can appear
through fluctuations of the vacuum will turn out to be virtual particles; we will explore
them below. We note that the physical vacuum, being a time average, is continuous. The
flat physical vacuum is also unique: it is the same for all observers. The strand model
thus contains both a minimum length and a continuous vacuum. The two aspects do
not contradict each other.
In nature, quantum particles move: quantum particles change position and phase over
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Strands, vacuum and particles
Motion Mountain – The Adventure of Physics
e show in this chapter that featureless strands that fluctuate, together
ith the fundamental principle – defining ℏ/2 as a crossing switch – imply
ithout alternative that matter is described by quantum theory. More precisely,
we deduce that tangles of fluctuating strands reproduce the spin 1/2 behaviour of matter
particles, allow us to define wave functions, and imply the Dirac equation for the motion of matter. In particular, we first show that the components and phases of the wave
function at a point in space are due to the orientation and phase of strand crossings at
that point. Then we show that the Dirac equation follows from the belt trick (or string
trick).
Furthermore, we show that strands imply the least action principle and therefore, that
tangles of fluctuating strands are described by the Lagrangian of relativistic quantum
particles. So far, it seems that the strand model is the only microscopic model of relativistic quantum theory that is available in the research literature.
In the present chapter, we derive the quantum theory of matter: we show that strands
reproduce all observations about fermions and their motion. We leave for later the derivation of the quantum theory of light and the nuclear interactions, the standard model of
elementary particles, and the quantum description of gravitation. As usual in quantum
theory, we work in flat space-time.
q ua ntum theory d ed uced from stra nds
A fermion
Strand model :
time average
of crossing
switches
tails
belt trick
orientation
core
positions
phases
crossing
orientations
173
Observed
probability
density :
spin
orientation
position
phase
F I G U R E 18 A fermion is described by a tangle of two or three strands. The crossings in the tangle core
and their properties lead, after averaging, to the wave function and the probability density.
⊳ An elementary matter particle, or fermion, is a tangle of two or more strands
that realizes the belt trick.
⊳ The position of a particle is given by the centre of the averaged tangle core.
The particle position is thus the average of all its crossing positions.
⊳ The phase of a matter particle is given by half the angle that describes the
orientation of the tangle core around the spin axis. The particle phase is thus
the average of all its crossing phases.
⊳ The spin orientation of a matter particle is given by the rotation axis of the
core. The spin orientation is thus the average of all its crossing orientations.
⊳ The wave function of a matter particle is a blurred rendering of the crossing
of its fluctuating strands.
Page 184
These definitions are illustrated in Figure 18 and will be explored in detail below. We
note that all these definitions imply a short-time average over tangle fluctuations. With
the definitions, we get:
⊳ Motion of any quantum particle is the change of the position and orientation
of its tangle core.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
The details of this definition will become clear shortly, including the importance of the
related tangle family. In every tangle, the important structure is the tangle core: the core
is the that region of the tangle that contains all the links and crossings. The core is connected to the border of space by the tails of the tangle.
Motion Mountain – The Adventure of Physics
time. We therefore must define these concepts. At this stage, as just explained, we concentrate on quantum matter particles and leave radiation particles for later on. As illustrated in Figure 18 and Figure 19, we define:
174
8 q ua ntum theory d ed uced from stra nds
In nature, quantum particle motion is described by quantum theory. The main property
of quantum theory is the appearance of the invariant quantum of action ℏ. In the strand
model, ℏ/2 is described by a single crossing switch; the value of the quantum of action
is thus invariant by definition.
We now explore in detail how the quantum of action ℏ determines the motion of
quantum particles. In particular, we will show that tangle fluctuations reproduce usual
textbook quantum theory. As an advance summary, we clarify that
⊳ Free quantum particle motion is due to fluctuations of tangle tails. The deformations of the tangle core are not important for free motion, and we can
neglect them in this case.
Rotation, spin 1/2 and the belt trick
⊳ Spin is core rotation.
Indeed, in the strand model, all quantum particles, including those with spin 1/2, differ
from everyday objects such as stones, and the essential difference is due to extension:
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
In nature, quantum particles are described by their behaviour under rotation and by their
behaviour under exchange. The behaviour of a particle under rotation is described by
its spin value, its spin axis and its phase. The behaviour of quantum particles under
exchange can be of two types: a quantum particle can be a fermion or a boson. In nature,
particles with integer spin are bosons, and particles with half-integer spin are fermions. This
is the spin–statistics theorem.
We now show that all properties of particle rotation and exchange follow from the
strand model. We start with the case of spin 1/2 particles, and first clarify the nature of
particle rotation. (We follow the usual convention to use ‘spin 1/2’ as a shorthand for
‘z-component of spin with value ℏ/2’.)
It is sometimes claimed that spin is not due to rotation. This misleading statement is
due to two arguments that are repeated so often that they are rarely questioned. First,
it is said, spin 1/2 particles cannot be modelled as small rotating stones. Secondly, it is
allegedly impossible to imagine rotating electric charge distributions with a speed of rotation below that of light and an electrostatic energy below the observed particle masses.
These statements are correct. Despite being correct, there is a way to get around them,
namely by modelling particles with strands; at the present stage, we focus on the first
argument: we will show that spin can be modelled as rotation.
In the strand model, for all quantum particles we have:
Motion Mountain – The Adventure of Physics
Page 222
In other words, when exploring quantum theory, we approximate tangle cores as being rigid. We will study core deformations in the next chapter, where we show that they
are related to interactions. Core deformations will lead to quantum field theory. In this
chapter we explore just the deformations of tangle tails; they produce the motion of free
(and stable) quantum particles. In short, tail deformations lead to quantum mechanics.
To deduce quantum mechanics from strands, we first study the rotation and then the
translation of free matter particles.
q ua ntum theory d ed uced from stra nds
175
The belt trick or string trick or plate trick or scissor trick explains the possibility of continuous
core rotation for any number of tails. A rotation by 4π is equivalent to none at all :
moving
tails
aside
moving
upper
tails
moving
lower
tails
moving
all
tails
The same equivalence can be shown with the original version of the belt trick :
or simply rearranging the belts,
independently of their number,
yields the other situation
F I G U R E 19 The belt trick – or string trick or plate trick or scissor trick – shows that a rotation by 4 π of a
central object with three or more tails (or with one or more ribbons) attached to spatial infinity is
equivalent to no rotation at all. This equivalence allows a suspended object, such as a belt buckle or a
tangle core, to rotate for ever. The belt trick thus shows that tangle cores made from two or more
strands behave as spin 1/2 particles.
⊳ Quantum particles are particles whose tails cannot be neglected.
For stones and other everyday objects, tails do not play an important role, because everyday objects are mixed states, and not eigenstates of angular momentum. In short, in
everyday objects, tails can be neglected. Therefore, everyday objects are neither fermions nor bosons. But for quantum particles, the tails are essential. Step by step we will see
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
rotating the buckle either by 4π,
Motion Mountain – The Adventure of Physics
core
(or
belt
buckle)
176
8 q ua ntum theory d ed uced from stra nds
F I G U R E 20 The belt trick: a double
rotation of the belt buckle is equivalent to
no rotation; te animation shows one way
in which the belt trick can be performed.
Not shown: the belt trick is also possible
with any number of belts attached to the
buckle. (QuickTime film © Greg Egan)
that
⊳ The tails of quantum particles explain their spin behaviour, their exchange
behaviour and their wave behaviour.
Ref. 161
In particular, we will see that in the strand model, wave functions are blurred tangles; we
can thus explore the general behaviour of wave functions by exploring the behaviour of
tangles.
The spin behaviour of quantum particles is a consequence of strand tails. Indeed, it
has been known for about a century that the so-called belt trick – illustrated in Figure 19,
Figure 20, Figure 21 and Figure 22 – can be used, together with its variations, to model
the behaviour of spin 1/2 particles under rotations. The belt trick is the observation that
a belt buckle rotated by two full turns – in contrast to a buckle rotated by only one full
turn – can be brought back into its original state without moving the buckle; only the
motion of the belt is necessary. The belt trick is also called the scissor trick, the plate trick,
the string trick, the Philippine wine dance or the Balinese candle dance. It is sometimes
incorrectly attributed to Dirac.
The belt trick is of central importance in the strand model of spin 1/2 particles. In
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
animation shows another way – another
direction – in which the trick can be
performed, for the same belt orientation
as in the previous figure. (QuickTime film
© Greg Egan)
Motion Mountain – The Adventure of Physics
F I G U R E 21 The belt trick again: this
q ua ntum theory d ed uced from stra nds
177
Page 198
Challenge 128 e
the strand model, all spin 1/2 particles are made of two (or more) tangled strands, and
thus have four (or more) tails to the ‘border’, as shown in Figure 19. For such tangles, a
rotation by 4π of the tangle core – thus a rotation by two full turns – can bring back the
tangle to the original state, provided that the tails can fluctuate. Any system that returns
to its original state after rotation by 4π is described by spin 1/2. In fact, the tails must be
unobservable for this equivalence to hold; in the strand model, tails are simple strands
and thus indeed unobservable. We will show below that the intermediate twisting of the
tails that appears after rotation by only 2π corresponds to a multiplication of the wave
function by −1, again as expected from a spin 1/2 particle.
If we replace each belt by its two coloured edges, Figure 22 shows how tails behave
when a spin 1/2 tangle is rotated. By the way, systems with tails – be they strands or
bands – are the only possible systems that realize the spin 1/2 property. Only systems with
tails to spatial infinity have the property that a rotation by 4π is equivalent to no rotation
at all. (Can you show this?) The fundamental connection between spin 1/2 and extension
is one of the properties that led to the strand model.
The animations show that the belt trick works with one and with two belts attached
to the buckle. In fact, belt trick works with any number of belts attached to the buckle.
The belt trick even works with infinitely many belts, and also with a full two-dimensional
sheet. The wonderful video www.youtube.com/watch?v=UtdljdoFAwg by Gareth Taylor
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 189
a particle. The animation then shows that such a particle (the square object) can return to the starting
position after rotation by 4π (but not after 2π). Such a ‘belted’ or ‘tethered’ particle thus fulfils the
defining property of a spin 1/2 particle: rotating it by 4π is equivalent to no rotation at all. The belted
square thus represents the spinor wave function; for example, a 2π rotation leads to a twist; this means
a change of the sign of the wave function. A 4π rotation has no influence on the wave function. The
equivalence is shown here with two attached belts, but the trick works with any positive number of
belts! You can easily repeat the trick at home, with a paper strip or one or several real belts. (QuickTime
film © Antonio Martos)
Motion Mountain – The Adventure of Physics
F I G U R E 22 Assume that the belt cannot be observed, but the square object can, and that it represents
178
8 q ua ntum theory d ed uced from stra nds
a rotating ball attached to a sheet
(QuickTime film © www.ariwatch.
com/VS/Algorithms/DiracStringTrick.
htm).
⊳ An object or a tangle core that is attached by (three or more) tails to the
border of space can rotate continuously.
Here we made the step from belts to strands. In other terms, the possibility of continuous
rotation allows us to describe spin 1/2 particles by rotating tangles. In other terms,
⊳ Rotating tangles model spin.
The tail fluctuations required to rearrange the tails after two full turns of the core can
be seen to model the average precession of the spin axis. We thus confirm that spin and
rotation are the same for spin 1/2 particles.
The belt trick is not unique
Ref. 161
Challenge 129 e
One aspect of the belt trick seems unmentioned in the research literature: after a rotation
of the belt buckle or tangle core by 4π, there are various options to untangle the tails. Two
different options are shown in Figure 20 and Figure 21. You can test this yourself, using a
real belt. In fact, there are two extreme ways to perform the belt trick, and a continuum
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
and the slightly different animation of Figure 23 both illustrate the situation. A sphere
glued to a flexible sheet can be rotated as often as you want: if you do this correctly, there
is no tangling and you can go on for ever.
The animations of the belt trick lead us to a statement on strands and tangles that is
central for the strand model:
Motion Mountain – The Adventure of Physics
F I G U R E 23 The belt trick realized as
q ua ntum theory d ed uced from stra nds
179
fresh blood in
welded seals
platelet-poor
blood out
plastic bag
rotates at over 100
revolutions per second
F I G U R E 24 In an apheresis machine, the central bag spins at high speed despite being connected with
tubes to a patient; this happens with a mechanism that continuously realizes the belt trick (photo
© Wikimedia).
Motion Mountain – The Adventure of Physics
of options in between. These options will be of central importance later on: the options
require a description of fermions with four complex functions. We will discover that
the various options of the belt trick are related to the difference between matter and
antimatter and to the parity violation of the weak interaction.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
F I G U R E 25 The basis of the
apheresis machine – and yet
another visualisation of the belt
trick, here with 6 belts
(QuickTime film © Jason Hise).
180
8 q ua ntum theory d ed uced from stra nds
Without the belt trick, the apheresis machines found in many hospitals would not work.
When a person donates blood platelets, the blood is continuously extracted from one
arm and fed into a bag in a centrifuge, where the platelets are retained. The platelet-free
blood then flows back into the other arm of the donor. This happens continuously, for
about an hour or two. In order to be sterile, tubes and bag are used only once and are
effectively one piece, as shown in Figure 24. Apheresis machines need tethered rotation
to work. Topologically, this set-up is identical to a fermion tangle: each tube corresponds
to one belt, or two strand tails, and the rotating bag corresponds to the rotating core.
In such apheresis machines, the centrifugation of the central bag takes place at over
100 revolutions per second, in the way illustrated in Figure 25. To avoid tangling up the
blood tubes, a bracket moves the tubes during each rotation, alternatively up and down.
This so-called anti-twister mechanism produces precisely the motion along which the
belt moves when it is untangled after the buckle is rotated by 4π. An apheresis machine
thus performs the belt trick 50 times per second, with each rotation of the centrifugation.
Due to the centrifugation, the lighter platelets are retained in the bag, and the heavier
components of the blood are pumped back to the donor. The retained platelets are then
used to treat patients with leukaemia or severe blood loss due to injury. A single platelet
donation can sustain several lives.
In short, without the belt trick, platelet donations would not be sterile and would thus
be impossible. Only the belt trick, or tethered rotation, allows sterile platelet donations
that save other people’s lives.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
An aside: the belt trick saves lives
Motion Mountain – The Adventure of Physics
F I G U R E 26 A version of the
antitwister mechanism, or belt
trick, with 96 belts attached to a
black and white ball that rotates
continuously (QuickTime film
© Jason Hise).
q ua ntum theory d ed uced from stra nds
181
Any two sufficiently complex tangles behave as fermions
under (single or double) exchange of their cores (try it) :
Motion Mountain – The Adventure of Physics
is possible to rearrange their tails to yield the original situation. This is not possible when the tangles
are only exchanged once. Spin 1/2 tangles are thus fermions. The figure presents common systems that
show this behaviour.
Fermions and spin
In nature, fermions are defined as those particles whose wave function changes sign when
they are exchanged. Does the strand model reproduce this observation?
We will see below that in the strand model, wave functions are blurred tangles. We
thus can explore exchange properties of quantum particles and of their wave functions
by exploring the exchange properties of their tangles. Now, if we exchange two tangle
cores twice, while keeping all tails connections fixed, tail fluctuations alone can return the
situation back to the original state! The exchange properties of spin 1/2 tangles are easily
checked by playing around with some pieces of rope or bands, as shown in Figure 27, or
by watching the animation of Figure 28.
The simplest possible version of the experiment is the following: take two coffee cups,
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
F I G U R E 27 When two spin 1/2 tangles each made of several strands or bands, are exchanged twice, it
182
8 q ua ntum theory d ed uced from stra nds
one in each hand, and cross the two arms over each other (once). Keeping the orientation
of the cups fixed in space, uncross the arms by walking around the cups. This is possible,
but as a result, both arms are twisted. If you are intrepid, you can repeat this with two (or
more) people holding the cups. And you can check the difference with what is possible
after a double crossing of arms: in this case, all arms return to the starting situation.
All these experiments show:
⊳ A simple exchange of two spin 1/2 particles (tangles, cups on hands, belt
buckles) is equivalent to a multiplication by −1, i.e., to twisted tangles, arms
or belts.
⊳ In contrast, a double exchange of two spin 1/2 particles can always be untwisted and is equivalent to no exchange at all.
Spin 1/2 particles are thus fermions. In other words, the strand model reproduces the
spin–statistics theorem for spin 1/2: all elementary matter particles are fermions. In summary, a tangle core made of two or more tangled strands behaves – both under rotations
and under exchange – like a spin 1/2 particle.
We note that it is sometimes claimed that the appearance of spin 1/2 can only be mod-
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
represent particles. We know from above that belted buckles behave as spin 1/2 particles. The
animation shows that two such particles return to the original situation if they are switched in position
twice (but not once). Such particles thus fulfil the defining property of fermions. (For the opposite case,
that of bosons, a simple exchange would lead to the identical situation.) You can repeat the trick at
home using paper strips. The equivalence is shown here with two belts per particle, but the trick works
with any positive number of belts attached to each buckle. This animation is the essential part of the
proof that spin 1/2 particles are fermions. This is called the spin–statistics theorem. (QuickTime film
© Antonio Martos)
Motion Mountain – The Adventure of Physics
F I G U R E 28 Assume that the belts cannot be observed, but the square objects can, and that they
q ua ntum theory d ed uced from stra nds
A boson
Strand model :
tail
spin
time average
of crossing
switches
183
Observed
probability
density :
spin
core
F I G U R E 29 A massive spin 1 particle in the strand model (left) and the observed probability density
Boson exchange
Starting situation, before exchange
Final situation, after exchange
crossings, and thus without changing the sign of the phase.
Challenge 130 e
elled with the help of a topology change of space or space-time. The various belt trick
animations given above prove that this is not correct: spin 1/2 can be modelled in three
dimensions in all its aspects. No topology change is needed. You might want to model
the creation of a spin 1/2 particle–antiparticle pair as a final argument.
B osons and spin
Vol. IV, page 117
For tangles made of one strand – thus with two tails to the border – a rotation of the
tangle core by 2π restores the original state. Such a tangle, shown in Figure 29, thus
behaves like a spin 1 particle. The figure also shows the wave function that results from
time averaging the crossings.
Bosons are particles whose combined state does not change phase when two particles
are exchanged. We note directly that this is impossible with the tangle shown in Figure 29; the feat is only possible if the boson tangle is made of unknotted strands. Indeed,
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
F I G U R E 30 In the strand model, unknotted boson tangles can switch states without generating
Motion Mountain – The Adventure of Physics
when averaging its crossings over long time scales (right).
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8 q ua ntum theory d ed uced from stra nds
for unknotted strands, the exchange process can easily switch the two deformations, as
illustrated in Figure 30.
⊳ Massive elementary particles thus can only be bosons if they also have an
unknotted tangle in the tangle family that represents them.
Page 314, page 331
Page 279
The simplest strand model for each elementary boson – the photon, the W boson, the Z
boson, the gluon and the Higgs boson – must thus be made of unknotted strands. We
will deduce the precise tangles below, in the chapter on the particle spectrum. The tangle
for the hypothetical graviton – also a boson, but in this case with spin 2 and invariant
under core rotations by π – will be introduced in the chapter on general relativity.
In summary, unknotted tangles realize the spin–statistics theorem for particles with
integer spin: radiation particles, which have integer spin, are automatically bosons.
Challenge 131 e
Tangle functions: blurred tangles
In the strand model, particle motion is due to the motion of tangle cores. But according to the fundamental principle, strands and tangles are not observable; only crossing
switches are. To explore the relation between crossing switches and motion, we first recall what a crossing is.
⊳ A crossing of strands is a local minimum of strand distance. The position,
orientation and the phase of a crossing are defined by the space vector corresponding to the local distance minimum, as shown in Figure 31. The sign
of the orientation is defined by arbitrarily selecting one strand as the starting strand. The even larger arbitrariness in the definition of the phase will
be of great importance later on, and lead to gauge invariance.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 162
We just saw that fluctuating strands reproduce the spin–statistics theorem for fermions
and for bosons, and thus for all elementary particles, if appropriate tangles are used.
Apart from this fundamental result, the strand model also implies that no spins lower
than ℏ/2 are possible, and that spin values are always an integer multiple of ℏ/2. All this
matches observations.
In the strand model, temporal evolution and particle reactions conserve spin, because
all interactions conserve the number of strands and tails. The details of the conservation
will become clear later on. Again, the result agrees with observations.
The strand model thus explains the origin of permutation symmetry in nature: permutation symmetry of particles is due the possibility to exchange tangle cores of identical
particles; and identical particles have tangle cores of identical topology. We have thus
already ticked off one item from the millennium list of unexplained properties of nature.
In summary, the strand model reproduces the rotation, the spin and the exchange
behaviour of elementary quantum particles – both fermions and bosons – in all its observed details. We now proceed to the next step: quantum mechanics of translational
motion.
Motion Mountain – The Adventure of Physics
Spin and statistics
q ua ntum theory d ed uced from stra nds
185
A strand crossing
phase
shortest distance
position
orientation
F I G U R E 31 The definition of a crossing, its position, its orientation and its phase.
⊳ The tangle function of a system described by a tangle is the short-time average of the positions and the orientations of its crossings (and thus not of
crossing switches and not of the strands themselves).
⊳ For the definition of the tangle function, the short-time average of crossings
is taken over the typical time resolution of the observer. This is a time that
is much longer than the Planck time, but also much shorter than the typical
evolution time of the system. The time resolution is thus what the observer
calls an ‘instant’ of time. Typically – and in all known experiments – this
will be 10−25 s or more; the typical averaging will thus be over a time interval
with a value between 10−43 s, the Planck time, and around 10−25 s.
There are two ways to imagine tangle fluctuations and to deduce the short-time average
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
The tangle function can be called the ‘oriented crossing density’ or simply the ‘blurred
tangle’. As such, the tangle function is a continuous function of space, similar to a cloud;
we will see below what its precise mathematical description looks like. The tangle function captures the short-time average of all possible tangle fluctuations. For a tangle made
of two strands, Figure 32 illustrates the idea. However, the right-hand side of the figure
does not show the tangle function itself, but its probability density. We will see shortly
that the probability density is the (square of the) crossing position density, whereas the
tangle function is a density that describes both position and orientation of crossings.
The tangle function at any given time is not observable, as its definition is not based
on crossing switches, but only on crossings. However, since crossing switches only occur
at places with crossings, the tangle function is a useful tool to calculate observables. In
fact, we will show that the tangle function is just another name for what is usually called
the wave function. In short, the tangle function, i.e., the oriented crossing density, will
turn out to describe the quantum state of a system.
In summary, the tangle function is a blurred image of the tangle – with the important
detail that the crossings are blurred, not the strands.
Motion Mountain – The Adventure of Physics
To describe the motion of tangles, we need concepts that allow us to take the step from
general strand fluctuations to the motion of tangle cores. As a mathematical tool to describe crossing fluctuations, we define:
186
8 q ua ntum theory d ed uced from stra nds
Strand model :
A slowly moving strand :
time average
of crossing
switches
Observed
probability
density, i.e.,
crossing
switch
density :
precession
of spin axis
A rapidly moving strand :
Challenge 132 e
from a given tangle. The first, straightforward way is to average over all possible strand
fluctuations during the short time. Each piece of strand can change in shape, and as a
result, we get a cloud. This is the common Schrödinger picture of the wave function and of
quantum mechanics. The second, alternative way to average is to imagine that the tangle
core as a whole changes position and orientation randomly. This is easiest if the core with
all its crossings is imagined to be tightened to a small, almost ‘point-like’ region. Then
all observables are also localized in that region. It is often simpler to imagine an average
over all position and orientation fluctuations of such a tightened core, that to imagine an
average over all possible strand fluctuations. This alternate view leads to what physicists
call the path integral formulation of quantum mechanics. (Can you show the equivalence
of the two averaging methods?) Of course, in both cases the final result is that the tangle
function is a cloud, i.e., a probability amplitude.
Details on fluctuations and averages
In the strand model, the strand fluctuations of particle strands are a consequence of the
embedding of all particles in a background which itself is made of fluctuating vacuum
strands. Fluctuations randomly add detours to particle strands and randomly shift the
core position. Fluctuations do not keep the strand length constant. Fluctuations do not
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
probability density that results when averaging crossing switches over time. (The black dots are not
completely drawn correctly.)
Motion Mountain – The Adventure of Physics
F I G U R E 32 Some strand configurations, some of their short time fluctuations, and the corresponding
q ua ntum theory d ed uced from stra nds
187
conserve strand shape nor any other property of strands, as there is no mechanism that
enforces such rules. Strand fluctuations are thus quite wild. What then can be said about
the details of the averaging procedure for strand fluctuations?
The fluctuations of the vacuum are those strand fluctuations that lead to the definition
of the background space. This definition is possible in a consistent manner only if the
fluctuations are homogeneous and isotropic. The vacuum state can thus be defined as that
state for which the fluctuations are (locally) homogeneous and isotropic. In particular,
the fluctuations imply
⊳ Flat vacuum has a tangle function that vanishes everywhere.
Challenge 133 e
Page 108, page 166
Tangle functions are wave functions
In the following, we show that the tangle function, the blurred image of tangle crossings,
is the same as what is usually called the wave function. We recall what we know from
textbook quantum theory:
⊳ A single-particle wave function is, generally speaking, a rotating and diffusing cloud.
The rotation describes the evolution of the phase, and the diffusion describes the evolution of the density. We now show that tangle functions have these and all other known
properties of wave functions. We proceed by deducing all the properties from the definition of tangle functions. We recall that, being a short-time average, a tangle function is
a continuous function of space and time.
⊳ Using the tangle function, we define the strand crossing position density,
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 202
Motion Mountain – The Adventure of Physics
Page 294
The proof is an interesting exercise. The existence of a homogeneous and isotropic background space then implies conservation of energy, linear and angular momentum of
particles travelling through it.
The fluctuations of a tangle lead, after averaging, to the tangle function, i.e., as we will
see, to the wave function. The conservation of energy and momentum implies that the
time average of the tangle fluctuations also conserves these quantities.
Therefore we can continue our discussion without yet knowing the precise details of
the tangle fluctuations themselves. (We will provide these details below, in the section
on general relativity.) Here we only require that the average of the fluctuations behaves
in such a way as to be consistent with the definition of the background used by the observer. We thus make explicit use of the conviction that a background-free description
of nature is impossible, and that a fundamental description of nature must contain a circular definition that makes an axiomatic description of nature impossible. Despite this
limitation, such a circular description of nature must be self-consistent.
We will also show below that the definition of the tangle function does not introduce
hidden variables, even though first impression might suggest the opposite. In fact, it is
possible to define something akin to a strand evolution equation. However, it does not
deepen our understanding of the evolution equation of the wave function.
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8 q ua ntum theory d ed uced from stra nds
or crossing density, for each point in space, by discarding the orientation
information, counting the crossings in a volume, and taking the square root.
The crossing density – more precisely, its square root – is a positive number,
more precisely, a positive real function ?(?, ?) of space and time.
We will see shortly that the crossing position density is the square root of what is usually
called the probability density.
⊳ A tangle function also defines an average crossing orientation and a average
phase at each point in space. The average crossing orientation and the average phase are related to the spin orientation and phase of the wave function.
The mathematical descriptions of these quantities depend on the approximation used.
⊳ The quantum phase of fermions is one half the core rotation angle ?.
Challenge 134 s
Page 202
If a system changes with time, the tangle function changes; this leads to crossing
switches; therefore, temporal evolution is expected to be observable through these crossing switches. As we will see shortly, this leads to an evolution equation for tangle functions.
Here is a fun challenge: how is the shortest distance between the strands, for a crossing
located at position ? and ?, related to the magnitude, i.e., the absolute value ?(?, ?), of
the wave function?
We note that if many particles need to be described, the many-particle tangle function
defines a separate crossing density for each particle tangle.
Tangle functions form a vector space. To show this, we need to define the linear combination or superposition ? = ?1 ?1 + ?2 ?2 of two tangle functions. This requires the
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Without the neglect of spin, and especially when the spin axis can change over space, the
description of orientation and phase averages require more details; we will study these
cases separately below. They will lead to the non-relativistic Pauli equation and to the
relativistic Dirac equation.
In short, in the simple approximation when spin effects can be neglected, the local
tangle function value can be described by one real number ? and by one quantum
phase ?. The tangle function can thus be described by a complex number ? at each point
in space and time:
?(?, ?) = ?(?, ?) e??(?,?)/2 .
(122)
Motion Mountain – The Adventure of Physics
The simplest approximation for a tangle function is to assume, in the physical situation
under study, that the spin direction is independent of spatial position and thus not taken
into consideration; this approximation will lead to the Schrödinger equation. In this
simplest approximation, at each point in space, the local average orientation of the fluctuations of the tangle core will just be described by a single angle. This quantum phase is
a function of time and space and describes how much the local average phase is rotated
around the fixed spin orientation.
q ua ntum theory d ed uced from stra nds
189
definition of two operations: scalar multiplication and addition. We can do this in two
ways. The first way is to define the operations for tangle functions directly, as is done in
quantum mechanics:
⊳ First, boring definition: The scalar multiplication ?? and the addition
?1 + ?2 of quantum states are taken by applying the relative operations on
complex numbers at each point in space, i.e., on the local values of the tangle
function.
The second way to deduce the vector space is more fun, because it will help us to visualize
quantum mechanics: we can define addition and multiplication for tangles, and take the
time average after the tangle operation is performed.
⊳ Second, fun definition: The addition of two tangles ?1 ?1 and ?2 ?2 , where
?1 and ?2 have the same topology and where ?12 + ?22 = 1, is defined by
connecting those tails that reach the border of space, and discarding all parts
of the tangles that were pushed to the border of space. The connection of
tangles must be performed in such a way as to maintain the topology of
the original tangles; in particular, the connection must not introduce any
crossings or linking. Time averaging then leads to the tangle function of the
superposition ? = ?1 ?1 + ?2 ?2 .
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
The scalar multiplication for strands is illustrated in Figure 33. The above definition of
scalar multiplication is only defined for factors ? ⩽ 1. Indeed, no other factors ever
appear in physical problems (provided all wave functions are normalized), so that scalar
multiplication is not required for other scalars.
The strand version of scalar multiplication is unique; indeed, even though there is
a choice about which fraction ? of a tangle is kept and which fraction 1 − ? is sent to
the border of space, the resulting tangle function, which is defined as an average over
fluctuations, is independent from this choice.
The scalar multiplication of strands behaves as expected for 1 and 0. By construction,
the strand version of scalar multiplication is associative: we have ?(??) = (??)?. The
strand multiplication by −1 is defined as the rotation of the full tangle core by 2π.
We also need to define the addition operation that appears in the linear combination
of two tangle functions. This is a straightforward complex addition at each point in space.
Again, for fun, we also define the operation on tangles themselves, and take the time
average that leads to the tangle function afterwards.
Motion Mountain – The Adventure of Physics
⊳ Second, fun definition: The scalar multiplication ?? of a state ? by a complex number ? = ?e?? is formed by taking a tangle underlying the tangle
function ?, then rotating the tangle core by the angle 2?, and finally pushing
a fraction 1 − ? of the tangle to the border of space, thus keeping the fraction
? of the original tangle at finite distances. Time averaging then leads to the
tangle function ??.
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8 q ua ntum theory d ed uced from stra nds
Strand multiplication :
Strand model :
?
time average
of crossing
switches
Observed
probability
density :
√0.2 ?
functions.
Page 191
To visualize the result of addition and superposition, it is easiest to imagine that the
strands reaching the border of space have fluctuated back to finite distances. This is
possible because by definition, these connections are all unlinked. An example of superposition, for the case of two quantum states at different positions in space, is shown
in Figure 34. We note that despite the wording of the definition, no strand is actually cut
or re-glued in the operation of addition.
The definition of linear combination requires that the final strand ? has the same topology and the same norm as each of the two strands ?1 and ?2 to be combined. Physically,
this means that only states for the same particle can be added and that particle number
is preserved; this automatically implements the so-called superselection rules of quantum
theory. This result is pretty because in usual quantum mechanics the superselection rules
need to be added by hand. This is not necessary in the strand model.
The sum of two tangle functions is unique, for the same reasons given in the case of
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
F I G U R E 33 Scalar multiplication of localized tangles, visualizing the scalar multiplication of wave
Motion Mountain – The Adventure of Physics
√0.8 ?
q ua ntum theory d ed uced from stra nds
191
Linear combination of strands :
Strand model :
time average
of crossing
switches
Observed
probability
densities :
The two quantum states localized at different positions :
?1
?2
x2
x2
x1
x2
A linear combination ? = √0.8 ?1 + √0.2 ?2 :
untangled
“addition
region”
x1
x2
F I G U R E 34 A linear combination of strands, in this case for two states representing a particle at two
different position in space, visualizing the linear combination of wave functions.
Challenge 135 e
scalar multiplication. The definition of addition can also be extended to more than two
terms. Addition is commutative and associative, and there is a zero state, or identity
element, given by no strands at all. The definition of addition also implies distributivity
with respect to addition of states and with respect to addition of scalars. It is also possible
to extend the definitions of scalar multiplication and of addition to all complex numbers
and to unnormed states, but this leads us too far from our story.
In short, tangle functions form a vector space. We now define the scalar product and
the probability density in the same way as for wave functions.
⊳ The scalar product between two states ? and ? is ⟨?|?⟩ = ∫?(x)?(x) dx.
⊳ The norm of a state is ‖?‖ = √⟨?|?⟩ .
⊳ The probability density ? is ?(?, ?) = ?(?, ?)?(?, ?) = ?2 (?, ?). It thus ignores the orientation of the crossings and is the crossing position density.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
x1
Motion Mountain – The Adventure of Physics
x1
192
Challenge 136 e
Challenge 137 e
8 q ua ntum theory d ed uced from stra nds
The scalar product and the probability density are observables, because their definitions
can be interpreted in terms of crossing switches. Indeed, the scalar product ⟨?|?⟩ can
be seen as the (suitably normed) number of crossing switches required to transform the
tangle ? into the tangle ?, where the tangle ? is formed from the tangle ? by exchanging
the orientation of each crossing. A similar interpretation is possible for the probability
density, which therefore is at the same time the crossing density squared and the crossing
switch density. We leave this confirmation as fun for the reader.
It is also possible to define the scalar product, the norm and the probability density
using tangles, instead of using tangle functions. This is left as a puzzle to the reader.
In summary, we have shown that tangle functions form a Hilbert space. The next
steps are now obvious: We must first show that tangle functions obey the Schrödinger
equation. Then we must extend the definition of quantum states by including spin and
special relativity, and show that they obey the Dirac equation.
⊳ Particles with high energy have rapidly rotating tangles.
⊳ Particles with low energy have slowly rotating tangles.
The energy of a rotating tangle is the number of crossing switches per time. Rotating a
tangle core leads to crossing switches in its tails. In the strand model, the kinetic energy
? of a particle is thus due to the crossing switches formed in its tails. In other words, the
kinetic energy ? is related to the (effective) angular frequency ? of the core rotation by
? = ℏ? .
(123)
The local phase of the tangle function ? changes with the rotation. This implies that
? = ?∂? ? .
(124)
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
The Schrödinger equation, like all evolution equations in the quantum domain, results
when the definition of the wave function is combined with the energy–momentum relation. As already mentioned, the Schrödinger equation for a quantum particle also assumes that the orientation of particle spin is constant for all positions and all times. In
this case, the spin can be neglected, and the tangle function is a single complex number
at each point in space and in time, usually written ?(?, ?). How does the tangle function
evolve in time? To answer this question, we will only need the fundamental principle
that crossing switches define the quantum of action ℏ.
We start with a free particle. We assume a fixed, but unspecified rotation direction of
its tangle. Now, in the strand model, a localized particle with constant speed is described
by a localized tangle that rotates and advances. In other words, the strand fluctuations
produce a peak of probability density that changes position with constant speed.
Every tangle rotation leads to crossing switches. A rapid tangle rotation leads to many
crossing switches per time, and slow rotation to few crossing switches per time. Now, the
fundamental principle tells us that crossing switches per time are naturally measured in
action per time, or energy. In other words, tangle rotation is related to tangle energy.
Motion Mountain – The Adventure of Physics
Deducing the S chrödinger equation from tangles
q ua ntum theory d ed uced from stra nds
193
Strand model :
time average
of crossing
switches
Localized particle at rest :
Observed
probability
density :
t1
t1
t2
t2
Slow motion :
t2
t1
Motion Mountain – The Adventure of Physics
t1
t2
rotation,
precession and
displacement
t2
t1
t1
t2
rotation,
precession and
displacement
F I G U R E 35 Examples of moving tangles of free particles.
We will need the relation shortly.
The linear motion of a tangle implies that it makes also sense to pay attention to the
number of crossing switches per distance.
⊳ Rapidly moving tangles show many crossing switches per distance.
⊳ Slowly moving tangles show few crossing switches per distance.
The fundamental principle tells us that the natural observable to measure crossing
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Rapid motion :
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8 q ua ntum theory d ed uced from stra nds
switches per distance is action per distance, or momentum. Linear motion of tangles
is thus related to momentum: The momentum of a moving tangle is the number of crossing switches per distance. The momentum ? is thus related to the (effective) wave number
? = 2π/? of the core motion by
? = ℏ? .
(125)
The local phase of the tangle function ? changes with the motion. This implies
? = −?∂? ? .
(126)
?=
?2
2?
and ? =
ℏ 2
? .
2?
(127)
?ℏ∂? ? = −
ℏ2
∂ ?.
2? ??
(128)
This is the famous Schrödinger equation for a free particle (written for just one space
dimension for simplicity). We thus have deduced the equation from the strand model
under the condition that spin can be neglected and that velocities are small compared to
the speed of light. In this way, we have also deduced, indirectly, Heisenberg’s indeterminacy relations.
We have thus completed the proof that tangle functions, in the case of negligible spin
effects and small velocities, are indeed wave functions. In fact, tangle functions are wave
functions also in the more general case, but then their mathematical description is more
involved, as we will see shortly. We can sum up the situation in a few simple terms: wave
functions are blurred tangles.
Mass from tangles
In quantum theory, particles spin while moving: the quantum phase rotates while a
particle advances. The coupling between rotation and translation has a name: it is called
the mass of a particle. We saw that the rotation is described by an average angular fre-
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
This is dispersion relation for masses moving at velocities much smaller than the speed
of light. The relation agrees with all experiments. The constant ? is a proportionality
factor that depends on the tangle core. We can now use the same argument that was
used already by Schrödinger. Substituting the tangle relations in the dispersion relation,
we get the evolution equation for the tangle function ? given by
Motion Mountain – The Adventure of Physics
This completes the description of matter wave functions without spin.
The belt trick for the fluctuating tails now has a fascinating consequence. To allow
the belt trick also for high linear momentum, the more the momentum increases, the
more the spin rotation axis has to align with the direction of motion. This is shown in
Figure 35. This leads to a quadratic increase of crossing switches with momentum ?:
one factor ? is due to the increase of the speed of rotation, the other factor is due to the
increase of the alignment. We thus get
q ua ntum theory d ed uced from stra nds
195
quency ?, and the translational motion is described by a wave number ?. The proportionality factor ? = ℏ?2 /2? = ?2 /2? is thus a quantity that relates rotation frequency
and wave number. In quantum theory,
⊳ The (inertial) mass ? describes the coupling between translation and rotation.
⊳ Linked and localized tangles have mass.
⊳ Unknotted or unlinked, unlocalized tangles, such as those of photons, are
predicted to be massless.
⊳ Particle masses are calculable – if the tangle topology is known.
Challenge 138 e
Page 358
This is an exciting prospect! To sum up, the strand model predicts that experiments in
viscous fluids can lead to a deeper understanding of the masses of elementary particles.
The tangle model also implies that the mass of elementary particles – thus of particles
made of few strands – will be much smaller than the Planck mass. This is the first hint
that the strand model solves the so-called mass hierarchy problem of particle physics.
At this point, however, we are still in the dark about the precise origin of particle
mass values. We do not know how to calculate them. Nevertheless, the missing steps
are clear: first, we need to determine the tangle topology for each elementary particle;
then we need to deduce their mass values, i.e., the relation between their rotation and
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 356
We also deduce that the more complicated a tangle is, the higher the mass value is.
In addition to the geometry effect due to the core, which is valid for massive bosons
and fermions, the rotation of fermions is also influenced by the tails. The effective volume
required by the belt trick will influence the coupling between translation and rotation.
This effective volume will depend on the topology of the tangle core, and on the number
of its tails. We again deduce that, for a given number of tails, a complicated core topology
implies a high mass value.
In other words, the strand model links the mass ? of a particle to its tangle topology:
large tangle cores have large mass. The strand model thus predicts
Motion Mountain – The Adventure of Physics
Ref. 162
We note that a large mass value implies, for a given momentum value, both a slow translation and a slow rotation.
In the strand model, particle translation and rotation are modelled by the translation
and rotation of the tangle core. Now, the strand model makes a point that goes beyond
usual quantum theory. The strand model explains why core translation and rotation are
coupled: When the core moves through the vacuum, the vacuum strands and the core
effectively push against each other, due to their impenetrability. The result is a motion
that resembles the motion of an asymmetrical body in a viscous fluid.
When an asymmetrical body is moved through a viscous fluid, it starts rotating. For
example, this happens when a stone falls through water or honey. The rotation results
from the asymmetrical shape of the body. All the tangle cores of elementary particles are
asymmetrical. The strand model thus predicts that tangle cores will rotate when they
move through vacuum. In other terms, the strand model predicts
196
Challenge 139 e
8 q ua ntum theory d ed uced from stra nds
translation. This is a central aim in the following.
An example of the issues that arise: How does the mass value depend on the number
of strands in a tangle? How does mass depend on the type of tangle?
Potentials
(?ℏ∂? − ??)? =
(129)
This equation is the simplest formulation of quantum theory. We saw in the fourth
volume that it describes and explains the size of atoms and molecules, and thus of all
objects around us; and we saw that it also explains the (relative) colours of all things.
The equation also explains interference, tunnelling and decay.
In summary, a non-relativistic fluctuating tangle reproduces the full Schrödinger
equation. An obvious question is: how does the strand model explain the influence of
interactions on the rotation speed and on the wavelength of tangles? In other words:
why do strands imply minimal coupling? We will answer this question in the following
chapter, on gauge interactions.
Q uantum interference from tangles
Page 189
Page 175
The observation of interference of quantum particles is due to the linear combination of
states with different phases at the same position in space. Tangle functions, being wave
functions, reproduce the effect. But again, it is both more fun and more instructive to
explain and visualize interference with the help of tangles.
As mentioned above, a pure change of phase of a state ? is defined by multiplication
by a complex number of unit norm, such as e?? . This corresponds to a rotation of the
tangle core by an angle 2?, where the factor 2 is due to the belt trick of Figure 19.
To deduce interference, we simply use the above definition of linear combinations
of tangles. This leads to the result shown in Figure 36. We find, for example, that a
symmetric sum of a tangle and the same tangle with the phase rotated by π/2 (thus a
core rotated by π) results in a tangle whose phase is rotated by the intermediate angle,
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 222
1
(−?ℏ∇ − ??)2 ? .
2?
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Page 232
In quantum mechanics, interactions are described by potentials. An electric potential
?(?, ?) changes the total energy of a particle with charge ? at position ?, since in quantum
mechanics, electric potentials influence the rotation velocity of the wave function. As a
result, with an electric potential, the left-hand side of the Schrödinger equation (128), the
energy term, is changed from ℏ??(?, ?) to (ℏ? − ??)?(?, ?).
Another possibility is a potential that does not change the rotation velocity, but that
changes the wavelength of a charged particle. Such a magnetic vector potential ?(?, ?)
thus changes the momentum term ℏ? on the right-hand side of Schrödinger’s equation
to (ℏ?−??)?(?, ?). This double substitution, the so-called minimal coupling, is equivalent
to the statement that quantum electrodynamics has a U(1) gauge symmetry. We will
deduce it in detail in the next chapter.
In the strand model of quantum mechanics, potentials are introduced in precisely
the same way as in usual quantum mechanics, so that the full Schrödinger equation for
charged particles in external fields is recovered:
q ua ntum theory d ed uced from stra nds
197
Observed
probability
density :
Strand model :
time average
of crossing
changes
The two quantum states with different phase at the same position :
?2 = ?1 e?π/2
?1
?
?
?
A linear combination :
? = (?1 + ?2 )/√2
?
Extinction (requires situations with space-dependent phase) :
no possible
tangle
topology
?
zero
density
?
F I G U R E 36 Interference: the linear combination of strands with different phase, but located at the
same position.
thus π/4.
The most interesting case of interference is that of extinction. Scalar multiplication
of a tangle function ? by −1 gives the negative of the tangle function, the additive inverse −?. The sum of a tangle function with its negative is zero. This gives extinction
in usual quantum theory. Let us check the result in the strand model, using the tangle
definition of linear combinations. We have seen above that the negative of a tangle is a
tangle whose core is rotated by 2π. Using the tangle definition of linear combination,
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
?
Motion Mountain – The Adventure of Physics
?
198
8 q ua ntum theory d ed uced from stra nds
we find that it is topologically impossible to draw or construct a localized tangle for the
sum of a quantum state with its negative. The resulting particle tangle therefore must
have vanishing crossing density in spatial regions where this operation is attempted. In
short, particle tangles do explain extinction. And as expected from quantum particles,
the explanation of extinction directly involves the tangle structure.
Deducing the Pauli equation from tangles
cos(?/2)e??/2
Ψ(?, ?) = √? e??/2 (
) ,
? sin(?/2)e−??/2
(130)
0 1
0 −?
1 0
? = ((
),(
),(
)) .
1 0
? 0
0 −1
(131)
The three matrices are the well-known Pauli matrices.
We now take the description of the axis orientation and the description of the spinning and insert both, as we did for the Schrödinger equation, into the non-relativistic
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
which is the natural description of a tangle that includes the orientation of the axis. As
before, the crossing density is the square root of the probability density ?(?, ?). The angle
?(?, ?), as before, describes the phase, i.e., (one half of) the rotation around the axis. The
local orientation of the axis is described by a two-component matrix and uses the two
angles ?(?, ?) and ?(?, ?). Due to the belt trick, the expression for the tangle function only
contains half angles. And indeed, due to the half angles, the two-component matrix is
not a vector, but a spinor. (The term ‘spinor’ was coined by well-known physicist Paul
Ehrenfest in analogy to ‘vector’ and ‘tensor’; the English pronunciation is ‘spinnor’.)
For ? = ? = 0, the previous wave function ? is recovered.
The other ingredient we need is a description of the spinning motion of the tangle.
In contrast to the Schrödinger case, the spinning motion itself must be added in the
description. A spinning tangle implies that the propagation of the wave is described by
the wave vector ? multiplied with the spin operator ?. The spin operator ?, for the wave
function just given, is defined as the vector of three matrices
Motion Mountain – The Adventure of Physics
Ref. 163
As we have seen, the Schrödinger equation describes the motion of quantum particles
when their spin is neglected, by assuming that spin is constant over space and time. The
next step is thus to include the variations of spin over space and time. This turns out to
be quite straightforward.
In the strand model, spin is modelled by the continuous rotation of a tangle. We also
saw that we get wave functions from tangles if we average over short time scales. At a
given position in space, a tangle function will have a local average density of crossings, a
local average phase, and new, a local average orientation of the rotation axis of the tangle.
To describe the axis and orientation of the tangle core, we use the Euler angles ?, ?
and ?. This yields a description of the tangle function as
q ua ntum theory d ed uced from stra nds
199
dispersion relation ℏ? = ? = ?2 /2? = ℏ2 ?2 /2?. We then get the wave equation
?ℏ∂? Ψ = −
Challenge 140 s
(132)
This is Pauli’s equation for the evolution of a free quantum particle with spin 1/2.
As final step, we include the electric and the magnetic potentials, as we did in the
case of the Schrödinger equation. We again use minimal coupling, substituting ?ℏ∂? by
?ℏ∂? − ?? and −?ℏ∇ by −?ℏ∇ − ??, thus introducing electric charge ? and the potentials
? and ?. A bit of algebra involving the spin operator then leads to the famous complete
form of the Pauli equation
(?ℏ∂? − ??)Ψ =
?ℏ
1
(−?ℏ∇ − ??)2 Ψ −
??Ψ ,
2?
2?
(133)
Rotating arrows, path integrals and interference
Ref. 164
Another simple way to visualize the equivalence between the strand model and the Pauli
equation uses the formulation of quantum theory with path integrals. We recall that
tangle tails are not observable, and that the tangle core defines the position and phase of
the quantum particle. The motion of the core thus describes the ‘path’ of the particle.
Different paths are due to different core motions.
The continuous rotation of the tangle core corresponds to Feynman’s rotating little
arrow in his famous popular book on QED. The different paths then correspond to different motions of the tangle core. The tangle model also reproduces the path integral
formulation of quantum mechanics.
Also interference can be visualized with strands. Because of its tails, a fermion tangle
obeys spinor statistics and spinor rotation behaviour. This leads to the correct interference behaviour for spin 1/2 particles. Indeed, interference for fermions is visualized in
Figure 37. The corresponding visualization for photon interference is given in Figure 38.
Measurements and wave function coll apse
In nature, a measurement of a quantum system in a superposition is observed to yield one
of the possible eigenvalues and to prepare the system in the corresponding eigenstate. In
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
where now the magnetic field ? = ∇ × ? appears explicitly. The equation is famous for
describing, among others, the motion of silver atoms, which have spin 1/2, in the Stern–
Gerlach experiment. This is due to the new, last term on the right-hand side, which does
not appear in the Schrödinger equation. The new term is a pure spin effect and predicts
a ?-factor of 2. Depending on the spin orientation, the sign of the last term is either
positive or negative; the term thus acts as a spin-dependent potential. The two options
for the spin orientation then produce the upper and the lower beams of silver atoms that
are observed in the Stern–Gerlach experiment.
In summary, a non-relativistic tangle that rotates continuously reproduces the Pauli
equation. In particular, such a tangle predicts that the ?-factor of an elementary charged
fermion is 2.
Motion Mountain – The Adventure of Physics
Vol. IV, page 83
ℏ2
(?∇)2 Ψ .
2?
200
8 q ua ntum theory d ed uced from stra nds
destructive
interference
constructive
interference
interference (right).
constructive
interference
destructive
interference
interference (right).
Vol. IV, page 143
nature, the probability of each measurement outcome depends on the coefficient of that
eigenstate in the superposition.
To put the issue into context, here is a short reminder from quantum mechanics. Every
measurement apparatus shows measurement results. Thus, every measurement apparatus is a device with memory. (In short, it is classical.) All devices with memory contain
one or several baths. Thus, every measurement apparatus couples at least one bath to the
system it measures. The coupling depends on and defines the observable to be measured
by the apparatus. Every coupling of a bath to a quantum systems leads to decoherence.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
F I G U R E 38 The double-slit experiment with photons: constructive interference (left) and destructive
Motion Mountain – The Adventure of Physics
F I G U R E 37 A fermion tangle passing a double slit: constructive interference (left) and destructive
q ua ntum theory d ed uced from stra nds
Spin
measurement
direction :
201
Strand model :
Observed spin :
Basis states :
always
`up’
always
`down’
untangled
“addition
region”
either
`up’
or `down’
inwards.
Decoherence leads to probabilities and wave function collapse. In short, collapse and
measurement probabilities are necessary and automatic in quantum theory.
The strand model describes the measurement process in precisely the same way as
standard quantum theory; in addition, it visualizes the process.
⊳ A measurement is modelled as a strand deformation induced by the measurement apparatus that ‘pulls’ a tangle towards the resulting eigenstate.
⊳ This pulling of strands models and visualizes the collapse of the wave function.
An example of measurement is illustrated in Figure 39. When a measurement is performed on a superposition, the untangled ‘addition region’ can be imagined to shrink into
disappearance. For this to happen, one of the underlying eigenstates has to ‘eat up’ the
other: that is the collapse of the wave function. In the example of the figure, the addition
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F I G U R E 39 Measurement of a spin superposition: the addition region disappears either outwards or
Motion Mountain – The Adventure of Physics
Superposition (one of
two equivalent states) :
202
8 q ua ntum theory d ed uced from stra nds
region can disappear either towards the outside or towards the inside. The choice is due
to the bath that is coupled to the system during measurement; the bath thus determines
the outcome of the measurement. We also deduce that the probability of measuring a
particular eigenstate will depend on the (weighed) volume that the eigenstate took up in
the superposition.
This visualization of the wave function collapse also makes clear that the collapse is
not limited by any speed limit, as no energy and no information is transported. Indeed,
the collapse happens by displacing strands and at most crossings, but does not produce
any crossing changes.
In summary, the strand model describes measurements in precisely the same way as
usual quantum theory. In addition, strands visualize the collapse of the wave function as
a shape deformation from a superposed tangle to an eigenstate tangle.
Hidden variables and the Ko chen–Specker theorem
Therefore, the strand model does not contradict the Kochen–Specker theorem.
In simple language, in quantum theory, hidden variables are not a problem if they are
properties of the environment, and not of the quantum system itself. This is precisely
the case for the strand model. For a quantum system, the strand model provides no
hidden variables. In fact, for a quantum system, the strand model provides no variables
beyond the usual ones from quantum theory. And as expected and required from any
model that reproduces decoherence, the strand model leads to a contextual, probabilistic
description of nature.
In summary, despite using fluctuating tangles as underlying structure, the strand
model is equivalent to usual quantum theory. The strand model contains nothing more
and nothing less than usual quantum theory.
Many-particle states and entanglement
In nature, the quantum states of two or more particles can be entangled. Entangled states
are many-particle states that are not separable. Entangled states are one of the most fascinating quantum phenomena; especially in the case of macroscopic entanglement, they
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
⊳ The evolution of strand shapes and crossing switches is contextual.
Motion Mountain – The Adventure of Physics
Ref. 165
At first sight, the strand model seems to fall into the trap of introducing hidden variables
into quantum theory. One could indeed argue that the shapes (and fluctuations) of the
strands play the role of hidden variables. On the other hand, it is well known that noncontextual hidden variables are impossible in quantum theory, as shown by the Kochen–
Specker theorem (for sufficiently high Hilbert-space dimensions). Is the strand model
flawed? No.
We recall that strands are not observable. In particular, strand shapes are not physical
observables and thus not physical variables either. Even if we tried promoting strand
shapes to physical variables, the evolution of the strand shapes would only be observable
through the ensuing crossing switches. And crossing switches evolve due to the influence of the environment, which consists of all other strands in nature, including those
of space-time itself. Thus
q ua ntum theory d ed uced from stra nds
Strand model :
203
Observation :
First separable basis state :
| ↑↓⟩
?1
?2
?1
?2
| ↓↑⟩
model.
are still being explored in many experiments. We will discover that the strand model
visualizes them simply and clearly.
To describe entanglement, we first need to clarify the notion of many-particle state.
In the strand model,
⊳ A many-particle state is composed of several tangles.
In this way, an ?-particle wave function defines ? values at every point in space, one
value for each particle. This is possible, because in the strand model, the strands of each
particle tangle are separate from these of other particles.
Usually, a ?-particle wave function is described by a single-valued function in 3?
dimensions. It is less known that a single-valued ?-particle wave function in 3? dimensions is mathematically equivalent to an ?-valued wave function in three dimensions. Usually, ?-valued functions are not discussed; we feel uneasy with the concept.
But the strand model naturally defines ? wave function values at each point in space:
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F I G U R E 40 Two examples of two distant particles with spin in separable states: observation and strand
Motion Mountain – The Adventure of Physics
Second separable basis state :
204
8 q ua ntum theory d ed uced from stra nds
Entangled state
√90 % | ↑↓⟩ + √10 % | ↓↑⟩
Strand model :
Observation yields
either this eigenstate (90%) :
untangled
“addition
region”
or this eigenstate (10%) :
?2
?1
?2
F I G U R E 41 An entangled spin state of two distant particles.
⊳ An entangled state is a non-separable superposition of separable manyparticle states. State are separable when their tangles can be pulled away
without their tails being entangled.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 164
each particle has its own tangle, and each tangle yields, via short-term averaging, one
complex value, with magnitude and phase, at each point in space. In this way, the strand
model is able to describe ? particles in just 3 dimensions.
In other words, the strand model does not describe ? particles with 1 function in
3? dimensions; it describes many-particle states with ? functions in 3 dimensions. In
this way, the strand model remains as close to everyday life as possible. Many incorrect
statements on this issue are found in the research literature; many authors incorrectly
claim the impossibility of many-particle quantum theory in 3 dimensions. Some authors
even claim, in contrast to experiment, that it is impossible to visualize many-particle
states in 3 dimensions. These arguments all fail to consider the possibility to define
completely separate wave functions for each particle in three dimensions. (It must be
said that this unusual possibility is hard to imagine if wave functions are described as
continuous functions.) However, clear thinkers like Richard Feynman always pictured
many-particle wave functions in 3 dimensions. Also in this domain, the strand model
provides an underlying picture to Feynman’s approach. This is another situation where
the strand model eliminates incorrect thinking habits and supports the naive view of
quantum theory.
Now that we have defined many-particle states, we can also define entangled states.
Motion Mountain – The Adventure of Physics
?1
q ua ntum theory d ed uced from stra nds
205
The Aspect experiment
First separable basis state :
Entangled state (50% + 50%) :
source
source
Second separable basis state :
the untangled
addition region
expands with time
in this situation
F I G U R E 42 The basis states and an entangled state of two distant photons travelling in opposite
directions, with total spin 0.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
We will now show that the above definitions of superpositions and of measurements
using strands are sufficient to describe entanglement.
As first example, we explore entangled states of the spin of two distant massive fermions. This is the famous thought experiment proposed by David Bohm. In the strand
model, two distant particles with spin 1/2 in a separable state are modelled as two distant, separate tangles of identical topology. Figure 40 shows two separable basis states,
namely the two states with total spin 0 given by | ↑↓⟩ and by | ↓↑⟩. Such states can also be
produced in experiments. We note that to ensure total spin 0, the tails must be imagined
to cross somewhere, as shown in the figure.
We can now draw a superposition √90 % | ↑↓⟩ + √10 % | ↓↑⟩ of the two spin-0 basis
states. We simply use the definition of addition and find the state shown in Figure 41. We
can now use the definition of measurement to check that the state is indeed entangled.
If we measure the spin orientation of one of the particles, the untangled addition region
disappears. The result of the measurement will be either the state on the inside of the addition region or the state on the outside. And since the tails of the two particles are linked,
after the measurement, independently of the outcome, the spin of the two particles will
always point in opposite directions. This happens for every particle distance. Despite
this extremely rapid and apparently superluminal collapse, no energy travels faster than
light. The strand model thus reproduces exactly the observed behaviour of entangled
spin 1/2 states.
A second example is the entanglement of two photons, the well-known Aspect experiment. Also in this case, entangled spin 0 states, i.e., entangled states of photons of
opposite helicity (spin), are most interesting. Again, the strand model helps to visualize
Motion Mountain – The Adventure of Physics
source
206
Page 227
Ref. 166
8 q ua ntum theory d ed uced from stra nds
the situation. Here we use the strand model for the photon that we will deduce only later
on. Figure 42 shows the strand model of the two separable basis states and the strand
model of the entangled state. Again, the measurement of the helicity of one photon in
the entangled state will lead to one of the two basis states. And as soon as the helicity of one photon is measured, the helicity of its companion collapses to the opposite
value, whatever the distance! Experimentally, the effect has been observed for distances
of many kilometres. Again, despite the extremely rapid collapse, no energy travels faster
than light. And again, the strand model completely reproduces the observations.
Mixed states
Mixed states are statistical ensembles of pure states. In the strand model,
⊳ A mixed state is a (weighted) temporal alternation of pure states.
The dimensionalit y of space-time
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 166
‘Nature consists of particles moving in empty space.’ Democritus stated this 2500 years
ago. Today, we know that is a simplified description of one half of physics: it is a simplified description of quantum theory. In fact, Democritus’ statement, together with
strands, allows us to argue that physical space must have three dimensions, as we will
see now.
Deducing the dimensionality of physical space from first principles is an old and difficult problem. The difficulty is also due to the lack of alternative descriptions of nature.
Our exploration of the foundations of the strand model has shown that humans, animals
and machines always use three spatial dimensions to describe their environment. They
cannot do otherwise. Humans, animals and machines cannot talk and think without
three dimensions as background space.
But how can we show that physical space – not the background space we need for thinking – is three-dimensional and must be so? We need to show that (1) all experiments
reproduce the result and that (2) no other number of dimensions yields a consistent description of nature.
In nature, and also in the strand model, as long as particles can be defined, they
can be rotated around each other and they can be exchanged. No experiment has ever
been performed or has ever been proposed that changes this observation. The observed
properties of rotations, of spin 1/2, of particle exchange and all other observations confirm that space has three dimensions. Fermions only exist in three dimensions. In
the strand model, the position and the orientation of a particle is intrinsically a threedimensional quantity; physical space is thus three-dimensional, in all situations where
it can be defined. (The only situations where this definition is impossible are horizons
and the Planck scales.) In short, both nature and the strand model are found to be threedimensional at all experimentally accessible energy scales. Conversely, detecting an additional spatial dimension would directly invalidate the strand model.
Motion Mountain – The Adventure of Physics
Mixed states are important in discussions of thermodynamic quantities. We mention
them to complete the equivalence of the states that appear in quantum theory with those
provided by the strand model. We do not pursue this topic any further.
q ua ntum theory d ed uced from stra nds
Page 162
Nature has three dimensions. The only way to predict this result is to show that no
other number is possible. The number of dimensions of nature can only result from a
self-consistency argument. And interestingly, the strand model produces such an argument.
In the strand model, knots and tangles are impossible to construct in physical spaces
with dimensions other than three. Indeed, mathematicians can show that in four spatial
dimensions, every knot and every tangle can be undone. (In this argument, time is not
and does not count as a fourth spatial dimension, and strands are assumed to remain onedimensional entities.) Worse, in the strand model, spin does not exist in spaces that have
more or fewer than three dimensions. Also the vacuum and its quantum fluctuations do
not exist in more than three dimensions. Moreover, in other dimensions it is impossible
to formulate the fundamental principle. In short, the strand model of matter and of
observers, be they animals, people or machines, is possible in three spatial dimensions
only. No description of nature with a background or physical space of more or less than
three dimensions is possible with strands. Conversely, constructing such a description
would invalidate the strand model.
The same type of arguments can be collected for the one-dimensionality of physical
time. It can be fun exploring them – for a short while. In summary, the strand model only
works in 3+1 space-time dimensions; it does not allow any other number of dimensions.
We have thus ticked off another of the millennium issues. We can thus continue with our
adventure.
Operators and the Heisenberg picture
?? − ?? = ℏ?
(134)
is related to a crossing switch. The present section confirms that speculation.
In quantum mechanics, the commutation relation follows from the definition of the
momentum operator as ? = ℏ?, ? = −?∂? being the wave vector operator. The factor
ℏ defines the unit of momentum. The wave vector counts the number of wave crests of
a wave. Now, in the strand model, a rotation of a state by an angle π is described by a
multiplication by ?. Counting wave crests of a propagating state is only possible by using
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 167
In quantum theory, Hermitean operators play an important role. In the strand model,
Hermitean or self-adjoint operators are operators that leave the tangle topology invariant.
Also unitary operators play an important role in quantum theory. In the strand model,
unitary operators are operators that deform tangles in a way that the corresponding wave
function retains its norm, i.e., such that tangles retain their topology and their core shape.
Physicists know two ways to describe quantum theory. One is to describe evolution
with time-dependent quantum states – the Schrödinger picture we are using here – and
the other is to describe evolution with time-dependent operators. In this so-called Heisenberg picture, the temporal evolution is described by the operators.
The two pictures of quantum theory are equivalent. In the Heisenberg picture, the
fundamental principle, the equivalence of a crossing switch with ℏ, becomes a statement
on the behaviour of operators. Already in 1987, Louis Kauffman had argued that the
commutation relation for the momentum and position operators
Motion Mountain – The Adventure of Physics
Challenge 141 e
207
208
8 q ua ntum theory d ed uced from stra nds
the factor ?, as this factor is the only property that distinguishes a crest from a trough.
In short, the commutation relation follows from the fundamental principle of the strand
model.
L agrangians and the principle of least action
Before we derive the Dirac equation, we show that the strand model naturally leads to
describe motion with Lagrangians.
In nature, physical action is an observable measured in multiples of the natural unit,
the quantum of action ℏ. Action is the fundamental observable about nature, because
action measures the total change occurring in a process.
In the strand model,
⊳ The physical action ? of a physical process is the observed number of crossing switches of strands. Action values are multiples of ℏ.
⊳ Energy is the number of crossing switches per time in a system.
⊳ The kinetic energy ? of a particle is the number of crossing switches per time
induced by shape fluctuations of the continuously rotating tangle core.
We call T the corresponding volume density: T = ?/?. In nature, the Lagrangian is a
practical quantity to describe motion. For a free particle, the Lagrangian density L = T
is simply the kinetic energy density, and the action ? = ∫L d?d? = ?? is the product
of kinetic energy and time. In the strand model, a free particle is a constantly rotating
and advancing tangle. We see directly that this constant evolution minimizes the action
? for a particle, given the states at the start and at the end.
This aspect is more interesting for particles that interact. Interactions can be described
by a potential energy ?, which is, more properly speaking, the energy of the field that
produces the interaction. In the strand model,
⊳ Potential energy ? is the number of crossing switches per time induced by
an interaction field.
We call U the corresponding volume density: U = ?/?. In short, in the strand model,
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
In nature, when free quantum particles move, their phase changes linearly with time. In
other words, the ‘little arrow’ representing the free particle phase rotates with constant
angular frequency. We saw that in the strand model, the ‘little arrow’ is taken as (half)
the orientation angle of the tangle core, and the arrow rotation is (half) the rotation of
the tangle core.
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We note that these multiples, if averaged, do not need to be integer multiples. We further
note that through this definition, action is observer-invariant. This important property is
thus automatic in the strand model.
In nature, energy is action per time. Thus, in the strand model we have:
q ua ntum theory d ed uced from stra nds
209
an interaction changes the rotation rate and the linear motion of a particle tangle.
In the strand model, the difference between kinetic and potential energy is thus a
quantity that describes how much a system consisting of a tangle and a field changes
at a given time. The total change is the integral over time of all instantaneous changes.
In other words, in the strand model we have:
⊳ The Lagrangian density L = T − U is the number of crossing switches per
volume and time, averaged over many Planck scales.
⊳ The physical action ? = ∫? d? = ∫∫L d?d? of a physical process is the observed number of crossing switches of strands. The action value ?if between
an initial state ?i and a final state ?f is given by
?if = ⟨?i |∫L d? | ?f ⟩ = ⟨?i |∫(T − U) d? | ?f ⟩ .
(135)
⊳ Evolution of tangles minimizes the action ?.
L=
iℏ
ℏ2
(? ∂? ? − ∂? ? ?) −
∇?∇? .
2
2?
(136)
In this way, the principle of least action can be used to describe the evolution of the
Schrödinger equation. The same is possible for situations with potentials, for the Pauli
equation, and for all other evolution equations of quantum particles.
We thus retain that the strand model explains the least action principle. It explains it
in the following way: quantum evolution minimizes the number of crossing switches.
Special rel ativit y: the vacuum
In nature, there is an invariant limit energy speed ?, namely the speed of light and of all
other massless radiation. Special relativity is the description of the consequences from
this observation, in the case of a flat space-time.
We remark that special relativity also implies and requires that the flat vacuum looks
exactly the same for all inertial observers. In the strand model, the idea of flat vacuum
as a set of fluctuating featureless strands that are unknotted and unlinked automatically
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Challenge 142 e
In the strand model, the least action principle appears naturally. In the strand model, an
evolution has least action when it occurs with the smallest number of crossing changes.
With this connection, one can also show that the strand model implies Schwinger’s
quantum action principle.
To calculate quantum motion with the principle of least action, we need to define the
kinetic and the potential energy in terms of strands. There are various possibilities for
Lagrangian densities for a given evolution equation; however, all are equivalent. In case
of the free Schrödinger equation, one possibility is:
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Since energy is related to crossing switches, it is natural that strand fluctuations that do
not induce crossing switches are favoured. In short, the strand model states
210
8 q ua ntum theory d ed uced from stra nds
implies that for any inertial observer the flat vacuum has no matter content, has no energy content, is isotropic and is homogeneous. The strand model thus realizes this basic
requirement of special relativity. In the strand model, vacuum is Lorentz-invariant.
Many models of the vacuum, even fluctuating ones, have difficulties reproducing
Lorentz invariance. The strand model differs, because the strands are not the observable entities; only their crossing switches are. This topological definition, together with
the averaging of the fluctuations, makes the vacuum Lorentz-invariant.
We note that in the strand model, the vacuum is unique, and the vacuum energy of
flat infinite vacuum is exactly zero. In the strand model, there is no divergence of the
vacuum energy, and there is thus no contribution to the cosmological constant from
quantum field theory. In particular, there is no need for supersymmetry to explain the
small energy density of the vacuum.
Special rel ativit y: the invariant limit speed
⊳ The Planck speed ? is the observed average speed of crossing switches due
to photons.
Page 351
Page 351
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Ref. 168
Because the definition uses crossing switches and a massless particle, the speed of light
? is an energy speed. The speed of light ? is also an average for long times. Indeed, as is
well-known in quantum field theory, due to the indeterminacy relation, single photons
can travel faster or slower than light, but the probability for large deviations is extremely
low.
The linear motion of a helically deformed photon strand through the vacuum strands
is similar to the motion of a bottle opener through cork. It differs from the linear motion
of a matter tangle through vacuum, which makes use of the belt trick. The belt trick slows
fermions down, though the details are not simple, as we will discover below. In short, we
find that matter tangles always move more slowly than light. The speed ? is a limit speed.
In fact, we see that ultrarelativistic tangles move, as shown in Figure 43, almost like
light. We thus find that matter can almost reach the speed of light. The speed ? is indeed
a limit speed.
However, one problem remains open: how exactly do tangles move through the web
that describes the vacuum? We will clarify this issue later on. In a few words, the motion
of a photon requires that the strands of the surrounding space make room for it. This
requires favourable fluctuations, thus a finite time. The motion process of photons thus
makes it clear that the speed of light is finite.
The speed of light ? is defined as an average, because, as well-known in quantum
field theory, there are small probabilities that light moves faster or slower that ?. But
the average result ? will be the same for every observer. The value of the speed ? is thus
invariant.
In 1905, Einstein showed that the mentioned properties of the speed of light – energy
Motion Mountain – The Adventure of Physics
Page 227
In the strand model, massless particles are unknotted and untangled. Even though we
will deduce the strand model for photons only later on, we use it here already, to speed
up the discussion. In the strand model, the photon is described by a single, helically
deformed unknotted strand, as shown in Figure 50. Therefore, we can define:
q ua ntum theory d ed uced from stra nds
211
Strand model :
time average
of crossing
switches
Localized particle at rest :
Observed
probability
density :
t1
t1
t2
t2
Slow motion :
t2
t1
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t1
t2
rotation,
precession and
displacement
t1
t2
t1
t2
rotation,
precession and
displacement
F I G U R E 43 Tangles at rest, at low speed and at relativistic speed.
speed, limit speed, finite speed and invariant speed – imply the Lorentz transformations.
In particular, the three properties of the speed of light ? imply that the energy ? of a
particle of mass ? is related to its momentum ? as
?2 = ?2 ?4 + ?2 ?2
Page 356
Page 150
or
ℏ2 ?2 = ?2 ?4 + ?2 ℏ2 ?2 .
(137)
This dispersion relation is thus also valid for massive particles made of tangled strands –
even though we cannot yet calculate tangle masses. (We will do this later on.)
Should we be surprised at this result? No. In the fundamental principle, the definition
of the crossing switch, we inserted the speed of light as the ratio between the Planck
length and the Planck time. Therefore, by defining the crossing switch in the way we did,
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Relativistic motion :
212
Page 206
Page 162
Page 194
8 q ua ntum theory d ed uced from stra nds
we have implicitly stated the invariance of the speed of light.
Fluctuating strands imply that flat vacuum has no matter or energy content, for every
inertial observer. Due to the strand fluctuations, flat vacuum is also homogeneous and
isotropic for every inertial observer. Therefore, together with the 3 + 1-dimensionality of
space-time deduced above, we have now definitely shown that flat vacuum has Poincaré
symmetry. This settles another issue from the millennium list.
The relativistic dispersion relation differs from the non-relativistic case in two ways.
First, the energy scale is shifted, and now includes the rest energy ?0 = ?2 ?. Secondly,
the spin precession is not independent of the particle speed any more; for relativistic
particles, the spin lies close to the direction of motion. Both effects follow from the
existence of a limit speed.
If we neglect spin, we can use the relativistic dispersion relation to deduce directly the
well-known Klein–Gordon equation for the evolution of a wave function:
(138)
In other words, the strand model implies that relativistic tangles follow the Klein–
Gordon equation. We now build on this result to deduce Dirac’s equation for relativistic
quantum motion.
Dirac ’ s equation deduced from tangles
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The relativistic Klein–Gordon equation assumes that spin effects are negligible. This approximation fails to describe most experiments. A precise description of relativistic elementary particles must include spin.
So far, we deduced the Schrödinger equation using the relation between phase and
the quantum of action, using the non-relativistic energy–momentum relation, and neglecting spin. In the next step we deduced the Pauli equation by including the properties
of spin 1/2. The following step was to deduce the Klein–Gordon equation using again
the relation between phase and the quantum of action, this time the relativistic energy–
momentum relation, but assuming zero spin. The final and correct description of elementary fermions, the Dirac equation, results from combining all three ingredients: (1)
the relation between the quantum of action and the phase of the wave function, (2) the
relativistic mass–energy relation, and (3) the effects of spin 1/2. Now we can reproduce
this derivation because all three ingredients are reproduced by the strand model.
We first recall the derivation of the Dirac equation found in textbooks. The main
observation about spin in the relativistic context is the existence of states of right-handed
and of left-handed chirality: spin can precess in two opposite senses around the direction
of momentum. In addition, for massive particles, the two chiral states mix. The existence
of two chiralities requires a description of spinning particles with a wave function that
has four complex components, thus twice the number of components that appear in the
Pauli equation. Indeed, the Pauli equation implicitly assumes only one, given sign for the
chirality, even though it does not specify it. This simple description is possible because
in non-relativistic situations, states of different chirality do not mix.
Consistency requires that each of the four components of the wave function of a relativistic spinning particle must follow the relativistic energy–momentum relation, and
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− ℏ2 ∂?? ? = ?2 ?4 − ?2 ℏ2 ∇2 ? .
q ua ntum theory d ed uced from stra nds
213
Motion Mountain – The Adventure of Physics
deduce the Dirac equation (© Springer Verlag, from Ref. 171).
Ref. 169
Ref. 170
thus the Klein–Gordon equation. This requirement is known to be sufficient to deduce
the Dirac equation. One of the simplest derivations is due to Lerner; we summarize it
here.
When a spinning object moves relativistically, we must take both chiralities into account. We call ? the negative chiral state and ? the positive chiral state. Each state is
described by two complex numbers that depend on space and time. The 4-vector for
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F I G U R E 44 The belt trick for a rotating body with many tails, as used by Battey-Pratt and Racey to
214
8 q ua ntum theory d ed uced from stra nds
probability and current becomes
?? = ?† ?? ? + ?† ?? ? .
(139)
We now introduce the four-component spinor ? and the 4 × 4 spin matrices ??
?
?=( )
?
0
?
and ?? = ( ?
) ,
0 ??
(140)
where ?? = (?, ?) and ?? = (?, −?) and ? is the 2 × 2 identity matrix. The 4-current can
then be written as
?? = ?† ?? ? .
(141)
The three requirements of current conservation, Lorentz invariance and linearity then
yield the evolution equation
iℏ∂? (?? ?) + ???5 ? = 0 .
This is the Dirac equation in the (less usual) spinorial representation.* The last term
shows that mass mixes right and left chiralities. The equation can be expanded to include
potentials using minimal coupling, in the same way as done above for the Schrödinger
and Pauli equations.
The above textbook derivation of the Dirac equation from usual quantum theory can
be repeated and visualized also with the help of strands. There is no difference in arguments or results. The derivation with the help of strands was performed for the first
time by Battey-Pratt and Racey, in 1980. They explored a central object connected by
unobservable strands (or ‘tails’) to the border of space, as shown in Figure 44. In their
approach, the central object plus the tails correspond to a quantum particle. The central
object is assumed to be continuously rotating, thus reproducing spin 1/2. They also assumed that only the central object is observable. (In the strand model, the central object
becomes the tangle core.) Battey-Pratt and Racey then explored a relativistically moving object of either chirality. They showed that a description of such an object requires
four complex fields. Studying the evolution of the phases and axes for the chiral objects
yields the Dirac equation. The derivation by Battey-Pratt and Racey is mathematically
equivalent to the textbook derivation just given.
We can thus say that the Dirac equation follows from the belt trick. We will visualize this connection in more detail in the next section. When the present author found
this connection in 2008, Lou Kauffman pointed out the much earlier paper by BatteyPratt and Racey. In fact, Paul Dirac was still alive when they found this connection, but
unfortunately he did not answer their letter asking for comment.
* The matrix ?5 is defined here as
0
?5 = (
?
where ? is the 2 × 2 identity matrix.
?
) ,
0
(143)
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Ref. 171
(142)
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Ref. 170
q ua ntum theory d ed uced from stra nds
215
In summary, tangles completely reproduce both the rotation and the linear motion
of elementary fermions. Therefore, the strand model provides a simple view on the
evolution equations of quantum theory. In the terms of the strand model, when spin
is neglected, the Schrödinger equation describes the evolution of crossing density. For
relativistic fermions, when the belt trick is included, the Dirac equation describes the
evolution of crossing density. In fact, strands visualize these evolution equations in the
most concrete way known so far.
Visualizing spinors and Dirac ’ s equation using tangles
Ref. 172
In total, these are eight real parameters; they correspond to one positive real number and
seven phases. They lead to the description of a spinor wave function as
? = √? e?? ?(?) ?(?/2, ?/2, ?/2) ,
Ref. 172
(144)
where the product ?? is an abbreviation for the boosted and rotated unit spinor and
all parameters depend on space and time. This expression is equivalent to the description with four complex parameters used in most textbooks. In fact, this description of a
spinor wave function and the related physical visualization of its density and its first six
phases dates already from the 1960s. The visualisation can be deduced from the study of
relativistic spinning tops or of relativistic fluids. Rotating tangles are more realistic, however. In contrast to all previous visualizations, the rotating tangle model explains also the
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
— There is an average density ?(?, ?); physically, this is the probability density. In the
strand model, this is the local crossing density.
— There is a set of three Euler angles ?, ? and ?; physically, they describe the average
local orientation and phase of the spin axis. In the strand model, this is the average
local orientation and phase of the tangle core.
— There is a second set of three parameters ? = (?? , ??, ?? ); physically, they describe, at
one’s preference, either the average local Lorentz boost or a second set of three Euler
angles. In the strand model, these parameters describe the average local deformation
of the core that is due to the Lorentz boost. It can also be seen as the axis around
which the belt trick is performed.
— There is a phase ?; physically, this represents the relative importance of particle and
antiparticle density. In the strand model, this phase describes with what probability
the average local belt trick is performed right-handedly or left-handedly.
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Despite its apparent complexity, the Dirac equation makes only a few statements: spin
1/2 particles are fermions, obey the relativistic energy–momentum relation, keep the
quantum of action invariant, and thus behave like a wave. Each statement is visualized by the tangle model of fermions: tangles behave as spinors, the relativistic energy–
momentum relation is built-in, the fundamental principle holds, and rotating tangle
cores reproduce the evolution of the phase. Let us look at the details.
Given a particle tangle, the short-time fluctuations lead, after averaging of the crossings, to the wave function. The tangle model of fermions also provides a visualization
of the spinor wave function. Indeed, at each point in space, the wave function has the
following parameters:
216
Ref. 164
Ref. 173
Ref. 175
1. Tangles support the view that elementary particles are little rotating entities, also in
the relativistic case. This fact has been pointed out by many scholars over the years.
The strand model provides a consistent visualization for these discussions.
2. The belt trick can be seen as the mechanism underlying the famous Zitterbewegung
that is part of the Dirac equation. The limitations in the observing the belt trick translate directly into the difficulties of observing the Zitterbewegung.
3. The belt trick also visualizes why the velocity operator for a relativistic particle has
eigenvalues ±?.
4. The Compton length is often seen as the typical length at which quantum field effects
take place. In the tangle model, it would correspond to the average size needed for
the belt trick. The strand model thus suggests that the mass of a particle is related to
the average size needed for the belt trick.
5. Tangles support the – at first sight bizarre – picture of elementary particles as little
charges rotating around a centre of mass. Indeed, in the tangle model, particle rotation requires a regular application of the belt trick of Figure 19, and the belt trick can
be interpreted as inducing the rotation of a charge, defined by the tangle core, around
a centre of mass, defined by the average of the core position. It can thus be helpful to
use the strand model to visualize this description.
6. The tangle model can be seen as a vindication of the stochastic quantization research
programme; quantum motion is the result of underlying fluctuations. For example,
the similarity of the Schrödinger equation and the diffusion equation is modelled and
explained by the strand model: since crossings can be rotated, diffusion of crossings
leads to the imaginary unit that appears in the Schrödinger equation.
In short, rotating tangles are a correct underlying model for the propagation of fermions.
And so far, tangles are also the only known correct model. Tangles model propagators.
This modelling is possible because the Dirac equation results from only three ingredients:
— the relation between the quantum of action and the phase of the wave function (the
wave behaviour),
— the relation between the quantum of action and spinor behaviour (the exchange behaviour),
— and the mass–energy relation of special relativity (the particle behaviour), itself due
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 174
Page 175
last, seventh phase. This is the phase that describes matter and anti-matter, that explains
the appearance of the quantum of action ℏ, and that explains the fermion behaviour.
In short, only rotating tangles together with the fundamental principle provide a
simple, complete and precise visualisation of spinor wave functions and their evolution.
The tangle model for spinning relativistic quantum particles remains a simple extension
of Feynman’s idea to describe a quantum particle as a rotating little arrow. The arrow
can be imagined as being attached to the rotating tangle core. The tails are needed to
reproduce fermion behaviour. The specific type of tangle core determines the type of
particle. The blurring of the crossings defines the wave function. Rotating arrows describe non-relativistic quantum physics; rotating tangles describe relativistic quantum
physics.
Visualizing spinor wave functions with tangles of strands helps the understanding of
the Dirac equation in several ways.
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Ref. 172
8 q ua ntum theory d ed uced from stra nds
q ua ntum theory d ed uced from stra nds
A hydrogen atom
Simplified
strand model :
time average
of crossing
switches
217
Observed
electron
probability
density :
proton
proton
electron
tangle
electron cloud
to the fundamental principle.
And all three ingredients are reproduced by the strand model. We see that the apparent
complexity of the Dirac equation hides its fundamental simplicity. The strand model
reproduces the ingredients of the Dirac equation, reproduces the equation itself, and
makes the simplicity manifest. In fact, we can say:
Page 178
The belt trick is fundamental for understanding the Dirac equation. In the strand model,
core rotations vary along two dimensions – the rotation is described by two angles – and
so does the belt trick. The resulting four combinations form the four components of the
Dirac spinor and of the Dirac equation.
In summary, tangles can be used as a precise visualization and explanation of
quantum physics. Wave functions, also those of fermions, are blurred tangles – with the
detail that not the strands, but their crossings are blurred.
Q uantum mechanics vs. quantum field theory
Quantum mechanics is the approximation to quantum physics in which fields are continuous and particles are immutable. In the strand model, quantum mechanics is thus
the approximation in which a particle is described by a tangle with a shape that is fixed
in time. This approximation allows us to derive the Dirac equation, the Klein–Gordon
equation, the Proca equation, the Pauli equation and the Schrödinger equation. In this
approximation, the strand model for the electron in a hydrogen atom is illustrated in
Figure 45. This approximation already will allow us to deduce the existence of the three
gauge interactions, as we will see in the next chapter.
In contrast, quantum field theory is the description in which fields are themselves described by bosons, and particles types can transform into each other. The strand model
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⊳ The Dirac equation describes the relativistic infinitesimal belt trick or string
trick.
Motion Mountain – The Adventure of Physics
F I G U R E 45 A simple, quantum-mechanical view of a hydrogen atom.
218
8 q ua ntum theory d ed uced from stra nds
allows us to deduce the existence of all known gauge bosons, as shown in the next chapter.
In the strand description of quantum field theory, particles are not tangles with a fixed
shape of their core, but for each particle, the shape varies. This variation leads to gauge
boson emission and absorption.
A fl ashback: set tling three parad oxes of Galilean physics
Page 108
“
Challenge 143 s
Urlaub ist die Fortsetzung des Familienlebens
unter erschwerten Bedingungen.*
Dieter Hildebrandt
”
Are the definitions for the addition and multiplication of Schrödinger wave functions
that were given above also valid for spinor tangle functions?
∗∗
* ‘Vacation is the continuation of family life under aggravated conditions.’ Dieter Hildebrandt (b. 1927
Bunzlau, d. 2013 Munich) was a cabaret artist, actor and author.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Fun challenges abou t quantum theory
Motion Mountain – The Adventure of Physics
In all descriptions of physics, space and time are measured, explained and defined using
matter. This occurs, for example, with the help of metre bars and clocks. On the other
hand, matter is measured, explained and defined using space and time. This occurs, for
example, by following a localized body over space and time. The circularity of the two
definitions is at the basis of modern physics.
As already mentioned above, the circularity is a natural consequence of the strand
model. Both matter and space-time turn out to be approximations of the same basic
building blocks; this common origin explains the apparent circular reasoning of Galilean
physics. Most of all, the strand model changes it from a paradox to a logical necessity.
The strand model defines vacuum, and thus physical space, as a result of averaging
strand crossings. Space is thus a relative concept. Newton’s bucket experiment is sometimes seen as a counter-argument to this conclusion and as an argument for absolute
space. However, the strand model shows that any turning object is connected to the rest
of the universe through its tails. This connection makes every rotation an example of relative motion. Rotation is thus always performed relatively to the horizon of the universe.
On the other hand, the detection of tangles among the tails allows a local determination
of the rotation state, as is observed. Strands thus confirm that rotation and space are relative concepts. Strands thus also explain why we can turn ourselves on ice by rotating an
arm over our head, without outside help. Strands lie to rest all issues around the rotating
bucket.
A long time ago, Zeno of Elea based one of his paradoxes – the flying arrow that
cannot reach the target – on an assumption that is usually taken as granted: he stated
the impossibility to distinguish a short-time image (or state) of a moving body from the
image (or state) of a resting body. The flattening of the tangles involved shows that the
assumption is incorrect; motion and rest are distinguishable, even in (imagined) photographs taken with extremely short shutter times. The argument of Zeno is thus not
possible, and the paradox disappears.
q ua ntum theory d ed uced from stra nds
Challenge 144 e
219
The definition of tangle functions, or wave functions, did not take into account the crossings of the vacuum strands, but only those of the particle tangle. Why is this allowed?
∗∗
Challenge 145 e
Modelling the measurement of action at the quantum level as the counting of full turns
of a wheel is a well-known idea that is used by good teachers to take the mystery out
of quantum physics. The strand model visualizes this idea by assigning the quantum of
action ℏ to a full turn of one strand segment around another.
∗∗
Challenge 146 s
Is any axiomatic system of quantum theory in contrast with the strand model?
∗∗
∗∗
Ref. 176
Challenge 148 e
If you do not like the deduction of quantum mechanics given here, there is an alternative:
you can deduce quantum mechanics in the way Schwinger did in his course, using the
quantum action principle.
∗∗
Challenge 149 r
Modern teaching of the Dirac equation replaces the spinor picture with the vector
picture. Hrvoje Nikolić showed that the vector picture significantly simplifies the understanding of Lorentz covariance of the Dirac equation. How does the vector picture clarify
the relation between the belt trick and the Dirac equation?
∗∗
Challenge 150 s
In the strand description of quantum mechanics, strands are impenetrable: they cannot
pass through each other (at finite distances). Can quantum mechanics also be derived if
the model is changed and this process is allowed? Is entanglement still found?
∗∗
Challenge 151 e
A puzzle: Is the belt trick possible in a continuous and deformable medium – such as a
sheet or a mattress – in which a coloured sphere is suspended? Is the belt trick possible
with an uncountably infinite number of tails?
∗∗
Page 180
Challenge 152 s
At first sight, the apheresis machine diagram of Figure 24 suggests that, using the belt
trick, animals could grow and use wheels instead of legs, because rotating wheels could
be supplied with blood and connected to nerves. Why did wheels not evolve nevertheless?
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 177
Motion Mountain – The Adventure of Physics
Challenge 147 s
In the strand model, tangle energy is related to tangle core rotation. What is the difference between the angular frequency for tangles in the non-relativistic and in the relativistic case?
220
8 q ua ntum theory d ed uced from stra nds
summary on quantum theory of matter:
experimenta l predictions
In this chapter, we used the fundamental principle – crossing switches define the
quantum of action ℏ and the other Planck units – to deduce that particles are tangles
of strands and that wave functions are time-averaged rotating tangles. In simple words,
⊳ Both non-relativistic and relativistic wave functions are blurred rotating
tangles.
⊳ The Dirac equation is essentially the infinitesimal version of the belt trick
(or string trick).
L = ? (?ℏ?/
D − ?2 ?) ? ,
where
Page 39
/ = ?? D? = ?? (∂? − ??? ? ) .
D
(145)
(146)
We thus conclude that strands reproduce the quantum theory of matter.
The strand model predicts deviations from the relativistic matter Lagrangian, and thus
from the Dirac equation, only in three cases: first, when quantum aspects of electrodynamic field play a role, second, when nuclear interactions play a role, and third, when
space curvature, i.e., strong gravity, plays a role. All this agrees with observation.
We will deduce the description of quantum electrodynamics and of the nuclear interactions in the next chapter. In the case of gravity, the strand model predicts that
deviations from quantum theory occur exclusively when the energy–momentum of an
elementary particle approaches the Planck value, i.e., for really strong gravity. Such deviations are not accessible to experiment at present. We will explore this situation in the
subsequent chapter.
In addition, the strand model predicts that in nature, the Planck values for momentum
and energy are limit values that cannot be exceeded by a quantum particle. All experi-
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
In other words, we have shown that strands reproduce the relativistic Lagrangian density
L of charged, elementary, relativistic fermions in an external electromagnetic field ?
Motion Mountain – The Adventure of Physics
More precisely, a wave function appears from the blurred crossings of a tangle. The components and phases of the wave function at a point in space are due to the orientation
and phase of crossings at that point. We also deduced that blurred tangles obey the least
action principle and the Dirac equation.
In other words, visualizing the quantum of action as a crossing switch implies
quantum theory. The strand model confirms Bohr’s statement: quantum theory is indeed a consequence of the quantum of action. Specifically, the strand model thus shows
that all quantum effects are consequences of extension and consequences of the three dimensions of space. More precisely, all quantum effects are due to tails, the tails of the
tangles that represent a quantum system. In particular, the strand model confirms that
sum m ary on q ua ntum theory of m at ter: experimental pred ictions
Page 162
221
ments agree with this prediction.
The deduction of quantum theory from strands given here is, at present, the only
known microscopic explanation for quantum physics. So far, no other microscopic
model, no different explanation nor any other Planck-scale deduction of quantum theory has been found. In particular, the extension of fundamental entities – together with
observability limited to crossing switches – is the key to understanding quantum physics.
Let us evaluate the situation. In our quest to explain the open issues of the millennium
list, we have explained the origin of Planck units, the origin of wave functions, the origin
of the least action principle, the origin of space-time dimensions, the Lorentz and Poincaré symmetries, the origin of particle identity, and the simplest part of the Lagrangian
of quantum field theory, namely, the Lagrangian of free fermions, such as the electron,
and that of fermions in continuous external fields. Therefore, for the next leg, we turn to
the most important parts of the standard model Lagrangian that are missing: those due
to gauge interactions.
Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Chapter 9
GAUGE INT ER ACT IONS DEDUCED
FROM ST R ANDS
Ref. 178
Ref. 179
W
Page 174
Ref. 164
Experiments in the quantum domain show that interactions change the phase of wave
functions. But how precisely does this happen? The strand model will give us a simple
answer: the emission and the absorption of gauge bosons is only possible together with
a phase change. To explain this connection, we need to study the phase of tangle cores in
more detail.
When we explored spin and its connection to the belt trick, we pictured the rotation
of the tangle core in the same way as the rotation of a belt buckle. We assumed that the
core of the tangle rotates like a rigid object; the rotation is completed through the shape
fluctuations of the tails only. Why did we assume this?
In Feynman’s description of quantum theory, free particles are advancing rotating arrows. In the strand model, free particle motion is modelled as the change of position of
the tangle core and spin as the rotation of the core. We boldly assumed that the core
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Interactions and phase change
Motion Mountain – The Adventure of Physics
Page 18
hat are interactions? At the start of this volume, when we summarized
hat relates the Planck units to relativity and to quantum theory,
e pointed out that the nature of interactions at Planck scales was still in the
dark. In the year 2000, it was known for several decades that the essential properties
of the electromagnetic, the weak and the strong nuclear interaction are their respective
gauge symmetries: all three interactions are gauge interactions. But the underlying
reason for this property was still unknown.
In this chapter we discover that fluctuating strands in three spatial dimensions explain the existence of precisely three gauge interactions, each with precisely the gauge
symmetry group that is observed. This is the first time ever that such an explanation
is possible. In other terms, we will deduce quantum field theory from strands. Indeed,
strands provide a natural mechanism for interactions that explains and implies Feynman
diagrams. The term ‘mechanism’ has to be taken with a grain of salt, because there is
nothing mechanical involved; nevertheless, the term is not wrong, because we shall discover a surprisingly simple result: Gauge interactions and gauge symmetries are due to
specific strand deformations.
In this chapter, we work in flat space-time, as is always done in quantum field theory. We leave the quantum aspects of curved space-time and of gravitation for the next
chapter. We thus start by exploring the non-gravitational interactions in the quantum
domain.
g aug e interactions d ed uced from stra nds
Strand model :
tail
time average
of crossing
changes
spin
223
Observed
probability
density :
spin
position
phase
position
phase
core
F I G U R E 46 In the chapter on quantum theory, the phase was defined assuming a rigidly rotating core;
position
phase
such core deformations lead to gauge interactions.
remained rigid, attached the phase arrow to it, and described spin as the rotation of the
core with its attached arrow, as shown again in Figure 46. This bold simplification led us
to the Dirac equation. In short, the assumption of a rigid core works.
However, we swept a problem under the rug: what happens if the core is not rigid? It
turns out that the answer to this question automatically leads to the existence of gauge
interactions. Now, we know from usual quantum theory that
⊳ An interaction is a process that changes the phase of a wave function, but
differs from a rotation.
In the strand model, shape deformations of tangle cores also lead to phase changes. In
fact, we will discover that core deformations automatically lead to precisely those three
gauge interactions that we observe in nature.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
F I G U R E 47 A magnified tangle core shows that the phase can also change due to core deformations;
Motion Mountain – The Adventure of Physics
this approximation was also used in the description of particle translation.
224
9 g auge interactions d ed uced from stra nds
Tail deformations versus core deformations
We can summarize the previous chapter, on the free motion of matter tangles, as the
chapter that focused on shape fluctuations of tails. Indeed, the belt trick completed the
proof that
⊳ Space-time symmetries are due to tail deformations.
All space-time symmetries – translation, rotation, boost, spin and particle exchange – are
due to tail deformations; in such tail deformations, the tangle core is assumed to remain
unchanged and rigid (in its own rest frame).
In contrast, the present chapter focuses on shape fluctuations in tangle cores.* We will
discover that
⊳ Gauge symmetries are due to core deformations.
⊳ When the phase of a core changes through core shape deformation, we speak
of interaction.
Ref. 179
Ref. 181
Ref. 180
We thus need to understand two things: First, what kinds of core deformation exist?
Secondly, how precisely is the phase – i.e., each arrow definition – influenced by core
deformations? In particular, we have to check the answers and deductions with experiment.
The first question, on the classification of the core deformations, is less hard than
it might appear. The fundamental principle – events are crossing switches of strands –
implies that deformations are observable only if they induce crossing switches. Other
deformations do not have any physical effect. (Of course, certain deformations will have
crossing switches for one observer and none for another. We will take this fact into consideration.) Already in 1926, the mathematician Kurt Reidemeister classified all those
tangle deformations that lead to crossing switches. The classification yields exactly three
classes of deformations, today called the three Reidemeister moves. They are shown in
Figure 48.
* The contrast between tail deformations and core deformations has a remote similarity to gravity/gauge
duality, or AdS/CFT correspondence, and to space-time duality. For example, in the strand model, the three
Reidemeister moves on tangle cores represent the three gauge interactions, whereas the three Reidemeister
moves on the vacuum represent (also) gravitational effects.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
⊳ When the phase of a core changes through rigid orientation change, we speak
of core rotation.
Motion Mountain – The Adventure of Physics
Let us explore the tangle core in more detail. Figure 47 shows a magnified view of the
core and its phase arrow. The phase of the core results from the phases of all its crossings. Thus, the figure makes it clear that the phase arrow will be sensitive to the shape
fluctuations and deformations of the strand segments that make up the core.
In nature, any phase change of the wave function that is not due to a space-time symmetry is due to an interaction. For the strand model, this connection implies:
g aug e interactions d ed uced from stra nds
Reidemeister move I
or twist
Reidemeister move II
or poke
225
Reidemeister move III
or slide
F I G U R E 48 The Reidemeister moves: the three types of deformations that induce crossing switches – if
the moves are properly defined in three dimensions.
⊳ The first Reidemeister move, or type I move, or twist, is the addition or removal of a twist in a strand.
⊳ The third Reidemeister move, or type III move, or slide, is the displacement
of one strand segment under (or over) the crossing of two other strands.
⊳ The first Reidemeister move corresponds to electromagnetism.
⊳ The second Reidemeister move corresponds to the weak nuclear interaction.
⊳ The third Reidemeister move corresponds to the strong nuclear interaction.
We will prove this correspondence in the following.
For each Reidemeister move we will explore two types of core deformation processes:
One deformation type are core fluctuations, which correspond, as we will see, to the emission and absorption of virtual interaction bosons. The other deformations are externally
induced core disturbances, which correspond to the emission and absorption of real interaction bosons. As the first step, we show that both for fluctuations and for disturbances,
the first Reidemeister move, the twist, is related to the electromagnetic interaction.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
The type number of each Reidemeister move is also the number of involved strands.
We will discover that despite appearances, each Reidemeister move induces a crossing
switch. To find this connection, we have to generalize the original Reidemeister moves,
which were defined in a two-dimensional projection plane, to the three-dimensional
situation of tangle cores.
The three Reidemeister moves turn out to be related to the three gauge interactions:
Motion Mountain – The Adventure of Physics
⊳ The second Reidemeister move, or type II move, or poke, is the addition or
removal of a bend of one strand under (or over) a second strand.
226
9 g auge interactions d ed uced from stra nds
photon
vacuum
twist
transfer
fermion
fermion
with
different
phase
F I G U R E 49 A single strand changes the rotation of a tangle: twist transfer is the basis of
electrodynamics and the first reidemeist er move
Strands and the t wist, the first R eidemeister move
In the strand model of electromagnetism, massless spin 1 bosons such as the photon are
made of a single strand. How can a single strand change the phase of a tangle? The
answer is given in Figure 49: a twisted loop in a single strand will influence the rotation
of a tangle because it changes the possible fluctuations of the tangle core. Due to the
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 5
Experiments show that all four fundamental interactions are described by potentials.
Experiments also show that potentials change the phase, the rotation frequency and the
wave number of wave functions. Experiments show that interactions result from the
absorption and the emission of gauge bosons. In particular, for electromagnetism, the
potentials are due to the flow of real and virtual, massless, uncharged spin-1 photons.
Photons are emitted from or absorbed by charged elementary particles; neutral elementary particles do not emit or absorb photons. There are two types of electric charge, positive and negative. The attraction and repulsion of static charges diminishes with the
inverse square of the distance. Charge is conserved. All charged particles are massive
and move slower than light. The Lagrangian of matter coupled to the electromagnetic
field has a U(1) gauge symmetry. Electromagnetism has a single fundamental Feynman
diagram. The electromagnetic coupling constant at low energy, the so-called fine structure constant, is measured to be ? = 1/137.035 999 139(31); its energy dependence is
described by renormalization.
The previous paragraph contains everything known about the electromagnetic interaction. For example, Maxwell’s field equations follow from Coulomb’s inverse square
relation, its relativistic generalization, and the conservation of charge. More precisely, all
experimental observations about electricity and magnetism follow from the Lagrangian
of quantum electrodynamics, or QED. In short, we now need to show that the Lagrangian
of QED follows from the strand model.
Motion Mountain – The Adventure of Physics
electromagnetism in the strand model. No strand is cut or reglued; the transfer occurs, statistically,
through the excluded volume due to the impenetrability of strands.
electrodynamics a nd the first reidemeister m ov e
227
The photon
Strand model:
Observation :
time average
of crossing
switches
electric
field
strength
phase
helicity
motion
wavelength
F I G U R E 50 The photon in the strand model.
Page 232
⊳ A photon is a twisted strand. An illustration is given in Figure 50.
⊳ The electromagnetic interaction is the transfer of twists, i.e., the transfer of
first Reidemeister moves, between two particles, as shown in Figure 49.
The transfer of a twist from a single strand to a tangle core thus models the absorption
of a photon. We stress again that this transfer results from the way that strands hinder
each other’s motion, because of their impenetrability. No strand is ever cut or reglued.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 174
impenetrability of strands, an approaching twisted loop will sometimes transfer its twist
to the tangle and thereby change its phase. The observed effect of an electromagnetic
field on the phase of a charged fermion is the time average of all such twist transfers.
Single strands represent bosons, as we saw above. Twisted loops are single strands and
can have two twist senses, or two polarizations. Single, twisted and unknotted strands
have no mass; in other words, twisted loops effectively move with the speed of light.
And twisted loops, being curved, carry energy.
Approaching twisted loops will change the phase, i.e., the orientation of a matter
tangle. Twisted loops correspond to a local rotation of a strand segment by π. But twists
can be generalized to arbitrary angles. These generalized twists can be concatenated. Because they are described by a single angle, and because a double twist is equivalent to no
twist at all, twists form a U(1) group. We show this in detail shortly.
In summary, twists behave like photons in all their properties. Therefore, the strand
model suggests:
Motion Mountain – The Adventure of Physics
or, equivalently:
228
9 g auge interactions d ed uced from stra nds
Can photons decay, disappear or break up?
Challenge 153 e
Surrounded by a bath of photon strands, not all fermion tangles will change their phase.
A tangle subject to randomly approaching virtual photons will feel a net effect over time
only if it lacks some symmetry. In other words, only tangles that lack a certain symmetry
will be electrically charged. Which symmetry will this be?
In a bath of photon strands, thus in a bath that induces random Reidemeister I moves,
only chiral fermion tangles are expected to be influenced. In other terms:
⊳ Electric charge is due to lack of mirror symmetry, i.e., to tangle chirality.
Conversely, we have:
⊳ Electrically charged particles randomly emit twisted strands. Due to the
tangle chirality, a random emission will lead to a slight asymmetry, so that
right-handed twists will be in the majority for particles of one charge, and
left-handed twists will be in the majority for particles of the opposite charge.
Equating electric charge with tangle chirality allows modelling several important observations. First, because chirality can be right-handed or left-handed, there are positive and
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Electric charge
Motion Mountain – The Adventure of Physics
Page 351
The strand model of the photon, as shown in Figure 50, might be seen to suggest that
photons can disappear. For example, if a photon strand is straightened out by pulling the
ends of the helical deformation, the helix might disappear. A helix might also disappear
by a shape fluctuation or transform into several helices. However, this is a fallacy.
A lone twist cannot disappear by pulling; ‘‘pulling’’ requires an apparatus that performs it. That is impossible. A lone twist cannot disappear by fluctuations either, because
a photon also includes the vacuum strands around it. In the strand model, the energy
of the photon is localized in the configuration formed by the photon strand and the surrounding vacuum strands. In the strand model, energy is localized in regions of strand
curvature. If the helical strands disappears, the surrounding vacuum strands are curved
instead, or more strongly, and the twist energy is taken up by these surrounding strands.
The net result is that the helix is transferred, permanently or for a short time, to another
strand. In other terms, in the strand model, photons can also move by hopping from one
strand to the next.
Also, a single photon strand cannot break up into several photon strands of smaller
helical diameters or of different rotation frequencies. Such a process is prevented by the
fundamental principle, when the vacuum is taken into account.
The only way in which a photon can disappear completely is by transferring its crossing, i.e., its energy to a tangle. Such a process is called the absorption of a photon by a
charged particle.
In short, due to energy and to topological restrictions, the strand model prevents the
decay, disappearance or splitting of photons, as long as no electric charge is involved.
Linear and angular momentum conservation also lead to the same conclusion. Photons
are stable particles in the strand model.
electrodynamics a nd the first reidemeister m ov e
The twist move (or first
Reidemeister move)
in textbook form :
The twist move (or first
Reidemeister move)
applied to an interacting
tangle and loop :
fermion
photon
229
The unique
generator
of the twist
move is a
rotation by π.
vacuum
The basic twist can be described
as a local rotation by π.
A full rotation, from -π to π,
produces a crossing switch.
affected
not affected
Emission of numerous random twists by
chiral tangles leads to Coulomb’s law :
strand
model
observed
time average
negative charges. Second, because strands are never cut or reglued in the strand model,
chirality, and thus electric charge, is a conserved quantity. Third, chirality is only possible for tangles that are localized, and thus massive. Therefore, chiral tangles – charged
particles – always move slower than light. Fourth, a chiral tangle at rest induces a twisted strand density around it that changes as 1/?2 , as is illustrated in Figure 51. Finally,
photons are uncharged; thus they are not influenced by other photons (to first order).
In short, all properties of electric charge found in nature are reproduced by the tangle
model. We now check this in more detail.
Challenge: What top ological invariant is electric charge?
Chirality explains the sign of electric charge, but not its magnitude in units of the elementary charge ?. A full definition of electric charge must include this aspect.
Mathematicians defined various topological invariants for knot and tangles. Topological invariants are properties that are independent of the shape of the knot or tangle, but
allow to distinguish knots or tangles that differ in the ways they are knotted or tangled up.
Several invariants are candidates as building blocks for electric charge: chirality ?, which
can be +1 or −1, minimal crossing number ?, or topological writhe ?, i.e., the signed min-
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
F I G U R E 51 Electromagnetism in the strand model: the electromagnetic interaction, electric charge and
Coulomb’s inverse square relation.
Motion Mountain – The Adventure of Physics
Large numbers of random twists
affect only chiral tangles :
230
Page 419
Page 385
9 g auge interactions d ed uced from stra nds
imal crossing number.
A definition of electric charge ?, proposed by Claus Ernst, is ? = ? (? mod 2). Another
option for the definition of charge is ? = ?/3. Equivalent definitions use the linking
number. At this point of our exploration, the issue is open. We will come back to the
detailed connection between charge, chirality and tangle topology later on.
Electric and magnetic fields and p otentials
The definition of photons with twisted strands leads to the following definition.
⊳ The electric field is the volume density of (oriented) crossings of twisted
loops.
⊳ The magnetic field is the flow density of (oriented) crossings of twisted
loops.
⊳ The magnetic potential is the flow density of twisted loops.
Challenge 154 e
The strand model thus reproduces electromagnetic energy.
We note that in the strand model, the definition of the fields implies that there is no
magnetic charge in nature. This agrees with observation.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 184
The simplest way to check these definitions is to note that the random emission of twisted
loops by electric charges yields Coulomb’s inverse square relation: the force between two
static spherical charges changes with inverse square of the distance. The strand model
implies that in this case, the crossing density is proportional to the square of the loop density; in other words, the potential falls of as the inverse distance, and the electric field as
the square distance.
The definition of the magnetic field simply follows from that of the electric field by
changing to moving frame of reference. The two field definitions are illustrated in Figure 52.
We note that the electric field is defined almost in the same way as the wave function:
both are oriented crossing densities. However, the electric field is defined with the crossing density of twisted loops, whereas the wave function is defined with the crossing density of tangles. The definitions differ only by the topology of the underlying strand structures.
In the strand model, energy, or action per time, is the number of crossing switches
per time. The electromagnetic field energy per volume is thus given by the density of
crossing switches per time that are due to twisted loops. Now, the strand model implies
that the crossing switch density per time is given by half the square of the crossing density
plus half the square of the crossing density flow. For twisted loops, we thus get that the
energy density is half the square of the electric plus half the square of the magnetic field.
Inserting the proportionality factors that lead from Planck units to SI units we get the
well-known expression
? ?0 2
1 2
? .
(147)
= ? +
?
2
2?0
Motion Mountain – The Adventure of Physics
⊳ The electric potential is the density of twisted loops.
electrodynamics a nd the first reidemeister m ov e
twisted
loop
average
loop
motion
t1
t2
small
volume
element
with
crossing
magnetic
field
B
charge
F I G U R E 52 Moving twists allow us to define electric fields – as the density of twisted loop crossings –
and magnetic fields – as the corresponding flow.
The strand model predicts limit values to all observables. They always appear when
strands are as closely packed as possible. This implies a maximum electric field value
?max = ?4 /4?? ≈ 1.9 ⋅ 1062 V/m and a maximum magnetic field value ?max = ?3 /4?? ≈
6.3 ⋅ 1053 T. All physical systems – including all astrophysical objects, such as gamma-ray
bursters or quasars – are predicted to conform to this limit. This strand model prediction
agrees with observations.
The L agrangian of the electromagnetic field
In classical electrodynamics, the energy density of the electromagnetic field is used to
deduce its Lagrangian density. The Lagrangian density describes the intrinsic, observerindependent change that occurs in a system. In addition, the Lagrangian density must
be quadratic in the fields and be a Lorentz-scalar.
A precise version of these arguments leads to the Lagrangian density of the electromagnetic field ?
?
1 2
1
LEM = 0 ?2 −
? =−
? ???
(148)
2
2?0
4?0 ??
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
velocity
Motion Mountain – The Adventure of Physics
electric
field
E
231
232
9 g auge interactions d ed uced from stra nds
Freedom of definition of the crossing phase / crossing orientation:
phase 1
phase 2
phase 3
F I G U R E 53 The definition of the phase or orientation of a single crossing is not unique: there is a
freedom of choice.
where the electromagnetic field ? is defined with the electromagnetic potential ? as
(149)
Since the strand model reproduces the electromagnetic energy, it also reproduces the
Lagrangian of classical electrodynamics. In particular, Maxwell’s equations for the electromagnetic field follow from this Lagrangian density. Maxwell’s field equations are thus
a consequence of the strand model. Obviously, this is no news, because any model that
reproduces Coulomb’s inverse square distance relation and leaves the speed of light invariant automatically contains Maxwell’s field equations.
In nature, the electromagnetic potential ? ? is not uniquely defined: one says that there
is a freedom in the choice of gauge. The change from one gauge to another is a gauge
transformation. Gauge transformations are thus transformations of the electromagnetic
potential that have no effect on observations. In particular, gauge transformations leave
unchanged all field intensities and field energies on the one hand and particle probabilities and particle energies on the other hand.
All these observations can be reproduced with strands. In the strand model, the following definitions are natural:
⊳ A gauge choice for radiation and for matter is the choice of definition of the
respective phase arrow.
⊳ A gauge transformation is a change of definition of the phase arrow.
In the case of electrodynamics, the gauge freedom is a result of allowing phase choices
that lie in a plane around the crossing orientation. (The other interactions follow from
the other possible phase choices.) The phase choice can be different at every point in
space. Changing the (local) phase definition is a (local) gauge transformation. Changing
the phase definition for a single crossing implies changing the phase of wave functions
and of the electromagnetic potentials. A schematic illustration of the choice of gauge is
given in Figure 53 and Figure 54.
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U(1) gauge invariance induced by t wists
Motion Mountain – The Adventure of Physics
??? = ∂? ? ? − ∂? ? ? .
electrodynamics a nd the first reidemeister m ov e
233
Freedom of definition for the crossing phase of tangles
Strand model :
Observation :
time average
of crossing
changes
Matter :
spin
spin
U(1) phase
definition
freedom
probability
amplitude
core
helicity
helicity
U(1) phase
definition
freedom
F I G U R E 54 The freedom in definition of the phase of crossings leads to the gauge invariance of
electrodynamics. Three exemplary choices of phase are shown.
Twists on tangle cores
form a U(1) group
axis
axis
π
The basic twist, or Reidemeister I
move, is a local rotation, by
an angle π around the axis,
of the core region enclosed
by a dashed circle.
Generalized to arbitary angles,
the basic twist generates
a U(1) group.
F I G U R E 55 How the set of generalized twists – the set of all local rotations of a single strand segment
around an axis – forms a U(1) gauge group.
We note that gauge transformations have no effect on the density or flow of crossings or crossing switches. In other words, gauge transformations leave electromagnetic
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
electromagnetic
potential
Motion Mountain – The Adventure of Physics
Photons :
234
9 g auge interactions d ed uced from stra nds
field intensities and electromagnetic field energy invariant, as observed. Similarly, gauge
transformations have no effect on the number of crossing switches of rotating tangles.
A rotation by 4π does not change the phase, independently of which definition of arrow is chosen. Therefore, gauge transformations leave probability densities – and even
observable phase differences – unchanged. This agrees with experiment.
A gauge transformation on a wave functions also implies a gauge transformation on
the electrodynamic potential. The strand model thus implies that the two transformations are connected, as is observed. This connection is called minimal coupling. In short,
minimal coupling is a consequence of the strand model.
U(1) gauge interactions induced by t wists
There is only a small step from a gauge choice to a gauge interaction. We recall:
The L agrangian of QED
Given the U(1) gauge invariance of observables, the Lagrangian of quantum electrodynamics, or QED, follows directly, because U(1) gauge invariance is equivalent to minimal
coupling. We start from the Lagrangian density L of a neutral, free, and relativistic fermion in an electromagnetic field. It is given by
L = Ψ(?ℏ?/∂ − ?2 ?)Ψ −
1
? ??? .
4?0 ??
(150)
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Challenge 155 e
In particular, electromagnetism results from the transfer of twists; twists are one of the
three types of core deformations that lead to a crossing switch.
The basic twist, or first Reidemeister move, corresponds to a local rotation of some
strand segment in the core by an angle π, as illustrated by Figure 55. Twists can be generalized to arbitrary angles: we simply define a generalized twist as a local rotation of a
strand segment by an arbitrary angle. The rotation axis is chosen as in shown by Figure 55. Generalized twists can be concatenated, and the identity twist – no local rotation
at all – also exists. Generalized twists thus form a group. Furthermore, a generalized twist
by 2π is equivalent to no twist at all, as is easily checked with a piece of rope: keeping
the centre region is it disappears by pulling the ends, in contrast to a twist by π. These
properties uniquely define the group U(1). In short, Figure 55 shows that generalized
twists define the group U(1), which has the topology of a circle.
In summary, the addition of a twist to a fermion tangle or to a photon strand changes
their phase, and thus represents a gauge interaction. We have shown that core fluctuations induced by twists produce a U(1) gauge symmetry. Electromagnetic field energy
and particle energy are U(1) invariant. In short, the strand model implies that the gauge
group of quantum electrodynamics is U(1). With this result, we are now able to deduce
the full Lagrangian of QED.
Motion Mountain – The Adventure of Physics
⊳ A gauge interaction is a change of phase resulting from a strand deformation
of the particle core.
electrodynamics a nd the first reidemeister m ov e
Strand model :
charged
fermion
Observation :
time average
of crossing
switches
photon
235
charged
fermion
t2
photon
t2
t1
charged
fermion
charged
fermion
t1
F I G U R E 56 The fundamental Feynman diagram of QED and its tangle version.
Page 220
LQED = Ψ(?ℏ?/
D − ?2 ?)Ψ −
1
? ??? .
4?0 ??
(151)
/ = ?? D? is the gauge covariant derivative that is defined through minimal coupHere, D
ling to the charge ?:
D? = ∂? − ??? ? .
(152)
Page 378
Minimal coupling implies that the Lagrangian density of QED is invariant under U(1)
gauge transformations. We will discuss the details of the charge ? later on.
We have thus recovered the Lagrangian density of quantum electrodynamics from
strands. Strands thus reproduce the most precisely tested theory of physics.
Feynman diagrams and renormalization
Feynman diagrams are abbreviations of formulas to calculate effects of quantum electrodynamics in perturbation expansion. Feynman diagrams follow from the Lagrangian of
QED. All Feynman diagrams of QED can be constructed from one fundamental diagram,
shown on the right-hand side of Figure 56. Important Feynman diagrams are shown on
the left-hand sides of Figure 57 and of Figure 58.
In the strand model, the fundamental Feynman diagram can be visualized directly
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
We deduced the fermion term in the chapter of quantum theory, and we deduced the
electromagnetic term just now, from the properties of twisted loops.
As we have seen, the strand model implies minimal coupling. This changes the Lagrangian density for a charged, i.e., interacting, relativistic fermion in the electromagnetic
field, into the Lagrangian density of QED:
Motion Mountain – The Adventure of Physics
(Only crossing switches
are observable, strands
are not.)
time
236
real
electron
9 g auge interactions d ed uced from stra nds
virtual
photon
real
electron
t2
t2
real
photon
virtual
electron
t2
t2
real
photon
t2
t2
vacuum
t1
t1
t1
real
electron
virtual
electron
time
virtual
electron
vacuum
real
photon
t1
vacuum
vacuum
t1
time
t1
virtual
electron
virtual
electron
t2
time
real
positron t
2
t2
t2
t1
t1
t1
real
electron
time
virtual
positron
t2
t2
t1
real
photon
real
electron
real
positron
virtual
electron
t2
t2
vacuum
time
real
positron t
2
t2
vacuum
t1
t1
real
photon
time
t1
t1
virtual
photon
time
t1
t1
virtual
photon
vacuum
time
F I G U R E 57 The different variations of the fundamental Feynman diagram of QED and their tangle
versions.
Page 314
in terms of strands, as shown on the left-hand side of Figure 56. This is the same diagram that we have explored right at the start of the section on electrodynamics, when we
defined electrodynamics as twist exchange. (The precise tangles for the charged fermions will be deduced later on.) Since all possible Feynman diagrams are constructed from
the fundamental diagram, the strand model allows us to interpret all possible Feynman
diagrams as strand diagrams. For example, the strand model implies that the vacuum is
full of virtual particle-antiparticle pairs, as shown in Figure 58.
In quantum field theory, Lagrangians must not only be Lorentz and gauge invariant, but must also be renormalizable. The strand model makes several statements on
this issue. At this point, we focus on QED only; the other gauge interactions will be
treated below. The strand model reproduces the QED Lagrangian, which is renormalizable. Renormalizability is a natural consequence of the strand model in the limit that
strand diameters are negligible. The reason for renormalizability that the strand model
reproduces the single, fundamental Feynman diagram of QED, without allowing other
types of diagrams.
The twist deformations underlying the strand model for QED also suggest new ways
to calculate higher order Feynman diagrams. Such ways are useful in calculations of ?factors of charged particles, as shown in the next section. In particular, the strand model
for QED, as shown in Figure 56, implies that higher order QED diagrams are simple strand
deformations of lower order diagrams. Taking statistical averages of strand deformations
up to a given number of crossings thus allows us to calculate QED effects up to a given
order in the coupling. The strand model thus suggests that non-perturbative calculations
are possible in QED. However, we do not pursue this topic in the present text.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 226
Motion Mountain – The Adventure of Physics
virtual
electron
vacuum
electrodynamics a nd the first reidemeister m ov e
237
Virtual pair creation:
t3
t3
t2
t2
electron
positron
electron
positron
t1
time
t1
Motion Mountain – The Adventure of Physics
Electron-positron annihilation:
t3
t3
t2
t2
t1
positron
t1
time
electron
positron
F I G U R E 58 Some Feynman diagrams of QED with their tangle versions.
For precise non-perturbative calculations, the effective diameter of the strands must
be taken into account. The diameter eliminates the Landau pole and all ultraviolet divergences of QED. In the strand model, the vacuum energy of the electromagnetic field
is automatically zero. In other words, the strand model eliminates all problems of QED;
in fact, QED appears as an approximation of the strand model for negligible strand diameter. In passing, we thus predict that perturbation theory for QED is valid and converges
if the strand model, and in particular the finite strand diameter, is taken into account.
(The diameter is the only gravitational influence predicted to affect QED.)
The strand model also suggests that the difference between renormalized and unrenormalized mass and charge is related to the difference between minimal and nonminimal crossing switch number, or equivalently, between tangle deformations with few
and with many crossings, where strands are deformed on smaller distance scales. In other
terms, unrenormalized quantities – the so-called bare quantities at Planck energy – can
be imagined as those deduced when the tangles are pulled tight, i.e., pulled to Planck
distances, whereas renormalized mass and charge values are those deduced for particles
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
electron
238
Page 382
9 g auge interactions d ed uced from stra nds
surrounded by many large-size fluctuations.
The strand model also suggests a visualization for the cut-off used in QED. The cutoff is a characteristic energy or length used in intermediate calculations. In the strand
model, the cut-off corresponds to the size of the image.
In summary, the strand model provides a new underlying picture or mechanism for
Feynman diagrams. The strand model does not change any physical result at any experimentally accessible energy scale. In particular, the measured change or ‘running’ with
energy of the fine structure constant and of the masses of charged particles are reproduced by the strand model, because Feynman diagrams of all orders are reproduced up
to energies just below the Planck scale. Deviations between QED and the strand model
are only expected near the Planck energy, when tangles of Planck diameter are pulled
tight.
The anomalous magnetic moment
Page 224
Here, ? is the magnetic moment and ? is the intrinsic angular momentum, or spin.
The mechanical or geometric rotation frequency is related to the ratio of the intrinsic
angular momentum ? and the mass ?. Using the definitions from classical physics, we
have ?/? = ? × ?. The magnetic rotation frequency is related to the ratio of the magnetic
moment ? and the electric charge ?. Classically, this ratio is ?/? = ? × ?. Therefore, in
classical physics – and also in the first order of the Pauli–Dirac description of the electron
– the two rotation frequencies coincide, and the factor ?/2 is thus equal to 1. However, as
mentioned, both experiment and QED show a slight deviation of ?/2 from unity, called
the anomalous magnetic moment.
In the strand model, the geometric or mechanical rotation of a charged elementary
particle is due to the rotation of the tangle core as a rigid whole, whereas the magnetic
rotation also includes phase changes due to the deformations of the tangle core. In particular, the magnetic rotation of a charged elementary particle includes phase changes
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
where ?/2 is half the so-called g-factor, with a measured value of 1.00116(1), and ? is
the fine structure constant, with a measured value of 1/137.036(1). Julian Schwinger
discovered this expression in 1948; the involved calculations that led Schwinger to this
and similar results in quantum field theory earned him the 1965 Nobel Prize in Physics.
The result is also inscribed on the memorial marker near his grave in Mount Auburn
Cemetery. The strand model proposes an intuitive explanation for this result.
Generally speaking, the factor ?/2 describes the ratio between the ‘mechanical’ or
‘geometric’ rotation frequency – the rotation of the particle mass that leads to spin –
and ‘magnetic’ rotation frequency – the rotation of the particle charge that leads to the
magnetic moment. More precisely, the definition of the g-factor of a particle with charge
? and mass ? is
?
?/?
=
.
(154)
2 ?/?
Motion Mountain – The Adventure of Physics
The anomalous magnetic moment ? of the electron and of the muon is given by the
well-known expression
?
?
(153)
=1+
− ?(?2 ) ,
2
2π
electrodynamics a nd the first reidemeister m ov e
239
The propagating electron
Conjecture 1
electrodynamics and the strand model for a propagating free electron. The lower strand model
configurations are shown for a single instant – marked in magenta – of the electron propagator drawn
above them. (For simplicity, the external field is not drawn.) In the first conjecture, the loops of the belt
trick are conjectured to correspond to the virtual photons in the propagator and to be responsible for
the anomalous magnetic moment. In the second conjecture, the deformations of the core correspond
to the virtual photons.
due to emission and reabsorption of virtual photons, i.e., of twisted loops.
In nature, the probability of the emission and reabsorption of a photon is determined by the fine structure constant ?. The emission and reabsorption process leads to an
additional angle that makes the ‘magnetic’ rotation angle differ from the ‘mechanical’
rotation angle. Since the fine structure constant describes the rotation of the phase due
to virtual photon exchange, the emission and reabsorption of a virtual photon leads to
an angle difference, and this angle difference is given by the fine structure constant itself.
The ratio between the purely mechanical or geometric and the full magnetic rotation
frequency is therefore not one, but increased by the ratio between the additional angle ?
and 2π. This is Schwinger’s formula.
In short, the strand model reproduces Schwinger’s celebrated formula for the an-
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
F I G U R E 59 Two conjectured correspondences between the Feynman diagrams of quantum
Motion Mountain – The Adventure of Physics
Conjecture 2
240
Challenge 156 e
omalous magnetic moment almost from thin air. The strand model also implies that
Schwinger’s formula is valid for all charged elementary particles, independently of their
mass; this is indeed observed. Higher order corrections also appear naturally in the
strand model. Finally, the strand model implies that the complete expression, with all
orders included, converges, because the full result is due to the shape and dynamics of
the tangle core. The discussions about the existence of the perturbation limit in QED are
thus laid to rest.
If we look into the details, it might be that the belt trick itself is at the origin of the
anomalous magnetic moment. A conjecture for this connection is proposed and illustrated in Figure 59: if the two loops formed by the belt trick are seen as virtual photons,
the factor 2?/4π arises naturally. So do the higher-order terms. This explanation would
relate the belt trick directly to the additional magnetic rotation angle. However, it might
also be that this correspondence of the strand images in the figure to the upper diagrams
is not fully correct. The topic is subject of research.
A second conjecture is also given in Figure 59. The virtual photons could correspond
to deformations of the tangle core. This conjecture is more in line with the distinction
between gravity and gauge interactions given above, where it was stated that gravity is
due to tail deformations and gauge interactions are due to core deformations. This conjecture is more in line with the distinction between a geometric and a magnetic rotation:
the geometric rotation would be due to the rigid rotation of the tangle core, and the
magnetic rotation would be due to an additional effect due to core deformation.
Both conjectures on the origin of the g-factor imply that 1 < ?/2 < 2; in fact, we can
even argue, using ? < 1, that the strand model implies
Page 389
1
.
2π
(155)
This is not a new result; it is already implied by ordinary quantum field theory. However,
the strand description of particle rotation suggests a way to calculate the g-factor and the
fine structure constant. We will explore this below.
Max well ’ s equations
The strand model also allows us to check Maxwell’s field equations of classical electrodynamics directly. The equations are:
?
,
?0
∇B =0 ,
∂B
∇×E=−
,
∂?
1 ∂E
∇×B = 2
+ ?0 J .
? ∂?
∇E=
(156)
The first of these equations is satisfied whatever the precise mechanism at the basis
of twisted loop emission by electric charges may be. Indeed, any mechanism in which a
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
1 < ?/2 < 1 +
Motion Mountain – The Adventure of Physics
Page 224
9 g auge interactions d ed uced from stra nds
electrodynamics a nd the first reidemeister m ov e
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 231
charge randomly sends out or swallows a twisted handle yields a 1/?2 dependence for the
electrostatic field and the required connection between charge and the divergence of the
electric field. This is not a deep result: any spherically-symmetric system that randomly
emits or swallows some entity produces the equation, including the underlying inversesquare dependence. The result can also be confirmed in another, well-known way. In any
exchange interaction between two charges, the exchange time is proportional to their
distance apart ?; in addition, quantum theory states that the exchanged momentum is
inversely proportional to the distance ?. Therefore, the force, or momentum per unit
time, varies as 1/?2 . This relation is valid independently of the underlying motion of the
twisted loops, because space has three dimensions: all localized sources automatically
fulfil the inverse square dependence.
The constant on the right-hand side of the first equation results from the definition
of the units; in the language of the strand model, the constant fixes the twisted loop
emission rate for an elementary charge.
The second of the field equations (156) expresses the lack of magnetic charges. This
equation is automatically fulfilled by the strand model, as the definition of the magnetic
field with strands does not admit any magnetic sources. In fact, strands suggest that no
localized entity can have a magnetic charge. Also this equation is valid independently of
the details of the motion of the strands. Again, this is a topological effect.
The third field equation relates the temporal change of the magnetic field to the curl
of the electric field. In the strand model, this is satisfied naturally, because a curl in
the electric field implies, by construction, a change of the magnetic field, as shown by
Figure 52. Again, this relation is valid independently of the details of the motion of the
strands, as long as the averaging scale is taken to be large enough to allow the definition
of electric and the magnetic fields.
The most interesting equation is the last of the four Maxwell equations (156): in particular, the second term on the right-hand side, the dependence on the charge current.
In the description of electrodynamics, the charge current J appears with a positive sign
and with no numerical factor. (This is in contrast to linearized gravity, where the current
has a numerical factor and a negative sign.) The positive sign means that a larger current
produces a larger magnetic field. The strand model reproduces this factor: strands lead
to an effect that is proportional both to charge (because more elementary charges produce more crossing flows) and to speed of movement of charge (large charge speed lead
to larger flows). Because of this result, the classical photon spin, which is defined as ?/?,
and which determines the numerical factor, namely 1, that appears before the charge current J, is recovered. Also this connection is obviously independent of the precise motion
of the underlying strands.
The first term on the right-hand side of the fourth equation, representing the connection between a changing electric field and the curl of the magnetic field, is automatically
in agreement with the model. This can again be checked from Figure 52 – and again,
this is a topological effect, valid for any underlying strand fluctuation. As an example,
when a capacitor is charged, a compass needle between the plates is deflected. In the
strand model, the accumulating charges on the plates lead to a magnetic field. The last
of Maxwell’s equations is thus also confirmed by the strand model.
In summary, the strand model reproduces Maxwell’s equations. However, this is not
a great feat. Maxwell-like equations appear in many places in field theory, for example
Motion Mountain – The Adventure of Physics
Page 231
241
242
9 g auge interactions d ed uced from stra nds
in solid-state physics and hydrodynamics. Mathematical physicists are so used to the
appearance of Maxwell-like equations in other domains that they seldom pay it much
attention. The real tests for any model of electrodynamics, quantum or classical, are the
deviations that the model predicts from electrodynamics, especially at high energies.
Curiosities and fun challenges abou t QED
Challenge 157 e
Can you show that the calculation of the vacuum energy density of an infinity flat vacuum, when using strands, yields exactly zero, as expected?
∗∗
Challenge 158 e
Can you confirm that the strand model of quantum electrodynamics does not violate
charge conjugation C nor parity P at any energy?
∗∗
Can you confirm that the strand model of quantum electrodynamics conserves colour
and weak charge at all energies, using the results of the next sections?
∗∗
Challenge 160 e
Can you determine whether the U(1) gauge group deduced here is that of electrodynamics or that of weak hypercharge?
∗∗
Challenge 161 d
Can you find a measurable deviation of the strand model from QED?
Page 162
Ref. 182
In the strand model, photons are single, helically twisted strands, randomly exchanged
between charges; charges are chiral tangles, and therefore they effectively emit and absorb real and virtual photons. This is the complete description of QED using strands.
In particular, we have shown that Reidemeister I moves – or twists – of tangle cores
lead to U(1) gauge invariance, Coulomb’s inverse square relation, Maxwell’s equations of
electrodynamics and to Feynman diagrams. In short, we have deduced all experimental
properties of quantum electrodynamics, except one: the strength of the coupling. Despite this open point, we have settled one line of the millennium list of open issues: we
know the origin of the electromagnetic interaction and of its properties.
Is there a difference between the strand model and quantum electrodynamics? The
precise answer is: there are no measurable differences between the strand model and QED.
For example, the ?-factor of the electron or the muon predicted by QED is not changed
by the strand model. The U(1) gauge symmetry and the whole of QED remain valid at all
energies. There are no magnetic charges. There are no other gauge groups. QED remains
exact in all cases – as long as gravity plays no role.
The strand model prediction of a lack of larger gauge symmetries is disconcerting.
There is thus no grand unification in nature; there is no general gauge group in nature,
be it SU(5), SO(10), E6, E7, E8, SO(32) or any other. This result indirectly also rules out
supersymmetry and supergravity. This unpopular result contrasts with many cherished
habits of thought.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Summary on QED and experimental predictions
Motion Mountain – The Adventure of Physics
Challenge 159 e
electrodynamics a nd the first reidemeister m ov e
243
Motion Mountain – The Adventure of Physics
In the strand model, the equivalence of Feynman diagrams and strand diagrams implies that deviations of the strand model from QED are expected only when gravity starts
to play a role. The strand model predicts that this will only happen just near the Planck
energy √ℏ?5/4? . At lower energies, QED is predicted to remain valid.
The strand model also confirms that the combination of gravity and quantum theory
turns all Planck units into limit values, because there is a maximum density of strand
crossings in nature, due to the fundamental principle. In particular, the strand model
confirms the maximum electric field value ?max = ?4 /4?? ≈ 1.9 ⋅ 1062 V/m and a maximum magnetic field value ?max = ?3 /4?? ≈ 6.3 ⋅ 1053 T. So far, these predictions are not
in contrast with observations.
Thus the strand model predicts that approaching the electric or magnetic field limit
values – given by quantum gravity – is the only option to observe deviations from QED.
But measurements are not possible in those domains. Therefore we can state that there
are no measurable differences between the strand model and QED.
Our exploration of QED has left open only two points: the calculation of the electromagnetic coupling constant and the determination of the spectrum of possible tangles for
the elementary particles. Before we clarify these points, we look at the next Reidemeister
move.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
244
9 g auge interactions d ed uced from stra nds
Reidemeister move II, or poke, in textbook form :
The basic poke
can be described
as a local rotation by π.
A full rotation, from -π to π,
produces crossing switches.
A poke transfer :
weak boson of
unbroken SU(2)
poke
transfer
fermion
with
different
phase
F I G U R E 60 Poke transfer is the basis of the weak interaction in the strand model. No strand is cut or
reglued; the transfer occurs only through the excluded volume due to the impenetrability of strands.
Page 253
In nature, the weak interaction is the result of the absorption and the emission of massive
spin-1 bosons that form a broken weak triplet. The W and the Z bosons are emitted or
absorbed by particles with weak charge; these are the left-handed fermions and righthanded antifermions. In other words, the weak interaction breaks parity P maximally. The W boson has unit electric charge, the Z boson has vanishing electric charge.
The emission or absorption of W bosons changes the particle type of the involved fermion. The weak bosons also interact among themselves. All weakly charged particles
are massive and move slower than light. The Lagrangian of matter coupled to the weak
field has a broken SU(2) gauge symmetry. There are fundamental Feynman diagrams
with triple and with quartic vertices. The weak coupling constant is determined by the
electromagnetic coupling constant and the weak boson masses; its energy dependence is
fixed by renormalization. The Higgs boson ensures full consistency of the quantum field
theory of the weak interaction.
The previous paragraph summarizes the main observations about the weak interaction. More precisely, all observations related to the weak interaction are described by
its Lagrangian. Therefore, we need to check whether the weak interaction Lagrangian
follows from the strand model.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
the weak nuclear interaction and the second
reidemeist er move
Motion Mountain – The Adventure of Physics
fermion
vacuum
the w ea k nuclea r interaction a nd the second reidemeister m ov e
Pokes on tangle cores
form an SU(2) group
245
The poke, or Reidemeister II move,
is a local rotation, by an angle π,
of the core region enclosed by a
dashed circle.
τ
The three basic pokes on tangle cores are
local rotations by an angle π around the three
coordinate axes of the core region (enclosed by
a dashed circle). The infinitesimal versions of
the three basic pokes generate an
SU(2) group. The SU(2) group appears
most clearly when the analogy to the
belt trick is highlighted.
axis
τx
τx
axis
τy
τy
π
τz
τz
π
F I G U R E 61 How the set of all pokes – the set of all deformations induced on tangle cores by the weak
interaction – forms an SU(2) gauge group: the three pokes lead to the belt trick, illustrated here with a
pointed buckle and two belts. For clarity, deformations of two strands are shown, instead of the
deformation of a single strand.
Strands, p okes and SU(2)
Page 222
As explained above, any gauge interaction involving a fermion is a deformation of the
tangle core that changes the phase and rotation of the fermion tangle. We start directly
with the main definition.
⊳ The weak interaction is the transfer of a poke, i.e., the transfer of a Reidemeister II move, between two particles. An illustration is given in Figure 60. Strands are not cut in this process; they simply transfer the deformation as a result of their impenetrability.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
axis
Motion Mountain – The Adventure of Physics
π
246
Challenge 162 e
Challenge 163 e
Page 189
9 g auge interactions d ed uced from stra nds
⋅
??
??
??
?? −1
??? −???
?? −??? −1
???
?? ??? −??? −1
(157)
Weak charge and parit y viol ation
A particle has weak charge if, when subject to many random pokes, a non-zero average
phase change occurs. Surrounded by a bath of strands that continuously induce random
pokes, not all tangles will change their phase on a long-time average: only tangles that
lack symmetry will. One symmetry that must be lacking is spherical symmetry. Therefore, only tangles whose cores lack spherical symmetry have the chance to be influenced
by random pokes. Since all tangles, independently of their core details, lack spherical
symmetry, all such tangles, i.e., all massive particles, are candidates to be influenced, and
thus are candidates for weakly charged particles. We therefore explore them in detail
now.
If a tangle is made of two or more linked strands, it represents a massive spin-1/2
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
In other terms, the three basic pokes – and in particular also their infinitesimal versions –
behave like the generators of an SU(2) group. Because pokes can be seen as local rotations
of a buckle region, they can be generalized to arbitrary angles. Such arbitrary pokes can
be concatenated. We thus find that arbitrary pokes form a full SU(2) group. This is the
reason for their equivalence with the belt trick.
The different gauge choices for a particle are not illustrated in Figure 61. The gauge
choices arise from the different ways in which the basic pokes ?? , ?? and ?? can be assigned to the set of deformations that describe the belt trick.
In summary, we can state that in any definition of the phase of a tangled fermion core,
there is an SU(2) gauge freedom; in addition, there exists an interaction with SU(2) gauge
symmetry. In other words, the strand model implies, through the second Reidemeister
move, the existence of the unbroken weak interaction with a gauge group SU(2).
Motion Mountain – The Adventure of Physics
Strands describe the weak interaction as exchange of pokes. In tangle cores, the basic
pokes induce local rotations by an angle π, as shown in Figure 61: each basic poke rotates
the region enclosed by the dotted circle. A full poke produces two crossings. There are
three, linearly independent, basic pokes, in three mutually orthogonal directions. The
three basic pokes ?? , ?? and ?? act on the local region in the same way as the three possible
mutually orthogonal rotations act on a belt buckle. For completeness, we note that the
following arguments do not depend on whether the two strands involved in a poke are
parallel, orthogonal, or at a general angle. The following arguments also do not depend
on whether the pokes are represented by deforming two strands or only one strand. Both
cases lead to crossing switches, for each possible poke type.
Figure 61 illustrates that the product of two different basic pokes gives the third basic
poke, together with a sign – which depends on whether the sequence is cyclic or not –
and a factor of ?. Using the definition of −1 as a local rotation of the buckle region by
2π, we also find that the square of each basic poke is −1. In detail, we can read off the
following multiplication table for the three basic pokes:
the w ea k nuclea r interaction a nd the second reidemeister m ov e
247
Pokes on cores generate an SU(2) group, like the belt trick does :
x
z
y
The three basic pokes
– shown here applied to
a tangle core – define, when
reduced to infinitesimal
angles, the 3 generators
of SU(2).
Random pokes affect only tangles of identical spin and handedness :
poke affects a
tangle and a
tangle differently.
F I G U R E 62 The three basic pokes and weak charge in the strand model.
Page 296
⊳ Non-vanishing weak charge for fermions appears only for tangle cores
whose handedness leads to average poke effects.
In other words, the strand model predicts that random pokes will only affect a core if
the core handedness and the randomly applied belt trick are of the same handedness. In
physical terms, random pokes will only affect left-handed particles or right-handed antiparticles. Thus, the strand model predicts that the weak interaction violates parity maximally, This is exactly as observed. In other terms, weak charge and the parity violation
of the weak interaction are consequences of the belt trick. This relation is summarized
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 217
particle (except for a simple twist, which represents the graviton). All such fermion cores
lack spherical and cylindrical symmetry. When a fermion spins, two things happen:
the core rotates and the belt trick occurs, which untangles the tails. Compared to the
direction of motion, the rotation and the untangling can be either left-handed or righthanded.
Every poke is a shape transformation of the core with a preferred handedness. The
chirality is of importance in the following.
A particle has weak charge if random pokes lead to a long-time phase change. In order
to feel any average effect when large numbers of random pokes are applied, a core must
undergo different effects for a poke and its reverse. As already mentioned, this requires
a lack of core symmetry. Whenever the core has no symmetry, non-compensating phase
effects will occur: if the core rotation with its tail untangling and the poke are of the
same handedness, the phase will increase, whereas for opposite handedness, the phase
will decrease a bit less.
Motion Mountain – The Adventure of Physics
A
248
9 g auge interactions d ed uced from stra nds
Weak bosons
⊳ Weak intermediate bosons are described by double strands. An illustration
is given in Figure 63.
Single strands that induce phase changes in fermions interacting weakly are shown on
the left side of Figure 63. They correspond to the three basic pokes ?? , ?? and ?? .
We note two additional points. First of all, the (unbroken) spin-1 bosons could also
be described by the motion of a single strand in a strand group. This makes them spin 1
particles.
Furthermore, unknotted tangles are massless. In the strand model, tangles that induce
pokes differ from the massive weak intermediate bosons, shown on the right of Figure 63.
This difference is due to the breaking of the SU(2) gauge symmetry, as we will find out
soon.
* Non-Abelian gauge theory was introduced by Wolfgang Pauli. In the 1950s, he explained the theory in
series of talks. Two physicists, Yang Chen Ning and Robert Mills, then wrote down his ideas. Yang later
received the Nobel Prize in Physics with Lee Tsung Dao for a different topic, namely for the violation of
parity of the weak interaction.
** This reworked strand model of the W and Z bosons arose in 2015.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Gauge bosons are those particles that are exchanged between interacting fermions: gauge
bosons induce phase changes of fermions. This implies that the (unbroken) weak bosons
are the particles** that induce the three poke moves:
Motion Mountain – The Adventure of Physics
in Figure 62.
If an elementary particle is described by a two tangled strands, we expect it to be influenced by average pokes. Such tangle cores are spin-1 bosons; their cores lack spherical
and cylindrical symmetry. The core rotation will induce a left-right asymmetry that will
lead to a higher effect of a poke than of its reverse. Two-stranded particles are thus predicted to carry weak charge. We therefore expect that quarks – to be explored below –
and the weak bosons themselves interact weakly.
Because the weak bosons interact weakly, the strand model implies that the weak interaction is a non-Abelian gauge theory, as is observed.*
If a tangle is made of a single unknotted strand, it is not affected by random pokes.
The strand model thus predicts that the photon has no weak charge, as is observed. The
same also holds for gluons.
The strand definition of weak charge leads to two conclusions that can be checked
by experiment. First, all electrically charged particles – having cores that are chiral and
thus lack cylindrical symmetry – are predicted to be weakly charged. Secondly, in the
strand model, only massive particles interact weakly; in fact, all massive particles interact weakly, because their cores lack cylindrical symmetry. In other words, all weakly
charged particles move more slowly than light and vice versa. Both conclusions agree
with observation.
In summary, all properties of weak charge found in nature are reproduced by the
tangle model.
the w ea k nuclea r interaction a nd the second reidemeister m ov e
249
Boson mass generation and SU(2) breaking :
weak bosons of
unbroken SU(2)
photon
a
+
b
Z boson
candidate
(massive)
c
Z0
photon
e
W0
B
A
+
Wx
W boson
candidate
(massive)
h
W–
F I G U R E 63 Poke-inducing strand motions (left) become massive weak vector tangles (right) through
symmetry breaking and tail braiding. Tail braiding is related to the Higgs boson, whose tangle model
will be clarified later on.
The L agrangian of the unbroken SU(2) gauge interaction
The energy of the weak field is given by the density of weak gauge boson strands. As
long as the SU(2) symmetry is not broken, the energy of the weak field and the energy of
fermions are both SU(2) invariant. As a consequence, we are now able to deduce a large
part of the Lagrangian of the weak interaction, namely the Lagrangian for the case that
the SU(2) symmetry is unbroken.
As long as SU(2) is unbroken, the vector bosons are described as unknotted tangles
that induce pokes, as shown on the left of Figure 63. There are three such bosons. Since
they can be described by a single strand that moves, they have spin 1; since they are
unknotted, they have zero mass and electric charge.
Energy is the number of crossing switches per time. As long as SU(2) is unbroken
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
g
B
bosons of U(1) and SU(2)
after symmetry breaking
Motion Mountain – The Adventure of Physics
–
f
+
W0
B
d
Higgs
250
Page 231
9 g auge interactions d ed uced from stra nds
and the weak bosons are massless, the energy of the weak boson field and thus their
Lagrangian density is given by the same expression as the energy of the photon field. In
particular, the strand model implies that energy density is quadratic in the field intensities. We only have to add the energies of all three bosons together to get:
1 3
L = − ∑ ?? ?? ?? ?? ,
4 ?=1
(158)
Lunbroken weak = ∑ Ψ? (?ℏ?/
D − ?? ?2 )Ψ? −
?
1 3
∑ ?? ?? ?? ?? ,
4 ?=1
(159)
SU(2) breaking
Page 255
In nature, the weak interaction does not have an SU(2) gauge symmetry. The symmetry
is only approximate; is said to be broken. The main effect of SU(2) symmetry breaking are
the non-vanishing – and different – masses for the W and Z bosons, and thus the weakness and the short range of the weak interaction. In addition, the symmetry breaking
implies a mixing of the weak and the electromagnetic interaction: it yields the so-called
electroweak interaction. This mixing is often called electroweak ‘unification’.
The strand model suggests the following description:
⊳ Mass generation for bosons and the related SU(2) symmetry breaking are
due to tail braiding at the border of space. Figure 63 illustrates the idea.
In this description, tail braiding* is assumed to occur at a distance outside the domain
of observation; in that region – which can be also the border of physical space – tail
braiding is not forbidden and can occur. The probability of tail braiding is low, because
* In the original strand model of the weak bosons, from the year 2008, the role of tail braiding was taken
by strand overcrossing.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
/ is now the SU(2) gauge covariant derivative and the first sum is taken over
where D
all fermions. In this Lagrangian, only the left-handed fermions and the right-handed
antifermions carry weak charge. This Lagrangian, however, does not describe nature:
the observed SU(2) breaking is missing.
Motion Mountain – The Adventure of Physics
This expression is SU(2) gauge invariant. Indeed, SU(2) gauge transformations have no
effect on the number of crossing switches due to weak bosons or to the motion of pokes.
Thus, gauge transformations leave weak field intensities and thus also the energy of the
weak fields invariant, as observed.
We can now write down the Lagrangian for weakly charged fermions interacting with
the weak vector bosons. Starting from the idea that tangle core deformations lead to
phase redefinitions, we have found that pokes imply that the unbroken weak Lagrangian
density for matter and radiation fields is SU(2) gauge invariant. In parallel to electrodynamics we thus get the Lagrangian
the w ea k nuclea r interaction a nd the second reidemeister m ov e
Page 356
Page 255
Page 254
Page 356
Page 358
Open issue: are the W and Z tangles correct?
In 2014, Sergei Fadeev raised an issue: A tangle version of the W and Z that does not
contain any knot and does not require an actual strand overcrossing process at spatial
infinity, the strand model would gain in simplicity and elegance. Thinking about the
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 319
the crossings have first to fluctuate to that distance and then fluctuate back. Nevertheless,
the process of tail braiding can take place.
Tail braiding appears only in the weak interaction. It does not appear in the other
two gauge interactions, as the other Reidemeister moves are not affected by processes at
the border of space. In the strand model, this is the reason that only SU(2) is broken
in nature. In short, SU(2) breaking is a natural consequence of the second Reidemeister
move.
Tail braiding transforms the unbraided, and thus massless, poke strands into the
braided, and thus massive W and Z strands. Tail braiding leads to particle cores: therefore is a mass-generating process. The precise mass values that it generates will be determined below. The strand model thus confirms that mass generation is related to the
breaking of the weak interaction.
Tail braiding mixes the W0 with the ‘original’ photon. This is shown in Figure 63. The
mixing is due to the topological similarities of the strand models of the two particles. The
resulting Z boson is achiral, and thus electrically neutral, as observed. We note that the
existence of a neutral, massive Z boson implies that elastic neutrino scattering in matter
occurs in nature, as was observed for the first time in 1974. Since any electrically charged
particle also has weak charge, the existence of a Z boson implies that any two electrically
charged particles can interact both by exchange of photons and by exchange of Z bosons.
In other words, SU(2) breaking implies electroweak mixing, or, as is it usually called,
electroweak ‘unification’.
Tail braiding takes place in several weak interaction processes, as shown in Figure 66.
Tail braiding thus can change particle topology, and thus particle type. The strand model
thus predicts that the weak interaction changes particle flavours (types), as is observed.
In fact, the strand model also predicts that only the weak interaction has this property.
This is also observed.
On the other hand, strands are never cut or glued back together in the strand model,
not even in the weak interaction. As a result, the strand model predicts that the weak interaction conserves electric charge, spin and, as we will see below, colour charge, baryon
number and lepton number. All this is observed.
Tail braiding also implies that the tangles for the Z boson and for the W boson shown
above are only the simplest tangles associated with each boson; more complicated tangles
are higher order propagating states of the same basic open knots. This will be of great
importance later on, for the proof that all gauge bosons of nature are already known
today.
In summary, the second Reidemeister move leads to tail braiding; tail braiding leads
to the observed properties of SU(2) symmetry breaking. (Equivalently, the strand model
implies that the simplest tangles of the weak interaction bosons show SU(2) symmetry,
whereas the more complicated, massive tangles break this symmetry.) The value of the
mixing angle and the particle masses have still to be determined. This will be done below.
Motion Mountain – The Adventure of Physics
Page 248
251
252
9 g auge interactions d ed uced from stra nds
Weak bosons of
unbroken SU(2)
Weak bosons of
broken SU(2)
Wx
Wy
W0
W+
W–
Z0
2008 (on the right side), now seen to be incorrect.
Challenge 164 ny
The electroweak L agrangian
Page 329
We can now use the results on SU(2) symmetry breaking to deduce the electroweak Lagrangian density. We have seen that symmetry breaking leaves the photon massless but
introduces masses to the weak vector bosons, as shown in Figure 63. The non-vanishing
boson masses ?? and ?? add kinetic terms for the corresponding fields in the Lagrangian.
Due to the symmetry breaking induced by tail braiding, the Z boson results from
the mixing with the (unbroken) photon. The strand model predicts that the mixing can
be described by an angle, the so-called weak mixing angle ?w . In particular, the strand
model implies that cos ?w = ?? /?? .
As soon as symmetry breaking is described by a mixing angle due to tail braiding,
we get the known electroweak Lagrangian, though at first without the terms due to the
Higgs boson. (We will come back to the Higgs boson later on.) We do not write down
the Lagrangian of the weak interaction predicted by the strand model, but the terms are
the same as those found in the standard model of elementary particles. There is one
important difference: the Lagrangian so derived does not yet contain quark and lepton
mixing. Indeed, experiments show that the weak fermion eigenstates are not the same as
the strong or electromagnetic eigenstates: quarks mix, and so do neutrinos. The reason
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 249
issue, it became clear that such a tangle could occur when vacuum strands were included,
as shown above.
In contrast, in 2008, in the first version of the strand model, the W boson after symmetry breaking was thought to be an open overhand knot, and the Z boson an open
figure-eight knot.
It might well be that the new, 2015/2016 strand models for the two intermediate vector
bosons, shown in Figure 63 are still not correct. The possibility remains intriguing and
a definitive issue still needs to be found.
Motion Mountain – The Adventure of Physics
F I G U R E 64 The supposed models for the massive weak gauge bosons after symmetry breaking, from
the w ea k nuclea r interaction a nd the second reidemeister m ov e
Neutral current processes
253
Charged current processes
quark 2 or
lepton 2
quark 1 or
lepton 1
Z
W
quark 1 or
lepton 1
quark 1 or
lepton 1
Triple boson coupling processes
W
Quartic coupling processes
W
Z, γ
W
W
W
Z, γ
W
W
Z, γ
F I G U R E 65 The fundamental Feynman diagrams of the weak interaction that do not involve the Higgs
boson.
for this observation, and the effect that mixing has on the weak Lagrangian, will become
clear once we have determined the tangles for each fermion.
In summary, the strand model implies the largest part of the Lagrangian of the weak
interaction. The issue of the Higgs boson is still open, and the electroweak Lagrangian
contains a number of constants that are not yet clarified. These unexplained constants are
the number of the involved elementary particles, their masses, couplings, mixing angles
and CP violation phases, as well as the value of the weak mixing angle.
The weak Feynman diagrams
Challenge 165 e
Page 382
In nature, the weak interaction is described by a small number of fundamental Feynman
diagrams. Those not containing the Higgs boson are shown in Figure 65. These Feynman
diagrams encode the corresponding Lagrangian of the weak interaction.
In the strand model, pokes lead naturally to strand versions of the fundamental Feynman diagrams. This happens as shown in Figure 66. We see again that the strand model
reproduces the weak interaction: each Feynman diagram is due to a strand diagram for
which only crossing switches are considered, and for which Planck size is approximated
as zero size. In particular, the strand model does not allow any other fundamental diagrams for the weak interaction.
The finite and small number of possible strand diagrams and thus of Feynman diagrams implies that the weak interaction is renormalizable. For example, the change or
‘running’ of the weak coupling with energy is reproduced by the strand model, because
the running can be determined through the appropriate Feynman diagrams.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 372
Motion Mountain – The Adventure of Physics
W
254
9 g auge interactions d ed uced from stra nds
Weak strand diagrams
electron
Weak Feynman diagrams
Z boson
time average of
crossing switches,
at lower magnification
t2
electron
Z
t2
vacuum
t1
t1
electron
time
W
neutrino
t2
W
t2
vacuum
W
electron
time
W
W
t2
W
t2
vacuum
γ
(photon)
W
W
t1
t1
γ
(photon)
time
W
t2
W
t2
t1
t1
Z
Z
time
Z
Z
F I G U R E 66 The strand model for the fundamental Feynman diagrams of the weak interaction. The
tangles for the fermions are introduced later on.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
electron
t1
t1
Motion Mountain – The Adventure of Physics
neutrino
electron
the w ea k nuclea r interaction a nd the second reidemeister m ov e
255
Fun challenges and curiosities abou t the weak interaction
Challenge 166 e
The W boson and its antiparticle are observed to annihilate through the electromagnetic
interaction, yielding two or more photons. The tangle model of the weak bosons has a
lot of advantages compared to the knot model: The annihilation is much easier to understand.
∗∗
The strand model, like the standard model of particle physics, predicts that everything
about the weak interaction is already known. Nevertheless, the most important weak
process, the decay of the neutron, is being explored by many precision experiments. The
strand model predicts that none of these experiments will yield any surprise.
∗∗
Page 329
We have deduced the main properties of the weak Lagrangian from the strand model.
We have shown that Reidemeister II moves – or pokes – in tangle cores lead to a broken
SU(2) gauge group and to massive weak bosons. We found that the deviation from
tangle core sphericity plus chirality is weak charge, and that the weak interaction is nonAbelian. We have also shown that the weak interaction naturally breaks parity maximally
and mixes with the electromagnetic interaction. In short, we have deduced the main experimental properties of the weak interaction.
Is there a difference between the strand model and the electroweak Lagrangian of the
standard model of particle physics? Before we can fully answer the question on deviations between the strand model and the standard model, we must settle the issue of the
Higgs boson. This is done later on.
In any case, the strand model predicts that the broken SU(2) gauge symmetry remains
valid at all energies. No other gauge groups appear in nature. The strand model thus
predicts again that there is no grand unification, and thus no larger gauge group, be it
SU(5), SO(10), E6, E7, E8, SO(32) or any other group. Also this result indirectly rules out
supersymmetry and supergravity.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Summary on the weak interaction and experimental predictions
Motion Mountain – The Adventure of Physics
Ref. 183
The strand model makes clear that the weak interaction and the electromagnetic interaction mix, but do not unify. There is only electroweak mixing, and no electroweak unification, despite claims to the contrary by the Nobel Prize committee and many other
physicists. In fact, Sheldon Glashow, who received the Nobel Prize in Physics for this
alleged ‘unification’, agrees with this assessment. So do Richard Feynman and, above
all, Martin Veltman, who was also involved in the result; he even makes this very point
in his Nobel Prize lecture. The incorrect habit to call electroweak mixing a ‘unification’
was one of the main reason for the failure of past unification attempts: it directed the
attention of researchers in the wrong direction.
In the strand model, the mixing of the electromagnetic and the weak interaction can
be seen as a consequence of knot geometry: the poke generators of the weak interaction
also contain twists, i.e., also contain generators of the electromagnetic interaction. In
contrast, generators of other Reidemeister moves do not mix among them or with pokes;
and indeed, no other type of interaction mixing is observed in nature.
256
Page 162
9 g auge interactions d ed uced from stra nds
Motion Mountain – The Adventure of Physics
The strand model also predicts that the combination of gravity and quantum theory
turns all Planck units into limit values, because there is a maximum density of strand
crossings in nature, due to the fundamental principle. Therefore, the strand model predicts a maximum weak field value given by the Planck force divided by the smallest weak
charge. All physical systems – including all astrophysical objects, such as neutron stars,
quark starts, gamma-ray bursters or quasars – are predicted to conform to this limit. So
far, no observed field value is near this limit, so that the prediction does not contradict
observation.
So far, our exploration of the weak interaction has left us with a few open issues: we
need to calculate the weak coupling constant and determine the tangle for each particle
of the standard model, including the Higgs boson. But we also need to explain weak fermion mixing, CP violation and the masses of all particles. Despite these open points, we
have settled another line of the millennium list: we know the origin of the weak interaction and of its main properties. Before we clarify the open issues, we explore the third
Reidemeister move.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
the strong nuclea r interaction a nd the third reid emeister m ov e
Reidemeister move III, or slide, in textbook form :
257
Slide in SU(3) form (one example) :
A slide transfer :
gluon
slide
transfer
fermion
with
different
phase
vacuum
F I G U R E 67 A gluon changes the phase of a tangle: slide transfer is the basis of the strong interaction in
the strand model. During the interaction, no strand is cut or reglued; the transfer occurs purely through
the excluded volume that results from the impenetrability of strands.
In nature, the strong interaction is the result of the absorption and the emission of massless, electrically uncharged, spin-1 gauge bosons that are called gluons. Gluons interact
with quarks, the only fermions with colour charge. Fermions can have three different colour charges, antifermions three different anticolours. Gluons form an octet, are themselves colour charged and therefore also interact among themselves. The Lagrangian
of quarks coupled to the gluon field has an unbroken SU(3) gauge symmetry. There
are three fundamental Feynman diagrams: one for quark-gluon interaction and two for
gluon-gluon interactions: a triple and a quartic gluon vertex. The strong coupling constant is about 0.5 at low energy; its energy dependence is determined by renormalization.
The previous paragraph summarizes the main observations about the strong interaction. All known observations related to the strong interaction, without any known
exception, are contained in its Lagrangian. Therefore, we need to show that the strong
interaction Lagrangian follows from the strand model.
Strands and the slide, the third R eidemeister move
Page 222
As explained above, interactions of fermions are deformations of the tangle core that
change its phase. We start directly by presenting the strand model for the strong interaction.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
the strong nuclear interaction and the third
reidemeist er move
Motion Mountain – The Adventure of Physics
fermion
258
9 g auge interactions d ed uced from stra nds
⊳ The strong interaction is the transfer of slides, i.e., the transfer of third Reidemeister moves, between a gluon and a particle. As shown in Figure 67,
strands are not cut in this process; they simply transfer deformations as a
result of their impenetrability.
Such a slide transfer will influence the phase of the affected particle tangle. Therefore,
slide transfers are indeed a type of interaction.
An introduction to SU(3)
8
? = e∑?=1 ?? ??? /2
(160)
0 1 0
cos ?/2 ? sin ?/2 0
?i?1 /2
1
0
0
?
?1 = (
) , ?1 (?) = e
= ( sin ?/2 cos ?/2 0) ,
0 0 0
0
0
1
0 ? 0
?1 = eπi?1 /2 = ( ? 0 0)
0 0 1
0 −? 0
cos ?/2 sin ?/2 0
? 2 = ( ? 0 0) , ?2 (?) = e?i?2 /2 = (− sin ?/2 cos ?/2 0) ,
0 0 0
0
0
1
πi?2 /2
?2 = e
0 1 0
= (−1 0 0)
0 0 1
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
where the eight real parameters ?? can be thought of as the eight coordinates of the group
elements on the group manifold. Since SU(3) is compact and simple, these coordinates
are best visualized as angles. Of course, ? is the imaginary unit. The generators ? ? are
complex, traceless and hermitian 3 × 3 matrices; they are used to define a basis for the
group elements. The eight generators are not group elements themselves. They describe
the structure of the group manifold near the identity matrix; for a Lie group, this local
structure defines the full group manifold. Like for any basis, also set of eight generators
? ? is not unique. Of the many possible choices for the generators, the Gell-Mann matrices
? 1 to ? 8 are the most commonly used in physics.
The Gell-Mann matrices ? ? , the corresponding group elements ?? for general angles,
and the group elements ?? for the finite angle π are given by:
Motion Mountain – The Adventure of Physics
Challenge 167 e
Before we show that slides are responsible for the strong nuclear interaction, we summarize the mathematical properties of the Lie group SU(3). This Lie group is the structure generated by the unitary 3 × 3 matrices with determinant +1. It is a group, because
matrices can be properly multiplied, because the identity matrix is included, and inverse
matrices exist. SU(3) is also a manifold; a quick check shows that it has eight dimensions.
In short, SU(3) is a Lie group: its elements behave like points on a manifold that can be
multiplied. The Lie bracket is the commutator. A general element ? of SU(3) can be
written as an exponential in the well-known way
the strong nuclea r interaction a nd the third reid emeister m ov e
259
1 0 0
cos ?/2 + ? sin ?/2
0
0
0
cos ?/2 − ? sin ?/2 0) ,
? 3 = (0 −1 0) , ?3 (?) = e?i?3 /2 = (
0 0 0
0
0
1
πi?3 /2
?3 = e
? 0 0
= (0 −? 0)
0 0 1
0 0 1
cos ?/2 0 ? sin ?/2
1
0 ),
? 4 = (0 0 0) , ?4 (?) = e?i?4 /2 = ( 0
1 0 0
? sin ?/2 0 cos ?/2
0 0 ?
?4 = eπi?4 /2 = (0 1 0)
? 0 0
0 0 1
?5 = eπi?5 /2 = ( 0 1 0)
−1 0 0
0 0 0
1
0
0
?i?6 /2
? 6 = (0 0 1) , ?6 (?) = e
= (0 cos ?/2 ? sin ?/2) ,
0 1 0
0 ? sin ?/2 cos ?/2
0 0 0
1
0
0
?i?7 /2
? 7 = (0 0 −?) , ?7 (?) = e
= (0 cos ?/2 sin ?/2 ) ,
0 ? 0
0 − sin ?/2 cos ?/2
1 0 0
?7 = eπi?7 /2 = (0 0 1)
0 −1 0
?8 =
1 0 0
1
(0 1 0 ) ,
√3 0 0 −2
√
?8 (?) = e 3 ?i?8 /2 = (
cos ?/2 + ? sin ?/2
0
0
0
cos ?/2 + ? sin ?/2
0
),
0
0
cos ? − ? sin ?
? 0 0
?8 = ?8 (π) = (0 ? 0 ) .
(161)
0 0 −1
The eight Gell-Mann matrices ? ? are hermitean, traceless and trace-orthogonal. The
corresponding group elements ?? and ?? can be thought as the unnormed and normed
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
1 0 0
?6 = eπi?6 /2 = (0 0 ? )
0 ? 0
Motion Mountain – The Adventure of Physics
0 0 −?
cos ?/2 0 sin ?/2
0
1
0 ),
? 5 = (0 0 0 ) , ?5 (?) = e?i?5 /2 = (
? 0 0
− sin ?/2 0 cos ?/2
260
9 g auge interactions d ed uced from stra nds
basis vectors of the group manifold. We note that the definition of ?8 differs from that of
the other group elements ??: it contains an extra factor √3 . The fourfold concatenation
of each matrix ?? ? is the identity matrix – except for the case ?? 8 . Instead, the generator
? 8 commutes with ? 1 , ? 2 and ? 3 – though not with the other generators.
There is no ninth or tenth Gell-Mann matrix. Such a matrix would not be linearly
independent from the first eight ones. Indeed, the two matrices deduced from ? 3 using
symmetry considerations, namely
−1 0 0
cos ?/2 − ? sin ?/2 0
0
0
1
0
),
? 9 = ( 0 0 0) , ?9 (?) = e?i?9 /2 = (
0 0 1
0
0 cos ?/2 + ? sin ?/2
−? 0 0
?9 = ?9 (π) = ( 0 1 0)
0 0 ?
?10
(162)
are linear combinations of ? 3 and ? 8 ; in particular, we have ? 3 + ? 9 + ? 10 = 0 and
√3 ? 8 + ? 9 = ? 10 . Therefore, ? 9 and ? 10 are not Gell-Mann matrices. (Also two further
matrices corresponding to ? 8 in the other two triplets can be defined. The sum of these
three matrices is 0 as well.)
The multiplication properties of the Gell-Mann generators ? 1 to ? 8 are listed in
Table 10. To make the threefold symmetry more evident, the table also lists the products
containing the linearly dependent matrices ? 9 and ? 10 . Writing the table with the commutators would directly show that the generators form a Lie algebra.
The centre of SU(3), the subgroup that commutes with all other elements of the group,
is ?3 ; its threefold symmetry is useful in understanding the behaviour of the group elements and of the generators in more detail.
The group elements ?1 to ?8 listed above share the property that their fourth powers
(??)4 are the identity matrix. The first matrix triplet ?1 , ?2 , ?3 , the second triplet
?4 , ?5 , ?9 and the third triplet ?6 , ?7 , ?10 each form a SU(2) subgroup. Reflecting the
threefold symmetry of its centre, SU(3) contains three linearly independent SU(2) subgroups. The group element ?8 commutes with the first triplet ?1 , ?2 , ?3 ; therefore, these
four elements generate a U(2) subgroup of SU(3). This U(2) subgroup, often sloppily
labeled as SU(2)xU(1), is given by those 3 by 3 matrices that contain a unitary 2 by 2
matrix in the upper left, contain zeroes in the remaining four off-diagonal elements, and
contain the inverse value of the determinant of the 2 by 2 matrix in the remaining, lower
right diagonal element. In short, SU(3) contains three linearly independent U(2) subgroups.
SU(3) is characterized by the way that the SU(2) triplets are connected. In particular,
the product ?3 ?9 ?10 is the identity, reflecting the linear dependence of the three corresponding generators ? ? . We also have ?8 ?9 = ?10 . Also the product of ?8 with its
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Vol. V, page 225
1 0 0
= ?10 (π) = (0 ? 0 )
0 0 −?
Motion Mountain – The Adventure of Physics
0 0 0
1
0
0
0
? 10 = (0 1 0 ) , ?10(?) = e?i?10 /2 = (0 cos ?/2 + ? sin ?/2
),
0 0 −1
0
0
cos ?/2 − ? sin ?/2
the strong nuclea r interaction a nd the third reid emeister m ov e
261
TA B L E 10 The multiplication table for the generators ? 1 to ? 8 of SU(3), and for the additional, linearly
dependent matrices ? 9 = −? 3/2 − ? 8√3 /2 and ? 10 = −? 3/2 + ? 8√3 /2 that are not generators. Note that,
despite the appearance, ?24 = ?25 = ?29 and ?26 = ?27 = ?210.
?1
?2
?3
?4
?5
?9
?6
?7
? 10
?8
?1
2/3
+? 8 /√3
?? 3
−?? 2
? 6 /2
+?? 7 /2
−?? 6 /2
+? 7 /2
−? 1 /2
+?? 2 /2
? 4 /2
+?? 5 /2
−?? 4 /2
+? 5 /2
? 1 /2
+?? 2 /2
? 1 /√3
?2
−?? 3
2/3
+? 8 /√3
?? 1
?? 6 /2
−? 7 /2
? 6 /2
+?? 7 /2
−?? 1 /2
−? 2 /2
−?? 4 /2
+? 5 /2
−? 4 /2
−?? 5 /2
−?? 1 /2
+? 2 /2
? 2 /√3
?3
?? 2
−?? 1
2/3
+? 8 /√3
? 4 /2
+?? 5 /2
−?? 4 /2 −1/3 − ? 3 /3 −? 6 /2
+? 5 /2
+? 9 /3
−?? 7 /2
?? 6 /2 −1/3 + ? 3 /3 ? 3 /√3
−? 7 /2
+? 10 /3
?4
? 6 /2 −?? 6 /2
−?? 7 /2 −? 7 /2
? 4 /2
−?? 5 /2
2/3 + ? 3 /2
−? 8 /2√3
−?? 9
?? 5
? 1 /2
+?? 2 /2
?? 1 /2
−? 2 /2
−? 4 /2
−?? 5 /2
−? 4 /2√3
−?√3 ? 5 /2
?5
?? 6 /2 ? 6 /2
+? 7 /2 −?? 7 /2
?? 4 /2
+? 5 /2
?? 9
2/3 + ? 3 /2
−? 8 /2√3
−?? 4
−?? 1 /2
+? 2 /2
? 1 /2
+?? 2 /2
?? 4 /2
−? 5 /2
?√3 ? 4 /2
−? 5 /2√3
?9
−? 1 /2 ?? 1 /2 −1/3 − ? 3 /3
−?? 2 /2 −? 2 /2
+? 9 /3
−?? 5
?? 4
?6
+? 4 /2 ?? 4 /2
−?? 5 /2 +? 5 /2
−? 6 /2
+?? 7 /2
? 1 /2
−?? 2 /2
?? 1 /2
+? 2 /2
? 6 /2
+?? 7 /2
2/3 − ? 3 /2
−? 8 /2√3
?? 10
−?? 7
−? 6 /2√3
−?√3 ? 7 /2
?7
?? 4 /2 −? 4 /2
+? 5 /2 +?? 5 /2
−?? 6 /2
−? 7 /2
−?? 1 /2
−? 2 /2
? 1 /2
−?? 2 /2
−?? 6 /2
+? 7 /2
−?? 10
2/3 − ? 3 /2
−? 8 /2√3
?? 6
?√3 ? 6 /2
−? 7 /2√3
? 10
−? 1 /2 −?? 1 /2 −1/3 + ? 3 /3 −? 4 /2
+?? 2 /2 −? 2 /2
−? 10 /3
+?? 5 /2
?? 7
−?? 6
2/3 − ? 3 /3
+? 9 /3
1
+? 9
?8
? 1 /√3 ? 2 /√3
1
+? 9
2/3
−? 8 /√3
? 3 /√3
2/3 + 2? 3 /3 ? 6 /2
+? 9 /3
−?? 7 /2
−?? 4 /2 −1/3 − ? 9 /3
−? 5 /2
+? 10 /3
−? 4 /2√3 −?√3 ? 4 /2
+?√3 ? 5 /2 −? 5 /2√3
−1
+? 10
?? 6 /2 −1/3 − ? 9 /3
+? 7 /2
+? 10 /3
−? 6 /2√3 −?√3 ? 6 /2
+?√3 ? 7 /2 −? 7 /2√3
−1
+? 10
companions from the other two triplets is the identity.
Finally, the product (?? ?? )3 for any ? taken from the set (1, 2, 4, 5, 6, 7) and any ? from
the same set, but from a different triplet, is also the identity matrix. This property of the
third powers – taken together with the threefold symmetry of its centre – can be seen as
the essential property that distinguishes SU(3) from other Lie groups. We now return to
the strand model and show that slides indeed define an SU(3) group.
From slides to SU(3)
The slide, or third Reidemeister move, involves three pieces of strands. The textbook version of the third Reidemeister move – which is called ?0 here and is illustrated in Figure 68 – moves or ‘slides’ one strand, taken to be the horizontal blue one in the figure,
262
9 g auge interactions d ed uced from stra nds
A textbook slide, or Reidemeister III move:
E0
F I G U R E 68 The textbook version ?0 of the slide move, or third Reidemeister move, is unobservable,
because it does not involve crossing switches.
?? = eπi?? /2 .
(163)
In the strand model, the generators ? ? describe the difference between an infinitesimal
generalized slide – thus a slide-rotation with a rotation by an infinitesimal angle – and
the identity. For slides, concatenation is equivalent to group multiplication, as expected.
Slides form a group. We will now show that the slide generators obey the multiplication
table already given in Table 10.
To see how the SU(3) multiplication table follows from Figure 69, we first note that
the starting strand configuration of the Reidemeister III move contains, if all spatial configurations are considered, the same threefold symmetry as the centre of SU(3). In particular, like the generators and the basis vectors of SU(3), also the slides of the figure can
be grouped into three triplets.
We now focus on the first triplet, the one formed by the three slides ?1 , ?2 and ?3 .
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
against a crossing of the other two. Equivalently, we can say that a slide pushes two
strands against the blue strand that is kept in place. This textbook slide – we also call
it a pure slide here – does not contain any crossing switch; following the fundamental
principle of the strand model, it is therefore unobservable, or, simply said, of no physical
relevance. However, related strand moves that do involve crossing switches do exist.
We introduce eight generalized slides, or slide-rotations, for a three-strand configuration; they are shown in Figure 69. We directly call these generalized slides ?1 to ?8 , because they will turn out to correspond to the SU(3) group elements with the same name
that were introduced above. In other words, we will show that the generalized slides ??
are elements of a Lie group SU(3); in particular, they obey all the properties expected
from the correspondence with the SU(3) generators ? ? in Gell-Mann’s choice:
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Colours and arrows are only added for clarity.
The textbook slide is unobservable, because it lacks crossing switches.
It is thus not physical and uninteresting.
the strong nuclea r interaction a nd the third reid emeister m ov e
263
The generalized slides, or Reidemeister III moves, acting on three strands, form an SU(3) group.
The 8 generalized slides are shown below, with grey background. They are
local slides and rotations by an angle π of an imaginary buckle formed
by (usually) two strands. The strands lie (mostly) in a paper plane.
λ2
For each SU(2) triplet inside SU(3), the rotation axes of the finite group
elements En are arranged at right angles to each other, as are those of
the generators λn shown on the right. The rotation axes for E3, E9 and
E10 are parallel; they are perpendicular to the paper plane. The three
imaginary belt buckles for the three SU(2) subgroups are also shown.
λ3
G
R
λ7
E1 = e i π λ 1 / 2
π
antigreen
E2 = e i π λ2 / 2
B
λ10
=
– λ3/2
+ λ8√3/2
λ4
red
green
E4 = e i π λ4 / 2
E6 = e i π λ6 / 2
π
π
antiblue
antired
E5 = e i π λ5 / 2
E7 = e i π λ7 / 2
π
antigreen
antiblue
E9 = e i π λ9 / 2
E10 = e i π λ10 / 2
π
π
π
antiblue
antired
antigreen
E3 = e i π λ3 / 2
E8 = e i π √3 λ8 / 2
π √3
F I G U R E 69 The strand deformations for the generalized slide moves ??. The corresponding generators
? ? lead to an SU(3) structure, as shown in the text. Note that the rotation vectors for the generators ? ?
and for the generalized slide moves ?? differ from each other. For clarity, the figure shows, instead of
the deformation of the strand under discussion, the complementary deformations of the other two
strands.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
π
antired
λ5
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blue
λ9 = – λ3 / 2
– λ8 √3 / 2
λ8
λ6
Starting
position
λ1
264
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
To makes things clear, these moves can be pictured as combined deformations and slides
of the red and green strands against the horizontal blue strand. We can imagine these
moves like those of the belt trick, but acting on an imaginary buckle formed only by the
red and green strands. These generalized slides do contain crossing changes; therefore
they are observable and are of physical relevance.
We note that ‘slide’ is not a perfect term for the generalized deformations ?1 to ?8 ;
in fact, we might prefer to call them slide-rotations, because they are slide-rotations by
an angle π that are applied to an imaginary belt buckle. Despite the involved construction, these generalized, observable moves remain modelled on the textbook slide ?0 ; in
particular, they require three strand segments. The generalized, observable moves just
defined differ from the twists and pokes discussed above, in the sections on the electromagnetic and weak interactions; thus they differ from Reidemeister I and II moves. As
a result, we will usually continue to call the generalized, observable moves simply slides.
For simplicity, we assume – similarly to what we did in the discussion about the weak
interaction – that the three strand segments are (roughly) in a plane. This is an idealized situation; in fact, the arguments given in the following apply also to all other threedimensional configurations of three strands. In particular, the same results appear if all
three strands segments are assumed perpendicular to each other, instead of lying in a
plane.
We note that the rotation axes of the generalized slides ?1 and ?2 are neither aligned
nor orthogonal to the paper plane. More precisely, the rotation axes of ?1 , ?4 and ?6 are
perpendicular to the sides of a cube. ?2 , ?5 and ?7 are perpendicular to them. For the
first triplet, the rotation axes ?1 , ?2 and ?3 form an orthonormal basis; the same is valid
for the other two triplets. We now show that the slides of the first triplet define an SU(2)
group.
The observable, generalized slides in the triplet ?1 , ?2 and ?3 can be concatenated.
We distinguish two cases. The first case is the concatenation of any such slide with itself.
The result corresponds to a rotation by 2π of the chosen strand pair and its imaginary
belt buckle, and thus induces a corresponding amount of tail twisting. In fact, when
any slide of the triplet is concatenated four times with itself, the result is the identity
operation. Comparing a twofold and a fourfold concatenation, we see that they differ
only by an entangling, or algebraically, by a minus sign for the imaginary buckle. This
already realizes half of the belt trick that visualizes SU(2).
The other case to be checked is the concatenation of two different slides of the triplet.
The result is always the third slide of the triplet (up to a sign that depends on whether the
combination is cyclical or not). This behaviour realizes the other half of the belt trick. In
short, we have shown that the triplet containing the first three generalized slides defines
an SU(2) group. More precisely, the infinitesimal slide-rotations ? 1 , ? 2 and ? 3 corresponding to the finite SU(3) elements ?1 , ?2 and ?3 generate the SU(2) Lie algebra of an
SU(2) Lie group. The SU(2) subgroup just found is just one of the three linearly independent SU(2) subgroups of SU(3). The generators of the first slide triplet thus reproduce
the nine results in the upper left of Table 10. We thus retain that we can indeed visualize
the first three generalized slides with the help of the three orthogonal rotations by π of
an imaginary belt buckle formed by the red and green strands.
For the visualization of SU(3) it is essential to recall that the direction in threedimensional space of the vectors visualizing ? ? and those visualizing ?? differ from each
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9 g auge interactions d ed uced from stra nds
the strong nuclea r interaction a nd the third reid emeister m ov e
265
Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
other. This already the case for U(1).
The remaining generalized slides that are possible in the three-strand configuration
are easily constructed using the threefold symmetry of the strand configuration; they
are illustrated in Figure 69. For each of the three strand segments there is a triplet of
observable slides; this yields a total of nine possible generalized slides for the observer
defined by the paper plane. In the second triplet, the slides corresponding to ?1 and ?2
are called ?4 and ?5 , and in the third triplet they are called ?6 and ?7 . For the three slides
corresponding to ?3 – we call the other two ?9 and ?10 – only two generators are linearly
independent. Indeed, the figure shows that ?3 ?9 ?10 – whose axes are all three parallel –
is the identity matrix; this expected from an SU(3) structure. The three operations ?3 , ?9
and ?10 also commute with all other operations; thus they form the centre of the group
defined by all ?. The second linearly independent, generalized slide of common use, ?8 ,
is also shown in the figure; it is a linear combination of ?9 and ?10 . We note that the
strand model also visualizes the factor √3 in the definition of ?8 . In total, we get eight
linearly independent generalized slides. All slides, except for ?8 , act on an imaginary
belt buckle that is formed by two strands.
We saw that the generators corresponding to the slides ?1 , ?2 and ?3 generate an
SU(2) subgroup. The same holds for the corresponding triplet ?4 , ?5 and the linear
combination ?9 = −?3 /2 − ?8 √3 /2 (corresponding to ?3 ), and for the triplet ?6 , ?7
and ?10 = −?3 /2 + ?8 √3 /2. For each of these slides, a fourfold concatenation yields the
identity; and inside each triplet, the concatenation of two different slides yields a multiple of the third slide. In short, for each triplet, the corresponding infinitesimal slides
generate an SU(2) group. These three SU(2) groups are linearly independent. We have
thus reproduced an important part of the structure of SU(3). In addition, we have found
a visualization of SU(3); since each SU(2) group can be represented by a separate imaginary buckle, the group SU(3) can be visualized – in many, but not all in aspects – with the
help of three imaginary buckles. The top right of Figure 69 illustrates this visualization.
The correspondence of the slides and the multiplication table increases further if we
change slightly the definition of the first triplet. In this first triplet we can take as imaginary buckle the set of all three central segments. Moving all three strands together simplifies the visualization, because for the first triplert, the blue strand is trapped between the
other two strands. In this way, generalized slide still consists of a rotation followed by a
slide. And we still have a SU(2) subgroup for the first triplet.
The slide ?8 differs from the other slides, as expected from SU(3). It describes a motion that rotates the red and green strands in opposite directions; this is illustrated in
Figure 69. ?8 is thus not well described with an imaginary belt buckle. It is straightforward to check that the slide ?8 commutes with ?1 , ?2 , ?3 and obviously with itself, but
not with the other generalized slides. Together, ?8 and the first triplet thus form a U(2)
Lie group, as expected. In addition, we find that ?8 commutes with ?9 and ?10 , and that
?8 ?9 = ?10 , as expected from SU(3). The strand model also implies that the product of
?8 with its two counterparts from the other triplets is the identity matrix, as expected
from SU(3).
The last step to show the equivalence of slides and SU(3) requires us to confirm the
multiplication properties – between slides ?? or between generators ? ? – from different
triplets. In fact, because of the three-fold symmetry of the centre, we only need to check
two multiplication results between slides from different triplets: one that either involves
266
The strand model for gluons
Physically, the eight slides corresponding to the Gell-Mann matrices represent the effects
of the eight gluons, the intermediate vector bosons of the strong interaction, that can act
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
? 3 or ? 8 , and one that does not.
We begin with products involving ? 3 and one of the first two elements of another
triplet. Such products yield a weighted sum of generators of the triplet. It is easier to
check these product properties by using the exemplary relation between finite group elements ?5 ?3 ?4 = ?3 . Note that only this specific permutation of 5, 3 and 4 yields this
result. Playing with the strand model confirms the relation. Similar comments apply to
?6 ?3 ?7 = ?3 – and to the corresponding products involving ?9 , such as ?1 ?9 ?2 = ?9 ,
or ?10 , such as ?1 ?10 ?2 = ?10 – as well as ?4 ?8 ?5 = ?8 and ?6 ?8 ?7 = ?8 . The strand
model allows anybody to check that these relations are satisfied.
We continue with the exemplary product ? 5 ? 7 , respectively ?5 ?7 . We note a basic difference between a product like ? 5 ? 7 and any product of two generators from the same
triplet. The product ? 5 ? 7 – like the other concatenations of generators from different
triplets – does not yield a single generator, but yields a combination, i.e., a sum of generators. The combination is not easy to visualize with strands; an easier way is to check
the SU(3) algebra using the properties of the product ?5 ?7 .
As mentioned above, in SU(3), for products involving the first two members from
different triplets, the threefold concatenation (?? ?? )3 is the identity. And indeed, Figure 69
confirms that (?2 ?4 )3 or (?5 ?7 )3 is the identity. Similarly, also the other products can be
tested with the help of three strands.
Using the visualization with three strands, we have thus confirmed all products of
generators from two different triplets that appear in Table 10. We note that Figure 69 also
illustrates that the three slides ?2 , ?5 and ?7 generate an SO(3) group, the rotation group
in three dimensions. In order to see this, we observe that the infinitesimal versions of
the three slides generate all possible rotations in three dimensions of the central triangle.
An SO(3) group also appears for the slides 1, 4 and 7, for the slides 1, 5 and 6, and for
the slides 2, 4 and 6. These are the four basic SO(3) subgroups of SU(3). The remaining
combinations of three operations from three different triplets – such as 1, 4 and 6, or the
combination 1, 5 and 7, or the combination 2, 4 and 7, or the combination 2, 5 and 6 –
do not generate any subgroup. This can be confirmed by exploring the corresponding
strand moves.
We can conclude: in a region with three strands crossing each other, the eight linearly
independent, generalized slides that can be applied to that region define the group SU(3).
In other words, the group SU(3) follows from the third Reidemeister move.
In the same way as for the other gauge groups, we find that particles whose strand
models contain configurations with three strand segments can be subject to an SU(3)
gauge interaction. In experiments, this interaction is called the strong nuclear interaction.
The strong interaction is due to the Reidemeister III move. Like for the other interactions, a particle will only interact strongly if its tangle is not too symmetric, because in
the symmetric case, averaged over time, there will be no net interaction. We will clarify
the details below, when we discuss the specific tangles and colour charges of the different
elementary matter particles.
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9 g auge interactions d ed uced from stra nds
the strong nuclea r interaction a nd the third reid emeister m ov e
267
on a particle.
⊳ Given that the eight slides ?1 to ?8 represent the effects of the eight gluons,
they also represent the gluons themselves.
Interactions are transfers of a tangle process to another tangle. Therefore
⊳ The absorption of a gluon is a slide that is transferred to another particle.
⊳ The emission of a gluon is a slide that is transferred to three vacuum strands.
Motion Mountain – The Adventure of Physics
Challenge 168 e
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
To visualize the concept of gluon even further, we can say that every gluon can be described as a strand structure that continuously performs an SU(3) operation, i.e., a generalized slide continuously repeating itself. We found a similar correspondence for the
other gauge interactions. In case of the electromagnetic interaction, the intermediate
vector boson, the photon, can be described as a strand that continuously performs a
U(1) operation, i.e., a rotation. In case of the weak interaction, a weak intermediate vector boson can be described as a strand that continuously performs an SU(2) operation,
i.e., an operation from the belt trick. This is most evident in the unbroken form of the
weak bosons.
Every gluon can also be seen as the deformation of a single strand that drags its surrounding with it. This single strand description of gluons implies that gluons have vanishing mass and vanishing charge. This single strand description of gluons also implies
that they have spin 1, as is observed. The strand model of the gluon also implies that free
gluons would have a huge energy.
The SU(3) multiplication table confirms that the eight gluons transform according to
the adjoint (and faithful) representation of SU(3). Therefore, each row or column in a
Gell-Mann matrix thus corresponds to one of the three colours of the strong interaction. The exploration of slide concatenation also showed that two general slides do not
commute and do not anticommute. The group SU(3) is non-Abelian. This implies that
gluons interact among themselves. Both the multiplication table and the strand model
for gluons imply that two interacting gluons can yield either one or two new gluons, but
not more. This is illustrated in Figure 70. The strand model, through its generation of
SU(3), thus implies that gluons interact among themselves, but only in triple and quartic
gluon vertices.
Slides – i.e., gluon emission or absorption – never change the topology of tangles,
and in particular, of matter tangles. Therefore, the strand model predicts that the strong
interactions conserve electric charge, baryon number, weak isospin, flavour, spin and all
parities. This is indeed observed. In particular, there is a natural lack of C, P and CP
violation by slides. This is precisely what is observed for the strong interaction.
Because gluons do not change the topology of the particle tangles they act upon, but
only change their shape, gluons are predicted to be massless in the strand model, despite interacting among themselves. And because gluons interact among themselves, free
gluons are predicted not to appear in nature. And of course, all these conclusions agree
with experiments.
268
9 g auge interactions d ed uced from stra nds
A triple gluon vertex :
green
anti-green
t2
red
t2
anti-blue
vacuum
t1
red
t1
time
anti-blue
The quartic gluon vertex :
anti-green green
t2
red anti-blue
t1
anti-green red
green
t1
time
anti-blue
In summary, we have shown that in the strand model, the strong nuclear interaction
and all its properties appear automatically form slides, i.e., from Reidemeister III moves.
In particular, the strand model implies that the Lagrangian of strongly interacting fermions has a SU(3) gauge invariance that is due to generalized slide deformations.
The gluon L agrangian
Gluons are massless particles with spin 1. As a result, the field intensities and the Lagrangian are determined in the same way as for photons: energy density is the square of
crossing density, i.e., the ‘square’ of field intensity. Since there are 8 gluons, the Lagrangian density becomes
1 8
Lgluons = − ∑ ???? ???
(164)
?
4 ?=1
where the gluon field intensities, with two greek indices, are given naturally as
???? = ∂? ??? − ∂? ??? − ????? ??? ??? ,
(165)
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
F I G U R E 70 The two types of self-interaction of gluons in the strand model.
Motion Mountain – The Adventure of Physics
t2
the strong nuclea r interaction a nd the third reid emeister m ov e
269
Strong charge, or colour :
Random slides affect
only rational tangles
with broken threefold
tail symmetry :
Random slides
do not affect
knotted tangles :
Random slides
affect gluons :
Random slides
do not affect photons :
Colour charge
Surrounded by a bath of gluons that randomly induce slides of all kinds, not all fermion
cores will change their rotation state. Generally speaking, particles have colour if a bath
of random gluons changes their phase. Only tangles which lack some symmetry will
therefore possess colour charge. Tangle that are symmetric will be neutral, or ‘white’.
Which symmetry is important here?
We see directly that the photon tangle is not sensitive to a gluon bath. The same is
valid for W and Z bosons. These tangles are too simple. The strand model predicts that
these particles are colour-neutral, i.e., that they are ‘white’, as is observed.
On the other hand, the multiplication properties given above shows that gluons interact among themselves and thus that they have colour charge. In fact, group theory
shows that their properties are best described by saying that gluons have a colour and an
anticolour; this is the simplest way to describe the representation to which they belong.
In short, the strand model of gluons automatically implies that they carry both a colour
and an anti-colour.
Fermions behave differently. In the strand model, a fermion has colour charge if the
corresponding triple belt model is affected by large numbers of random gluons. The
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 231
and ???? are the structure constants of SU(3) that can be deduced from the multiplication table given above. The quantities ??? , with one greek index, are the gluon vector
potentials. The last term in the definition of the field intensities corresponds to the triple
and quartic vertices in the Feynman diagrams of gluon interactions. They are shown
in Figure 70. The Lagrangian is simply the natural generalization from the U(1) case of
photons to the SU(3) case of gluons. In short, we obtain the usual free gluon Lagrangian
from the strand model.
Motion Mountain – The Adventure of Physics
F I G U R E 71 Tangles with and without colour charge. (This figure needs to be updated.)
270
9 g auge interactions d ed uced from stra nds
Strong interaction diagram :
real
quark
+
Feynman diagram :
time average
of crossing
switches
virtual
gluon
real
quark
t2
virtual
gluon
t2
core is
rotated by
2π/3 around
vertical axis
t1
t1
quark
time
correspond to the three belts.
⊳ A fermion tangle has colour charge if its three-belt model is not symmetric
for rotations by ±2π/3.
Coloured rational tangles automatically have three possible colours:
⊳ The three colour charges are the three possibilities to map a tangle to the
three belt model.* Each colour is thus a particular orientation in ordinary
space.
Page 319
Challenge 169 ny
If we want to explore more complicated types of tangles of two strands, such as prime
tangles or locally knotted tangles, we recall that such tangles are not part of the strand
* Can you define a geometric or even a topological knot invariant that reproduces colour charge?
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
first tangles that come to mind are tangles made of three strands, such as the simple
tangles shown in Figure 69. But a short investigation shows that such tangles are colourneutral, or ‘white’. We will see below that this implies that leptons are colour-neutral, or
‘white’. In contrast, a rational fermion tangle does not suffer this fate. (We recall that
a so-called rational tangle is by definition made of exactly two strands; a two-stranded
tangle is rational if the two strands can be untangled just by moving the tails around.) In a
bath of gluon strands that induce slides, i.e., third Reidemeister moves, a general rational
tangle made of two strands is expected to be influenced, and thus to be colour-charged.
Rational tangles made of two strands are the simplest possible tangles with colour. A
tangle is called rational if it can be untangled just by moving the tails around. An example
of a rational tangle is shown in Figure 72. Such tangles break the three-fold symmetry
of the three-belt structure, and are thus colour-charged. We will show below how these
tangles are related to quarks. We can thus say:
Motion Mountain – The Adventure of Physics
F I G U R E 72 The Feynman diagram of the strong interaction for a quark. The upper triplet of tails
the strong nuclea r interaction a nd the third reid emeister m ov e
271
model. The strand model thus predicts that rational tangles made of two strands are
the basic colour states. And indeed, in nature, quarks are the only fermions with colour
charge.
We can summarize that colour charge is related to orientation in space. The three
possible colours and anticolours are consequences of the possible orientations along the
three dimensions of space.
Properties of the strong interaction
The L agrangian of Q CD
/ − ?? ?2 ???? )Ψ?? −
LQCD = ∑ Ψ? (?ℏ?D
?
1 8 ?
∑ ? ? ?? ,
4 ?=1 ?? ?
(166)
where the index ? counts the coloured fermion, i.e., the quark. In this Lagrangian density,
/ is now the SU(3) gauge covariant derivative
D
/ = ∂/ − ? ?? ??? ? ? ,
D
(167)
where ? is the gauge coupling, ? ? are the generators of SU(3), i.e., the Gell-Mann
matrices given above, and the ??? are, as before, the gluon vector potentials. The last
term in the covariant derivative corresponds to the Feynman diagram and the strand
diagram of Figure 72. This is the Lagrangian density of QCD.
In summary: the strand model reproduces QCD. However, we have not yet deduced
the number and masses ?? of the quarks, nor the strong gauge coupling ?.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
We started from the idea that tangle core deformations lead to phase redefinitions. We
then found that slides imply that the strong interaction Lagrangian for matter and for
radiation fields is SU(3) gauge invariant. If we include these two gauge invariances into
the fermion Lagrangian density from the Dirac equation, we get
Motion Mountain – The Adventure of Physics
In the strand model, all interactions are deformations of the tangle core. Specifically, the
strong interaction is due to exchange of slides. Particles have strong charge, or colour, if
their tangles lack the three-belt symmetry just specified. In the case of coloured fermions, colour change is a change of the mapping to the three-belt model, i.e., a change of
orientation of the tangle in space.
If we use the strand definition of the strong interaction, visual inspection shows us
that slide exchanges, and thus gluon exchanges, are deformations that conserve topology;
therefore gluon exchange conserves colour. Since the strong interaction conserves the
topology of all involved tangles and knots, the strong interaction also conserves electric
charge, parity, and, as we shall see below, all other quantum numbers – except colour
itself, of course. All these results correspond to observation.
272
9 g auge interactions d ed uced from stra nds
R enormalization of the strong interaction
Page 382
Page 334, page 341
Curiosities and fun challenges abou t SU(3)
Vol. V, page 288
Ref. 184
∗∗
Challenge 170 ny
We have discussed the shape deformations that lead to the SU(3) group. But what are the
precise phase choices for a crossing that lead to SU(3) invariance?
∗∗
Challenge 171 ny
Do the two linear independent gluons with lined-up tails have the same properties as the
other six gluons?
∗∗
Challenge 172 s
Three strands can cross each other also in another way, such that the three strands are
interlocked. Why can we disregard the situation in this section?
∗∗
Deducing the Lie groups U(1), SU(2) and SU(3) directly from a basic principle contradicts another old dream. Many scholars hoped that the three gauge groups have something to do with the sequence complex numbers, quaternions and octonions. The strand
model quashes this hope – or at least changes it in an almost unrecognizable way.
∗∗
Challenge 173 e
The tangles for the W and Z bosons have no colour charge. Can you confirm this?
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Deducing the Lie group SU(3) from a three-dimensional model is a new result. In particular, deducing the gauge group SU(3) as a deformation gauge group is new. Frank
Wilczek, Alfred Shapere, Alden Mead, Jerry Marsden and several others have confirmed
that before this discovery, only the geometric Lie group SO(3) and its subgroups had
been found in deformations. The fundamental principle of the strand model shows its
power by overcoming this limitation. (Apparently, nobody had even realized that the
belt trick already implies the possibility of an SU(2) gauge group for deformations.)
Motion Mountain – The Adventure of Physics
Page 338
The slide move description of the strong interaction implies that only three Feynman
diagrams are possible: one QCD Feynman diagram is possible for quarks, and only the
triple and the quartic vertices are possible among gluons. This limited range of options
allowed us to deduce the QCD Lagrangian. The limited range of options is also essential
for the renormalization of QCD. The strand model thus automatically ensures that the
strong interaction is renormalizable.
In short, the strand model provides a new underlying picture for the Feynman diagrams of the strong interaction, but does not change the physical results at any energy
scale accessible in the laboratory. In particular, the measured running of the strong coupling constant is reproduced. Indeed, in the strand model, a flux-tube–like bond between
the quarks appears automatically, as we will see when exploring hadrons. At high kinetic energies, the bond has little effect, so that quarks behave more like free particles. In
short, we find that the strand model reproduces asymptotic freedom and also provides an
argument for quark confinement. We will return to the issue in more detail below.
the strong nuclea r interaction a nd the third reid emeister m ov e
273
∗∗
Challenge 174 ny
The Lie group SU(3) is also the symmetry group of the three-dimensional harmonic oscillator. What is the geometric relation to the Lie group SU(3) induced by slides?
∗∗
Challenge 175 e
Confirm that the strand model does not contradict the Coleman–Mandula theorem on
the possible conserved quantities in quantum field theory.
∗∗
Challenge 176 e
Confirm that the strand model does not contradict the Weinberg–Witten theorem on
the possible massless particles in quantum field theory.
∗∗
Are the Wightman axioms of quantum field theory fulfilled by the strand model with
interactions? The Haag–Kastler axioms? Is Haag’s theorem circumvented?
∗∗
Ref. 185
Challenge 178 ny
Show that the BCFW recursion relation for tree level gluon scattering follows from the
strand model.
Summary on the strong interaction and experimental predictions
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 353
We have deduced the Lagrangian density of QCD from the strand model with the help of
slides. Is there a difference between the strand model and QCD? No, not as long as gravity
plays no role. The strand model predicts that gravitation only comes into play near the
Planck energy √ℏ?5 /4? . And indeed, accelerator experiments have not yet found any
effect that contradicts QCD, and therefore no effect that contradicts the strand model of
the strong interaction.
The strand model also predicts that the strong interaction is naturally CP-invariant.
This means that axions – particles invented to explain the invariance – are unnecessary:
as shown below, the strand model even predicts that they do not to exist. Both predictions agree with experiment.
The strand model of the strong interaction implies that the SU(3) gauge symmetry is
valid at all energies. No other gauge group plays a role in the strong interaction. The
strand model thus predicts again that there is no grand unification in nature, and thus
no larger gauge group. Often discussed groups such as SU(5), SO(10), E6, E7, E8 or
SO(32) are predicted not to apply to nature. Also this prediction is not contradicted by
experiment.
The strand model further predicts that the combination of gravity and quantum theory turns all Planck units into limit values. The strand model thus predicts a maximum
strong field value given by the Planck force divided by the strong charge of the quark. All
physical systems – including all astrophysical objects, such as neutron stars, quark stars,
gamma-ray bursters or quasars – are predicted to conform to this field limit. So far, this
prediction is validated by experiment.
In summary, we have shown that Reidemeister III moves – or slides – in tangle cores
lead to an SU(3) gauge invariance and a Lagrangian that reproduces the strong interac-
Motion Mountain – The Adventure of Physics
Challenge 177 d
274
Page 162
9 g auge interactions d ed uced from stra nds
tion. Colour charge is related to the topology of certain rational tangles. In this way, we
have deduced the origin and most observed properties of the strong interaction. We have
thus settled another issue of the millennium list. However, we still need to deduce the
tangles and the number of quarks, their masses and the strength of the strong coupling.
Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
sum m ary and pred ictions ab ou t g auge interactions
275
summary and predict ions abou t gauge interactions
Page 150
Page 162
Ref. 181
Already in 1926, Kurt Reidemeister proved an important theorem about possible deformations of knots or tangles that lead to changes of crossings. When tangles are described
with two-dimensional diagrams, all possible deformations can be reduced to exactly three
moves, nowadays called after him. In the strand model, the two-dimensional tangle diagram describes what an observer sees about a physical system. Together with the equivalence of interactions as crossing-changing deformations, Reidemeister’s theorem thus
proves that there are only three gauge interactions in nature. In particular, there is no fifth
force. Searches for additional gauge interactions are predicted to fail. And indeed, they
have all failed up to now.
Unification of interactions
Ref. 142
We can also state that there is only one Reidemeister move. This becomes especially clear
if we explore the three-dimensional shape of knots instead of their two-dimensional diagrams: all three Reidemeister moves can be deduced from the same deformation of a
single strand. Only the projection on a two-dimensional diagram creates the distinction
between the three moves. In the terms of the strand model, this means that all gauge
interactions are in fact aspects of only one basic process, a fluctuation of strand shape,
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Predicting the number of interactions in nature
Motion Mountain – The Adventure of Physics
Page 147
At this point of our adventure, we have deduced gauge theory and the three known gauge
interactions from strands. Using only the fundamental principle, we explained the dimensions of space-time, the Planck units, the principle of least action, the appearance
of the gauge groups U(1), broken SU(2) and SU(3), of renormalization, of Lorentz symmetry and of permutation symmetry. Thus we have deduced all the concepts and all the
mathematical structures that are necessary to formulate the standard model of elementary particles.
In particular, the strand model provides a description and explanation of the three
gauge interactions at Planck scales that is based on deformations of strands. The deduction of the three gauge interactions given in this text, with the help of the Reidemeister
moves, is the first and, at present, the only explanation of the three gauge forces. No other
explanation or deduction has ever been given.
We have shown that quantum field theory is an approximation of the strand model.
The approximation appears when the strand diameter is neglected; quantum field theory
is thus valid for all energies below the Planck scale. In other words, in contrast to many
other attempts at unification, the strand model is not a generalization of quantum field
theory. The strand model for the three gauge interactions is also unmodifiable. These
properties are in agreement with our list of requirements for a final theory.
We have not yet deduced the complete standard model: we still need to show which
types of particles exist, which properties they have and what couplings they produce.
However, we have found that the strand model explains all the mathematical structures
from the millennium list that occur in quantum field theory and in the standard model
of particle physics. In fact, the strand explanation for the origin of the gauge interactions
allows us to make several definite predictions.
276
Page 380
9 g auge interactions d ed uced from stra nds
No divergences
Page 150
Grand unification, supersymmetry and other dimensions
Page 391
Page 348
The three gauge interactions are due to the three Reidemeister moves. Therefore, the
strand model asserts that there is no single gauge group for all interactions. In short, the
strand model asserts that there is no so-called grand unification. The absence of grand
unification implies the absence of large proton decay rates, the absence of additional, still
undiscovered gauge bosons, the absence of neutron–antineutron oscillations, and the
absence of sizeable electric dipole moments in elementary particles. All these searches
are ongoing at present; the strand model predicts that they yield null results.
Supersymmetry and approaches based on it assume gauge group unification. However, as just explained, the strand model predicts that there is no supersymmetry and
therefore no supergravity. The strand model also predicts the absence of all conjectured
‘superparticles’. In 2016 and again in 2017, the numerous experiments at CERN confirmed the prediction: there is no sign of supersymmetry in nature.
Reidemeister moves are confined to three spatial dimensions. Indeed, the strand
model is based on exactly three spatial dimensions. It predicts that there are no other,
undetected dimensions of space. The strand model also predicts the absence of noncommutative space-time, even though, with some imagination, strands can be seen as
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
The strand model implies that there are no divergences in the quantum description of
nature. This lack of divergence occurs because all measurement values appear after
strand effects have been averaged out. As mentioned above, strand effects on space-time
disappear through ‘shivering’ and strand effects on particles disappear through wavefunctions.
In summary, in the strand model, no interaction implies or contains divergences:
neither gravity nor the gauge interactions. There are neither ultraviolet nor infrared divergences. The strand model avoids divergences, infinities and singularities of any kind
from its very start.
Motion Mountain – The Adventure of Physics
and that the three gauge interactions are only distinguished by their projections. In this
way, the three gauge interactions are thus unified by the strand model.
The plane of projection used in a strand diagram defines a mapping from strand fluctuations to Reidemeister moves. The projection plane is defined by the observer, i.e.,
by the frame of reference. Depending on the projection plane, a general deformation
is mapped into different Reidemeister moves. At first sight, the nature of an interaction – whether electromagnetic, strong or weak – seems to depend on the observer. In
nature, however, this is not the case. But this contradiction is only apparent. In the
strand model, the nature of interaction of a particle results from the type of asymmetry
of its tangle core. Certain strand deformations do not lead to interactions, because their
effects are suppressed by the averaging of short-time fluctuations underlying every observation. In other words, the averaging process at the basis of observations also ensures
that interactions are effectively observer-independent at low energy.
In short, the strand model provides a natural unification of the interactions. And this
unification of the interactions differs completely from any past proposal. The final test,
of course, can only be provided by experiment.
sum m ary and pred ictions ab ou t g auge interactions
Page 147
Page 395
277
remotely related to that approach. Finally, the strand model predicts the lack of different
vacua: the vacuum is unique.
In short, the strand model differs both experimentally and theoretically from the unification proposals made in the twentieth century. In particular, the strand model predicts the absence of additional symmetries, of additional energy scales, and of additional
space-time properties at high energy. The strand model predicts that unification is not
achieved by searching for higher symmetries, nor for higher dimensions, nor for concepts that contain both. This lack of complex mathematical or symmetry concepts in
nature is disappointing; the hopes and search activities in the last fifty years are predicted
to have been misguided. In other words, the predictions of the strand model are unpopular. However, these predictions agree with our list of requirements for a final theory;
and so far, all these predictions agree with experiment.
No new observable gravit y effects in particle physics
In the ‘cube’ structure of physics shown in Figure 1, the transition from the final, unified
description to quantum field theory occurs by neglecting gravity, i.e., by assuming flat
space-time. The same transition occurs in the strand model, where neglecting gravity in
addition requires neglecting the strand diameter. In this way, the gravitational constant
? disappears completely from the description of nature.
We can summarize our findings on quantum field theory also in the following way:
Page 311
This result will be complemented below by a second, equally restrictive result that limits the observable quantum effects in the study of gravity. In short, the strand model
keeps particle physics and general relativity almost completely separated from each other.
This is a consequence of the different effects produced by tail deformations and by core
deformations. And again, the prediction of a lack of additional gravitational effects in
particle physics agrees with all experiments so far.
The status of our quest
Page 162
In this chapter, we have deduced that strands predict exactly three interactions. Interactions are deformations of tangle cores and just three classes of such core deformations
exist. The three classes of deformations are given by the three Reidemeister moves. Because of the properties of the Reidemeister moves, the three interactions are described
by a U(1), a broken SU(2) and a SU(3) gauge symmetry, respectively.
Strands also show that the three interactions are renormalizable, relativistically invariant, and that they follow the least action principle. Strands thus imply the three interaction Lagrangians of the standard model of particle physics. In addition, strands predict
the absence of other interactions, symmetries and space-time structures.
If we look at the millennium list of open issues in fundamental physics, we have now
solved all issues concerning the mathematical structures that appear in quantum field
theory and in the standard model of particle physics.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
⊳ The strand model predicts that particle masses are the only observable effect
of gravity in quantum physics and in particle physics.
Motion Mountain – The Adventure of Physics
Page 8
278
9 g auge interactions d ed uced from stra nds
⊳ All mathematical structures found in quantum physics result from the fundamental principle of the strand model.
Equivalently, extension contains all quantum effects. This is an intriguing result that
induces us to continue our exploration. Only two groups of issues are still unexplained:
the theory of general relativity and the spectrum of elementary particles. We proceed in
this order.
Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
C h a p t e r 10
GENER AL R EL AT IVIT Y DEDUCED
FROM ST R ANDS
G
Page 206
Page 209
We have seen above that any observer automatically introduces a 3+1-dimensional background space-time. We have also seen that in the case of quantum theory, physical spacetime, the space-time that is formed by the fluctuations of the vacuum strands, is naturally
3+1-dimensional and flat. In the absence of gravity, physical space and background space
coincide.
Using strands, we have deduced:
⊳ ? is the invariant limit for all energy speeds.
This limit is achieved only by free massless particles, such as photons. Strands also
showed us that massive particles move more slowly than light. In short, strands reproduce special relativity.
The strand model thus predicts that pure special relativity is correct for all situations
and all energies in which gravity and quantum theory play no role. The strand model
also predicts that when gravity or quantum effects do play a role, general relativity or
quantum theory must be taken into account. This means that there is no domain of
nature in which intermediate descriptions are valid.
It is sometimes suggested that the invariant Planck energy limit for elementary
particles might lead to a ‘doubly special relativity’ that deviates from special relativity
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Fl at space, special rel ativit y and its limitations
Motion Mountain – The Adventure of Physics
eneral relativity describes the deformations of the vacuum. In everyday life,
ravitation is the only such effect that we observe. But on astronomical scale,
ravity shows more phenomena: vacuum can deflect light, producing gravitational lenses, can wobble, giving gravitational waves, and can accelerate, yielding the
darkness of the sky and the fascinating black holes. All these observations require general
relativity for their description. Therefore, general relativity must be part of any unified
description of nature.
In the following, we explain the existence of gravity as a consequence of strands. Then
we deduce the field equations of general relativity, the entropy of black holes and relativistic cosmology from the strand model. We also predict the outcome of many quantum
gravity experiments. Finally, we deduce the consequences of strands for cosmology, including a new experimental prediction. Of all Planck-scale models of space or spacetime, strands seem to be the simplest one that provides these deductions.
280
10 g eneral rel ativity d ed uced from stra nds
first mass
gravitational
interaction
~1/r2
second mass
distance r
Ref. 85
Ref. 186
Page 8
Cl assical gravitation
In nature, at low speeds and in the flat space limit, gravitation is observed to lead to
an acceleration ? of test masses that changes as the inverse square distance from the
gravitating mass;
?
?=? 2 .
(168)
?
This acceleration is called universal gravitation or classical gravitation. It is an excellent
approximation for the solar system and for many star systems throughout the universe.
In the strand model, every space-time effect, including gravitation, is due to the behaviour of tangle tails. In the strand model, every mass, i.e., every system of tangles, is
connected to the border of space by tails. The nearer a mass is to a second mass, the
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 64
at high particle energy. However, this suggestion is based on two assumptions: that at
Planck energy point masses are a viable approximation to particles, and that at Planck
energy vacuum and matter differ. In the strand model, and in nature, both assumptions
are incorrect. Nature, as general relativity shows, does not allow the existence of point
masses: the densest objects in nature are black holes, and these are not point-like for any
mass value.
In addition, quantum theory implies the fuzziness of matter and space. As a result,
near Planck energy, matter and vacuum cannot be distinguished. Put simply, no system
near Planck energy can be described without general relativity and without quantum
gravity. In short, the strand model predicts that the approach of ‘doubly special relativity’ cannot be correct. Also Figure 1 makes this point: there is no description of nature
besides the usual ones.
To sum up, the strand model reproduces special relativity when masses are approximated as point-like in flat space. But at the same time, the strand model states that a
negligibly small, light and localizable mass cannot exist – neither in flat nor in curved
space. This matches observations.
Motion Mountain – The Adventure of Physics
F I G U R E 73 Gravitational attraction as result of strands.
g enera l rel ativit y d educed from stra nds
281
more frequently the tails of the two masses cross and get tangled. Figure 73 illustrates
the situation. The strand model states:
⊳ Gravitation is due to the fluctuations of tail crossings.
Challenge 179 e
Ref. 187
Page 287
Black holes have entropy; this implies universal gravitation. There are at least two ways
to explain this connection.
An especially concise explanation was recently given by Erik Verlinde. In this view,
gravity appears because any mass ? generates an effective vacuum temperature around it.
A gravitating mass ? attracts test masses because during the fall of a test mass, the total
entropy decreases. It is not hard to describe these ideas quantitatively.
Given a spherical surface ? enclosing a gravitating mass ? at its centre, the acceleration ? of a test mass located somewhere on the surface is given by the local vacuum
temperature ?:
2π ??
?=?
,
(169)
ℏ
where ? is the Boltzmann constant. This relation is called the Fulling–Davies–Unruh
effect and relates vacuum temperature and local acceleration. Thus, an inertial or a freely
falling mass (or observer) measures a vanishing vacuum temperature.
In the strand model, the vacuum temperature at the surface of the enclosing sphere is
given by the crossing switches induced by the tails starting at the mass. We can determine
the vacuum temperature by dividing the energy ? contained inside the sphere by twice
the maximum possible entropy ? for that sphere. This maximum value is the entropy that
the sphere would have if it were a black hole horizon; it can be calculated by the strand
model, as we will see shortly.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Deducing universal gravitation from bl ack hole properties
Motion Mountain – The Adventure of Physics
Page 356
Around a mass, the tail crossings fluctuate; averaged of time, the fluctuations lead to a
crossing switch density around every mass. The resulting potential energy – where energy is action per time and thus given by the number of crossing switches per time –
changes like the inverse distance from the central mass. This is the reason for the 1/?dependence of the gravitational potential and the 1/?2 -dependence of gravitational acceleration. (This applies to all those cases where spatial curvature is negligible.) In simple
words, in the strand model, the inverse square dependence of gravitational acceleration
is due to the three-dimensionality of space combined with the one-dimensionality of
strands.
The strand model also shows that masses and energies are always positive: every
tangle contains curved strands. The model also shows qualitatively that larger masses
produce stronger attraction, as they generally produce more crossing switches. We will
show below that the number density of crossing switches is indeed given by the mass.
In the strand model, crossing switches are not only related to action and energy; they
are also related to entropy. A slightly different – but equivalent – view on gravitation
therefore appears when we put the stress on the entropic aspect.
282
10 g eneral rel ativity d ed uced from stra nds
The temperature ? is thus given by the expression
?=
?
? 2?ℏ
=
.
2? ? ??
(170)
The factor 2 needs explanation: it might be due to the combination of the effects of space
and matter.
Neglecting spatial curvature, we can set ? = 4π?2 ; this gives a temperature at the
enclosing sphere given by
? ?ℏ
?= 2
.
(171)
? 2π ??
Page 281
The change of entropy d?/d? when a test mass ? moves by a distance ? can be determined from the strand model in a simple manner. When the test mass ? moves by a
(reduced) Compton wavelength, in the strand model, the mass has rotated by a full turn:
the entropy change is thus 2π? per (reduced) Compton wavelength. Thus we have
d?
2π ??
=?
.
d?
ℏ
(174)
Using the temperature ? found in expression (171), we get an expression for the gravitational force given by
??
?=? 2 .
(175)
?
Page 35
This is universal gravitation again. This time we have thus deduced universal gravitation
from the entropy and temperature generated by gravitating masses.
We note that the temperature and entropy of black holes are limit values. We can thus
state that universal gravitation is a consequence of nature’s limit values.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
This is universal gravitation, as discovered by Robert Hooke and popularized by Isaac
Newton. Since spatial curvature was neglected, and the central mass was assumed at rest,
this expression is only valid for large distances and small speeds. We have thus deduced
universal gravity from the effects of gravitating masses on vacuum temperature. Below,
we show that in the relativistic case this sequence of arguments – which was given by
Jacobson fifteen years before Verlinde – leads to the field equations of general relativity.
An alternative deduction of universal gravitation from black hole entropy is the following. The gravitational force ? on a test mass ? is given by the vacuum temperature ?
created by the central mass ? and by the change of entropy ? per length that is induced
by the motion of the test mass:
d?
?=?
.
(173)
d?
Motion Mountain – The Adventure of Physics
Page 293
Inserting this expression into the expression (169) for the Fulling–Davies–Unruh acceleration ?, we get
?
?=? 2 .
(172)
?
g enera l rel ativit y d educed from stra nds
283
The partial link :
2R
axis
other vacuum strands
Summary on universal gravitation from strands
Ref. 188
Vol. I, page 218
Curved space
In nature, observation shows that physical space is not flat around masses, i.e., in the
presence of gravity. Near mass and energy, physical space is curved. Observations also
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 287
Universal gravitation is due to the temperature and entropy of the (curved) vacuum
around masses. The limit case is the temperature and entropy of black holes. In the
strand model, these temperature and entropy values are a consequence of the underlying strand crossing switches; we will show this shortly. Universal gravitation thus (again)
appears as an effect of the crossing switches induced by masses.
We can say that we have several explanations of universal gravitation using strands.
We have deduced universal gravitation from the energy of strands, from the temperature
of strands and from the entropy of strands around a mass. We have also have deduced
universal gravitation from the maximum force, which strands fulfil as well. In short,
strands explain the origin of universal gravitation.
Incidentally, modelling mass as a source for strand crossing switches is remotely reminiscent of Georges-Louis Lesage’s eighteenth-century model of gravitation. Lesage
proposed that gravity appears because many tiny, usually unnoticed corpuscules push
masses together. In fact, as we will see shortly, there is a certain similarity between
these assumed tiny corpuscules and virtual gravitons. And interestingly, all criticisms
of Lesage’s model then cease to hold. First, there is no deceleration of free masses in
inertial motion, thanks to the built-in special-relativistic invariance. Secondly, there is
no heating of masses, because the entangled tails represent virtual gravitons that scatter elastically. Thirdly, and most of all, by replacing the corpuscules ultra-mondains of
Lesage by virtual gravitons – and finally by strands – we can predict an additional effect
of gravity that is not described by the inverse square dependence: space-time curvature.
Motion Mountain – The Adventure of Physics
F I G U R E 74 A schematic model of the fundamental defect, and thus the fundamental type of curvature:
the partial link.
284
10 g eneral rel ativity d ed uced from stra nds
show that curved space-time remains 3+1-dimensional. The observation of this type of
curvature was predicted long before it was measured, because curvature follows unambiguously when the observer-invariance of the speed of light ? and the observerinvariance of the gravitational constant ? are combined.
We continue directly with the strand model of spatial curvature and show that all
observations are reproduced.
⊳ Curvature (of physical space-time) is due to simple, unknotted and weakly
localized defects in the tangle of strands that make up the vacuum. An example is shown in Figure 74.
⊳ Mass is a localized defect in space and is due to tangled strands. Thus mass
curves space around it.
⊳ Energy in a volume is the number of crossing switches per unit time. As a
result, mass is equivalent to energy. As a second result, energy also curves
space.
⊳ Gravitation is the space-time curvature originating from compact regions
with mass or energy.
Vol. II, page 284
The structure of horizons and bl ack holes
In general relativity, another concept plays a fundamental role. In the strand model we
have:
⊳ A horizon is a tight, one-sided weave of strands.
Therefore, there are no strands behind the horizon. This implies that behind a horizon,
there is no matter, no light, no space and no time – just nothing. Indeed, this is the
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 356
These natural definitions show that curvature is due to strand configurations. In particular, curvature is built of unknotted – i.e., massless – defects. The massless defects leading
to curvature are usually dynamic: they evolve and change. Such curvature defects – virtual gravitons – originate at regions containing matter or energy. In fact, the curvature of
space around masses is a natural result of fluctuations of the strands that make up matter
tangles.
We note that curved space, being a time average, is continuous and unique. Vacuum or
curved space, more precisely, curved physical space, thus differs from background space,
which is flat (and drawn in grey in the figures).
Incidentally, the distinction between physical and background space also avoids Einstein’s hole argument; in fact, the distinction allows discussing it clearly, as only physical
space describes nature.
Motion Mountain – The Adventure of Physics
⊳ In the case of curvature, physical space-time, which is due to averaged strand
crossing switches, differs from flat background space-time, which usually
corresponds to the tangent or to the asymptotic space-time. In Figure 74,
the grey background colour can be taken as visualization of the background
space.
g enera l rel ativit y d educed from stra nds
General horizon :
(side view)
285
Black hole :
F I G U R E 75 A schematic model of a general and a spherical horizon as tight weaves, as pictured by a
distant observer. In the strand model there is nothing, no strands and thus no space, behind a horizon.
⊳ A black hole is a tight, one-sided and closed weave of strands.
Is there something behind a horizon?
A drawing of a horizon weave, such as the one of Figure 75, clearly points out the difference between the background space and the physical space. The background space is
the space we need for thinking, and is the space in which the drawing is set. The physical
space is the one that appears as a consequence of the averaging of the strand crossings.
Physical, curved space exists only on the observer side – usually outside – of the horizon.
The physical space around a black hole is curved; it agrees with the background space
only at infinite distance from the horizon. Inside the horizon, there is background space,
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In principle, closed horizons can have any shape. The simplest case is the spherical, nonrotating horizon, which defines the Schwarzschild black hole. It is illustrated on the righthand side of Figure 75.
If an observer is located outside a spherical horizon, the strand model states that there
is nothing inside the horizon: no matter, no light and no vacuum. The strand model
thus provides a simple and drastic view of black hole horizons. Figure 75 also illustrates
that the concept of radius (or size) of a black hole has to be approached with the (wellknown) care. In general, the size of a structure made of strands is the number of crossings
encountered when travelling through it. However, an observer cannot travel through a
black hole: there are no strands inside, thus there is no vacuum there! The size of a black
hole must therefore be defined indirectly. The simplest way is to take the square root
of the area, divided by 4π, as the radius. Thus the strand model, like general relativity,
requires that the size of a compact horizon be defined by travelling around it.
We note that the strand model also provides an intuitive explanation for the differences between a rotating and a non-rotating black hole.
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experience of any observer about a horizon. A horizon is thus a structure that limits
physical space. It does not limit background space.
One particular type of horizon is well-known:
286
10 g eneral rel ativity d ed uced from stra nds
but no physical space. In short, the strand model implies that – for an observer at spatial
infinity – there is nothing, not even a singularity, inside a black hole horizon.
⊳ There is no physical space, no matter and no singularity inside a horizon.
Energy of bl ack hole horizons
?=
Challenge 181 e
?cs 4π?2 ?4
?4
=
=?
.
?
2π? 4?
2?
(176)
Strands thus imply the well-known relation between energy (or mass) and radius of
Schwarzschild black holes.
How do the crossing switches occur at a horizon of a black hole? This interesting
puzzle is left to the reader.
The tight-weave model of horizons also illustrates and confirms both the hoop conjecture and the Penrose conjecture. For a given mass, because of the minimum size of crossings, a spherical horizon has the smallest possible diameter, compared to other possible
shapes. The strand model naturally implies that, for a given mass, spherical black holes
indeed are the densest objects in nature.
The nature of bl ack holes
The strand model naturally implies the no-hair theorem. Since all strands are the same,
independently of the type of matter that formed or fell into the horizon, a black hole
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Challenge 180 e
The strand model allows us to calculate the energy content of a closed horizon. Energy is
action per unit time. In the strand model, the energy of a non-rotating spherical horizon
is thus given by the number ?cs of crossing switches per time unit. In a tight weave,
crossing switches cannot happen in parallel, but have to happen sequentially. As a result,
a crossing switch ‘propagates’ to the neighbouring Planck area on the surface. Since the
horizon weave is tight and the propagation speed is one crossing per crossing switch
time, this happens at the speed of light. In the time ? that light takes to circumnavigate
the spherical horizon, all crossings switch. We thus have:
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Horizons are observer-dependent. Both the existence and the shape of a horizon depends on the observer. As we will see, this happens in precisely the same way as in usual
general relativity. In the strand model, there is no contradiction between the one observer at spatial infinity who says that there is nothing behind a horizon, not even physical space, and another, falling observer, who does not observe a horizon and thus states
that there is something there. In the strand model, the two statements naturally transform into each other under change of viewpoint. Indeed, the transformation between
the two viewpoints contains a deformation of the involved strands.
We note that the equivalence of viewpoints and the statement that there is nothing
behind a horizon is based on the combination of general relativity and quantum theory. If we would continue thinking that space and time is a manifold of points – thus
disregarding quantum theory – these statements would not follow.
In summary, one-sided tight weaves are a natural definition of horizons.
g enera l rel ativit y d educed from stra nds
287
has no characteristics other than mass, angular momentum and charge. Here we used a
result from the next chapter, when it will become clear that all elementary particles are
indeed made of the same featureless strands. Taking that result as given, we deduce that
flavour quantum numbers and particle number do not make sense for black holes. We
also deduce that weak and strong charge are not defined for black holes. Strands explain
naturally why neutral black holes made of antimatter and neutral black holes made of
matter do not differ, if their masses and angular momenta are the same. In short, the
strand model of nature implies the no-hair theorem: strands, not hairs.
Horizons and black holes are borderline systems between space and matter. This borderline property must be fulfilled by every final theory. The strand model fulfils this
requirement: in the strand model, black holes can either be described as curved space or
as tightly packed particles in permanent free fall.
Entropy of vacuum and mat ter
⊳ The flat and infinite vacuum has vanishing entropy, because the number of
crossing switches is zero on average.
At the same time,
The entropy of vacuum and of horizons differs from that of matter. In the absence of
gravity, the number of microstates of matter is determined – as in usual thermodynamics
(thermostatics) – by the behaviour of tangle cores.
In strong gravity, when the distinction between matter and vacuum is not so clear-cut,
the number of microstates is determined by the possible crossing switches of the strands.
In strong gravity, only tails play a role. This becomes clear when we calculate the entropy
of black holes.
Entropy of bl ack holes deduced from the strand model
Despite the tight weaving, the strands making up a horizon are fluctuating and moving:
the weave shape fluctuates and crossing switch all the time. This fluctuating motion is
the reason why horizons – in particular those of black holes – have entropy.
The weave model of a horizon, illustrated in detail in Figure 76, allows us to calculate
the corresponding entropy. Since the horizon is a tight weave, there is a crossing on each
Planck area. To a first approximation, on each (corrected) Planck area of the horizon, the
strands can cross in two different ways. The fundamental principle of the strand model
thus yields two microstates per Planck area. The number ? of Planck 2areas is given by
?2 = ??3 /4?ℏ. The resulting number of black hole microstates is 2? . The entropy is
given by the natural logarithm of the number of the possible microstates times ?. This
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⊳ Curved space and horizons have non-vanishing entropy.
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Both vacuum and matter are made of fluctuating strands. We note directly:
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10 g eneral rel ativity d ed uced from stra nds
F I G U R E 76 The entropy of black holes results from the number of possible crossing states above a
Planck area.
?=?
Ref. 189
??3
ln 2 .
4?ℏ
(177)
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This result is the well-known first approximation of black hole entropy: one bit per corrected Planck area. In the strand model, the proportionality of entropy and area is thus
a direct consequence of the extension of the strands. This proportionality is also well
known from studies of quantum gravity and of strings. In those approaches however,
the relation between the area proportionality and extension is less obvious.
For Schwarzschild black holes, the entropy value of expression (177) is not correct.
In the strand model, this incorrect value is explained as a consequence of neglecting
the effects of the strand tails. Indeed, additional contributions to the entropy appear at
a finite distance from the horizon, due to the crossing of the tails on their way to the
border of space, as shown in Figure 76. The actual entropy will thus be larger than the
first approximation, but still be proportional to the area ?.
The correct proportionality factor between the area and the entropy of a black hole
results when the strand tails are taken into account. (The correction factor is called the
Barbero–Immirzi parameter in the research literature on quantum gravity.) The calculation is simplest for Schwarzschild black holes. By construction, a black hole with macroscopic radius ?, being a tight weave, has ?/?Pl tails. For each given Planck area, there
are, apart from the basic, or lowest crossing, additional crossings ‘above it’, along the
radial direction, as shown in Figure 76. These additional crossings are due to the tails
from neighbouring and distant Planck areas.
Taking into effect all strand tails allows us to calculate the average number of crossings
above a given Planck area. The main point is to perform this calculation for all those
tails that start in a circular strip of Planck width centred around the Planck area under
consideration. We then add the probabilities for all possible circular strips. One such
circular strip is drawn in Figure 76.
The definition of horizons as tight weaves implies that a horizon with ?2 Planck areas
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approximation gives an entropy of a horizon of
g enera l rel ativit y d educed from stra nds
Challenge 182 e
289
is made of ? strands. This means that for each circular strip of radius ??Pl, there is only
one strand that starts there and reaches spatial infinity as a tail. For this tail, the average
probability ? that it crosses above the central Planck area under consideration is
?=
1
.
?!
(178)
∞
Summing over all strips, i.e., over all values ?, we get a total of ∑?=0 1/?! = e = 2.71828...
microstates on and above the central Planck area under consideration. Thus the number
e replaces the number 2 of the first approximation:
the number of horizon microstates
2
2
of a Schwarzschild black hole is not 2? , but e? . As a consequence, the entropy of a
macroscopic Schwarzschild horizon becomes
?=?
(179)
This is the Bekenstein–Hawking expression for the entropy of Schwarzschild black holes.
The strand model thus reproduces this well-known result. With this explanation of the
difference between 2 and e = 2.71828..., the strand model confirms an old idea:
The above calculation, however, counts some states more than once. Topologically
identical spherical horizons can differ in the direction of their north pole and in their
state of rotation around the north–south axis. If a spherical horizon is made of ? strands,
it has ?2 possible physical orientations for the north pole and ? possible angular2 orientations around the north–south axis. The actual number of microstates is thus ?? /?3 .
Using the relation between ?2 and the surface area ?, namely ? = ?2 4?ℏ/?3 , we get
the final result
??3 3? ? ?3
?=?
−
ln
.
(180)
4?ℏ 2
4?ℏ
Ref. 190
The strand model thus makes a specific prediction for the logarithmic correction of
the entropy of a Schwarzschild black hole. This final prediction of the strand model
agrees with many (but not all) calculations using superstrings or other quantum gravity
approaches.
In summary, the entropy value (179), respectively (180), of black holes is due to the
extension of the fundamental entities in the strand model and to the three dimensions of
space. If either of these properties were not fulfilled, the entropy of black holes would
not result. This is not a surprise; also our deduction of quantum theory was based on the
same two properties. In short: like every quantum effect, also the entropy of black holes
is a result of extension and three-dimensionality. Only a three-dimensional description
of nature agrees with observation.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
⊳ The entropy of a black hole is located at and near the horizon.
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??3
.
4?ℏ
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10 g eneral rel ativity d ed uced from stra nds
Temperature, radiation and evap oration of bl ack holes
The strands that make up a horizon fluctuate in shape. Since every horizon contains
energy, the shape fluctuations imply energy fluctuations. In other words, horizons are
predicted to have a temperature. The value of the temperature can be deduced from
the strand model by noting that the characteristic size of the fluctuations for a spherical
horizon is the radius ? of the horizon. Therefore we have
?? =
ℏ?
.
2π?
(181)
Using the definition of surface gravity as ? = ?2 /?, we get
?=
(182)
The strand model predicts that horizons have a temperature proportional to their surface
gravity. This result has been known since 1973.
All hot bodies radiate. The strand model thus predicts that Schwarzschild black holes
radiate thermal radiation of the horizon temperature, with power and wavelength
? = 2πℏ?2/?2 , ? ≈ ? .
(183)
Bl ack hole limits
In many ways, black holes are extreme physical systems. Not only are black holes the
limit systems of general relativity; black holes also realize various other limits. As such,
black holes resemble light, which realizes the speed limit. We now explore some of these
limits.
For a general physical system, not necessarily bound by a horizon, the definitions of
energy and entropy with strands allow some interesting conclusions. The entropy of a
system is the result of the number of crossing possibilities. The energy of a system is
the number of crossing changes per unit time. A large entropy is thus only possible if a
system shows many crossing changes per time. Since the typical system time is given by
the circumference of the system, the entropy of a physical system is therefore limited:
? ⩽ ?? 2π?/ℏ? .
(184)
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This confirms a well-known consequence of the temperature of black holes.
Like all thermal systems, horizons follow thermodynamics. In the strand model, black
hole radiation and evaporation occur by reduction of the number of strands that make
up the horizon. The strand model thus predicts that black holes evaporate completely,
until only elementary particles are left over. In particular, the strand model implies that
in black hole radiation, there is no information loss.
In short, strands reproduce all aspects of black hole evaporation. The strand model
also shows that there is no information loss in this process.
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Ref. 57, Ref. 58
ℏ?
.
2π??
g enera l rel ativit y d educed from stra nds
Ref. 191
291
This relation is known as Bekenstein’s entropy bound; the precise definitions of the quantities in the bound need some care, as Don Page explains. The bound thus also follows
from the strand model. Strands imply that the equality is realized only for black holes.
In the strand model, horizons are tight, one-sided weaves. For example, this implies
that any tangle that encounters a horizon is essentially flat. Because of tangle flatness and
the extension of the tails, at most one Planck mass can cross a horizon during a Planck
time. This yields the mass rate limit
d?/d? ⩽ ?3 /4?
(185)
that is valid in general relativity and in nature.
Black holes can rotate. The strand model states that there is a highest angular frequency possible; it appears when the equator of the black hole rotates with the speed of
light. As a result, the angular momentum ? of a black hole is limited by
Ref. 192
This limit is well known from general relativity.
The electric charge of a black hole is also limited. The force limit in nature implies
that the electrical forces between two charged black holes must be lower than their gravitational interaction. This means that
?2
??2
⩽
,
4π?0 ?2
?2
(187)
?2 ⩽ 4π?0 ??2 .
(188)
This is the well-known charge limit for (static) black holes given by the Reissner–
Nordström metric. The maximum charge of a black hole is proportional to its radius.
It follows directly from the maximum force principle.
To explain the charge limit, we deduce that the extremal charge surface density ?/?
of a black hole is proportional to 1/?. The higher the horizon curvature, the more charge
per Planck area is possible. In the strand model, a horizon is a tight weave of strands. We
are thus led to conjecture that at Planck scale, electric charge is related to and limited by
strand curvature. We will explore this connection in more detail below.
The strand model limits energy density to the Planck energy per Planck volume, or to
the value ?7 /(16?2 ℏ). This limit implies a lower size limit for black holes, particles and
any localized system. Therefore, the strand model does not allow singularities, be they
dressed or naked. And indeed, no singularity has ever been observed.
In summary, the strand model reproduces the known limit properties of horizons.
And all these results are independent of the precise fluctuation details of the strands.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
or
Page 378
(186)
Motion Mountain – The Adventure of Physics
? < 2??2 /? .
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10 g eneral rel ativity d ed uced from stra nds
Curvature around bl ack holes
The tails of a black hole extend up to the border of space; the density of tails is highest at
the horizon. A black hole is therefore surrounded by partial links at any finite distance
from the horizon. In other words, the space around a black hole is curved. The value of
the space-time curvature increases as one approaches the horizon, because of the way
in which the partial links hinder each other in their motion. The nearer they are to the
horizon, the more they hinder each other. The curvature that appears is proportional to
the density of partial links and to their average strand curvature.
At the horizon, the curvature radius is the horizon radius ?. By construction, the
number of tails departing from a non-rotating black hole is proportional to ?. The spatial
curvature is given by the average crossing density gradient. Hence at a radial distance ?
from a static black hole, the spatial curvature ? is
?
.
?3
(189)
So at the horizon itself, the curvature ? is (of the order of) the inverse square of the horizon radius; further away, it decreases rapidly, with the third power of the distance. This
result is a well-known property of the Schwarzschild solution and is due to the extension
of the strands. The rapid decay with radius is the reason why in everyday situations there
is no noticeable curvature of space-time. In short, strands allow us to deduce the correct
curvature of space-time around black holes and spherical masses.
The strand model also explains and visualizes the importance of spherical horizons in
nature. First of all, strands illustrate the non-existence of (uncharged) one-dimensional
or toroidal horizons in 3 + 1 space-time dimensions. Such configurations are unstable,
in particular against transverse shear and rearrangement of the strands.
The strand model also implies that non-rotating, closed horizons are spherical. Obviously, spheres are the bodies with the smallest surface for a given volume. The minimum
horizon surface appears because the strands, through their fluctuations, effectively ‘pull’
on each Planck area of the horizon. As a result, all non-rotating macroscopic horizons
will evolve to the spherical situation in a few Planck times. (Deviations from the spherical
shape will mainly occur near Planck scales.) With the definition of gravity waves given
below, it also becomes clear that strongly deformed, macroscopic and non-spherical horizons are unstable against emission of gravity waves or of other particles. In short,
⊳ All non-rotating horizons of non-spherical shape are unstable.
The strand model thus confirms that spherical horizons are favoured and that the most
compact bodies with a given mass. The reasoning can be extended to rotating horizons,
yielding the well-known shapes.
In summary, strands reproduce all known qualitative and quantitative properties
of horizons and of black holes, and thus of general systems with strong gravitational
fields. All predictions from strands agree with observations and with other approaches
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The shape of non-rotating bl ack holes
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?∼
g enera l rel ativit y d educed from stra nds
293
to quantum gravity. These hints already suggest that strands imply the field equations.
The field equations of general rel ativit y
Page 280
Ref. 22
The field equations can be deduced from the fundamental principle in two different, but
related ways. Essentially, both derivations repeat the reasoning for universal gravitation
given above, but for the relativistic case. The first deduction of the field equations is based
on an old argument on the thermodynamics of space-time. Strands show that horizons
have three thermodynamic properties:
— an area–entropy relation of ? = ? ??3 /4?ℏ,
— a curvature–temperature relation of ? = ? ℏ/2π??,
— a relation between heat and entropy of ?? = ???.
Using these three properties, and using the relation
(190)
that is valid only in case of horizons, we get the first principle of horizon mechanics
?? =
(191)
From this relation, using the Raychaudhuri equation, we obtain the field equations of
general relativity. This deduction was given above.*
In other words, the field equations result from the thermodynamics of strands. It is
worth noting that the result is independent of the details of the fluctuations or of the
microscopic model of space, as long as the three thermodynamic properties just given
are valid. In fact, these properties must be fulfilled by any model of space-time; and
indeed, several competing models of space claim to fulfil them.
* Here is the argument in a few lines. The first principle of horizon mechanics can be rewritten, using the
energy–momentum tensor ???, as
?2
∫ ??? ?? dΣ? =
? ??
8π?
where dΣ? is the general surface element and ? is the Killing vector that generates the horizon. The
Raychaudhuri equation allows us to rewrite the right-hand side as
∫ ????? dΣ? =
?4
∫ ??? ?? dΣ?
8π?
where ??? is the Ricci tensor describing space-time curvature. This equality implies that
??? =
Vol. II, page 98
?4
(? − (?/2 + Λ)???)
8π? ??
where Λ is an undetermined constant of integration. These are Einstein’s field equations of general relativity.
The field equations are valid everywhere and for all times, because a suitable coordinate transformation can
put a horizon at any point and at any time. To achieve this, just change to a suitable accelerating frame, as
explained in the volume on relativity.
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Page 32
?2
? ?? .
8π?
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?? = ?? ,
294
Page 186
Ref. 19
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10 g eneral rel ativity d ed uced from stra nds
We can use the relation between fluctuations and strands to settle an issue mentioned
above, in the section on quantum theory. Strand fluctuations must obey the thermodynamic properties to allow us to define space-time. If they obey these properties, then
space-time exists and curves according to general relativity.
A second derivation of the field equations of general relativity follows the spirit of the
strand model most closely. It is even shorter. Strands imply that all physical quantities
are limited by the corresponding Planck limit. These limits are due to the limit to the
fundamental principle, in other words, they are due to the packing limit of strands. In
particular, the fundamental principle limits force by ? ⩽ ?4 /4? and power by ? ⩽ ?5 /4?.
We have already shown above that this limit implies the field equation.
In other words,
⊳ Given that black holes and thus horizons are thermodynamic systems, so is
curved space.
⊳ Since black holes have thermodynamic aspects, so has gravity.
And since black holes are built from microscopic degrees of freedom, so is curved space.
Or, in simple words:
⊳ Space is made of many small entities.
⊳ Space is made of strands, because strands are the simplest entities that yield
black hole entropy.
Strands are the simplest way to incorporate quantum effects into gravitation. If we take
into consideration that strands are the only way known so far to incorporate gauge interactions, we can even conclude that strands are the only way known so far to incorporate
all quantum effects into gravitation.
In summary, the strand model asserts that the field equations appear as consequences
of fluctuations of impenetrable, featureless strands. In particular, the strand model implies and confirms that a horizon and a particle gas at Planck energy do not differ. However, the value of the cosmological constant is not predicted from strand thermodynamics.
E quations from no equation
Page 147
The strand model asserts that the field equations of general relativity are not the result of another, more basic evolution equation, but result directly from the fundamental
principle. To say it bluntly, the field equations are deduced from a drawing – the fundamental principle shown in Figure 10. This strong, almost unbelievable statement is due
to a specific property of the field equations and to two properties of the strand model.
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And finally we can state:
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The reason: both can be transformed into each other. Therefore:
g enera l rel ativit y d educed from stra nds
295
The Hilbert action of general rel ativit y
Page 208
We have just shown that the strand model implies the field equations of general relativity.
We have also shown above that, in the strand model, the least action principle is a natural
property of all motion of strands. Combining these two results, we find that a natural
way to describe the motion of space-time is the (extended) Hilbert action given by
?=
Page 283
?4
∫(? − 2Λ) d? ,
16π?
(192)
where ? is the Ricci scalar, d? = √det? d4 ? is the invariant 4-volume element of the
metric ?, and Λ is the cosmological constant, whose value we have not determined yet.
As is well known, the description of evolution with the help of an action does not add
anything to the field equations; both descriptions are equivalent.
For a curved three-dimensional space, the Ricci scalar ? is the average amount, at a
given point in space, by which the curvature deviates from the zero value of flat space.
In the strand model, this leads to a simple statement, already implied by Figure 74:
⊳ The Ricci scalar ? is the ratio of additional or missing crossings per spatial
volume, compared to flat space.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
⊳ Any principle that allows deducing the field equations cannot itself be an
evolution equation.
Motion Mountain – The Adventure of Physics
First of all, the field equations are, above all, consequences of the thermodynamics of
space-time. In the strand model, the thermodynamic properties are deduced as a consequence of the strand fluctuations. This deduction does not require underlying evolution equations; the field equations follow from the statistical behaviour of strands.
The second, essential property of the strand model is its independence from the underlying motion of the strands. In the strand model we obtain the evolution equations
of the vacuum – the field equations of general relativity – without deducing them from
another equation. We do not need an evolution equation for the strand shape; the deduction of the field equations works for any underlying behaviour of strand shapes, as
long as the thermodynamic properties of the strand fluctuations are reproduced.
The third and last essential property that allows us to deduce the field equations directly from a graph, and not from another equation, is the relation between the graph and
natural physical units. The relation with natural units, in particular with the quantum
of action ℏ and the Boltzmann constant ?, is fundamental for the success of the strand
model.
In summary, the fundamental principle of the strand model contains all the essential
properties necessary for deducing the field equations of general relativity. In fact, the discussion so far makes another important point: unique, underlying, more basic evolution
equations for the tangle shape cannot exist. There are two reasons. First, an underlying
equation would itself require a deduction, thus would not be a satisfying solution to unification. Secondly, and more importantly, evolution equations are differential equations;
they assume well-behaved, smooth space-time. At Planck scales, this is impossible.
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10 g eneral rel ativity d ed uced from stra nds
The graviton :
wavelength
F I G U R E 77 The graviton in the strand model.
As usual, the averaging is performed over all spatial orientations. A similar statement
can be made for the cosmological constant Λ. In short, we can say: the Hilbert action
follows directly from the fundamental principle of the strand model.
⊳ Space-time foam is made of fluctuating strands.
Ref. 193
At everyday scales, the foam is not noticed, because background space and physical space
are indistinguishable. At Planck scales, space-time is not fundamentally different from
everyday space-time. No unusual topology, no additional dimensions, and no new or
unusual properties appear at Planck scales. Above all, the strand model predicts that
there are no observable effects of space-time foam; for example, ‘space-time noise’ or
‘particle diffusion’ do not exist. The strand model of space-time foam is both simple and
unspectacular.
Gravitons, gravitational waves and their detection
In the strand model, gravitons can be seen as a special kind of partial links. An example
is shown in Figure 77. As a twisted pair of parallel strands, the graviton returns to itself
after rotation by π; it thus behaves like a spin-2 boson, as required.
Can single gravitons be observed? The strand model implies that the absorption of
a single graviton by an elementary particle changes its spin or position. However, such
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Quantum physics implies that at scales near the Planck length and the Planck time,
space-time fluctuates heavily. John Wheeler called the situation space-time foam; the
term quantum foam is also used. In a sense, quantum gravity can be defined, if at all, as
the description of space-time foam. This reduced view arises because no separate theory
of quantum gravity is possible in nature.
Historically, there have been many speculations on the details of space-time foam.
Apart from its fluctuations, researchers speculated about the appearance of topology
changes – such as microscopic wormholes – about the appearance of additional dimensions of space – between six and twenty-two – or about the appearance of other unusual
properties – such as microscopic regions of negative energy, networks or loop structures.
The strand model makes a simple prediction that contradicts most previous speculations:
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Space-time foam
g enera l rel ativit y d educed from stra nds
297
A candidate
tangle for an
extended
defect :
F I G U R E 78 A speculative, highly schematic model for a cosmic string, a one-dimensional defect in
space-time
⊳ Single gravitons cannot be detected.
Ref. 194
Open challenge: Improve the argument for the graviton tangle
Challenge 184 ny
The argument that leads to the graviton tangle is too much hand-waving. Can you make
the argument more compelling? Could the four tails form a cross and thus span a plane?
O ther defects in vacuum
The strand model provides a quantum description of gravitation. The strand model does
so by explaining physical space as the average of the crossing switches induced by strand
fluctuations among untangled strands. Matter, radiation and horizons are defects in the
‘sea’ of untangled strands.
So far, we have been concerned with particles, i.e., localized, zero-dimensional defects,
and with horizons, i.e., two-dimensional defects. Now, modelling of the vacuum as a set
of untangled strands also suggests the possible existence of one-dimensional – equivalent
to dislocations and disclinations in solids – of additional two-dimensional defects, or
of three-dimensional defects. Such defects could model cosmic strings, domain walls,
wormholes, toroidal black holes, time-like loops and regions of negative energy.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Challenge 183 e
The situation changes for gravitational waves. Such waves are coherent superpositions
of large numbers of gravitons and are observable classically. In such a case, the argument against the detection of single gravitons does not apply. In short, the strand model
predicts that gravitational waves can be observed. (This prediction, made by many since
1915 and repeated in this text on the basis of the strand model in 2008, came true in
February 2016. The observations also produced the extremely low mass limit of at most
1.2 × 10−22 eV/c2 for any possible mass of the photon.)
Motion Mountain – The Adventure of Physics
a change cannot be distinguished from a quantum fluctuation, because the graviton is
predicted to be massless. Furthermore, the strand model predicts that gravitons do not
interact with photons, because they have no electric charge. In summary, the strand
model predicts:
298
10 g eneral rel ativity d ed uced from stra nds
Vol. IV, page 143
Page 200
What is the gravitational field of a quantum system in a macroscopic superposition? The
issue has been raised by many scholars as an important step towards the understanding
of how to combine gravitation and quantum theory.
The strand model deflates the importance of the issue. The model shows – or predicts,
if one prefers – that the gravitational field of a superposition is the temporal and spatial
average of the evolving quantum system, possibly under inclusion of decoherence.
What is the gravitational field of a single quantum particle in a double-slit experiment? As Figure 38 shows, the gravitational field almost always appears in both slits,
and only very rarely in just one slit.
In summary, in the strand model, the combination of gravitation and quantum theory is much simpler than was expected by most researchers. For many decades it was
suggested that the combination was an almost unattainable goal. In fact, in the strand
model we can almost say that the two descriptions combine naturally.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
The gravit y of superp ositions
Motion Mountain – The Adventure of Physics
An example of such a possible new defect is illustrated in Figure 78. The illustration can be seen as the image of a one-dimensional defect or as the cross section of a
two-dimensional defect. Are such defects stable against fluctuations? The strand model
suggests that they are not. These defects are expected to decay into a mixture of gravitons, black holes, matter and radiation particles. However, this issue is still a topic of
research, and will not be covered here.
Exploring the stability of wormholes, time-like loops and toroidal black holes leads
to similar results. It seems that the strand model should not allow time-like loops of
macroscopic size, since any configuration that cannot be embedded locally into three
flat spatial dimensions is either a particle or a black hole. Alternatively, macroscopic
time-like loops would collapse or decay because of the fluctuations of the strands. In
the same way, wormholes or black holes with non-trivial topology should be unstable
against more usual strand structures, such as particles or black holes.
We also note the strand model does not allow volume defects (black holes being
surface-like defects). The most discussed types of volume defects are macroscopic regions of negative energy. Energy being action per unit time, and action being connected
to crossing changes, the model does not allow the construction of negative-energy regions. However, the strand model does allow the construction of regions with lower
energy than their environment, as in the Casimir effect, by placing restrictions on the
wavelengths of photons.
The strand model thus predicts the absence of additional defects and tangle types.
The final and general connection between tangle types and defects is shown (again) in
Table 11. The next chapter will give details of the tangles corresponding to each particle.
In summary, the strand model reproduces the results of modern quantum gravity and
predicts that the more spectacular defects conjectured in the past – linear defects such
as cosmic strings, surface defects such as wormholes, volume defects such as negativeenergy regions – do not appear in nature.
g enera l rel ativit y d educed from stra nds
299
TA B L E 11 Correspondences between physical systems and mathematical tangles.
Strands
Ta n g l e t y p e
Vacuum
Dark energy
Elementary vector boson
Quark
Lepton
Meson, baryon
many infinite unknotted strands
many fluctuating infinite strands
one infinite strand
two infinite strands
three infinite strands
three or more infinite strands
Higher-order propagating
fermion
Virtual particles
Composed systems
Graviton
Gravity wave
Horizon
Young universe
two or more infinite strands
unlinked
unlinked
a curve
rational tangle
braided tangle
composed of rational
tangles
general rational tangle
open or unlinked strands
many strands
two infinite twisted strands
many infinite twisted strands
many tightly woven infinite strands
closed strand(s)
trivial tangles
separable tangles
specific rational tangle
many graviton tangles
web-like rational tangle
knot (link)
Torsion, curiosities and challenges abou t quantum gravit y
On the one hand, the strand model denies the existence of any specific effects of torsion
on gravitation. On the other hand, the strand model of matter describes spin with the
belt trick. The belt trick is thus the strand phenomenon that is closest to the idea of
torsion. Therefore, exaggerating a bit in the other direction, it could also be argued that
in the strand model, torsion effects are quantum field theory effects.
∗∗
Ref. 196
Ref. 197
Ref. 156
Ref. 198
Ref. 157
Ref. 199
The strand model describes three-dimensional space as made of tangled strands. Several
similar models have been proposed in the past.
The model of space as a nematic world crystal stands out as the most similar. This
model was proposed by Hagen Kleinert in the 1980s. He took his inspiration from
the famous analogy by Ekkehart Kröner between the equations of solid-state elasticity
around line defects and the equations of general relativity.
Also in the 1980s, the mentioned posets have been proposed as the fundamental structure of space. Various models of quantum gravity from the 1990s, inspired by spin networks, spin foams and by similar systems, describe empty space as made of extended
constituents. These extended constituents tangle, or bifurcate, or are connected, or sometimes all of this at the same time. Depending on the model, the constituents are lines,
circles or ribbons. In some models their shapes fluctuate, in others they don’t.
Around the year 2000, another type of Planck-scale crystal model of the vacuum has
been proposed by David Finkelstein. In 2008, a specific model of space, a crystal-like
network of connected bifurcating lines, has been proposed by Gerard ’t Hooft.
All these models describe space as made of some kind of extended constituents in
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 195
Motion Mountain – The Adventure of Physics
Physical system
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10 g eneral rel ativity d ed uced from stra nds
a three-dimensional background. All these models derive general relativity from these
constituents by some averaging procedure. The lesson is clear: it is not difficult to derive
general relativity from a Planck-scale model of space. It is not difficult to unify gravity
and quantum theory. As Luca Bombelli said already in the early 1990s, the challenge for
a Planck-scale model of nature is not to derive gravity or general relativity; the challenge
is to derive the other interactions. So far, the strand model seems to be the only model
that has provided such a derivation.
∗∗
Challenge 185 e
The Planck force is the force value necessary to produce a change ℏ in a Planck time over
a Planck length. The Planck force thus appears almost exclusively at horizons.
∗∗
Already in the 1990s, Leonard Susskind speculated that black holes could be formed by
a single wound-up string. Strands differ from strings; they differ in the number of dimensions, in their intrinsic properties, in their symmetry properties, in the fields they
carry and in the ways they generate entropy. Nevertheless, the similarity with the strand
model of black holes is intriguing.
∗∗
Page 161
Ref. 201
∗∗
Challenge 186 e
The first version of the strand model assumed that space is not defined at the cosmic
horizon, and that therefore, strand impenetrability does not hold there. The same was
thought to occur at black hole horizons. The newest version of the strand model does
not seem to need this exception to impenetrability. Can you explain black hole entropy
without it?
∗∗
Page 35
The strand model also allows us to answer the question whether quantum particles are
black holes: no, they are not. Quantum particles are tangles, like black holes are, but
particles do not have horizons. As a side result, the mass of all particles is lower than a
Planck mass, or more precisely, lower than a Planck mass black hole.
Strands imply that gravity is weaker than the three gauge interactions. This consequence, like the low particle mass just mentioned, is due to the different origins of
gravity and gauge interactions. Gravity is due to the strand tails, whereas gauge interactions are due to the tangle cores. Thus gravity is the weakest interaction in everyday
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 154
In September 2010, two years after the strand model appeared, independent research
confirmed its description of physical space, as already mentioned above. In an extended article exploring the small scale structure of space from several different research
perspectives in general relativity, Steven Carlip comes to the conclusion that all these
perspectives suggest the common idea that ‘space at a fixed time is thus threaded by
rapidly fluctuating lines’.
In 2011, also independently, Marcelo Botta Cantcheff modelled space as a statistic ensemble of one-dimensional ‘strings’. He explained the main properties of space, including the thermodynamic properties of black holes.
Motion Mountain – The Adventure of Physics
Ref. 200
g enera l rel ativit y d educed from stra nds
Ref. 202
Page 8
301
life. The observation of the weakness of gravity at everyday and other energy scales is
sometimes called the weak gravity conjecture. It is naturally valid in the strand model.
The conjecture is also part of the Bronshtein cube shown in Figure 1.
∗∗
For an observer at spatial infinity, a black hole horizon is an averaged-out tight web of
strands. What does a falling observer experience? The question will still capture the
imagination in many years. Such an observer will also see strands; above all, a falling
observer will never hit any singularity. The details of the fall are so involved that they
are not discussed here, because the fall affects both the black hole appearance and the
observer.
∗∗
∗∗
Ref. 203
The strand model makes the point that entanglement and the vacuum – and thus
quantum gravity – have the same nature: both are due to crossing strands. This idea
has been explored independently by Mark van Raamsdonk.
∗∗
⊳ Quantum gravity effects cannot be distinguished from ordinary quantum
fluctuations.
Despite many attempts to disprove it, all experiments so far confirm the conjecture. Because both quantum gravity effects and quantum effects are due to tail fluctuations, the
strand model seems to imply the conjecture.
∗∗
Ref. 204
The strand model of black holes also confirms a result by Zurek and Thorne from the
1980s: the entropy of a black hole is the logarithm of the number of ways in which it
could have been made.
∗∗
Challenge 187 s
Argue that because of the strand model, no black hole can have a mass below the (corrected) Planck mass, about 11 μg, and thus that microscopic black holes do not exist. Can
you find a higher lower limit for the mass?
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
As we have seen, the strand model predicts no observable violation of Lorentz-invariance
– even though it predicts its violation at Planck scale. Strands predict the lack of dispersion, birefringence and opacity of the vacuum. Strands predict that the vacuum has three
dimensions whenever it is observed and that it is unique, without phase transitions. We
already mentioned the impossibility of detecting single gravitons.
All these negative predictions are examples of the ‘no avail’ conjecture:
Motion Mountain – The Adventure of Physics
Can black hole radiation be seen as the result of trying to tear vacuum apart? Yes and no.
The answer is no, because physical vacuum cannot be torn apart, due to the maximum
force principle. But the answer is also yes in a certain sense, because the maximum force
is the closest attempt to this idea that can be realized or imagined.
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10 g eneral rel ativity d ed uced from stra nds
∗∗
Do atoms or the elementary fermions moving inside matter emit gravitational radiation,
and why? The question was already raised by Albert Einstein in 1916. The strand model
answers the issue in the same way as textbook physics. Elementary particles in atoms –
in the ground state – do not emit gravitational waves for the same reason that they do not
emit electromagnetic waves: for atoms in the ground state, there is no lower state into
which they could decay. Excited atomic states do not emit gravitational waves because of
the extremely low emission probability; it is due to the extremely low mass quadrupole
values.
∗∗
Ref. 205
∗∗
Page 58
It is often stated that general relativity does not allow the description of fermions if the
topology of space is kept fixed. This is wrong: the strand model shows that fermions can
be included in the case that space is seen as an average of extended fundamental entities.
∗∗
Predictions of the strand model abou t gravit y
As just presented, the strand model makes several verifiable predictions about general
relativity and quantum gravity.
— The maximum energy speed in nature is ?, at all energy scales, in all directions, at all
times, at all positions, for every physical observer. This agrees with observations.
— No deviations from special relativity appear for any measurable energy scale, as long
as gravity plays no role. No ‘double’ or ‘deformed special relativity’ holds in nature,
even though a maximum energy-momentum for elementary particles does exist in
nature. Whenever special relativity is not valid, general relativity, or quantum field
theory, or both together need to be used. This agrees with observations.
— There is a maximum power or luminosity ?5 /4?, a maximum force or momentum
flow ?4 /4?, and a maximum mass change rate ?3 /4? in nature. The limits hold for
all energy scales, in all directions, at all times, at all positions, for every physical observer. These predictions agree with observations, though only few experimental ob-
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 206
Following the fundamental principle of the strand model, ? is the fundamental constant
that describes gravitation. The strand model predicts that gravity is the same for all energy scales; in other words, the constant ? is not expected to change with energy. This
agrees with recent results from quantum gravity and distinguishes the behaviour of ?
from that of the coupling constants in the gauge interactions of particle physics.
Motion Mountain – The Adventure of Physics
Challenge 188 ny
In 2009 Mikhail Shaposhnikov and Christof Wetterich argued that if gravitation is
‘asymptotically safe’, there is no physics beyond the standard model and the Higgs mass
must be around 126 GeV – exactly the value that was found experimentally a few years
afterwards. A quantum field theory is called asymptotically safe if it has a fixed point
at extremely high energies. Does the strand model imply that gravity is – maybe only
effectively – asymptotically safe?
g enera l rel ativit y d educed from stra nds
Vol. V, page 146
All listed predictions are unspectacular; they are made also by other approaches that
contain general relativity as limiting cases. In particular, the strand model, like many
other approaches, predicts:
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
servations are close to these limit values.
— There is a minimum distance and a minimum time interval in nature. There is a maximum curvature and a maximum mass density in nature. There are no singularities in
nature. All this agrees with observations, including the newly discovered black hole
mergers.
— The usual black hole entropy expression given by Bekenstein and Hawking holds.
The value has never been measured, but is consistently found in all calculations performed so far. In fact, black hole entropy is related to the Fulling–Davies–Unruh effect, which itself is related to the Sokolov–Ternov effect. This latter effect has already
been observed in several accelerators, for the first time in 1971. However, it now seems
that this observation does not actually prove black hole entropy.
— There are no deviations from general relativity, as described by the Hilbert action, for
any measurable scale. The only deviations appear in situations with a few strands,
i.e., in situations where quantum theory is necessary. This agrees with observations,
including those of black hole mergers, but experimental data are far from sufficient;
undetected deviations could still exist.
— There is no modified Newtonian dynamics, or MOND, with evolution equations that
differ from general relativity. The rotation curves of stars in galaxies are due to dark
matter, to other conventional explanations, or both.
— There is no effect of torsion that modifies general relativity. This agrees with observations.
— There is no effect of higher derivatives of the metric on the motion of bodies. This
agrees with observations, but experimental data are far from sufficient.
— Observations are independent of the precise strand fluctuations. Mathematical consistency checks of this prediction are possible.
— No wormholes, no negative energy regions and no time-like loops exist. This agrees
with observations, but experimental data are far from covering every possible loophole.
— The Penrose conjecture and the hoop conjecture hold. Here, a mathematical consistency check is possible.
— There are no cosmic strings and no domain walls. This agrees with observations, but
experimental data are far from exhaustive.
— Gravitons have spin 2; they return to their original state after a rotation by π and are
bosons. This agrees with expectations.
— Gravitons cannot be detected, due to the indistinguishability with ordinary quantum
fluctuations of the detector. This agrees with data so far.
— Atoms emit neither gravitational waves nor gravitons.
— Gravitational waves exist and can be detected. This agrees with experiment; the final
confirmation occured in late 2015.
— The gravitational constant ? does not run with energy – as long as the strand diameter
can be neglected. In this domain, ? is not renormalized. This prediction agrees with
expectations and with data, though the available data is sparse.
Motion Mountain – The Adventure of Physics
Ref. 207
303
304
Page 307
Ref. 97
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⊳ With the exception of the cosmological constant and of particle mases (and
possibly the Sokolov–Ternov effect), no quantum gravity effects will be observed.
Gravity will not yield new measurable quantum effects. So far, this prediction agrees
with experiment – and with almost all proposed models of quantum foam in the research literature. In other words, we have found no unexpected experimental predictions
from the strand model in the domain of quantum gravity. This is the so-called ‘no avail’
conjecture; and it is not a surprise.
In fact, the Bronshtein cube of Figure 1 also implies:
⊳ There is no separate theory of quantum gravity that includes relativity but
does not include the other interactions.
cosmolo gy
The finiteness of the universe
In the strand model, cosmology is based on the following idea:
⊳ The universe is made of one fluctuating strand that criss-crosses from and to
the horizon. Fluctuations increase the complexity of the strand tangledness
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Cosmology is an active field of research, and new data are collected all the time. We start
with a short summary.
The sky is dark at night. This and other observations about the red shift show that
the universe is surrounded by a horizon and is of finite size and age. Precise measurements show that cosmic age is around 13 800 million years. The universe expands; the
expansion is described by the field equations of general relativity. The universe’s expansion accelerates; the acceleration is described by the cosmological constant Λ – also called
dark energy – that has a small positive value. The universe is observed to be flat, and, averaged over large scales, homogeneous and isotropic. At present, the observed average
matter density in the universe is about 18 times smaller than the energy density due to
the cosmological constant. In addition, there is a large amount of matter around galaxies
that does not radiate; the nature of this dark matter is unclear. Galaxy formation started from early density fluctuations; the typical size and amplitude of the fluctuations are
known. The topology of space is observed to be simple.
The strand model, like any unified description of nature, must reproduce and explain
these measurement results. Otherwise, the strand model is wrong.
Motion Mountain – The Adventure of Physics
There is no room for a theory of relativistic quantum gravity in nature.
In short, strands lead us to expect deviations from general relativity only in two domains: in cosmology (such as changes of the cosmological constant) and in particle physics. The rest of this chapter deals with cosmology. The subsequent chapters focus on
particle physics.
cosm olo gy
Universe’s
horizon
or
‘border
of space’
(pink)
305
Universal
tangle
(blue
lines)
Particle
tangle
(tangled
blue
lines)
Physical
space or
vacuum
(white)
Background
space
(grey)
Background
space
(grey)
Physical space (white) matches background space (grey) only inside the horizon. Physical space thus
only exists inside the cosmic horizon.
over time.
In other words, the strands of all particles are woven into the sky. The existence of finite
size and of finite age then follows automatically:
Ref. 208
Ref. 209
Page 307
Page 285
The strand model thus has a simple explanation for the finiteness of the universe and the
horizon that bounds it: The universe’s horizon is the weave that joins all strand tails. A
schematic illustration of the cosmic horizon is given in Figure 79.
The strand model predicts that the horizon of the universe is an event horizon, like
that of a black hole. Until 1998, this possibility seemed questionable; but in 1998, it was
discovered that the expansion of the universe is accelerating. This discovery implies that
the cosmic horizon is indeed an event horizon, as required by the strand model.
In fact, the strand model predicts that all horizons in nature are of the same type. This
also means that the universe is predicted to saturate Bekenstein’s entropy bound. More
precisely, the strand model predicts that the universe is a kind of inverted back hole. Like
for any situation that involves a horizon, the strand model thus does not allow us to make
statements about properties ‘before’ the big bang or ‘outside’ the horizon. As explained
above, there is nothing behind a horizon.
In particular, the strand model implies that the matter that appears at the cosmic horizon during the evolution of the universe appears through Bekenstein–Hawking radiation. This contrasts with the ‘classical’ explanation form general relativity that new matter appears simply because it existed behind the horizon beforehand and then crosses the
horizon into the ‘visible part’ of the universe.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
⊳ The universe’s horizon appears at the age or distance at which the strand
crossings cannot be embedded any more into a common three-dimensional
background space. The horizon expands over time.
Motion Mountain – The Adventure of Physics
F I G U R E 79 In the strand model, the universe is limited by a horizon, as schematically illustrated here.
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10 g eneral rel ativity d ed uced from stra nds
‘horizon’
‘time’
F I G U R E 80 An extremely simplified view of how the universe evolved near the big bang. In this
evolution, physical time, space and the surrounding horizon are in the process of getting defined.
“
Or cette liaison ou cet accommodement de
toutes les choses créées à chacune, et de
chacune à toutes les autres, fait que chaque
substance simple a des rapports qui expriment
toutes les autres, et qu’elle est par conséquent
un miroir vivant perpétuel de l’univers.*
Gottfried Wilhelm Leibniz, Monadologie, 56.
The big bang – withou t infl ation
Ref. 210
Page 291
”
Any expanding, homogeneous and isotropic matter distribution had earlier stages of
smaller size and higher density. Also the universe has been hotter and denser in the past.
But the strand model also states that singularities do not appear in nature, because there
is a highest possible energy density. As a result, the big bang might be imagined as illustrated in Figure 80. Obviously, physical space and time are not well defined near that
* ‘Now this connexion or adaptation of all created things to each and of each to all, means that each simple
substance has relations which express all the others, and, consequently, that it is a perpetual living mirror
of the universe.’
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Challenge 189 e
Motion Mountain – The Adventure of Physics
Page 101
We note that modelling the universe as a single strand implies that it contains tangles.
In other words, the strand model makes the prediction that the universe cannot be empty,
but that it must contain particles. Strand cosmology also confirms that the question of
initial conditions for the universe does not really make sense: particles appear at the
horizon.
We also note that describing the universe as made of a single strand is a natural,
but somewhat unusual way to incorporate what particle physicists and cosmologists like
to call holography. Holography is the idea that all observables of a physical system are
defined on a boundary enclosing the system. In other words, if we would know, at Planck
scale, everything that happens on the walls of a room, we could know everything that is
and goes on inside the room. Instead of holography, we could also call it the NSA dream.
Holography is a consequence of the extension of the fundamental constituents of nature
and is a natural consequence of the strand model. As a consequence, strand cosmology
naturally reproduces holographic cosmology – though not fully, as is easy to check.
cosm olo gy
307
situation, so that the figure has to be taken with a grain of salt. Nevertheless, it shows how
the evolution of the universe can be seen as resulting from the increase in tangledness of
the strand that makes up nature.
The strand model leads to the conjecture that the evolution of the universal strand just
after the big bang automatically yields both a homogeneous and isotropic matter distribution and a flat space. Also the scale invariance of early density fluctuations seems natural in the strand model. In short, the strand model looks like a promising alternative
to inflation: the hypothesis of inflation becomes unnecessary in the strand model, because strand cosmology directly makes the predictions that seem so puzzling in classical
cosmology. This issue is still subject of research.
The cosmological constant
For a spherical system, this yields
?⩽?
Ref. 212
??3
.
4?ℏ
(195)
The application of this inequality to the universe is called the Fischler–Susskind holographic conjecture. Using the energy–entropy relation ? = ?? valid for any holographic
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 211
In particular, the strand model predicts a small positive cosmological constant, i.e., a
constant that leads to a small repulsion of masses.
The relation between the cosmological constant and the radius of the universe can
be found also with another, more precise argument, based on holography, and given by
Balázs and Szapudi. Bekenstein’s holographic entropy bound states that for all systems
of size ? and energy ? we have
2π?
? ⩽ ??
.
(194)
ℏ?
Motion Mountain – The Adventure of Physics
Page 210
The cosmological constant is due to strands. When three mutually orthogonal strands
come together at a point, they cannot be completely straight; they are slightly bent. Equivalently, because the strands of the vacuum touch each other, due to the statistics of the
fluctuations, there is a slight effective repulsion between them. This is the strand model
for the cosmological constant.
In short, in the strand model, vacuum energy, or dark energy, is due to the cosmological constant, which itself is due to strand fluctuations. As we saw above, the strand
model predicts that the cosmological constant Λ for infinitely extended flat space vanishes, because the vacuum energy density vanishes in that case. But the strand model
also predicts that for finite extension, the cosmological constant does not vanish. Indeed, in the strand model, a finite size limits the fluctuations of the strands. Fluctuations
with sizes larger than the size of space are frozen out; this leads to an effective repulsion of strands that in turn leads to a cosmological constant given by (the square of) the
extension of space:
1
Λ= 2 .
(193)
?max
308
10 g eneral rel ativity d ed uced from stra nds
system, and introducing the energy density ?E , we get the limit given by
?E ⩽
Page 290
(196)
Using the formula for temperature ? = ℏ?/2π?? for a horizon – deduced above from the
strand model – we get
1 3?4
1 3?4
?E ⩽
=
.
(197)
? 2? 4π?2 2?
The strand model predicts that the universe saturates the entropy bound. In other words,
assuming that ? is ? times the age of the universe ?0 , the strand model predicts that the
total energy density of the universe is equal to the so-called critical energy density.
The equality of the measured total energy density and the critical density is well
known. These measurements show that the present total energy density of the universe
is about
?E vac ≈ 8.5 ⋅ 10−10 J/m3 or ?m vac = 0.94(9) ⋅ 10−26 kg/m3 .
(198)
In other words, the strand model, like the holographic argument, predicts that the cosmological constant is limited by
3
Λ⩽ 2 2 .
(199)
? ?0
Ref. 214
Ref. 215
Challenge 190 s
The result confirms the result of expression (193). Modern measurements yield 74 % of
the maximum possible value.
The argument for the value of the cosmological constant can be made for any age of
the universe. Therefore,
⊳ The strand model predicts that the cosmological constant Λ(?) decreases
with increasing radius of the universe.
In particular, there is no need for a scalar field that makes the cosmological constant
decrease; the decrease is a natural result of the strand model. The strand model states
that the cosmological constant appears in the field equations as a quantum effect due to
the finite size of the universe. The strand model thus implies that there is no separate
equation of motion for the cosmological constant, but that the constant appears as a
large-scale average of quantum effects, as long as the size of the universe is limited.
In summary, the strand model predicts that not only the field equations of general relativity, but also the amount of dark energy, the expansion of the universe and its acceleration result from strand fluctuations. The cosmological constant changes roughly with
the inverse square of time. In particular, the strand model implies that the effect proposed by Wiltshire – that the cosmological constant is an artefact of the inhomogeneity
of matter distribution – is not fundamental, but may at most influence the value somewhat. (Could the difference between the maximum possible and the measured value of
the cosmological constant be due to this effect?)
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 213
Motion Mountain – The Adventure of Physics
Ref. 213
? 3??3
.
? 4ℏ?
cosm olo gy
309
The value of the mat ter densit y
The strand model predicts that horizons emit particles. As a consequence, the strand
model predicts an upper limit for the number ?b of baryons that could have been emitted
by the cosmic horizon during its expansion. For a horizon shining throughout the age
of the universe ?0 while emitting the maximum power ?5 /4?, we get
?b0 ⩽
Ref. 213
(200)
Equality would hold only if the contributions of photons, electrons, neutrinos and dark
matter could be neglected. In short, using the age ?0 = 13.8 Ga, the strand model predicts
that at most 2.6 ⋅ 1079 baryons exist in the universe at present. Modern measurements
indeed give values around this limit.
In other terms, the strand model states that the sum of all particle energies in the
universe is at most ?0 ?5 /4?, or 50 % of the critical density; this includes observable matter
as well as dark matter. The experimental value for the total matter density is about 26 %
of the critical density. In observations, 4 % of the matter density is observed, and 22 % is
dark. We will discuss the nature of dark matter later on.
The strand model also makes a statement on the change of matter density with time.
As explained, the number of baryons is predicted to increase with time ?, due to their
appearance at the horizon. Since the radius will also increase (roughly) with time, we get
This unexpected prediction contrasts with the usually assumed 1/?3 dependence in a
matter-dominated universe. The prediction has yet to be tested with observations. We
note that the strands imply that the ratio between matter density and vacuum energy
density is related to the details of the radius increase during the history of the universe.
Open challenge: What are the effects of dark mat ter?
Challenge 191 ny
Challenge 192 ny
Challenge 193 ny
In the arguments above, is there a factor of order 2 missing that induces incorrect conclusions about dark matter density? Might the prediction of dark matter increase, decrease
or even disappear after correction of this missing numerical factor?
Conventionally, it is argued that cold dark matter exists for two reasons: First, it is
necessary to grow the density fluctuations of the cosmic microwave background rapidly
enough to achieve the present-day high values. Secondly, it is needed to yield the observed amplitudes for the acoustic peaks in the cosmic background oscillations. Can the
strand model change these arguments?
Later on, it will be argued that in the strand model, dark matter is a mixture of conventional matter and black holes. How does this dark matter prediction explain the galaxy
rotation curves? This leads to a really speculative issue: Could tangle effects at the scale
of a full galaxy be related to dark matter?
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
⊳ The strand model predicts that matter density decreases roughly as 1/?2 .
Motion Mountain – The Adventure of Physics
Ref. 213
Page 353
?0 ?5 /4?
= 2.6 ⋅ 1079 .
?b ?2
310
10 g eneral rel ativity d ed uced from stra nds
The top ology of the universe
In the strand model, physical space-time, whenever it is defined, cannot be multiply connected. Also all quantum gravity approaches make this prediction, and the strand model
confirms it: because physical space-time is a result of averaging strand crossing switches,
non-trivial topologies (except black holes) do not occur as solutions. For example, the
strand model predicts that wormholes do not exist. In regions where space-time is undefined – at and beyond horizons – it does not make sense to speak of space-time topology. In these regions, the fluctuations of the universal strand determine observations.
In short, the strand model predicts that all searches for non-trivial macroscopic (and microscopic) topologies of the universe, at both high and low energies, will yield negative
results. So far, this prediction agrees with all observations.
Predictions of the strand model abou t cosmology
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
— The universe is not empty. (Agrees with observation.)
— Its integrated luminosity saturates the power limit ?5 /4?. (Agrees with observation.)
— The universe’s energy density saturates the entropy bound. (Agrees with observation.)
— The are no singularities in nature. (Agrees with observation.)
— Dark energy results from vacuum strand fluctuations. (Agrees with observation.)
— Dark energy, or vacuum energy, is completely described by a cosmological constant Λ(?)
that is positive and changes with the radius ?(?) of the universe as 1/?(?)2. (This prediction differs from the usual cosmological models, which assume that Λ is constant
or changes with time in other ways. The strand prediction might be checked in the
near future by testing whether the minimum acceleration around galaxies changes
with distance – if this minimum is related to Λ.)
— The number of baryons in nature is limited by the maximum luminosity times the
age of the universe, thus by 2.6 ⋅ 1079 baryons. (Agrees with observation.)
— The matter density of the universe decreases with age, roughly as ?? ∼ 1/?2 . (Checks
are under way. This prediction differs from the usual cosmological models.)
— There is nothing behind the cosmic horizon. Matter, energy and space appear at the
horizon. (Agrees with observations and requirements of logic.)
— Early density fluctuations are scale-invariant. (Agrees with observation.)
— The universe is flat and homogeneous. (Agrees with observation.)
— Apart from the cosmological constant Λ(?), all other fundamental constants of nature
are constant over time and space. (Agrees with observation, despite claims of the
contrary.)
— Inflation is unnecessary.
— The universe’s topology is trivial. There are no wormholes, no time-like loops, no
cosmic strings, no toroidal black holes, no domain walls and no regions of negative
energy. (Agrees with observation.)
— The above statements are independent of the precise fluctuation details. (Can be
tested with mathematical investigations.)
Motion Mountain – The Adventure of Physics
In the domain of cosmology, the strand model makes the following testable predictions.
sum m ary on m illennium issues a b ou t rel ativity a nd cosm olo gy
311
All these predictions can and will be tested in the coming years, either by observation or
by computer calculations.
summary on millennium issues abou t rel ativit y and
cosmolo gy
Page 8
Page 147
Page 162
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 277
Motion Mountain – The Adventure of Physics
Page 353
We have deduced special relativity, general relativity and cosmology from the strand
model. The fundamental principle of the strand model implies the invariant Planck units,
the Lagrangian and action of general relativity, the finiteness of the universe and, above
all, it explains in simple terms the entropy of black holes.
Space-time foam is replaced by the strand model of the vacuum: empty space is the
time-average of untangled strands. More precisely, space is the thermodynamic average
of crossing switches that are due to shape fluctuations of untangled strands.
The strand model – and in particular, the strand model of the vacuum – explains the
number of space-time dimensions, the vacuum energy density, the matter density and
the finiteness of the universe. The cosmological constant is a consequence of the finite
size of the universe. The issue of the initial conditions of the universe has been defused.
The macroscopic and microscopic topology of the universe is simple. And dark matter
is predicted to be, as shown in the next chapter, a combination of conventional matter
and black holes.
The most important predictions of the strand model are the decrease of the cosmological constant with time and the absence of inflation. Various experiments will test these
predictions with increased precision in the coming years. So far, measurements do not
contradict these predictions.
The strand model confirms that the speed of light ? and the corrected Planck force
?4 /4? are limit values. The strand model also predicts that no variation in space and
time of ?, ?, ℏ and ? can be detected, because they define all measurement units.
The strand model predicts that the cosmological constant and the masses of the elementary particles are the only quantum effects that will be observed in the study of
gravitation. Strands strongly suggest that additional effects of quantum gravity cannot
be measured. In particular, no effects of space-time foam will be observed.
The strand model is, at present, the simplest – but not the only – known model of
quantum gravity that allows deducing all these results. In particular, the strands’ explanation of black hole entropy is by far the simplest one known.
General relativity is an approximation of the strand model. The approximation appears when the quantum of action and, in particular, the strand diameter are neglected.
General relativity and cosmology thus appear by approximating ℏ as 0 in the strand
model – as required by the Bronshtein cube of physics that is shown in Figure 1. Strands
imply that general relativity is valid for all energies below the Planck energy. In other
words, the strand model is not a generalization of general relativity. This conforms to
the list of requirements for the final theory.
If we look at the millennium list of open issues in physics, we see that – except for the
issue of dark matter – all issues about general relativity and cosmology have been settled.
The strand model explains the mathematical description of curved space-time and of
312
Page 275
10 g eneral rel ativity d ed uced from stra nds
general relativity. The strand model also provides a simple model of quantum gravity –
maybe the simplest known one. Above, we had already shown that the strand model explains all mathematical structures that appear in quantum theory and in particle physics.
Together with the results from this chapter we can now say: the strand model explains all
concepts, i.e., all mathematical structures that appear in physical theories. In particular,
strands explain the metric, curvature, wave functions, field intensities – and the probabilistic behaviour of all of them. They all result from averaging crossing switches.
In summary, starting from the fundamental principle of the strand model, we have
understood that strands are the origin of gravitation, general relativity, quantum gravity
and cosmology. We have also understood the mathematical description of gravitation –
and, before, that of quantum physics – found in all textbooks. These results encourage us
to continue our quest. Indeed, we are not done yet: we still need to deduce the possible
elementary particles and to explain their properties.
Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
C h a p t e r 11
T HE PART ICLE SPECT RUM DEDUCED
FROM ST R ANDS
Ref. 216
Page 161
”
Particles and quantum numbers from tangles
In nature, we observe three entities: vacuum, horizons and particles. Of these,
(quantum) particles are localized entities with specific intrinsic properties, i.e., properties that do not depend on their motion.
In nature, all the intrinsic properties of every particle, every object and every image
are completely described by three types of basic properties: (1) the elementary particles
they contain, (2) their behaviour under space-time transformations, (3) their interactions. The full list of these basic intrinsic properties of particles is given in Table 12.
Given the basic intrinsic properties for each elementary particle, physicists can deduce
all those intrinsic particle properties that are not listed; examples are the half life, decay
modes, branching ratios, electric dipole moment, T-parity, gyromagnetic ratio or electric
polarizability. Of course, the basic intrinsic properties also allow physicists to deduce
every property of every object and image, such as size, shape, colour, brightness, density,
** Voltaire (b. 1694 Paris, d. 1778 Paris) was an influential philosopher, politician and often satirical writer.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
trands describe quantum theory, gauge interactions and general relativity. But do
trands also settle all issues left open by twentieth-century physics? Do they
ettle the origin of all the elementary particles, their quantum numbers, their masses
and their mixing angles? How does the infinite number of possible tangles lead to a finite
number of elementary particles? And finally, do strands explain the coupling constants?
In the millennium list of open issues in fundamental physics, these are the issues that
remain. The strand model is correct only if these issues are resolved.
In this chapter, we show that the strand model indeed explains the known spectrum of
elementary particles, including the three generations of quarks and leptons. The strand
model is the first approach of modern physics that can provide such an explanation.
It should be stressed that from this point onwards, the ideas are particularly speculative. In the chapters so far, the agreement of the strand model with quantum field
theory and general relativity has been remarkable. The following chapters assign specific
tangles to specific particles. Such assignments are, by nature, not completely certain. The
speculative nature of the ideas now becomes particularly apparent.
Motion Mountain – The Adventure of Physics
S
“
No problem can withstand the assault of
sustained thinking.
Voltaire**
314
11 the pa rticle spectrum d ed uced from stra nds
Tangles made of one strand :
1a
curve
1b
prime long
knot
1c
composed
long knot
1a’
unknot
1b’
prime
knot
not in the
strand model
found in the early universe and at horizons
1c’
composed
knot
Correspondence :
not in the
strand model
F I G U R E 81 Examples for each class of tangles made of one strand.
particles made of one strand
Page 174
In the strand model, all particles made of one strand have spin 1, are elementary, and are
bosons. Conversely, all massless elementary spin-1 bosons can only have two tails, and
thus must be made of a single strand. Such one-stranded tangles return to the original
strand after a core rotation by 2π. Massive elementary spin-1 bosons can have one or
more strands. Tangles of more than one strand can only have spin 1 if they represent
massive elementary or composed particles. In short, classifying one-stranded tangles
allows classifying all elementary gauge bosons.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 173
elasticity, brittleness, magnetism or conductance.
In short, understanding all properties of matter and images thus only requires understanding the basic properties of quantum particles; and understanding the basic properties of quantum particles only requires understanding the basic properties of the elementary particles.
The strand model states that all elementary (and all composed) particles are tangles
of strands. This leads us to ask: Which tangle is associated to each elementary particle?
What kinds of elementary particles are possible? Do these tangles reproduce, for each
elementary particle, the observed values of the basic properties listed in Table 12?
It turns out that the strand model only allows a limited number of elementary particles.
In addition, the tangles of these elementary particle have intrinsic properties that match
the observed properties. To prove these strong statements, we first recall that all massive
elementary particles are represented by an infinite sequence of tangles. We now explore
tangles according to the number of strands they are made of.
Motion Mountain – The Adventure of Physics
photon,
W and Z,
gluon,
vacuum
pa rticles m a d e of one stra nd
315
TA B L E 12 The full list of basic intrinsic properties of quantum particles, from which all other observed
intrinsic properties of particles, objects and images can be deduced.
P roperty
P o s s i b l e Va l u e
Determines
Quantum numbers due to space-time symmetries:
Spin ? or ?
P parity
C parity
integer or half-integer
multiple of ℏ
even (+1) or odd (−1)
even (+1) or odd (−1)
statistics, rotation behaviour, conservation
behaviour under reflection, conservation
behaviour under charge conjugation,
conservation
Interaction properties:
Mass ?
Weak charge
Mixing angles
CP-violating phases between 0 and π/2
rational multiple of strong
coupling constant
Lorentz force, coupling to photons,
conservation
weak scattering and decays, coupling to W
and Z, partial conservation
mixing of quarks and neutrinos, flavour
change
degree of CP violation in quarks and
neutrinos
confinement, coupling to gluons,
conservation
Flavour quantum numbers, describing elementary particle content:
Lepton number(s) ?? integer(s)
Baryon number ?
integer times 1/3
Isospin ?? or ?3
+1/2 or −1/2
Strangeness ??
integer
Charmness ??
integer
Bottomness ??
integer
Topness ??
integer
conservation in strong and e.m.
interactions
conservation in all three gauge
interactions
up and down quark content, conservation
in strong and e.m. interactions
strange quark content, conservation in
strong and e.m. interactions
charm quark content, conservation in
strong and e.m. interactions
bottom quark content, conservation in
strong and e.m. interactions
top quark content, conservation in strong
and e.m. interactions
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Strong charge, i.e.,
colour
gravitation, inertia
Motion Mountain – The Adventure of Physics
Electric charge ?
between 0 and the Planck
mass
integer multiples of one
third of electron or proton
charge
rational multiple of weak
coupling constant
between 0 and π/2
316
11 the pa rticle spectrum d ed uced from stra nds
Photon γ :
Weak vector bosons W – before SU(2) breaking :
Wx
Wy
W0
Gluons g :
Motion Mountain – The Adventure of Physics
though, for clarity, the gluons are shown here using their complementary two-strand moves.
Mathematicians have already classified one-stranded tangles; they are usually called
open knots or long knots. To get an overview, we list an example for each class of onestranded tangles on the left-hand side of Figure 81. For completeness, closed curves are
shown on the right-hand side of the figure. We now explore each of these classes.
Unknot ted curves
Page 250
The simplest type of tangle made of one strand is an unknotted curve, shown as example
1a in Figure 81. The study of gauge interactions has shown that unknotted strands are,
depending on their precise average shape, either vacuum strands or gauge bosons.
The time-average of a vacuum strand is straight. A single strand represents a particle
if the time-averaged strand shape is not a straight line.
In the strand model, vacuum strands in flat space are, on average, straight. In this
property, vacuum strands differ from gauge bosons, which, on average, have curved
strands, and thus carry energy.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
F I G U R E 82 The gauge bosons in the strand model. All differ from vacuum by one curved strand –
pa rticles m a d e of one stra nd
317
Gauge bosons – and R eidemeister moves
Page 222
Gauge bosons are the carrier particles of the interactions. In the strand model, the gauge
interactions are due to the three Reidemeister moves. The electromagnetic, the weak and
the strong interaction correspond to respectively the first, second and third Reidemeister
move. As we have seen above, when the three Reidemeister moves deform fermion tangle
cores they generate U(1), SU(2) and SU(3) gauge symmetries. The detailed exploration
of the correspondence between tangle deformation and gauge theory led us to the gauge
boson tangles shown in Figure 82.
⊳ All gauge bosons – before symmetry breaking when applicable – are single,
curved strands.
Page 257
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 244
Motion Mountain – The Adventure of Physics
Page 226
A single strand represents a particle if the time-averaged strand shape is not a straight
line. The lack of straightness implies non-vanishing energy. A single-strand particle can
thus be either a strand with a bulge or a strand whose tails are not aligned along a straight
line.
As explained above, the first Reidemeister move, the twist, leads to the modelling of
photons as helical strands. Therefore, photons have vanishing mass and two possible polarizations. Photons do not have tangled, localized family members; photons are massless. Their specific unknotted and twisted strand shapes also imply that photons generate
an Abelian gauge theory and that photons do not interact among themselves. Automatically, photons have no weak and no strong charge. The strand model further implies
that photons have negative P-parity and C-parity, as is observed.
The study of the second Reidemeister move, the poke, showed that deformations induced by pokes can also involve braiding of tangle tails; this leads to the symmetry breaking of the weak interaction. As a result, the observed W and the Z boson strands become
massive. The tangle of the W is chiral, and thus it is electrically charged; the tangle of the
Z is achiral and thus electrically neutral. Being tangled, the W and the Z also carry weak
charge and thus interact among themselves, generating a non-Abelian gauge theory. The
strand model also implies that the W and the Z have no P-parity, no C-parity and no
colour charge, as is observed.
The study of the third Reidemeister move, the slide, led us to the existence of eight
gluons. The eight gluons are unknotted, thus they carry no mass, no electric charge and
no weak charge. Each gluon tangle has two possible polarizations. The strand model
of gluons also implies that they have negative P-parity and no C-parity, as is observed.
Gluons tangles carry colour and interact among themselves, thus they generate a nonAbelian gauge theory. In contrast to the other two interactions, free, single gluons are
short-lived, because their structure induces rapid hadronization: when gluons act on
the vacuum, quark–antiquark pairs are produced. Gluons do not have tangled family
members; they are massless in the high energy limit, when their tails are aligned.
For completeness we mention that by assignment, all gauge bosons differ from vacuum by a single curved strand, have vanishing lepton and baryon numbers, and thus
also lack all flavour quantum numbers. All this is as observed.
The strand model explains the lack of classical SU(2) field waves as a consequence
of the breaking of the SU(2) symmetry and the consequent mass of the weak bosons.
318
11 the pa rticle spectrum d ed uced from stra nds
Strands explain the lack of classical SU(3) waves, also called gluonic waves, as a consequence of the topological impossibility to produce such waves, which is related to the
infinite mass of single free gluons.
In somewhat sloppy language we can say that the shape and the effects of photons are
one-dimensional, those of the unbroken weak bosons are two-dimensional, and those
of the gluons are three-dimensional. This is the essential reason that they reproduce the
U(1), SU(2) and SU(3) groups, and that no higher gauge groups exist in nature.
In summary, Reidemeister’s theorem implies that the list of known gauge bosons with
spin 1 is complete. But the list of possible tangles made of a single strand is much more
extensive; we are not done yet.
Open or long knots
Page 251
Closed tangles: knots
Summary on tangles made of one strand
Page 275
In summary, a single strand represents a particle if the strand shape is, on average, not a
straight line. This distinguishes a vacuum strand from a particle strand. A particle strand
can thus be a strand with a bulge or a strand whose tails are not aligned along a straight
line. All tangles made of one open strand represent elementary particles of spin 1, thus
elementary vector bosons.
Massless elementary spin-1 particles are made of one open strand also because other
tangles cannot reproduce both zero mass and the spin-1 behaviour under rotations: only
one-stranded tangles return to the original strand after a core rotation by 2π and allow
vanishing mass at the same time.
In the strand model, all tangles made of one open curved strand are assigned to the
known gauge bosons. The strand model correctly reproduces and thus explains the gauge
boson spectrum and the quantum numbers for each gauge boson. In short, there is no
room for additional elementary gauge bosons.
In other words, the strand model predicts that all gauge bosons and thus all interactions are already known. We have thus a second argument – after the non-existence of
other gauge groups – stating that no other gauge interaction exists in nature. (Both arguments against the existence of other gauge interactions are related; in particular, both
are due to the three-dimensionality of space.) In particular, we find again that grand
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Figure 81 shows, on the right-hand side, examples for all classes of closed tangles of one
strand, i.e., of tangles without tails. They are usually just called knots in mathematics. In
the strand model knots may appear only in the early universe, and maybe near horizons.
They do not seem to have physical relevance and we do not explore them here.
Motion Mountain – The Adventure of Physics
Single strands could also contain knotted regions. We have explained earlier on that all
such possibilities – mathematically speaking, all so-called open knots or long knots – have
no relation to particles. In the strand model, they cannot appear and thus play no role.
The original strand model from 2008 did include such configurations as particles (for
example as W and Z bosons), but it now – i.e., after 2014 – seems that this inclusion is
an unnecessary complication.
pa rticles m a d e of t wo stra nds
319
Tangles made of two strands :
2a
trivial
tangle
2b
simple
crossing
2c
rational
tangle
(locally
unknotted)
2d
prime
tangle
(locally
unknotted)
2e
locally
knotted
tangle
Particle correspondence :
elementary:
quark or
graviton,
with higher
orders
not in
the strand
model
not in
the strand
model
2a’
unlink
2b’
Hopf
link
2c’
rational
link
2d’
prime
link
2e’
composed
link
2f
mixed
open-closed
tangle
Particle correspondence : none - only found in the early universe or near horizons.
F I G U R E 83 Possible tangles made of two strands.
unification and supersymmetry are not allowed in nature.
particles made of t wo strands
In the strand model, particle tangles can also be made of two strands. Examples for all the
classes of two-stranded tangles are given in Figure 83. Each class has a physical particle
assignment.
— The simplest tangle made of two strands is the trivial tangle, shown as example 2a
in Figure 83. In the strand model, the trivial tangle, like all separable tangles, is a
composite system. Each of the two strands can represent either the vacuum or a gauge
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
composed
of vacuum
or gauge
bosons; see
also 2c
Motion Mountain – The Adventure of Physics
composed
of vacuum
or gauge
bosons
320
11 the pa rticle spectrum d ed uced from stra nds
boson. Simply stated, the trivial tangle of two strands is not an elementary particle.
— The simplest non-trivial tangle made of two strands is the crossing, shown as 2b in
Figure 83. In the strand model, the crossing appears as part of the W and Z bosons;
in addition, for certain tail configurations, it can represent a graviton or the simplest
state of a down quark, as we will see below.
— A new class of tangles are the rational tangles, represented by example 2c in the figure.
A rational tangle is a tangle that can be untangled by moving its tails around. (Also
example 2b is a rational tangle.) Rational tangles are distinct from prime and from
locally knotted tangles, shown as examples 2d and 2e, which require pulling the tail
through the tangle to untangle it. Rational tangles are thus weakly tangled. As we will
see,
⊳ Rational tangles of two strands represent the graviton and the quarks.
Q uarks
Page 269
Page 378
Page 334, page 341
The exploration of the strand model and of the strong interaction showed: the tangle of
a coloured fermion, thus of a quark, must be rational, must reproduce the three possible
colour options, and must break the three-belt symmetry.
The simplest tangles that realize these requirements are shown in Figure 84: quark
tangles are rational tangles made of two strands. Higher quark generations have larger
crossing numbers. The four tails form the skeleton of a tetrahedron. A particle with two
strands tangled in this way automatically has spin 1/2. The electric charges of the quarks
are 1/3 and −2/3, an assignment that is especially obvious for up and down quarks and
that will become clearer later on, in the study of hadrons. Parity is naturally assigned as
done in Figure 84. Baryon number and the other flavour quantum numbers – isospin,
strangeness, charm, bottomness, topness – are naturally assigned as usual. The flavour
quantum numbers simply ‘count’ the number of corresponding quark tangles. Like all
localized tangles, quarks have weak charge. We will explore weak charge in more detail
below. Antiquarks are mirror tangles and have opposite quantum numbers. We will see
below that these assignments reproduce the observed quantum numbers of all mesons
and baryons, as well as all their other properties.
We note that the simplest version of the down quark is a simple crossing; nevertheless, it differs from its antiparticle, because the simple crossing mixes with the braid with
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
In short, the only two-stranded tangles of interest in the strand model are the rational
tangles. We now explore them in more detail.
Motion Mountain – The Adventure of Physics
We will discuss them in detail in the next two sections. More complicated rational
tangles are higher-order propagating states of the simpler ones.
— Another class of tangles are prime tangles, for which the tangle 2d is an example.
Like knotted one-stranded tangles, we conclude that prime tangles are not part of
the strand model.
— Still another class of tangles are locally knotted tangles, shown as example 2e. Also
this class is not part of the strand model.
— Finally, closed tangles, links and mixed tangles, shown in the lower row of Figure 83,
have no role in the strand model – except for the one closed strand that makes up the
whole universe. horizons.
pa rticles m a d e of t wo stra nds
321
Antiquarks :
Parity P = +1, B = 1/3, spin S = 1/2
Q = –1/3
Q = +2/3
P = –1, B = –1/3, S = 1/2
Q = +1/3
Q = –2/3
d
d
u
5.0 ± 1.6 MeV
2.5 ± 1.1 MeV
s
c
105 ± 35 MeV
1.27 ± 0.11 GeV
b
t
4.20 ± 0.17 GeV
171.3 ± 2.3 GeV
s
b
u
c
t
Seen from a larger distance, the tails follow (on average) the skeleton of a tetrahedron :
s
s
Page 323
Page 265
Page 334
seven crossings, 13 crossings, etc.; this mixing is due to the leather trick, as shown below.
And for every quark type, these more complicated braids differ from those of their antiparticles.
For each quark, the four tails form the skeleton of a tetrahedron. In Figure 84 and
Figure 85, the tetrahedral skeletons are drawn with one tail in the paper plane; of the
other three tails, the middle one is assumed to be above the paper plane, and the outer two
tails to be below the paper plane. This is important for the drawing of quark compounds
later on. The three tails allow us to reproduce the strong interaction and the colour charge
of the quarks: each colour is one of three possible orientations in space; more precisely,
the three colours result from the three possible ways to map a quark tangle to the three
belt structure. Each colour corresponds to a different choice for the tail that lies above
the paper plane, as shown in Figure 85. The colour interaction of quarks will be clarified
in the section on mesons.
In the strand model, the quark tangles thus carry colour. In nature, no free coloured
particle has been observed. The strand model reproduces this observation in several
ways. First of all, all leptons and baryons are colour-neutral, as we will see shortly.
Secondly, only free quark tangles, as shown in Figure 84, have a definite colour state, because they have a fixed orientation in space. Thirdly, free quark states, thus quark states
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
F I G U R E 84 The simplest tangles assigned to the quarks and antiquarks. For reference, the experimental
mass values are also given.
Motion Mountain – The Adventure of Physics
Quarks :
322
11 the pa rticle spectrum d ed uced from stra nds
The three colour states for a strange quark :
red
green
F I G U R E 85 The three colour charges correspond to the three possible spatial orientations; the centre
tail on the right is always above the paper plane, the other two tails on the right are below the paper
plane.
Page 338
Page 326
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 334
in the tetrahedral configuration of Figure 84, do not fit into vacuum even at large distances from the core; thus free quarks carry infinitely high energy. In practice, this means
that free quark states do not occur in nature. Indeed, a free, coloured quark tangle can
reduce its energy by interacting with one or several other quarks. The result is a strong
colour attraction between quarks that leads to colourless composites.
In short, also in the strand model, only colourless composites of quarks exist as stable
free particles. We will explore quark composites and the issue of confinement of quarks
in more detail shortly.
In nature, quarks are weakly charged and interact with W bosons. In the strand
model, the absorption or the emission of a W boson is the operation that takes a quark
tangle and adds or subtracts a braiding step. This process is illustrated in Figure 86, which
shows that a braiding (unbraiding) operation corresponds to the emission (absorption)
of an W boson before symmetry breaking. It is straightforward to check that this operation fulfils all conservation laws and properties that are observed for these so-called
flavour-changing charged currents. The absorption or emission of an (unbroken) Z boson
has no braiding effect. The strand model thus reproduces the result that only the charged
weak bosons can change quark flavours, as is observed.
For completeness, we mention that quarks, being tangles of two strands, have vanishing lepton number. Indeed, as we will see below, lepton tangles are made of three strands.
In summary, all quantum numbers of quarks are reproduced by the strand model, as
long as quarks are modelled as braids of two strands with ends directed along the corners
of a tetrahedron.
Motion Mountain – The Adventure of Physics
blue
pa rticles m a d e of t wo stra nds
323
An example for a flavour-changing charged current :
Strand model :
Observation :
s quark
s quark
W boson
W boson (unbroken)
t2
t2
t1
c quark
vacuum
t1
c quark
time
The leather trick :
F I G U R E 87 The leather trick is the deformation process that changes these two structures into each
Q uark generations
Page 363
Page 331
We stress that the quark tangles shown Figure 84 represent only the simplest tangle for
each quark. First of all, longer braids are mapped to each of the six quarks. This might
seem related to the leather trick shown in Figure 87. This trick is well-known to all people
in the leather trade: if a braid of three strands has ? ⩾ 6 crossings, it can be deformed
into a braid with ? − 6 crossings. We might conjecture that, due to the leather trick, there
is no way to introduce more than 6 quarks in the strand model.
In fact, the leather trick argument assumes that the braid end – and thus the ends
of the strands – can be moved through the braids. In the strand model, this can only
happen at the horizon, the only region where space (and time) are not well-defined, and
where such manipulations become possible. The low probability of such a process will
be important in the determination of quark masses.
Instead of resting on the leather trick, it is simpler to assume that braids with large
numbers of crossings are mapped modulo 6 to the braids with the smallest number of
crossings. This is consistent, because in the strand model, a braid with six additional
crossings is mapped to a particle together with a virtual Higgs boson. The modulo 6 rule
thus represents the Yukawa mass generation mechanism in the strand model.
In summary, in the strand model, each quark is not only represented by the tangles
shown in Figure 84, but also by tangles with 6 additional crossings, with 12 additional
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
other. The leather trick limits structures made of three-stranded braids to six basic types.
Motion Mountain – The Adventure of Physics
F I G U R E 86 Absorption or emission of a W boson changes quark flavour.
324
11 the pa rticle spectrum d ed uced from stra nds
The graviton :
F I G U R E 88 The graviton in the strand model.
The graviton
Page 296
Glueballs
Ref. 217, Ref. 218
Challenge 195 ny
Ref. 219
Ref. 220
There is no observational evidence for glueballs yet, even though simulations of QCD on
the lattice predict the existence of several such states in the 1.5 GeV/?2 mass range. The
lack of experimental confirmation is usually explained by the strong background noise
in the reaction that produces glueballs, and by the expected strong mixing with mesons
of similar quantum numbers. The experimental search for glueballs is still ongoing.
The lowest-mass glueball is usually expected to be made of two gluons. In the strand
model, a glueball made of two gluons would be made of two curved strands. However,
the strand model of gluons does not seem to allow such a tangle.
Could a situation in which two gluons are linked in such a way that the four tails are
perpendicular and span a plane lead, through averaging, to a zero spin value? The issue
of glueballs needs a more precise investigation.
Whatever the situation for glueballs might be, the strand model of gluons seems in
contrast with the models of glueballs as knots that were proposed by Buniy and Kephart
or by Niemi. These models are based on closed knots, not on tangles with tails. The strand
model does not seem to allow real particles of zero spin that are composed of gluons. On
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 279
One rational tangle made of two strands is special. This special tangle is shown (again)
in Figure 88. It differs from a quark tangle in one property: the tails are parallel (and
near) to each other, and thus lie (almost completely) in a plane. Its tangle core returns
to its original state after rotation by π, and therefore models a spin-2 particle. The tangle
is not localized along its propagation direction; thus it has no mass, no electric and no
weak charge. It also has no colour charge. The tangle represents the graviton. Similar
tangles with higher winding numbers represent higher orders in the perturbation theory
of gravitation.
The chapter on gravitation has already shown how gravitons lead to curvature, horizons and the field equations of general relativity.
Motion Mountain – The Adventure of Physics
Challenge 194 e
crossings, etc.
As a mathematical check, we can also ask whether all other rational tangles are
mapped to quarks. Rational tangles of higher complexity arise by repeatedly twisting
any pair of tails of a quark tangle. This process produces an infinite number of complex
two-stranded tangles. In the strand model, these tangles are quarks surrounded by virtual particles. Equivalently, we can say that all the more complex rational tangles that do
not appear in Figure 84 are higher-order propagators of quarks.
pa rticles m a d e of t wo stra nds
325
the other hand, if closed knots were somehow possible in the strand model, they would
imply the existence of glueballs.
In summary, the issue of glueballs is not settled; a definitive solution might even lead
to additional checks of the strand model.
The mass gap problem and the Cl ay Mathematics Institute
Ref. 221
Challenge 196 s
The topic of two-stranded tangles also requires to solve the puzzle of Figure 89. To which
physical states do the three pictured tangles correspond?
Summary on t wo-stranded tangles
In summary, the strand model predicts that apart from the six quarks and the graviton, no other two-stranded elementary particle exists in nature. Concerning composite
particles, the two-stranded glueball issue is not completely settled, but points towards
non-existence.
Quarks and the graviton, the elementary particles made of two strands, are rational
tangles. Their strand models are thus not tangled in a complicated way, but tangled in
the least complicated way possible. This connection will be of importance in our search
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
A puzzle
Motion Mountain – The Adventure of Physics
Page 275
The Clay Mathematics Institute offers a large prize to anybody who proves the following
statement: For any compact simple non-Abelian gauge group, quantum gauge theory exists
in continuous, four-dimensional space-time and produces a mass gap. This is one of their
so-called millennium problems.
The strand model does not allow arbitrary gauge groups in quantum field theory. According to the strand model, the only compact simple non-Abelian gauge group of interest is SU(3), the gauge group of the strong nuclear interaction. And since the strand
model does not seem to allow for glueballs, for SU(3) an effective mass gap of the order
of the Planck mass is predicted. (If glueballs would exist in the strand model, the mass
gap would still exist but be smaller.) Indeed, the strand model explains the short range
of the strong interaction as a consequence of the details of Reidemeister III moves and
the quark tangle topology.
The strand model further states that space-time and gauge groups are low-energy approximations, because neither points nor fields exist at a fundamental level; points and
fields are approximations to strands. According to the strand model, the quantum properties of nature result from the extension of strands. As a consequence, the strand model
denies the existence of any quantum gauge theory as a separate, exact theory on continuous space-time.
In summary, the strand model does predict a mass gap for SU(3); but the strand model
also denies the existence of quantum gauge theory for any other compact simple nonAbelian gauge group. And even in the case of SU(3) it denies – like for any other gauge
groups – the existence of a quantum gauge theory on continuous space-time. As deduced
above, the strand model allows only the three known gauge groups, and allows their existence only in the non-continuous strand model of space-time. In short, it is impossible
to realize the wish of the Clay Mathematics Institute.
326
11 the pa rticle spectrum d ed uced from stra nds
achiral 6-fold
crossing tangle
chiral 6-fold
crossing tangle
chiral 8-fold
crossing tangle
Motion Mountain – The Adventure of Physics
F I G U R E 89 Which particle states are described by these tangles?
Tangles made of three strands :
3b
simple
crossings
3c
braided
tangle
3d
rational
tangle
3e
prime
tangle
3f
locally
knotted
tangle
3g
closed or
mixed
open-closed
tangles
not part of
the strand
model
near the
big bang
or
horizons
Particle correspondence :
composed
of vacuum
and gauge
bosons
composed
of vacuum
and gauge
bosons
elementary: composed:
not part of
leptons and mesons of
the strand
model
Higgs boson spin 0 and
other particles
F I G U R E 90 Examples for all the classes of tangles made of three strands.
for elementary particles that are still undiscovered.
particles made of three strands
In the strand model, the next group are particles made of three strands. Examples for
all classes of three-stranded tangles are given in Figure 90. Several classes of three-
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
3a
trivial
tangle
pa rticles m a d e of three stra nds
327
Lepton tangles, all with spin S = 1/2 , parity P = +1 , lepton number L’ = 1 and baryon number B = 0 :
e
0.5 MeV
105 MeV
νμ
τ
all three with
Q = –1
1.77 GeV
ντ
all three with
Q=0
1±1 eV
1±1 eV
Seen from a larger distance, the tails follow (on average) the x, y and z axes of a coordinate system.
F I G U R E 91 The simplest tangles of the leptons, with the experimental mass values. Antileptons are
mirror tangles.
stranded tangles turn out to be composites of two-stranded particles. However, a number
of tangles are new and represent elementary particles.
L eptons
The candidate tangles from 2008 for the leptons shown in Figure 91 are the simplest
possible non-trivial tangles with three strands. These lepton tangles are simple braids
with tails reaching the border of space. The six tails probably point along the coordinate
axes. These braided tangles have the following properties.
— Each lepton is localized. Each lepton has mass: its three tails con be braided, thus
have non-vanishing Yukawa coupling, thus generate mass. And each lepton has spin
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
1±1 eV
Motion Mountain – The Adventure of Physics
νe
μ
328
11 the pa rticle spectrum d ed uced from stra nds
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 222
In summary, tangles of three strands have precisely the quantum numbers and most
properties of leptons. In particular, the strand model predicts exactly three generations
of leptons, and predicts that all leptons have mass.
This implies that searches for the neutrino-less double beta decay should yield negative results, that the magnetic moments of the neutrinos should have the exceedingly
Motion Mountain – The Adventure of Physics
1/2. Each lepton thus follows the Dirac equation.
— Each lepton has weak charge.
— Charged leptons and antileptons differ. Each has two possible chiralities.
— Three of the tangles are topologically chiral, thus electrically charged, and three other
tangles are topologically achiral, thus uncharged.
— The spatial parity ? of the charged lepton tangles is opposite to that of their antiparticles.
— Being made of three strands, lepton tangles have vanishing colour charge and vanishing baryon number.
— In contrast to quarks, lepton tangles can be inserted in the vacuum using a localized,
i.e., finite amount of energy and are thus predicted to exist as free particles.
— The three types of lepton (flavour) numbers can be assigned as usual; the lepton numbers are conserved in reactions, apart for neutrino mixing effects, as we will see below.
— The strand model predicts that the electron, the charged tangle with the lowest mass,
is stable, as there is no way for it to decay and conserve charge and spin. The other
two generations are predicted to be unstable, due to weak decays that simplify their
topology.
— The three generations are reproduced by the strand model, as every more complicated
braid can be seen as equivalent to one of the first six braids, with the same braiding
argument that limits the number of quarks.
— There is a natural mapping between the six quarks and the six leptons. It appears
when the final bend of the ‘longer’ quark strand is extended to the border of space,
thus transforming a two-stranded quark braid into a three-stranded lepton braid.
Thus we get three common generations for quarks and leptons.
— The neutrino strands differ by tail braiding; the strand model thus predicts that the
weak interaction mixes neutrinos.
— All lepton tangles differ from each other. Thus the mass values are different for each
lepton.
— Due to the small amount of tangling, the strand model predicts that the masses of
the leptons are much smaller than those of the W and Z boson. This is indeed observed. (This also suggests a relation between the mass and the total curvature of a
tight tangle.)
— The simplest tangle for the electron neutrino also suggests that the mass values for
the electron neutrino is naturally small, as its tangle is almost not tangled.
— The strand model predicts that lepton masses increase with the generation number.
Since the neutrino masses are not precisely known, this prediction cannot yet be
checked.
— Neutrinos and antineutrinos are both massive and differ from each other. If the tangle
of the electron neutrino is correct, the electron neutrino of opposite chirality is expected to be seen only rarely – as is observed.
pa rticles m a d e of three stra nds
329
A candidate tangle for the Higgs boson :
or
F I G U R E 92 A candidate tangle for the Higgs boson in the strand model: the open version (left) and the
corresponding closed version (right). For the left version, the tails approach the six coordinate axes at
infinity.
Ref. 223
Open issue: are the lepton tangles correct?
Page 386
The Higgs boson – the mistaken section from 2009
Page 250
The existence of the Higgs boson is predicted from the standard model of elementary
particle physics using two arguments. First of all, the Higgs boson prevents unitarity
violation in longitudinal W–W and Z–Z boson scattering. Secondly, the Higgs boson
confirms the symmetry breaking mechanism of SU(2) and the related mass generation
mechanism of fermions. Quantum field theory predicts that the Higgs boson has spin
0, has no electric or strong charge, and has positive C and P parity. In other words, the
Higgs boson is predicted to have, apart from its weak charge, the same quantum numbers
as the vacuum.
In the strand model, there seems to be only one possible candidate tangle for the Higgs
boson, shown on the left of Figure 92. The tangle has positive C and P parity, and has
vanishing electric and strong charge. The tangle also corresponds to the tangle added by
the leather trick; it thus could be seen to visualize how the Higgs boson gives mass to the
quarks and leptons. However, there are two issues with this candidate. First, the tangle
is a deformed, higher-order version of the electron neutrino tangle. Secondly, the spin
value is not 0. In fact, there is no way at all to construct a spin-0 tangle in the strand
model. These issues lead us to reconsider the arguments for the existence of the Higgs
boson altogether.
We have seen that the strand model proposes a clear mechanism for mass generation:
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Challenge 197 ny
The argument that leads to the lepton tangles is vague. The tangle assignments might
need corrections. There are two issues.
First, there is an aesthetic issue: in most particle tangles, the electric charge unit is
given by three crossings of the same sign. It seems odd that leptons should form an exception.
Secondly, the candidate tangles suggest that the muon neutrino is more massive than
the electron. Most probably therefore, the tangles need amends. Can you improve the
situation, either by finding better tangles or by finding better arguments?
Motion Mountain – The Adventure of Physics
small values predicted by the standard model of particle physics, and that rare muon and
other decays should occur at the small rates predicted by the standard model.
330
11 the pa rticle spectrum d ed uced from stra nds
Strand model for a longitudinal massive spin 1 boson :
Observation :
spin
precession
orthogonal
spin
time average
of crossing
switches
precession
motion
motion
time
Strand model for a transversal massive spin 1 boson :
time average
of crossing
switches
spin
parallel
spin
motion
motion
F I G U R E 93 In the strand model, transverse and longitudinal W and Z bosons differ. (Note added in
2012: this statement is mistaken.)
⊳ Mass is due to strand braiding.
Page 207
Ref. 224
Ref. 225
This mechanism, due to the weak interaction, explains the W and Z boson mass ratio,
as we will see below. The leather trick that explains fermion masses can be seen as the
addition of a sixfold tail braiding. In particular, the rarity of the braiding process explains
why particle masses are so much smaller than the Planck mass. In short, the strand model
explains mass without a Higgs boson.
If the Higgs boson does not exist, how is the unitarity of longitudinal W and Z boson
scattering maintained? The strand model states that interactions of tangles in particle
collisions are described by deformations of tangles. Tangle deformations in turn are described by unitary operators. Therefore, the strand model predicts that unitarity is never
violated in nature. In particular, the strand model automatically predicts that the scattering of longitudinal W or Z bosons does not violate unitarity.
In other terms, the strand model predicts that the conventional argument about unitarity violation, which requires a Higgs boson, must be wrong. How can this be? There are
at least two loopholes available in the research literature, and the strand model realizes
them both.
The first known loophole is the appearance of non-perturbative effects. It is known
for a long time that non-perturbative effects can mimic the existence of a Higgs boson in
usual, perturbative approximations. In this case, the standard model could remain valid
at high energy without the Higgs sector. This type of electroweak symmetry breaking
would lead to longitudinal W and Z scattering that does not violate unitarity.
The other loophole in the unitarity argument appears when we explore the details of
the longitudinal scattering process. In the strand model, longitudinal and transverse W
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 358
Motion Mountain – The Adventure of Physics
core
pa rticles m a d e of three stra nds
Ref. 226
331
Vol. V, page 253
Vol. V, page 258
In July 2012, CERN researchers from two different experiments announced the observation of a new neutral boson with a mass of 125 GeV. Additional data analysis showed
that the boson has spin 0 and positive parity. All experimental checks confirm that the
boson behaves like the Higgs boson predicted in 1963 by Peter Higgs and a number of
other researchers.
The results lead to question several statements made in 2009 in the previous section.
— Is the tangle on the left-hand side of Figure 92 really a higher order version of the
electron neutrino? It turns out that this statement is wrong: in contrast to the tangle
* If the arguments against the Higgs boson turn out to be wrong, then the strand model might be saved with
a dirty trick: we could argue that the tangle on the left-hand side of Figure 92 might effectively have spin 0.
In this case, the ropelength of the Borromean rings, 29.03, together with the ropelengths of the weak bosons,
lead to a Higgs mass prediction, to first order, in the range from (29.03/10.1)1/3 ⋅ 80.4 GeV = 114 GeV to
(29.03/13.7)1/3 ⋅ 91.2 GeV = 117 GeV, plus or minus a few per cent.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
The Higgs boson – the corrected section of 2012
Motion Mountain – The Adventure of Physics
or Z bosons are modelled as shown in Figure 93. For longitudinal bosons, spin and its
precession leads to a different situation than transversal bosons: longitudinal bosons are
more delocalized than transversal bosons. This is not the case for fermions, where the
belt trick leads to the same delocalization for longitudinal and transverse polarization.
Interestingly, it is also known for a long time that different delocalization for longitudinal
and transversal bosons maintains scattering unitarity, and that in the case of delocalization the conventional argument for the necessity of the Higgs boson is wrong. These
are well-known consequences of the so-called non-local regularization in quantum field
theory. The strand model thus provides a specific model for this non-locality, and at the
same time explains why it only appears for longitudinal W and Z bosons.
The issue of different scattering behaviour for longitudinal and transverse weak bosons also raises the question whether the mass of the longitudinal and the transversal
bosons are precisely equal. The possibility, triggered by Figure 93, might seem appealing
at first sight in order to solve the unitarity problem. However, the strand model forbids
such a mass difference. In the strand model, mass is due to tangle fluctuations, but does
not depend on spin direction.
In other words, the strand model predicts that the scattering of longitudinal W and Z
bosons is the first system that will show effects specific to the strand model. Such precision scattering experiments might be possible at the Large Hadron Collider in Geneva.
These experiments will allow checking the non-perturbative effects and the regularization
effects predicted by the strand model. For example, the strand model predicts that the
wave function of a longitudinal and a transversally polarized W or Z boson of the same
energy differ in cross section.
In summary, the strand model predicts well-behaved scattering amplitudes for longitudinal W and Z boson scattering in the TeV region, together with the absence of the
Higgs boson.* The strand model explains mass generation and lack of unitarity violations in longitudinal W or Z boson scattering as consequences of tail braiding, i.e., as
non-perturbative and non-local effects, and not as consequences of an elementary spin0 Higgs boson. The forthcoming experiments at the Large Hadron Collider in Geneva
will test this prediction.
332
11 the pa rticle spectrum d ed uced from stra nds
Higgs boson :
of the neutrino, the tangle of Figure 92 is not twisted.
— Does the tangle of Figure 94 have spin 1/2 or spin 0? As mentioned already in 2009,
an effective spin 0 might be possible, in a similar way that it is possible for spin-0
mesons. Spin 0 behaviour might appear because the tangle can be oriented in different directions or because of the Borromean property: no two strands have more
crossings than two vacuum strands; the time average of these situations has the same
symmetry as the vacuum, and thus implies spin 0.
— Does the tangle of Figure 94 have the correct, positive, C and P values expected for a
Higgs boson? It seems so.
— Is the mentioned non-locality effect for W and Z bosons real? If the effect were real, it
should also appear for other spin-1 particles. In the strand model, mass values should
not depend on spin orientation, but only on tangle core topology. The statements
made in 2009 on delocalization and longitudinal scattering seem wrong in retrospect.
— Would the Higgs boson tangle assignment of Figure 94 be testable? Yes; any tangle
assignment must yield the observed mass value and the observed branching ratios
and decay rates. This is a subject of research. But already at the qualitative level, the
proposed tangle structure of the Higgs boson suggests decays into leptons that are
similar to those observed at CERN.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
be oriented in different directions, so that the time average has spherical symmetry. The tangle has 9
crossings: 3 crossings appear already in the vacuum configuration of three strands, and the additional 6
crossings (see Figure 92) are due to the Higgs boson.
Motion Mountain – The Adventure of Physics
F I G U R E 94 The tangle of the Higgs boson in the strand model. Spin 0 appears because the braid can
pa rticles m a d e of three stra nds
Page 366
— Is the tangle of Figure 94 elementary? Yes.
— Are there other possible Higgs boson tangles? This issue is open. The braid structure
seems the most appealing structure, as it embodies the effect of tail braiding, an effect
that is important for the appearance of mass.
— Are knots and links, i.e., closed tangles, really forbidden? The discussion about the
Higgs boson concerns the open tangle shown in Figure 94, not the Borromean link
shown on the right-hand side of Figure 92. So far, there is no evidence for closed
tangles in the strand model. Such evidence would mean a departure from the idea
that nature is a single strand.
— Does the Higgs boson issue put into question the strand model as a whole? First of all,
SU(2) breaking is unaffected. Secondly, a mistaken tangle–particle assignment can be
accommodated in the strand model; new forces or symmetries cannot. Therefore the
strand model is not put into question.
— Could several, possibly charged, Higgs bosons exist? No such tangles seem possible
– as long as a tangle with two Figure 94 Higgs cores in sequence is not a separate
particle.
— Has some other strand model effect been overlooked? Could other elementary or
composed particles exist? For example, the structure of the Higgs boson might be
seen to suggest that lepton families reappear (roughly) every 125 GeV. Is that the
case? The issue is not completely settled. It seems more probable that those higher
tangles simply yield corrections to the Higgs mass.
2012 predictions abou t the Higgs
— The Higgs tangle implies a Higgs boson with vanishing charge, positive parity, being
elementary – as is observed.
— The Higgs tangle allows us to estimate the Higgs/Z mass ratio. Using the new, unknotted, tangle model for the W and Z bosons, the estimates are in the region of the
observed values. Improving the estimates is still subject of research.
— The Higgs tangle and the strand model imply that the standard model is correct up to
Planck energy, and that the Higgs mass value should reflect this. The observed Higgs
mass of 125 GeV complies also with this expectation.
— Therefore, the strand model suggests that no deviations between the standard model
and data should ever be observed in any experiment.
— The strand model again and consistently predicts the lack of supersymmetry.
— In the case that several Higgs bosons exist or that the braided Higgs tangle does not
apply, the strand model is in trouble.
— In the case that effects, particles or interactions beyond the standard model are observed, the strand model is in trouble.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
In short, the existence of the standard model Higgs boson seems compatible with the
strand model. The 2009 mistake about the Higgs also shows that the exploration of the
strand model is not yet complete. In any case, the strand model has not been falsified by
the discovery of the Higgs boson.
Assuming that the Higgs tangle shown in Figure 94 is correct, we have an intuitive
proposal for the mechanism that produces mass, namely tail braiding. The proposed
Higgs tangle also allows a number of experimental predictions.
Motion Mountain – The Adventure of Physics
Page 332
333
334
11 the pa rticle spectrum d ed uced from stra nds
Pseudoscalar and vector mesons made of up and down quarks :
Spin S = 0, L = 0
Parity P = –1
η : 548 MeV
C = +1
π+ : 140 MeV
π– : 140 MeV
η,
π0
ω0, ρ0
uu
uu
π+
ρ+
ud
ud
π–
ρ–
ud
ud
ω0
: 783 MeV
C = –1
ρ+ : 775 MeV
ρ– : 775 MeV
π0, η
ρ0, ω0
ρ0 : 775 MeV
dd
dd
C = –1
Circles indicate
pairs of tails to the
border of space
crossed tail pairs to the border of space), with the observed mass values.
Q uark-antiquark mesons
Ref. 227
Ref. 228
Ref. 229
In the strand model, all three-stranded tangles apart from the leptons, as well as all fourstranded tangles represent composite particles. The first example are mesons.
In the strand model, rational tangles of three strands are quark-antiquark mesons with
spin 0. The quark tangles yield a simple model of these pseudoscalar mesons, shown on
the left-hand sides of Figure 95, Figure 97 and Figure 98. The right-hand sides of the
figures show vector mesons, thus with spin 1, that consist of four strands. All tangles are
rational. Inside mesons, quarks and antiquarks ‘bond’ at three spots that form a triangle
oriented perpendicularly to the bond direction and to the paper plane. To increase clarity, the ‘bonds’ are drawn as circles in the figures; however, they consist of two crossed
(linked) tails of the involved strands that reach the border of space, as shown in Figure 96.
With this construction, mesons made of two quarks are only possible for the type ? ?.
Other combinations, such as ? ? or ? ?, turn out to be unlinked. We note directly that
this model of mesons resembles the original string model of hadrons from 1973, but also
the Lund string model and the recent QCD string model.
To compare the meson structures with experimental data, we explore the resulting
quantum numbers. As in quantum field theory, also in the strand model the parity of
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
F I G U R E 95 The simplest strand models for the light pseudoscalar and vector mesons (circles indicate
Motion Mountain – The Adventure of Physics
π0 : 135 MeV
C = +1
Spin S = 1, L = 0
Parity P = –1
Note: the larger circle
is above the paper plane.
pa rticles m a d e of three stra nds
335
Simplified drawing :
Spin S = 0, L = 0
Parity P = –1
π0 : 135 MeV
C = +1
Spin S = 1, L = 0
Parity P = –1
Note: the large circle
is above the paper plane.
π0, η
ρ0, ω0
ρ0 : 775 MeV
dd
dd
C = –1
Complete drawing :
Simplification used :
F I G U R E 96 The meaning of the circles used in the tangle graphs of mesons and baryons.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
a particle is the product of the intrinsic parities and of wave function parity. The states
with orbital angular momentum ? = 0 are the lowest states. Experimentally, the lightest
mesons have quantum numbers ??? = 0−+ , and thus are pseudoscalars, or have ??? =
1−− , and thus are vector mesons. The strand model reproduces these observed quantum
numbers. (We note that the spin of any composite particle, such as a meson, is lowenergy quantity; to determine it from the composite tangle, the tails producing the bonds
– drawn as circles in the figures – must be neglected. As a result, the low-energy spin of
mesons and of baryons is correctly reproduced by the strand model.)
In the strand model, the meson states are colour-neutral, or ‘white’, by construction,
because the quark and the antiquark, in all orientations, always have opposite colours
that add up to white.
In the strand model, the electric charge is an integer for all mesons. Chiral tangles are
charged, achiral tangles uncharged. The charge values deduced from the strand model
thus reproduce the observed ones.
In experiments, no mesons with quantum numbers 0−− , 0+− , or 1−+ are observed.
Also this observation is reproduced by the quark tangles, as is easily checked by direct
inspection. The strand model thus reproduces the very argument that once was central
to the acceptance of the quark model itself.
It is important to realize that in the strand model, each meson is represented by a
tangle family consisting of several tangle structures. This has three reasons. First, the
‘circles’ can be combined in different ways. For example, both the ? ? and the ? ? have
as alternate structure a line plus a ring. This common structure is seen as the underlying
Motion Mountain – The Adventure of Physics
Circles indicate
pairs of tails to the
border of space
336
11 the pa rticle spectrum d ed uced from stra nds
Pseudoscalar and vector mesons containing
strange and charm quarks :
K–
su
494 MeV
Note: the large circle
is above the paper plane.
K*–
su
892 MeV
K0
sd
498 MeV
K*0
sd
899 MeV
K0
sd
498 MeV
K*0
sd
899 MeV
η'
ss
958 MeV
ϕ'
ss
1020 MeV
D0
cu
1864 MeV
D*0
cu
2007 MeV
D0
cu
1864 MeV
D*0
cu
2007 MeV
D+
cd
1870 MeV
D*+
cd
2010 MeV
D–
cd
1870 MeV
D*–
cd
2010 MeV
Ds+
cs
1970 MeV
Ds*+
cs
2112 MeV
Ds–
cs
1968 MeV
Ds*–
cs
2112 MeV
ηc
cc
2981 MeV
J/ψ
cc
3097 MeV
F I G U R E 97 The simplest strand models for strange and charmed mesons with vanishing orbital angular
momentum. Mesons on the left side have spin 0 and negative parity; mesons on the right side have
spin 1 and also negative parity. Circles indicate crossed tail pairs to the border of space; grey boxes
indicate tangles that mix with their antiparticles and which are thus predicted to show CP violation.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
K*+
su
892 MeV
Motion Mountain – The Adventure of Physics
K+
su
494 MeV
pa rticles m a d e of three stra nds
337
Pseudoscalar and vector mesons containing a bottom quark :
Spin S = 0, L = 0
Parity P = –1
Spin S = 1, L = 0
Parity P = –1
Note: the large circle
is above the paper plane.
B*–
bu
5325 MeV
B0
bd
5279 MeV
B*0
bd
5325 MeV
B0s
bs
5366 MeV
B*s0
bs
5412 MeV
Bc–
bc
6286 MeV
B*c–
bc
not yet
discovered
ηb (C=+1)
bb
9300 MeV
Y (C=–1)
bb
9460 MeV
F I G U R E 98 The simplest strand models for some heavy pseudoscalar and vector mesons, together with
their experimental mass values. Antiparticles are not drawn; their tangles are mirrors of the particle
tangles. Circles indicate crossed tail pairs to the border of space; grey boxes indicate tangles that mix
with their antiparticles and which are thus predicted to show CP violation.
reason that these two quark structures mix, as is indeed observed. (The same structure is
also possible for ? ?, and indeed, a full description of these mesons must include mixing
with this state as well.) The second reason that mesons have several structures are the
mentioned, more complicated braid structures possible for each quark, namely with 6,
12, etc. additional braid crossings. The third reason for additional tangle structures is the
occurrence of higher-order Feynman diagrams of the weak interaction, which add yet
another group of more complicated topologies that also belong to each meson.
In short, the mesons structures of Figure 95, Figure 97 and Figure 98 are only the
simplest tangles for each meson. Nevertheless, all tangles, both the simplest and the more
complicated meson tangles, reproduce spin values, parities, and all the other quantum
numbers of mesons. Indeed, in the strand model, the more complicated tangles automatically share the quantum numbers of the simplest one.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Circles indicate
pairs of tails to the
border of space
Motion Mountain – The Adventure of Physics
B–
bu
5279 MeV
338
11 the pa rticle spectrum d ed uced from stra nds
Meson form factors
Ref. 230
The strand model also predicts directly that all mesons from Figure 95, Figure 97 and
Figure 98, in fact all mesons with vanishing orbital momentum, are prolate. This (unsurprising) result is agreement with observations. Mesons with non-vanishing orbital
momentum are also predicted to be prolate. This latter prediction about meson shapes
is made also by all other meson models, but has not yet been checked by experiment.
There is another way to put what we have found so far. The strand model makes
the following prediction: When the meson tangles are averaged over time, the crossing densities reproduce the measured spatial, quark flavour, spin and colour part of the
meson wave functions. This prediction can be checked against measured form factors
and against lattice QCD calculations.
Meson masses, excited mesons and quark confinement
? = ?0 + ?1 ?2
Ref. 232
(201)
with an (almost) constant factor ?1 for all mesons, about 0.9 GeV/f m. These relations,
the famous Regge trajectories, are explained in quantum chromodynamics as deriving
from the linear increase with distance of the effective potential between quarks, thus
from the properties of the relativistic harmonic oscillator. The linear potential itself is
usually seen as a consequence of a fluxtube-like bond between quarks.
In the strand model, the fluxtube-like bond between the quarks is built-in automatically, as shown in Figure 99. All mesons have three connecting ‘bonds’ and these three
bonds can be seen as forming one common string tube. In the simplified drawings, the
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 356
Motion Mountain – The Adventure of Physics
Ref. 231
The strand model also allows us to understand meson masses. We recall that a topologically complicated tangle implies a large mass. With this relation, Figure 95 predicts that
the π0 , ? and π+/− have different masses and follow the observed meson mass sequence
?(π0 ) < ?(π+/− ) < ?(?). The other mass sequences can be checked with the help of Figure 95, Figure 97 and Figure 98; there are no contradictions with observations. However,
there is one limit case: the strand model predicts different masses for the ?0 , ?, and ?+/− .
So far, observations only partly confirm the prediction. Recent precision experiments
seem to suggest that ?0 and ?+/− have different mass; this result has not been confirmed
yet.
More precise mass determinations will be possible with numerical calculations. This
will be explored in more detail later on. In any case, the strand model for mesons suggests that the quark masses are not so important for the determination of meson masses,
whereas the details of the quark-antiquark bond are. Indeed, the light meson and baryon
masses are much higher than the masses of the constituent quarks.
The relative unimportance of quark masses for many meson masses is also confirmed
for the case of excited mesons, i.e., for mesons with orbital angular momentum ?. It is
well known that mesons of non-vanishing orbital angular momentum can be grouped
into sets which have the same quark content, but different total angular momentum ? =
? + ?. These families are observed to follow a well-known relation between total angular
momentum ? and mass ?, called Regge trajectories:
pa rticles m a d e of three stra nds
339
Vanishing orbital angular momentum
Simplified drawing :
Spin S = 0, L = 0
Parity P = –1
π0 : 135 MeV π0, η
C = +1
dd
Spin S = 1, L = 0
Parity P = –1
Note: the large circle
is above the paper plane.
ρ0, ω0
dd
ρ0 : 775 MeV
C = –1
Complete drawing :
Simplified drawing :
Complete drawing :
bond or string tube is the region containing the circles. In orbitally excited mesons, the
three bonds are expected to lengthen and thus to produce additional crossing changes,
thus additional effective mass. The strand model also suggests a linear relation. Since the
mechanism is expected to be similar for all mesons, which all have three bonding circles,
the strand model predicts the same slope for all meson (and baryon) Regge trajectories.
This is indeed observed.
In summary, the strand model reproduces meson mass sequences and quark confinement in its general properties.
CP viol ation in mesons
Ref. 233
Ref. 231
In the weak interaction, the product CP of C and P parity is usually conserved. However,
rare exceptions are observed for the decay of the ?0 meson and in various processes
that involve the ?0 and ?0s mesons. In each of these exceptions, the meson is found to
mix with its own antiparticle. CP violation is essential to explain the matter–antimatter
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
F I G U R E 99 The strand model for mesons without (top) and with (bottom) orbital angular momentum.
Motion Mountain – The Adventure of Physics
With orbital angular momentum
340
O ther three-stranded tangles and glueballs
Page 324
Challenge 198 s
In the strand model, the aditted complicated tangles made of three strands are either
higher-order propagating versions of the tangles just presented or composites of onestranded or two-stranded particles.
The often conjectured glueball could also be made of three gluons. In the strand
model, such a structure would be a simple tangle made of three strands. However, the
masslessness of gluons does not seem to allow such a tangle. The argument is not watertight, however, and the issue is, as mentioned above, still subject of research.
We note that the three-strand analog of the graviton – three parallel, but twisted
strands – is not an elementary particle, but a composed structure.
Spin and three-stranded particles
Why do three strands sometimes form a spin 0 particle, such as the elementary Higgs
boson, sometimes a spin 1/2 particle, such as the elementary electron, and sometimes a
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 353
asymmetry of the universe.
The strand model allows us to deduce whether the mixing of a meson with its own
antiparticle is possible or not. As expected, only neutral mesons are candidates for such
mixing, because of charge conservation. In the strand model, particle–antiparticle mixing is possible whenever the transition from a neutral meson to its antiparticle is possible
in two ways: by taking the mirror of the meson tangle or by shifting the position of the
binding strands. All mesons for which this is possible are shown in grey boxes in Figure 95, Figure 97 and Figure 98. The strand model also makes it clear that such mixing
requires shifting of the bonds; this is a low-probability process that is due to the weak
interaction. The strand model thus predicts that the weak interaction violates CP invariance in mesons that mix with their antiparticles.
Since the spin 1 mesons decay strongly and thus do not live long enough, the small
effect of CP violation is de facto only observed in pseudoscalar, spin-0 mesons. The
strand model thus predicts observable mixings and CP violation for the mesons pairs
?0 − ?0 , ?0 − ?0 , ?0 − ?0 , ?0s − ?0s . The prediction by the strand model corresponds
precisely to those systems for which CP violation is actually observed. (CP violation in
? mesons was finally discovered at CERN in 2011, after it was predicted both by the
standard model and the strand model, in earlier editions of this volume.)
In the strand model, meson–antimeson mixing is possible because the various quarks
are braided strands. Because of this braid structure, the existence of meson–antimeson
mixing is a consequence of the existence of three quark generations. The meson structures also make it clear that such mixings would not be possible if there were no third
quark generation. The strand model thus reproduces the usual explanation of CP violation as the result of three quark generations.
For the strong and the electromagnetic interaction, the strand model predicts that
there is no mixing and no CP violation, because gluons and photons do not change
particle topology. Therefore, the strand model suggests the absence of axions. The lack of
a suitable tangle for axions, shown later on, then turns this suggestions into a prediction.
In summary, the existence of CP violation in the weak interactions and the lack of CP
violation in the strong interaction are natural consequences of the strand model.
Motion Mountain – The Adventure of Physics
Ref. 231
11 the pa rticle spectrum d ed uced from stra nds
ta ng les of four a nd m ore stra nds
341
spin 1 particle, such as a composed meson? The answer depends on how the strands are
free to move against each other.
The Higgs tangle appears through tangling of vacuum strands, and inherits the zero
spin of vacuum. The W and Z tangles have a special property: two strands can rotate
around the third; this makes them bosons as well, but of spin 1. Fermion tangles have
neither property; their core can only rotate through the belt trick; thus they are fermions.
Summary on three-stranded tangles
tangles of four and more strands
Baryons
In the strand model, rational tangles made of five or six strands are baryons. The quark
tangles of the strand model yield the tangles for baryons in a natural way, as Figure 100
shows. Again, not all quark combinations are possible. First of all, quark tangles do not
allow mixed ? ? ? or ? ? ? structures, but only ? ? ? or ? ? ? structures. In addition, the
tangles do not allow (fully symmetric) spin 1/2 states for ? ? ? or ? ? ?, but only spin 3/2
states. The model also naturally predicts that there are only two spin 1/2 baryons made of
? and ? quarks. All this corresponds to observation. The tangles for the simplest baryons
are shown in Figure 100.
The electric charges of the baryons are reproduced. In particular, the tangle topologies imply that the proton has the same charge as the positron. Neutral baryons have
topologically achiral structures; nevertheless, the neutron differs from its antiparticle, as
can be deduced from Figure 100, through its three-dimensional shape. The Δ baryons
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If we add one or more strand to a three-strand tangle, no additional class of tangles
appears. The tangle classes remain the same as in the three-strand case. In other words,
no additional elementary particles arise in the strand model. To show this, we start our
exploration with the rational tangles.
We saw above that the rational tangles made of four strands represent the vector
mesons. We have already explored them together with the scalar mesons. But certain
more complicated rational tangles are also important in nature, as we consist of them.
Motion Mountain – The Adventure of Physics
Page 331
Compared to two-stranded tangles, one new class of elementary particles appears for
three strands; the new class is somewhat less tangled than general rational tangles but still
more tangled than the trivial vacuum tangle: the braided tangles. Braided tangles represent the Higgs boson and the leptons; the tangles reproduce all their observed quantum
numbers. The braided tangles also imply that neutrinos and anti-neutrinos differ, are
massive, and are Dirac particles.
The strand model (corrected in 2012) also predicts that, apart from the six leptons and
one Higgs boson, no other elementary particle made of three strands exist in nature.
In the case of composite particles made of three strands, the strand model proposes
tangles for all pseudoscalar mesons; the resulting quantum numbers and mass sequences
match the observed values. In the spectrum of composite particles, the glueball issue is
not completely settled.
342
11 the pa rticle spectrum d ed uced from stra nds
The proton has two basic graphs,
corresponding to u↑ u↓ d↑ and u↑ u↑ d↓ :
Spin S = 1/2, L = 0,
d parity P = +1
u
u
u
uud = p
938 MeV
u
d
The neutron has two basic graphs,
corresponding to d↑ d↓ u↑ and d↑ d↑ u↓ :
d
d
u
d
u
d
d
uud=Δ+
1232 MeV
Circles indicate
pairs of tails to the
border of space
u
u
uuu=Δ++
1232 MeV
u
u
u
d
udd=Δ0
1232 MeV
d
d
ddd=Δ–
1232 MeV
d
d
F I G U R E 100 The simplest strand models for the lightest baryons made of up and down quarks (circles
indicate linked tail pairs to the border of space), together with the measured mass values.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
The four Δ baryons have one graph each, corresponding
to u↑ u↑ u↑, u↑ u↑ d↑, u↑ d↑ d↑ and d↑ d↑ d↑ :
Spin S = 3/2, L = 0,
parity P = +1
u
Motion Mountain – The Adventure of Physics
udd = n
940 MeV
ta ng les of four a nd m ore stra nds
343
Baryons with spin S = 1/2 and angular momentum L = 0 made of up, down and strange quarks :
s
s
s
dss=Ξ−
1322 MeV
s
uss=Ξ0
1315 MeV
d
d
dds=Σ−
1197
MeV
d
s
u
s
u
uds=Λ0
1116
MeV
Parity P = +1 expected
for both Ξ baryons, but
not yet measured.
d
u
uds=Σ0
1192 MeV
u
u
uus=Σ+
1189 MeV
d
s
s
d
d
Circles indicate
pairs of tails to the
border of space
udd=n
940 MeV
u
u
P = +1 for both
neutron and proton.
uud=p
938 MeV
u
d
Page 385
Ref. 230
have different electric charges, depending on their writhe.
Baryons are naturally colour-neutral, as observed. The model also shows that the baryon wave function usually cannot be factorized into a spin and quark part: the nucleons
need two graphs to describe them, and tangle shapes play a role. Baryon parities are reproduced; the neutron and the antineutron differ. All this corresponds to known baryon
behaviour. Also the observed baryon shapes (in other words, the baryon quadrupole
moments) are reproduced by the tangle model.
The particle masses of proton and neutron differ, because their topologies differ. However, the topological difference is ‘small’, as seen in Figure 100, so the mass difference is
small. The topological difference between the various Δ baryons is even smaller, and
indeed, their mass difference is barely discernible in experiments.
The strand model naturally yields the baryon octet and decuplet, as shown in Figure 101 and Figure 102. In general, complicated baryon tangles have higher mass than
simpler ones, as shown in the figures; this is also the case for the baryons, not illustrated
here, that include other quarks. And like for mesons, baryon Regge trajectories are due
to ‘stretching’ and tangling of the binding strands. Since the bonds to each quark are
again (at most) three, the model qualitatively reproduces the observation that the Regge
slope for all baryons is the same and is equal to that for mesons. We note that this also
implies that the quark masses play only a minor role in the generation of hadron masses;
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
F I G U R E 101 One tangle (only) for each baryon in the lowest J=L+S=1/2 baryon octet (circles indicate
linked tail pairs to the border of space), together with the measured mass values.
Motion Mountain – The Adventure of Physics
P = +1 for all four.
344
11 the pa rticle spectrum d ed uced from stra nds
Baryons with spin S = 3/2 and angular momentum L = 0 made of up, down and strange quarks :
s
s
sss=Ω−
1672 MeV
d
s
dss=Ξ∗–
1535 MeV
dds=Σ∗–
1387
MeV
d
d
u
uds=Σ∗0
1384
MeV
d
u
udd=Δ0
1232
MeV
d
s
s
u
uus=Σ∗+
1383
MeV
d
d
u
uud=Δ+
1232
MeV
u
s
u
u
uuu=Δ++
1232
MeV
u
u
F I G U R E 102 One tangle for each baryon in the lowest J=3/2 baryon decuplet (circles indicate linked
tail pairs to the border of space), together with the measured mass values.
this old result from QCD is thus reproduced by the strand model.
The arguments presented so far only reproduce mass sequences, not mass values. Actual hadron mass calculations are possible with the strand model: it is necessary to compute the number of crossing changes each tangle produces. There is a chance, but no
certainty, that such calculations might be simpler to implement than those of lattice QCD.
Tetraquarks and exotic mesons
Ref. 218
Ref. 234
Among the exotic mesons, tetraquarks are the most explored cases. It is now widely believed that the low-mass scalar mesons are tetraquarks. In the strand model, tetraquarks
are possible; an example is given in Figure 103. This is a six-stranded rational tangle. Spin,
parities and mass sequences from the strand model seem to agree with observations. If
the arrangement of Figure 103 would turn out to be typical, the tetraquark looks more
like a bound pair of two mesons and not like a state in which all four quarks are bound in
equal way to each other. On the other hand, a tetrahedral arrangement of quarks might
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
ddd=Δ–
1232
MeV
d
d
s
uss=Ξ∗0
1532 MeV
s
s
u
Motion Mountain – The Adventure of Physics
d
Circles indicate
pairs of tails to the
border of space
s
ta ng les of four a nd m ore stra nds
345
The scalar σ meson as a tetraquark (ud)(ud), c. 0.5 GeV :
d
u
d
u
Circles indicate
pairs of tails to the
border of space
F I G U R E 103 The strand model for a specific tetraquark (circles indicate linked tail pairs to the border of
space).
Page 324, page 340
O ther tangles made of four or more strands
Summary on tangles made of four or more strands
By exploring all possible tangle classes in detail, we have shown that every localized structure made of strands has an interpretation in the strand model. In particular, the strand
model makes a simple statement on any tangle made of four or more strands: such a
tangle is composite of the elementary tangles made of one, two or three strands. In other
terms, there are no elementary particles made of four or more strands in nature.
The strand model states that each possible tangle represents a physical particle sys-
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
We do not need to explore other prime tangles or locally knotted tangles made of four
or more strands. They are either not allowed or are higher-order versions of rational
tangles, as explained already in the case of two and three strands. We also do not need to
explore separable tangles. Separable tangles are composite of tangles with fewer strands.
One class of tangles remains to be discussed: braided tangles of four or more strands.
Now, a higher-order perturbation of the weak interaction can always lead to the topological entanglement of some vacuum strand with a tangle of fewer strands. Braided
tangles of four or more strands are thus higher-order propagating states of three-stranded
leptons or hadrons.
We can also state this in another way. There are no tangles of four or more strands
that are more tangled than the trivial tangle but less tangled than the lepton tangles.
Therefore, no additional elementary particles are possible. In short, the tangle model
does not allow elementary particles with four or more strands.
Motion Mountain – The Adventure of Physics
Ref. 220
also be possible. The details of this topic are left for future exploration.
The strand model makes an additional statement: knotted (hadronic) strings in
quark–antiquark states are impossible. Such states have been proposed by Niemi. In the
strand model, such states would not be separate mesons, but usual mesons with one or
several added virtual weak vector bosons. This type of exotic mesons is therefore predicted not to exist.
The situation for glueballs, which are another type of exotic mesons, has already been
discussed above.
346
11 the pa rticle spectrum d ed uced from stra nds
TA B L E 13 The match between tangles and particles in the strand model.
Strands
Ta n g l e
Pa r t i c l e
Ty pe
1
1
unknotted
knotted
elementary
–
vacuum, (unbroken) gauge boson
not in the strand model
2
2
2
unknotted
rational
prime, knotted
composed
elementary
–
composed of simpler tangles
quark or graviton
not in the strand model
3
3
3
unknotted
braided
rational
composed of simpler tangles
lepton
leptons
3
prime, knotted
composed
elementary
elementary or
composed
–
4 & more
like for 3 strands
all composed
composed of simpler tangles
not in the strand model
Challenge 199 s
In the strand model, mass appears due to tail braiding. But mass is also due to tangle
rotation and fluctuation. How do the two definitions come together?
∗∗
The following statement seems absurd, but is correct:
⊳ The tangle model implies that all elementary particles are point-like, without
internal structure.
Indeed, if at all, the strand model implies deviations from point-like behaviour only at
Planck scale; particles are point-like for all practical purposes.
∗∗
Challenge 200 e
In the strand model, only crossing switches are observable. How then can the specific tangle structure of a particle have any observable effects? In particular, how can
quantum numbers be related to tangle structure, if the only observables are due to crossing changes?
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
fun challenges and curiosit ies abou t particle
tangles
Motion Mountain – The Adventure of Physics
tem: an overview is given in Table 13. The mapping between tangles and particles is
only possible because (infinitely) many tangles are assigned to each massive elementary
particle.
The result of this exploration is that the strand model limits the number of elementary
particles to those contained in the standard model of particle physics.
fun cha llenges a nd curi osities a b ou t pa rticle ta ng les
347
∗∗
No neutral weak currents that change strangeness or other flavours are observed. In the
strand model this observation is a consequence of the tangle shape of the Z boson.
∗∗
Challenge
Ref.201
235r
In 2014, Marek Karliner predicted the existence of six-quark states. Can the strand model
reproduce them? Can it settle whether they are molecules of three mesons or genuine
six-quark states?
∗∗
Challenge 202 e
Can you use the strand model to show that pentaquarks do not exist?
∗∗
Ref. 236
What is the relation of the model shown here to the ideas of Viro and Viro on skew lines?
Ref. 237
The most prominent proponent of the idea that particles might be knots was, in 1868,
William Thomson–Kelvin. He proposed the idea that different atoms might be differently
‘knotted vortices’ in the ‘ether’. The proposal was ignored – and rightly so – because
it did not explain anything: neither the properties nor the interactions of atoms were
explained. The proposal simply had no relation to reality. In retrospect, the main reason
for this failure was that elementary particles and quantum theory were unknown at the
time.
Purely topological models for elementary particles have been proposed and explored by
various scholars in the past. But only a few researchers ever proposed specific topological
structures for each elementary particle. Such proposals are easily criticized, so that it is
easy to make a fool of oneself; any such proposal thus needs a certain amount of courage.
Ref. 238
Ref. 143
Ref. 239
Ref. 139
Ref. 141
Ref. 140
— Herbert Jehle modelled elementary particles as closed knots already in the 1970s.
However, his model did not reproduce quantum theory, nor does it reproduce all
particles known today.
— Ng Sze Kui has modelled mesons as knots. There is however, no model for quarks,
leptons or bosons, nor a description for the gauge interactions.
— Tom Mongan has modelled elementary particles as made of three strands that each
carry electric charge. However, there is no connection with quantum field theory or
general relativity.
— Jack Avrin has modelled hadrons and leptons as Moebius bands, and interactions
as cut-and-glue processes. The model however, does not explain the masses of the
particles or the coupling constants.
— Robert Finkelstein has modelled fermions as knots. This approach, however, does not
explain the gauge properties of the interactions, nor most properties of elementary
particles.
— Sundance Bilson-Thompson, later together with his coworkers, modelled elementary
fermions and bosons as structures of triple ribbons. The leather trick is used, like in
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
∗∗
Motion Mountain – The Adventure of Physics
∗∗
348
11 the pa rticle spectrum d ed uced from stra nds
the strand model, to explain the three generations of quarks and leptons. This is by
far the most complete model from this list. However, the origin of particle mass, of
particle mixing and, most of all, of the gauge interactions is not explained.
∗∗
Ref. 240
Strands do not require higher dimensions. On the other hand, it can be argued that
strands do produce an additional non-commutative structure at each point in space. In
a sense, when strands are averaged over time, a non-commutative inner space is created
at each point in space. As a result, when we focus at a specific spatial position over
somewhat longer times scales than the Planck time, we can argue that, at that point of
space, nature is described by a product of three-dimensional space with an internal, noncommutative space. Since many years, Alain Connes and his colleagues have explored
such product spaces in detail. They have discovered that with an appropriately chosen
non-commutative inner space, it is possible to reproduce many, but not all, aspects of the
standard model of particle physics. Among others, choosing a suitable non-commutative
space, they can reproduce the three gauge interactions; on the other hand, they cannot
reproduce the three particle generations.
Connes’ approach and the strand model do not agree completely. One way to describe the differences is to focus on the relation of the inner spaces at different points of
space. Connes’ approach assumes that each point has its own inner space, and that these
spaces are not related. The strand model, instead, implies that the inner spaces of neigh-
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
∗∗
Motion Mountain – The Adventure of Physics
Ref. 144
Strands are not superstrings. In contrast to superstrings, strands have a fundamental
principle. (This is the biggest conceptual difference.) The fundamental principle for
strands is not fulfilled by superstrings. In contrast to superstrings, strands have no tension, no supersymmetry and no own Lagrangian. (This is the biggest physical difference.)
Because strands have no tension, they cannot oscillate. Because strands have no supersymmetry, general relativity follows directly. Because strands have no own Lagrangian,
particles are tangles, not oscillating superstrings, and quantum theory follows directly.
In fact, the definitions of particles, wave functions, fields, vacuum, mass and horizons
differ completely in the two approaches.
In contrast to superstrings, strands describe the number of gauge interactions and of
particle generations. In contrast to superstrings, strands describe quarks, hadrons, confinement, Regge behaviour, asymptotic freedom, particle masses, particle mixing and
coupling constants. In the strand model, in contrast to ‘open superstrings’, no important configuration has ends. In contrast to open or closed superstrings, strands move in
three spatial dimensions, not in nine or ten; strands resolve the anomaly issue without
higher dimensions or supersymmetry, because unitarity is automatically maintained, by
construction; strands are not related to membranes or supermembranes. In the strand
model, no strand is ‘bosonic’ or ‘heterotic’, there is no E(8) or SO(32) gauge group, there
are no general ‘pants diagrams’ for all gauge interactions, there is no AdS/CFT duality,
there is no ‘landscape’ with numerous vacuum states, and there is no ‘multiverse’. In
contrast to superstrings, strands are based on Planck units. And in contrast to superstrings, strands yield the standard model of elementary particles without any alternative.
In fact, not a single statement about superstrings is applicable to strands.
fun cha llenges a nd curi osities a b ou t pa rticle ta ng les
Hanging
situation
349
A
B
bouring points are related; they are related by the specific topology and entanglement of
the involved strands. For this very reason the strand model does allow to understand the
origin of the three particle generations and the details of the particle spectrum.
There are further differences between the two approaches. Connes’ approach assumes
that quantum theory and general relativity, in particular, the Hilbert space and the spatial manifold, are given from the outset. The strand model, instead, deduces these structures from the fundamental principle. And, as just mentioned, Connes’ approach is not
unique or complete, whereas the strand model seems to be. Of the two, only the strand
model seems to be unmodifiable, or ‘hard to vary’.
∗∗
The strand model implies that there is nothing new at small distances. At small distances,
or high energies, nature consists only of strands. Thus there are no new phenomena
there. Quantum theory states that at small scales, nothing new appears: at small scales,
there are no new degrees of freedom. For example, quantum theory states that there is no
kingdom Lilliput in nature. The strand model thus confirms the essence of quantum theory. And indeed, the strand model predicts that between the energy scale of the heaviest
elementary particle, the top quark, 173 GeV, and the Planck energy, 1019 GeV, nothing
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
motion along the chain, when holding ring B
while dropping ring A.
Motion Mountain – The Adventure of Physics
F I G U R E 104 A ring chain gives an impression of
350
F I G U R E 105 The ring chain trick produces an illusion of motion (mp4 film © Franz Aichinger). Can
more rings be added in horizontal directions?
Translational motion of a photon :
t2
Translational motion of an electron :
t1
t2
F I G U R E 106 Motion of photons and electrons through strand hopping.
is to be found. There is a so-called energy desert – empty of interesting features, particles
or phenomena – in nature.
∗∗
Most ropes used in sailing, climbing or other domains of everyday life are produced
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
t1
Motion Mountain – The Adventure of Physics
Challenge 203 e
11 the pa rticle spectrum d ed uced from stra nds
fun cha llenges a nd curi osities a b ou t pa rticle ta ng les
351
F I G U R E 107 A discarded candidate tangle for the W boson.
∗∗
Challenge 204 e
Not all tangle assignments are self-evident at first sight. Figure 107 shows a tangle whose
status in the strand model is not clear. Can you explain what the tangle represents?
∗∗
Page 150
Challenge 205 ny
What is the effect of shivering on braiding, and thus on weak particle mixing, on particle
tangle families and on the number of generations?
Challenge 206 e
Are all bosons made of strands whose ends are exactly opposite to each other at spatial
infinity? Photon, graviton, gluon, W, Z and the virtual Higgs comply. The unbroken ones
are axial, the broken ones are flat. Is there a reason or a sense for this issue?
CPT invariance
CPT invariance is a fundamental property of quantum field theory. In the strand model,
charge conjugation C is modelled as a mirror transformation of the tangle; parity P is
modelled as the change of sign of the belt trick of the tangle core; and motion inversion
T is modelled as the inverse motion of the core of a particle tangle.
In other words, CPT invariance is natural in the strand model. Therefore, the strand
model predicts that particles and antiparticles have the same g-factor, the same dipole
moment, the same mass, the same spin, exactly opposite charge value, etc. All this is also
predicted by quantum field theory, and is confirmed by experiment.
Motion through the vacuum – and the speed of light
Ref. 241
Up to now, one problem was left open: How can a particle, being a tangle of infinite
extension, move through the web of strands that makes up the vacuum? An old trick,
known already in France in the nineteenth century, can help preparing for the idea of
particle motion in space. Figure 104 shows a special chain that is most easily made with a
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
∗∗
Motion Mountain – The Adventure of Physics
by braiding. Searching for ‘braiding machine’ on the internet yields a large amount of
videos. Searching for ‘LEGO braiding machine’ shows the most simple and beautiful
examples and allows you to see how they work.
352
11 the pa rticle spectrum d ed uced from stra nds
few dozen key rings. If the ring B is grabbed and the ring A released, this latter ring seems
to fall down along the whole chain in a helical path, as shown in the film of Figure 203.
If you have never seen the trick, try it yourself; the effect is astonishing. In reality, this is
an optical illusion. No ring is actually falling, but the sequence of rings moves in a way
that creates the impression of ring motion. And this old trick helps us to solve a number
of issues about particle motion that we swept under the carpet so far.
The main idea on particle motion in the strand model is the following:
⊳ Translational particle motion is also due to strand substitution, or ‘strand
hopping’.
”
summary on millennium issues abou t particles and
the vacuum
We have discovered that the strand model makes a strong statement: elementary particles
can only be made of one, two or three tangled strands. Each elementary particle is represented by an infinite family of rational tangles of fixed strand number. The family members are related through various degrees of tangling, such as tail braiding or the leather
trick.
For one-stranded particles, the strand model shows that the photon, the W, the Z and
the gluons form the full list of spin-1 bosons. For two-stranded particles, the strand model
shows that there are precisely three generations of quarks. For three-stranded elementary
particles, the strand model shows that there is a Higgs boson and three generations of
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
“
The ground of science was littered with the
corpses of dead unified theories.
Freeman Dyson
Motion Mountain – The Adventure of Physics
A schematic illustration of translational motion is given in Figure 106. In the strand
model, contrary to the impression given so far, a tangle does not always need to move
as a whole along the strand. This is seen most easily in the case of a photon. It is easy to
picture that the tangle structure corresponding to a photon can also hop from strand to
strand. At any stage, the structure is a photon; but the involved strand is never the same.
The idea of motion through strand hopping also works for massive particles. The
motion of a massive particle, such as an electron, is shown schematically in Figure 106.
The figure shows that through a tail unbraiding, the structure that describes an electron
can get rid of one strand and grab a new one. This process has a low probability, of
course. In the strand model, this is one reason that massive particles move more slowly
than light, even if the first approximation yields a zero mass value.
We note that this explanation of motion is important also for the mapping from strand
diagrams to Feynman diagrams. For many such diagrams, for example for the annihilation of particles and antiparticles in QED, strand hopping and tail unbraiding play a role.
Without them, the mapping from strands to quantum field theory would not be possible.
In summary, tangles of massive particles can move through the vacuum using hopping – via tail unbraiding – and this naturally happens more slowly than the motion of
photons, which do not need any process at the border of space to hop. The speed of
photons is thus a limit speed for massive particles; special relativity is thus recovered.
sum m ary on m illennium issues a b ou t pa rticles a nd the vacuum
Page 153
Page 162
leptons. Neutrinos and antineutrinos differ and are massive Dirac particles. The strand
model thus predicts that the neutrino-less double-beta decay will not be observed. Glueballs probably do not exist.
The strand model explains the origin of all quantum numbers of the observed elementary particles. Also all predicted quantum numbers for composed particles agree
with observations. Therefore, we have also completed the proof that all observables in
nature are due to crossing switches.
The strand model reproduces the quark model, including all the allowed and all the
forbidden hadron states. For mesons and baryons, the strand model predicts the correct
mass sequences and quantum numbers. Tetraquarks are predicted to exist. A way to
calculate hadron form factors is proposed.
In the strand model, all tangles are mapped to known particles. The strand model
predicts that no elementary particles outside the standard model exist, because no tangles
are left over. For example, there are no axions, no leptoquarks and no supersymmetric
particles in nature. The strand model also predicts the lack of other gauge bosons and
other interactions. In particular, the strand model – corrected in 2012 – reproduces the
existence of the Higgs boson. In fact, any new elementary particle found in the future
would contradict and invalidate the strand model.
In simple words, the strand model explains why the known elementary particles exist
and why others do not. We have thus settled two further items from the millennium list
of open issues. In fact, the deduction of the elementary particle spectrum given here is,
the first and, at present, also the only such deduction in the research literature.
The omnipresent number 3
Predictions abou t dark mat ter, the LHC and the vacuum
Astrophysical observations show that galaxies and galaxy clusters are surrounded by
large amounts of matter that does not radiate. This unknown type of matter is called
dark matter.
In the strand model, the known elementary particles are the only possible ones.
Therefore, the galactic clouds made of dark matter must consist of those particles mentioned up to now, or of black holes.
⊳ The strand model thus predicts that dark matter is a mixture of particles of
the standard model and black holes.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 206
The strand model shows that the number 3 that appears so regularly in the standard
model of particle physics – 3 generations, 3 interactions, charge values ?/3 and 2?/3
of quarks (as shown below), 3 colours and SU(3) – is, in each case, a consequence of
the three-dimensionality of space. In fact, the strand model adds a further, but related
number 3 to this list, namely the maximum number of strands that make up elementary
particles.
The three-dimensionality of space is, as we saw already above, a result of the existence
of strands: only three dimensions allow tangles of strands. In short, all numbers 3 that
appear in fundamental physics are explained by strands.
Motion Mountain – The Adventure of Physics
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353
354
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11 the pa rticle spectrum d ed uced from stra nds
This statement settles a further item from the millennium list of open issues.
The prediction from 2008 of a lack of new elementary particles in dark matter is at
odds with the most favoured present measurement interpretations, but cannot yet be
ruled out. The detection of black hole mergers in 2015 can even be seen as a partial
confirmation. However, the issue is obviously not yet settled. In fact, the prediction
provides another hard test of the model: if dark matter is found to be made of yet unknown particles, the strand model is in trouble.
We can condense all the results on particle physics found so far in the following statement:
⊳ There is nothing to be discovered about nature outside general relativity and
the standard model of particle physics.
Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Strands predict that there is no hidden aspect of nature left. In particular, the strand
model predicts a so-called high-energy desert: it predicts the lack of any additional elementary particle. Equivalently, the strand model predicts that apart from the Planck
scale, there is no further energy scale in particle physics. Researchers blinded by beliefs
sometimes call this the nightmare scenario.
In other words, there is no room for discoveries beyond the Higgs boson at the Large
Hadron Collider in Geneva, nor at the various dark matter searches across the world. If
any new elementary particle is discovered, the strand model is wrong. More precisely, if
any new elementary particle that contradicts the strand model is discovered, the strand
model is wrong. That some unknown elementary particle has been missed in the present
exploration of tangle classes is still a logical possibility.
Because the strand model confirms the standard model and general relativity, a further prediction can be made: the vacuum is unique and stable. There is no room for other
options. For example, there are no domains walls between different vacuum states and
the universe will not decay or change in any drastic manner.
In summary, the strand model predicts a lack of any kind of science fiction in modern
physics.
C h a p t e r 12
PART ICLE PR OPERT IES DEDUCED
FROM ST R ANDS
”
** ‘Everything you see, I owe it to spaghetti.’ Sofia Villani Scicolone is an Italian actress and Hollywood
star.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
he Planck units, via strands and the fundamental principle, explain almost all
hat is known about motion: strands explain what moves and how it moves. But
he strand model is only correct if it also explains every measured property of every
elementary particle. So far, we only deduced the quantum numbers of the elementary
particles. Three kinds of particle properties from the millennium list remain open: the
masses, the mixing angles and the couplings. These measured particle properties are important, because they determine the amount of change – or physical action – induced by
the motion of each elementary particle.
So far, the strand model has answered all open questions on motion that we explored.
In particular, the strand model has explained why quantum field theory, the interactions,
the particle spectrum, general relativity and cosmology are what they are. But as long
as we do not understand the measured properties of elementary particles, we do not
understand motion completely.
In short, the next step is to find a way to calculate these particle properties – and
obviously, to show that the calculations agree with the measurements. The step is particularly interesting; so far, no other unified model in the research literature has ever
achieved such calculations – not even calculations that disagree with measurements.
Because the strand model makes no experimental predictions that go beyond general
relativity and the standard model of particle physics, explaining the properties of elementary particles is the only way to confirm the strand model. Many ways to test and
to refute the strand model are possible; but only a calculation of the measured particle
properties can confirm it.
The ideas in this chapter are more speculative than those of the past chapters, because
the reasoning depends on the way that specific tangles are assigned to specific particles.
Such assignments are never completely certain. We continue keeping this in mind.
Motion Mountain – The Adventure of Physics
T
“
Tutto quel che vedete, lo devo agli spaghetti.**
Sophia Loren
356
12 pa rticle properties d ed uced from stra nds
TA B L E 14 The measured elementary particle masses, as given by the Particle Data Group in 2016.
Electron neutrino
Muon neutrino quark
Tau neutrino
< 2 eV/c2
< 2 eV/c2
< 2 eV/c2
Electron
Muon
Tau
0.510 998 9461(31) MeV/c2
105.658 3745(24) MeV/c2
1776.86(12) MeV/c2
Up quark (? = 2/3 ?)
Down quark (? = −1/3 ?)
Strange quark (? = −1/3 ?)
Charm quark (? = 2/3 ?)
Bottom quark (? = −1/3 ?)
Top quark (? = 2/3 ?)
2.2(6) MeV/c2
4.7(5) MeV/c2
96(8) MeV/c2
1.27(3) GeV/c2
4.18(4) GeV/c2
173.21(1.22) GeV/c2
W boson
Z boson
Higgs boson
80.385(15) GeV/c2
91.1876(21) GeV/c2
125.09(24) GeV/c2
Photon
Gluons
Graviton
not detectable
not detectable
not detectable
For comparison:
the corrected Planck mass
0.61 ⋅ 1019 GeV/c2
the masses of the elementary particles
The mass describes the inertial and gravitational effects of a body. The strand model
must reproduce all mass values observed in nature; if it doesn’t, it is wrong.
To reproduce the masses of all bodies, it is sufficient that the strand model reproduces
the measured masses, the mixing angles and the coupling strengths of the elementary
particles. We start with their masses. The measured values are given in Table 14; all these
values are unexplained. They are part of the millennium list of open issues in fundamental physics.
In nature, the gravitational mass of a particle is determined by the space curvature
that it induces around it. In the strand model, this curvature is due to the modified
fluctuations that result from the presence of the tangle core; in particular, the curvature
is due to the modified fluctuations of the particle tails and to the modified vacuum strand
fluctuations just around the particle position. The modified strand fluctuations produce
a crossing switch distribution around the tangle core; the crossing switch distribution
leads to spatial curvature; at sufficiently large distances, this curvature distribution is
detected as a gravitational mass.
In contrast, inertial mass appears in the Dirac equation. In the strand model, iner-
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
M a s s va l u e
Motion Mountain – The Adventure of Physics
E l e m e n tary pa rt i c l e
the m asses of the elem enta ry pa rticles
Page 280
357
tial mass is determined by the frequency and the wavelength of the helix drawn by the
rotating phase vector. These quantities in turn are influenced by the type of tangle, by
the fluctuations induced by the particle charges, by the topology changes induced by the
weak interaction, and, in the case of fermions, by the average frequency and size of the
belt and leather tricks. All these processes are due to strand fluctuations.
In short, both gravitational and inertial particle mass are due to strand fluctuations.
More specifically, the mass seems mainly due to the fluctuations of the tails of the particle
tangle. The strand model thus suggests that gravitational and inertial mass are automatically equal. In particular, the strand model suggests that every mass is surrounded by
fluctuating crossing switches whose density decreases with distance and is proportional
to the mass itself. As discussed above, this idea leads to universal gravity.
General properties of particle mass values
The general properties of particle masses are thus reproduced by the strand model.
Therefore, continuing our exploration makes sense. We start by looking for ways to
determine the mass values from the tangle structures. We discuss each particle class
separately, first looking at mass ratios, then at absolute mass values.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 370
— The strand model implies that the masses of elementary particles are not free parameters, but that they are determined by the topology, or tangledness, of the underlying
tangles and their tangle families. Particle masses are thus fixed and discrete in the
strand model – as is observed. Of course, we have to take into account the many
members in each tangle family.
— The strand model implies that masses are always positive numbers.
— The strand model implies that the more complex a tangle is, the higher its mass value
is. This follows from the behaviour of tangle tail fluctuations around the tangle core.
— Because particle masses are due to strand fluctuations, the strand model also implies
that all elementary particle masses are much smaller than the Planck mass, as is observed. Also this result follows from the behaviour of tangle tail fluctuations around
the tangle core.
— Because particle masses are due to strand fluctuations, particle and antiparticle
masses – their tangles are mirrors of each other – are always equal, as is observed.
— Because particle masses are due to strand fluctuations, particle masses do not depend
on the age of the universe, nor on their position in the universe, nor on any other state
variable: The strand model predicts that particle masses are constant and invariant,
as is observed.
— Because particle masses are due to strand fluctuations, and the fluctuations differ
somewhat for tight and loose tangles of the same shape, the strand model predicts
that particle masses change – or run – with energy, as is observed.
Motion Mountain – The Adventure of Physics
So far, our adventure allows us to deduce several results on the mass values of elementary
particles:
358
12 pa rticle properties d ed uced from stra nds
Tight W boson tangle candidate
Tight Z boson tangle candidate
B oson masses
Three elementary particles of integer spin have non-vanishing mass: the W boson, the Z
boson and the Higgs boson. Mass calculations are especially simple for bosons, because
in the strand model, they are clean systems: each boson is described by a relatively simple
tangle family; furthermore, bosons do not need the belt trick to rotate continuously.
We expect that the induced curvature, and thus the gravitational mass, of an elementary boson is due to the disturbance it introduces into the vacuum. At Planck energy,
this disturbance will be, to a large extent, a function of the ropelength introduced by the
corresponding tight tangle. Let us clarify these concepts.
Tight or ideal tangles or knots are those tangles or knots that appear if we imagine
strands as being made of a rope of constant diameter that is infinitely flexible, infinitely
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
F I G U R E 108 Tight tangle candidates (of 2015/2016) for the simplest tangles of the W, the Z and the
Higgs bosons.
Motion Mountain – The Adventure of Physics
Tight Higgs tangle candidate
the m asses of the elem enta ry pa rticles
359
slippery and pulled as tight as possible. Examples of tight tangles are shown in Figure 108.
With physical ropes from everyday life, tight knots and tangles can only be approximated, because they are not infinitely flexible and slippery; tight tangles are mathematical
idealizations. But tight tangles of strands are of special interest: if we recall that each
strand has an effective diameter of one Planck length, tight tangles realize the Planck
limit of the strand model.
— The ropelength of a tight closed knot is the length of a perfectly flexible and slippery rope of constant diameter required to tie the tight knot. In other words, the
ropelength is the smallest amount of idealized rope needed to tie a knot.
— The ropelength of a tight open knot is the length by which a very long rope tied into a
tight knot is shortened.
— With a bit of care, the concept of ropelength can be also be defined for tangles of
several strands.
W/Z boson mass ratio and mixing angle (in the 2016 tangle model)
Page 358
Ref. 244
Candidates for the simplest tangles of the W boson and of the Z boson families are shown
in Figure 108. The corresponding ropelength values for tight tangles, determined numerically, are ? W = 4.28 and ? Z = 7.25 rope diameters. The strand model estimates the W/Z
mass ratio by the cube root of the ropelength ratio:
1/3
?W
?
≈ ( W ) = 0.84 .
?Z
?Z
Ref. 231
(202)
This value has to be compared with the experimental ratio of 80.4 GeV/91.2 GeV=0.88.
The agreement between experiment and strand model is not good, but acceptable, for
two reasons.
On the one hand, the strand model reproduces the higher value of the neutral Z bo-
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 242
Motion Mountain – The Adventure of Physics
Page 357
In the following, the ropelength is assumed to be measured in units of the rope diameter.
Measuring ropelength in units of the rope radius is less common.
In the strand model, the ropelength measures, to a large extent, the amount by which
a tight knot or tangle disturbs the vacuum around it. The ropelength fulfils all the properties of particle mass mentioned above: the ropelength is discrete, positive, increases with
tangle complexity, is equal for particles and antiparticles, and is a constant and invariant
quantity. The ropelength will thus play an important role in any estimate of a particle
mass.
It is known from quantum field theory that the masses of W and Z bosons do not
change much between Planck energy and everyday energy, whatever renormalization
scheme is used. This allows us, with a good approximation, to approximate the weak
boson masses at low, everyday energy with their mass values at Planck energy. Thus we
can use tight tangles to estimate boson masses.
In the strand model, the gravitational mass of a spin 1 boson is proportional to the
radius of the disturbance that it induces in the vacuum. For a boson, this radius, and
thus the mass, scales as the third root of the ropelength of the corresponding tight tangle.
360
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Ref. 162
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 194, page 212
son’s mass: a tangle with spatial symmetry is more complex than one without. On the
other hand, is also clear why the calculated mass ratio does not match the experimental
result. First of all, the simple tangles represent and approximate W and Z bosons only to
the first order. As mentioned above, in the strand model, every massive particle is represented by an infinite family of tangles. The strand model thus also predicts that the match
between the calculated and the measured ratio ?W /?Z should improve when higherorder Feynman diagrams, and thus more complicated tangle topologies, are taken into
account. Improving the calculation is still a subject of research. Secondly, approximating
the tight knot effects with an effective radius, thus just using the ropelength to determine
the mass, implies neglecting the actual shape, and effectively approximating their shape
by a sphere. Thirdly, as already mentioned, this calculation assumes that the low energy
mass ratio and the mass ratio at Planck energy are equal.
Despite the used approximations, the tight tangle estimate for the W/Z mass ratio
gives an acceptable agreement with experiment. The main reason is that we expect the
strand fluctuations from the various family members to be similar for particles with the
same number of strands. For these mass ratios, the tail braiding processes cancel out.
Also the other two approximations are expected to be roughly similar for the two weak
bosons. This similarity explains why determining the W/Z boson mass ratio is possible
with acceptable accuracy.
The W/Z mass ratio also determines the weak mixing angle ?w of the weak interaction
Lagrangian, through the relation cos ?w = ?W /?Z. The strand model thus predicts the
value of the weak mixing angle to the same accuracy as it predicts the W/Z mass ratio.
This argument leads to a puzzle: Can you deduce from the strand model how the W/Z
mass ratio changes with energy?
Also the inertial masses of the W and Z bosons can be compared. In quantum theory, the inertial mass relates the wavelength and the frequency of the wave function. In
the strand model, a quantum particle that moves through vacuum is a tangle core that
rotates while advancing. The frequency and the wavelength of the helix thus generated
determine the inertial mass. The process is analogous to the motion of a body moving
at constant speed in a viscous fluid at small Reynolds numbers. Despite the appearance
of friction, the analogy is possible. If a small body of general shape is pulled through a
viscous fluid by a constant force, such as gravity, it follows a helical path. This analogy
implies that, for spin 1 particles, the frequency and the wavelength are above all determined by the effective radius of the small body. The strand model thus suggests that the
inertial mass – inversely proportional to the path frequency and the path wavelength
squared – of the W or the Z boson is approximately proportional to its tight knot radius.
This yields again a cube root of the ropelength and thus gives the same result as for the
gravitational mass.
Also the inertial mass is not exactly proportional to the average tight knot radius;
the precise shape of the tight knot and the other tangle family members play a role. The
strand model thus predicts that a more accurate mass calculation has to take into account
these effects.
In summary, the strand model predicts a W/Z mass ratio and thus a weak mixing
angle close to the observed ratio, and explains the deviation of the approximation from
the measured value – provided that the tangle assignments are correct.
Motion Mountain – The Adventure of Physics
Challenge 207 ny
12 pa rticle properties d ed uced from stra nds
the m asses of the elem enta ry pa rticles
361
The g-factor of the W boson
Ref. 231
Ref. 243
Experiments show that the W boson has a g-factor with the value ?? = 2.2(2). This result
– whose limited accuracy does not allow to detect any anomalous magnetic moment yet
– can be compared to the prediction of the strand model. In particular, the observation
might be used to eliminate certain tangle candidates for the W boson.
The strand model makes a simple prediction for charged elementary particles: because mass rotation and charge rotation are both due to the rotation of the particle core,
the g-factor of all such particles is 2 – in the approximation that neglects Feynman diagrams of higher order. In particular, the g-factor of the W boson is predicted to be 2 in
this approximation. Also this prediction thus agrees both with experiment and with the
standard model of particle physics.
The Higgs/Z boson mass ratio
Page 358
Ref. 244
(203)
Starting with the W boson yields an estimate for the Higgs mass of 128 GeV. Both estimates are not good but acceptable, given that the non-sphericity of the W, Z and Higgs
boson tangles have not been taken into account. (The strand model suggests that for a
strongly non-spherical shape – such as the shape of the W, Z and Higgs tangle – the effective mass is higher than the value deduced from ropelength alone.) Deducing better
mass ratio estimates for the W, Z and Higgs tangles is still a subject of research.
In summary, the strand model predicts a Higgs/Z, a Higgs/W and a W/Z mass ratio
close to the observed values; and the model suggests explanations for the deviations of
the approximation from the observed value – provided that the tangle assignments for
the three bosons are correct.
A first approximation for absolute boson mass values
The tangles for the W, Z and Higgs bosons also provide a first approximation for their
absolute mass values. The tangles are rational; in particular, each tangle is made of strands
that can be pulled straight. This implies, for each strand separately, that a configuration
with no extra strand length and no net core rotation is possible. As a result, in the first
approximation, the gravitational mass and the inertial mass of the elementary bosons
both vanish.
A better approximation for mass values requires to determine, for each boson, the
probability of crossing switches in and around its tangle core. This probability depends
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
?Higgs ≈ 121 GeV .
Motion Mountain – The Adventure of Physics
Page 332
The observed mass value of the Higgs boson is 125(1) GeV. The observed mass value for
the Z boson is 91.2(1) GeV. Like for the other bosons, the strand model suggests using
the ropelength to estimate the mass of the Higgs boson tangle. The candidate tangle
for the Higgs boson was illustrated above in Figure 94, and its tight version is shown in
Figure 108.
The ropelength of the tight Higgs tangle turns out to be 17.1 diameters, again determined by numerical approximation. This value yields a naive mass estimate for the Higgs
boson of (17.1/7.25)1/3 ⋅ 91.2 GeV, i.e.,
362
12 pa rticle properties d ed uced from stra nds
Antiquarks :
Parity P = +1, B = 1/3, spin S = 1/2
Q = –1/3
Q = +2/3
P = –1, B = –1/3, S = 1/2
Q = +1/3
Q = –2/3
d
u
d
u
5.0 ± 1.6 MeV
2.5 ± 1.1 MeV
5.0 ± 1.6 MeV
2.5 ± 1.1 MeV
s
c
s
c
105 ± 35 MeV
1.27 ± 0.11 GeV
105 ± 35 MeV
1.27 ± 0.11 GeV
b
t
b
t
4.20 ± 0.17 GeV
171.3 ± 2.3 GeV
4.20 ± 0.17 GeV
171.3 ± 2.3 GeV
Seen from a larger distance, the tails follow (on average) the skeleton of a tetrahedron :
s
s
are also given.
on the probabilities for tail braiding and for core rotation. These probabilities are low,
because, sloppily speaking, the corresponding strand fluctuations are rare. The rarity is
a due to the specific tangle type: tangles whose strands can be pulled straight have low
crossing switch probabilities at their core or at their tails when they propagate.
The strand model thus predicts that elementary boson masses, like all other elementary particle masses, are much smaller than the Planck mass, though not exactly zero. This
prediction agrees with observation: experimentally, the three elementary boson mass
values are of the order of 10−17 Planck masses. We will search for more precise mass
estimates below.
Q uark mass ratios
Quarks are fermions. In the strand model, mass estimates for fermions are more difficult
than for bosons, because their propagation involves the belt trick. Still, using Figure 109,
the strand model allows several predictions about the relations between quark masses.
— The quark masses are predicted to be the same for every possible colour charge. This
is observed.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
F I G U R E 109 The simplest tangles assigned to the quarks and antiquarks. The experimental mass values
Motion Mountain – The Adventure of Physics
Quarks :
the m asses of the elem enta ry pa rticles
Page 320
Page 372
363
— Furthermore, the progression in ropelength of the tight basic tangles for the six
quarks suggests a progression in their masses. This is observed, though with the exception of the up quark mass. For this exceptional case, effects due to tail braiding
and to quark mixing are expected to play a role, as argued below.
? ∼ e??
Ref. 245
Ref. 247
where ? is an unknown number of order 1. We note directly that such a relation promises
general agreement with the observed quark mass ratios.
Actual ropelength calculations by Eric Rawdon and Maria Fisher show that the
ropelength of quark tangles increases roughly linearly with ?, as expected from general
knot theoretic arguments. Their results are given in Table 15. Comparing these calculated ropelength differences with the known low-energy quark masses confirms that the
number ? has an effective value in the range between 0.4 and 0.9, and thus indeed is of
order one.
The results of Table 15 suggest that the top quark should be particularly heavy – as
is observed. The results of Table 15 also suggest that something special is going on for
the ?-? quark pair, which is out of sequence with the other quarks. Indeed, the strand
model predicts a very small mass, – at the Planck scale – for the down quark. However,
in nature, the down mass is observed to be larger than the up mass. (We note that despite
this issue, meson mass sequences are predicted correctly.) It could well be that the mirror
symmetry of the simplest down quark tangle is the reason that the braiding, i.e., the
mixing with more massive the family members with six and more additional crossings,
is higher than that for the up quark.
The experimental values for the quark masses are given in Table 16; the table also
includes the values extrapolated to Planck energy for the pure standard model. The calculation of the strand model does not agree with the data. The only encouraging aspect
is that the ropelength approximation provides an approximation for older speculations
on approximately fixed mass ratios between the up-type quarks u, c, t and fixed mass ratios between down-type quarks d, s, b. The attempted strand model estimate shows that
ropelength alone is not sufficient to understand quark mass ratios. Research has yet to
show which effect has to be included to improve the correspondence with experiment.
In fact, the strand model predicts that everyday quark masses result from a combin-
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 246
(204)
Motion Mountain – The Adventure of Physics
Let us try to extract numerical values for the quark mass ratios. We start by exploring the
tight quark tangles, thus Planck-scale mass values. For each quark number ?, the quark
mass will be the weighted average over the mass of its family tangles with ?, ? + 6, ? + 12,
... crossings, where the period 6 is due to the leather trick. Each tight tangle has a certain
ropelength. The mass of each tangle will be determined by the frequency of crossing
changes at the core, including those due to the belt trick. The quark mass then is the
average over all family tangles; it will be determined by the frequency of tail braiding
and of all other fluctuations that generate crossing switches.
For determining mass ratios, the frequency of the crossing switches in the core are
the most important. Given that the particles are fermions, not bosons, this frequency
is expected to be an exponential of the ropelength ?. Among quarks, we thus expect a
general mass dependence of the type
364
12 pa rticle properties d ed uced from stra nds
TA B L E 15 Calculated ropelengths, in units of the rope diameter, of tight quark tangles of Figure 84
(Page 321) with tails oriented along the skeleton of a tetrahedron.
Ta n g l e
Length
Ropelength
Difference
skeleton (vacuum)
simplest d
simplest u
simplest s
simplest c
simplest b
simplest t
138.564065
139.919533
142.627837
146.175507
149.695643
153.250364
157.163826
base value
1.355468
4.063773
7.611443
11.131578
14.686299
18.599761
1.355468
2.708305
3.547670
3.520136
3.554721
3.913462
TA B L E 16 For comparison: the quark masses at Planck energy,, calculated from the measured quark
Low energy
mass
Planck energy
mass
u (? = 2/3?)
d (? = −1/3?)
s (? = −1/3?)
c (? = 2/3?)
b (? = −1/3?)
t (? = 2/3?)
2.5(1.1) MeV
5.0(1.6) MeV
105(35) MeV
1270(110) MeV
4200(170) MeV
171300(2300) MeV
0.45(0.16) MeV
0.97(0.10) MeV
19.4(1.2) MeV
213(8) MeV
883(10) MeV
66993(880) MeV
ation of three effects: the effect of ropelength and of tangle core shape on rotation and
the belt trick, the effect of sixfold tail braiding, and the effect of the energy dependence
of mass between Planck energy and everyday energy, due to core loosening.
Even though an analytic calculation for quark masses seems difficult, better approximations are possible. With sufficient computer power, it is possible to calculate the effects of the core shape rotations, including the belt trick, and of the energy dependence
of the quark masses. The most difficult point remains the calculation of the probabilities
for tail braiding. More research is needed on all these points.
L epton mass ratios
Mass calculations for leptons are as involved as for quarks. Each lepton, being a fermion,
has a large family of associated tangles: there is a simplest tangle and there are the tangles
that appear through repeated application of tail braiding. Despite this large tangle family,
some results can be deduced from the simplest lepton tangles alone, disregarding the
higher-order family members.
Both for neutrinos and for charged leptons, the progression in ropelength of the tight
versions of the basic tangles predicts a progression in their masses. This is indeed observed. This is valid for all candidate set of lepton tangles.
For each lepton tangle with ? crossings, knot theory predicts a ropelength ? that in-
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Q ua rk
Motion Mountain – The Adventure of Physics
masses using the standard model of particle physics – assuming that it is correct up to Planck energy.
the m asses of the elem enta ry pa rticles
365
Lepton tangles, all with spin S = 1/2 , parity P = +1 , lepton number L’ = 1 and baryon number B = 0 :
e
0.5 MeV
105 MeV
νμ
τ
all three with
Q = –1
1.77 GeV
ντ
all three with
Q=0
1±1 eV
1±1 eV
Seen from a larger distance, the tails follow (on average) the x, y and z axes of a coordinate system.
F I G U R E 110 Simple candidate tangles for the leptons. Antileptons are mirror tangles. The experimental
mass values are also given.
creases roughly proportionally to the crossing number: ? ∼ ?. Each lepton mass value
will again be given by the frequency of crossing switches due to rotations, including the
belt trick, and of tail braiding. We thus expect a general relation of the type
?? ∼ e?? ?
(205)
where ? is a number of order 1 that takes into account the shape of the tangle core. Such
a relation is in general agreement with the observed ratios between lepton masses. Research on these issues is ongoing; calculations of ropelengths and other geometric properties of the lepton tangles will allow a more detailed analysis. The most important challenges are, first, to deduce the correct mass sequence among the muon neutrino and the
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
1±1 eV
Motion Mountain – The Adventure of Physics
νe
μ
366
Page 331
Ref. 248
12 pa rticle properties d ed uced from stra nds
electron, and second, to estimate the neutrino masses.
We note that the lepton mass generation mechanism of the strand model differs from
other proposals in the research literature. It agrees with the Higgs mechanism but goes
beyond it. For neutrinos, the mechanism contradicts the see-saw mechanism but confirms the Yukawa mechanism directly. From a distance, the mass mechanism of the
strand model also somewhat resembles conformal symmetry breaking.
On the absolu te values of particle masses
Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
In nature, the masses of elementary particles are observed to be much lower than the
Planck mass: the observed values lie between about 10−30 for neutrinos and 10−17 for the
top quark. Particle masses are constant over space and time. Antiparticles have the same
mass as particles. Gravitational and inertial masses are the same. Following the standard
model, particle masses are due to the Higgs mechanism. Finally, elementary particles
masses run with energy.
All qualitative observations about mass are reproduced by the strand model. The however, the explanation of the numerical values is still lacking.
In the strand model, the gravitational mass of elementary particles is due to disturbance of the vacuum, in particular to the disturbance of the vacuum fluctuations. Larger
masses are due to more complex tangles. Since rest mass is localized energy, rest mass is
due to crossing switches per time. Larger masses have more crossing switches per time
than lower masses.
In the strand model, the inertial mass of elementary particles is their reluctance to
rotate. Inertial mass describes the relation between rotation frequency and wavelength;
in other terms, inertial mass described the steepness of the helix drawn by the rotating
phase arrow of a propagating particle. Larger masses low steepness, smaller masses have
higher steepness. Larger masses are due to more complex tangles.
As we just saw, the strand model predicts mass sequences and ratios of elementary
particle masses that corroborate or at least do not contradict observations too much.
The next step is to determine absolute mass values from the strand model. So far we
only found that elementary particle masses are much smaller than a Planck mass. But to
validate the strand model, we need more precise statements. To determine gravitational
mass values, we need to count those crossing switches that occur at rest; to determine
inertial mass values, we need to look for crossing switches in the case of a moving particle
– or, if we prefer, to understand the origin of the steepness of the helix drawn by the phase
arrow. All these methods should first lead to mass value estimates and then to mass value
calculations.
In general, the strand model reduces mass determination to the calculation of the
details of a process: How often do the fluctuations of strands lead to crossing switches?
There are various candidates for the crossing switches that lead to particle mass.
The first candidate for mass-producing crossings is tail switching. In general however,
tail switching leads to different particle types. Only for the Higgs process, i.e., the addition of a full Higgs braid to a particle, is this process expected to be relevant. We can also
say that tail braid addition are the strand model’s version of the Yukawa coupling terms.
The next candidate for mass-producing crossings is the belt trick. However, the belt
trick cannot be the full explanation, as there are also massive bosons, and the belt trick
the m asses of the elem enta ry pa rticles
Page 331
⊳ Mass is crossing switch rate.
* What is a Planck mass? In the strand model, a Planck mass corresponds to a structure that produces one
crossing switch for every Planck time, constantly, without interruption. But the strand model predicts that
such structures do not appear as localized particles, because every localized particle – i.e., every tangle –
has, by construction, a much smaller number of induced crossing switches per time. Following the strand
model, elementary particles with Planck mass do not exist. This conclusion agrees with observation. But
the strand model also implies that black holes with a Planck mass do not exist. Indeed, such Planck-scale
black holes, apart from being extremely short-lived, have no simple strand structure. We can state that a
Planck mass is never localized. Given these results, we cannot use a model of a localized Planck mass as a
unit or a benchmark to determine particle masses.
The impossibility of using Planck mass as a unit is also encountered in everyday life: no mass measurement in any laboratory is performed by using this unit as a standard.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
only applies to fermions.
The third candidate for mass-producing crossings could appear when particles shed
one strand and grap a new one. The influence of this process is not clear yet.
A fourth candidate for mass-producing crossings is the leather trick. However, the
leather trick cannot be realized for strands that reach spatial infinity; therefore it is expected that it play no role.
A fifth candidate for mass-producing crossings are those crossings that occur above
the core, similar to the crossing that occur above the horizon of a black hole. This candidate includes the belt trick in the case of fermions, includes the Higgs mechanism, and
thus seems most promising.
It might be that some other mass-producing switching processes are being overlooked. Nevertheless, in the following we explore the fifth candidate in more detail, the
crossings around a given tangle core.
Before looking for estimates, we note that in the past, various researchers have reached
the conclusion that all elementary particle masses should be due to a common process
or energy scale. Among theoretical physicists, the breaking of conformal symmetry has
always been a candidate for the associated process. Among experimental physicists, the
Higgs mechanism – now confirmed by experiment – is the favourite explanation of all
elementary particle masses. In the strand model, crossing switches around tangles are
related to the Higgs boson. At the same time we can also argue that tangles break the
conformal symmetry of vacuum. With a bit of distance, we can thus say that the strand
model agrees with both research expectations.
We now continue with the quest for absolute mass estimates. In the strand model,
absolute mass values are not purely geometric quantities that can be deduced directly
from the shapes of tangle knots. Particle masses are due to dynamical processes. Absolute
mass values are due to strand fluctuations; and these fluctuations are influenced by the
core topology, the core shape, the core ropelength and core tightness.
To determine absolute particle mass values, we need to determine the ratio between
the particle mass and the Planck mass. This means to determine the ratio between the
crossing switch probability for a given particle and the crossing switch probability for a
Planck mass, namely one switch per Planck time.*
Energy is action per time. Mass is localized energy. In other words, the absolute mass
of a particle is given by the average number of crossing switches it induces per time:
Motion Mountain – The Adventure of Physics
Ref. 248, Ref. 249
367
368
12 pa rticle properties d ed uced from stra nds
three polymer strands
monomers
with
Planck length
diameter
tangled
core
region
R
F I G U R E 111 Determining lepton
liquid solution
More precisely, the crossing switch rate of a particle at rest is its gravitational mass, and
the crossing switch rate induced by propagation is its inertial mass. Let us explore the
relations.
Given that mass is determined by the crossing switch rate, we deduce that particle
mass values are determined by tangle topology, are fixed, are discrete, are positive, increase with tangle core complexity, are identical for particle and antiparticles, are constant over time, and are much smaller than the Planck mass. Because all these properties
match observations, the local crossing switch rate indeed realizes all qualitative requirements for absolute particle mass values. We can thus proceed with the hope to learn
more. In order to calculate absolute particle masses, we just need to determine the number of crossing switches per time that every particle tangle induces.
One general way to perform a particle mass calculation is to use a computer, insert
a strand model of the fluctuating vacuum plus the strand model of the particle under
investigation, and count the number of crossing switches per time. The basis for one such
approach, using the analogy of the evolution of a polymer in liquid solution, is shown in
Figure 111. In contrast to polymers, also the change of strand length has to be taken into
account. By determining, for a given core topology, the average frequency with which
crossing switches appear for a tethered core, we can estimate the masses of the leptons,
quarks and bosons. In such a mass calculation, the mass scale is set indirectly, through
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 357
mass values with the help of a
polymer analogy of strands. After
rescaling, the probability of
crossing switches around the
tangle core yields an estimate for
the mass of the elementary
particle with that tangle.
Motion Mountain – The Adventure of Physics
tails
the m asses of the elem enta ry pa rticles
369
the time scale of the fluctuation spectrum. This is tricky but feasible. One would first
need to find the parameter space and the fluctuation spectrum for which the polymer
tangle follows the Schrödinger equation. Calculations with different tangles should then
yield the different mass values. Such a simulation would also of interest for exploring the
strand model of quantum mechanics.
A more precise computer simulation would also model the vacuum itself with strands.
This approach would even allow to explore gravitational and inertial mass separately. In
such a simulation, the particle mass appears when the helical motion of a tangle moving
through a strand vacuum is observed. The required effort can be reduced by using the
most appropriate computer libraries.
A further general way to determine particle masses is to search for analytical approximations. This is a fascinating conceptual and mathematical challenge. The main issue is
to clarify which crossing switches contribute most to particle mass.
A first analytical attempt is the following. We assume that the inertial mass for a moving
fermion is proportional to the fluctuation-induced appearance of the belt trick. If the
tight core has a diameter of, say, three Planck lengths – and thus a circumference of
around 9 Planck lengths – then the probability ? of the belt trick for a particle with six
tails will be in the range
? ≈ (e−9 )6 ≈ 10−24 .
(206)
Challenge 208 ny
Challenge 209 r
Challenge 210 ny
⊳ What is the numerical probability of the belt trick for a tethered core of given
topology with fluctuating tails?
So far, several experts on polymer evolution have failed to provide even the crudest estimate for the probability of the belt trick in a polymer-tethered ball. Can you provide
one?
A second analytical approach starts from the following question:
⊳ How often does a tail cross above the tangle core?
This question is loosely related to the previous one; in addition, this approach illustrates
why complex cores have larger mass. The probability of such crossings, when squared,
would be an estimate for the crossing switch rate, and thus for the particle mass. (There
are additional details to the calculation.) We note directly that the number of tails will
have a smaller impact on mass than the complexity of the tangle. So far, a reliable estimate for the crossing number, as a function of the tangle core properties, is still missing
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
This value would be the order of magnitude for the mass estimate, in Planck units. Such
an estimate is only very rough, and the exponent can be quite different. Nevertheless, we
do get an explanation for the large difference between the Planck mass and the typical
fermion mass. A more precise analytical approximation for the belt trick probability –
not an impossible feat – will therefore solve the so-called mass hierarchy problem. We
thus want to know:
Motion Mountain – The Adventure of Physics
Analytical estimates for particle masses
370
Challenge 211 ny
12 pa rticle properties d ed uced from stra nds
– even a crude one. Can you find one?
Open issues abou t mass calcul ations
Challenge 212 e
Calculating absolute particle masses from tangle fluctuations, either numerically or with
an analytical approximation, will allow the final check of the statements in this section.
The strand model predicts that the resulting values will match experiments. For these
calculations, it is essential that the tangle assignment for each elementary particle is correct. In 2017, the tangles for the W tangle and for the lepton tangles are still doubtful.
∗∗
Because the strand model predicts a lack of new physics beyond the standard model of
particle physics, the calculation of neutrino masses, and thus their mass sequence, is one
of the few possible predictions – in contrast to retrodictions – that are left over in the
strand model.
Challenge 213 s
Is the mass of a tangle related to the vacuum density of strands?
∗∗
Challenge 214 s
Do particle masses depend on the cosmological constant?
∗∗
∗∗
Challenge 215 s
Can the concept of total curvature of a tangle help to calculate particle masses?
∗∗
Challenge 216 d
Does the effect of tail braiding confirm the conjecture that every experiment is described
by a small energy scale, determining the resolution or precision, and a large energy scale,
less obvious, that determines the accuracy?
∗∗
If tail braiding is due to the weak interaction, and if the Higgs is a tail-braided vacuum,
can we deduce that the Higgs interaction is a higher order effect of the weak interaction?
Can we deduce a concrete experimental prediction from this relation?
On fine-tuning and naturalness
It has become fashionable, since about a decade, to state that the standard model of elementary particle physics is ‘fine-tuned’. The term expresses several ideas. First of all,
the extremely low value of the vacuum energy is not obvious when all the zero-point
field contributions from the various elementary particles of the standard model are included. A low vacuum energy seems only possible if the masses and the particle types
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
The mass of an elementary particle does not depend on the spin direction. In particular,
the W and Z bosons have equal longitudinal and transversal mass. The strand model
does not allow an influence of spin orientation on mass.
Motion Mountain – The Adventure of Physics
∗∗
the m asses of the elem enta ry pa rticles
of the standard model are somehow interrelated. In other words, the term ‘fine tuning’
expresses, above all, the lack of understanding of the origin of the masses, mixings and
coupling constants of elementary particles.
The term ‘fine tuning’ is also used to state that the universe would be very different
if the fundamental constants would be different. But this statement lacks deep truth.
In this usage, the term ‘fine tuning’ states that particle masses are not parameters that
can be varied at will. In common usage, ‘parameters’ are variable constants; but the low
value of the vacuum energy – as well as many other observations – shows that the masses
of elementary particles cannot be varied without destroying the validity of the standard
model of particle physics.
Some people suggest that ‘fine-tuning’ implies that the standard model of particle
physics is ‘unnatural’, whatever this might mean in detail. Some even suggest that the
parameters of the standard model lack any explanation. The strand model – but also
common sense – show that this suggestion is false.
The strand model naturally has a low vacuum energy, because the unknotted strands
of flat space naturally have a zero energy density, and the particle masses, mixings and
coupling constants are not variable or random, but naturally unique and fixed in value.
Any correct description of nature must be ‘fine-tuned’. If the standard model would not
be ‘fine-tuned’, it would not describe nature.
In short, the fashionable term ‘fine-tuned’ is equivalent to the terms ‘unmodifiable’
and ‘hard to vary’ that were discussed above. All these terms highlight the lack of alternatives to the world as we observe it, the existence of explanations for the processes
around us, and our ability to discover and grasp them. This is part of the wonders of
nature. And the strand model makes those wonders apparent at the Planck scale.
Page 369
Page 162
The strand model implies that masses are dynamic quantities fixed by processes due to
the geometric and topological properties of specific tangle families. As a result, strands
explain why the masses of elementary particles are not free parameters, but fixed and
unique constants, and why they are much smaller than the Planck mass by many orders
of magnitude. Strands also reproduce all known qualitative properties of particle masses.
Strands provide estimates for a number of elementary particle mass ratios, such as
?W /?Z and ?Higgs /?W . Most quark and lepton mass sequences and first rough estimates of mass ratios agree with the experimental data. All hadron mass sequences are
predicted correctly. The strand model also promises to calculate absolute mass values, including their change or ‘running’ with energy. Such future calculations will allow either
improving the match with observations or refuting the strand model.
The results are encouraging for two reasons. First of all, no other unified model that
agrees with experiment explains the qualitative properties of mass and mass sequences.
Secondly, no research on statistical tangles exists; an understanding of the parameters of
nature might be lacking because results in this research field are still few.
In the millennium list of open issues we have thus seen how to settle the origin of
particle masses – though we have not calculated them yet. Because a few even more
interesting challenges are awaiting us, we continue nevertheless. In the next leg, we investigate how elementary particle states mix.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Summary on elementary particle masses and millennium issues
Motion Mountain – The Adventure of Physics
Page 163
371
372
12 pa rticle properties d ed uced from stra nds
A quark mass eigenstate :
A quark weak eigenstate :
s
s‘
two tails
above, one tail
below paper plane
all three tails
in paper plane
magnified core
F I G U R E 112 Tail shifting leads to quark mixing: mass eigenstates and weak eigenstates differ.
mixing angles
Q uark mixing – the experimental data
Vol. V, page 250
??
?
( ?? ) = (??? ) ( ? ) .
?
??
Ref. 231
(207)
where, by convention, the states of the +2/3 quarks ?, ? and ? are unmixed. Unprimed
quarks names represent strong (and electromagnetic) eigenstates, primed quark names
represent weak eigenstates. In its standard parametrization, the mixing matrix reads
?12 ?13
?13 e−??
?12 ?13
? = (−?12 ?23 − ?12 ?23 ?13 e??
?12 ?23 − ?12 ?23 ?13 e?? ?23 ?13 )
??
?12 ?23 − ?12?23 ?13 e
−?12 ?23 − ?12 ?23 ?13 e?? ?23 ?13
(208)
where ??? = cos ???, ??? = sin ??? and ? and ? label the generation (1 ⩽ ?, ? ⩽ 3). The mixing
matrix thus contains three mixing angles, ?12 , ?23 and ?13 , and one phase, ?. In the limit
?23 = ?13 = 0, i.e., when only two generations mix, the only remaining parameter is
the angle ?12 , called the Cabibbo angle; this angle is Cabibbo’s original discovery. The
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 231
In nature, the quark mass eigenstates and their weak eigenstates differ. This difference
was discovered in 1963 by Nicola Cabibbo and is called quark mixing. The values of the
elements of the quark mixing matrix have been measured in many experiments, and
more experiments aiming to increase the measurement precision are under way.
The quark mixing matrix, also called CKM mixing matrix, is defined by
Motion Mountain – The Adventure of Physics
In nature, the mass eigenstates for fermions differ from their weak eigenstates: quarks
mix among themselves, and so do neutrinos. Quarks also show CP violation; for neutrinos, the issue is still open. These effects are described by two so-called mixing matrices.
The two mixing matrices contain fundamental constants of nature. For the strand model
to be correct, it must allow calculating the measured values of all components of the two
mixing matrices.
m ixing a ng les
Challenge 217 e
Ref. 231
373
last parameter, the so-called CP-violating phase ?, by definition between 0 and 2π, is
measured to be different from zero; it expresses the observation that CP invariance is
violated in the case of the weak interactions. The CP-violating phase only appears in
the third column of the matrix; therefore CP violation requires the existence of (at least)
three generations.
The present 90 % confidence values for the measured magnitude of the complex quark
mixing matrix elements are
0.97427(14) 0.22536(61) 0.00355(15)
|?| = (0.22522(61) 0.97343(15) 0.0414(12) ) .
0.00886(33) 0.0405(12) 0.99914(5)
(209)
Q uark mixing – expl anations
Page 250
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 331
In the standard model of particle physics, the quark mixing matrix is usually seen as due
to the coupling between the vacuum expectation value of the Higgs field and the lefthanded quark doublets or the right handed quark singlets. However, this description
does not lead to a numerical prediction.
A slightly different description of quark mixing is given in the strand model. In the
strand model, the Higgs field and its role as mass generator and unitarity maintainer is
a special case of the process of tail braiding. And braiding is related to the weak interaction. Because the various quarks are differently tangled rational tangles, tail braiding
can reduce or increase the crossings in a quark tangle, and thus change quark flavours.
We thus deduce from the strand model that quark mixing is an automatic result of the
strand model and related to the weak interaction. We also deduce that quark mixing is
due to the same process that generates quark masses, as expected. But we can say more.
In the strand model, the mass eigenstate – and colour eigenstate – is the tangle shape
in which colour symmetry is manifest and in which particle position is defined. The
mass eigenstates of quarks correspond to tangles whose three colour-tails point in three
directions that are equally distributed in space. The shape in which the tails point in
three, equally spaced directions is the shape that makes the SU(3) representation under
core slides manifest.
In contrast, the weak eigenstates are those shapes that makes the SU(2) behaviour
of core pokes manifest. For a quark, the weak eigenstate appears to be that shape of a
tangle for which all tails lie in a plane; for such plane configuration, the tails and the core
mimic a belt and its buckle, the structure that generates SU(2) behaviour. The two types
of eigenstates are illustrated in Figure 112.
In the strand model, masses are dynamical effects related to tangle shape. In the case
of quarks, the two configurations just mentioned will thus behave differently. We call
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All these numbers are unexplained constants of nature, like the particle masses. Within
experimental errors, the matrix ? is unitary.
A huge amount of experimental work lies behind this short summary. The data have
been collected over many years, in numerous scattering and decay experiments, by thousands of researchers. Nevertheless, this short summary represents all the data that any
unified description has to reproduce about quark mixing.
374
12 pa rticle properties d ed uced from stra nds
the transformation from a mass eigenstate to a weak eigenstate or back tail shifting. Tail
shifting is a deformation: the tails as a whole are rotated and shifted. On the other hand,
tail shifting can also lead to untangling of a quark tangle; in other words, tail shifting
can lead to tail braiding and thus can transform quark flavours. The process of tail shifting can thus explain quark mixing. (Tail shifting also explains the existence of neutrino
mixing, and the lack of mixing for the weak bosons.)
Tail shifting can thus be seen as a partial tail braiding; as such, it is due to the weak
interaction. This connection yields the following predictions:
Page 207
Ref. 231
A challenge
Ref. 251
Can you deduce the approximate expression
tan ?u mix = √
Challenge 218 r
?u
?c
(210)
for the mixing of the up quark from the strand model?
CP viol ation in quarks
Ref. 231
Page 339
Page 336
The CP violating phase ? for quarks is usually expressed with the Jarlskog invariant,
2
defined as ? = sin ?12 sin ?13 sin ?23
cos ?12 cos ?13 cos ?23 sin ?. This involved expression
is independent of the definition of the phase angles and was discovered by Cecilia Jarlskog, an important Swedish particle physicist. Its measured value is ? = 3.06(21) ⋅ 10−5 .
Because the strand model predicts three quark generations, the quark model implies
the possibility of CP violation. In the section on mesons we have seen that the strand
model actually predicts the existence CP violation. In particular, Figure 97 shows that
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Performing a precise calculation of mixing angles and their running with energy is still
a subject of research.
Motion Mountain – The Adventure of Physics
— Tail shifting, both with or without tail braiding at the border of space, is a generalized
deformation. Therefore, it is described by a unitary operator. The first result from the
strand model is thus that the quark mixing matrix is unitary. This is indeed observed.
— For quarks, tail braiding is a process with small probability. As a consequence, the
quark mixing matrix will have its highest elements on the diagonal. This is indeed
observed.
— Tail shifting also naturally predicts that quark mixing will be higher between neighbouring generations, such as 1 and 2, than between distant generations, such as 1 and
3. This is also observed.
— The connection between mixing and mass also implies that the 1–2 mixing is stronger
than the 2–3 mixing, as is observed.
— Finally, tail shifting predicts that the numerical values in the quark mixing matrix
can be deduced from the difference between the shapes of the two kinds of tangles
shown in Figure 112. In particular, tail shifting also predicts that the quark mixing
angles change, or run, with energy. In addition, the effect is predicted to be small.
On the other hand, so far there is no reliable experimental data on the effect.
m ixing a ng les
375
A neutrino mass eigenstate :
A neutrino weak eigenstate :
νe’
νe
each set of three tails
follows three mutually
perpendicular directions
all tails lie in one plane
F I G U R E 113 Tail shifting leads to neutrino mixing: mass eigenstates and weak eigenstates differ.
Ref. 252
Ref. 231
The observation, in 1998, of neutrino mixing is comparably recent in the history of
particle physics, even though the important physicist Bruno Pontecorvo predicted the
effect already in 1957. Again, the observation of neutrino mixing implies that also for
neutrinos the mass eigenstates and the weak eigenstates differ. The values of the mixing matrix elements are only known with limited accuracy so far, because the extremely
small neutrino mass makes experiments very difficult. Experimental progress across the
world is summarized on the website www.nu-fit.org. The absolute value of the so-called
PMNS mixing matrix ? is
0.82(1) 0.54(2) −0.15(3)
−0.35(6)
0.70(6) 0.62(6) ) .
|?| = (
0.44(6) −0.45(6) 0.77(6)
(211)
Again, these numbers are unexplained fundamental constants of nature. Within experimental errors, the matrix ? is unitary. The mixing among the three neutrino states is
strong, in contrast to the situation for quarks. Neutrino masses are known to be positive; however, present measurements are not precise and only yield values of the order of
1 ± 1 eV.
In the strand model, the lepton mass eigenstates correspond to tangles whose tails
point along the three coordinate axes. In contrast, the weak eigenstates again correspond to tangles whose tails lie in a plane. The two kinds of eigenstates are illustrated
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Neu trino mixing
Motion Mountain – The Adventure of Physics
with the help of tail shifting, ?0 and ?0 mesons mix, and that the same happens with
certain other neutral mesons. Figure 98 shows a further example. As just mentioned,
the possibility of tail shifting implies that CP violation is small, but non-negligible – as
is observed.
The strand model thus predicts that the quark mixing matrix has a non-vanishing CPviolating phase. The value of this phase is predicted to follow from the geometry of the
quark tangles, as soon as their shape fluctuations are properly accounted for. This topic
is still a subject of research.
376
12 pa rticle properties d ed uced from stra nds
in Figure 113. Again, the transition between the two eigenstates is due to tail shifting, a
special kind of strand deformation.
We thus deduce that neutrino mixing, like quark mixing, is an automatic result of the
strand model and is related to the weak interaction. Given that the neutrino masses are
small and similar, and that neutrinos do not form composites, the strand model predicts
that the mixing values are large. This is a direct consequence of tail shifting, which in
the case of similar masses, mixes neutrino tangles leads to large mixings between all
generations, and not only between neighbouring generations. In the strand model, the
large degree of neutrino mixing is thus seen as a consequence of their low and similar
masses, of their tangle structure, and of their existence as free particles.
Like for quarks, the strand model predicts a unitary mixing matrix for neutrinos. The
strand model also predicts that the geometry of the neutrino tangles and their fluctuations will allow us to calculate the mixing angles. More precise predictions are still
subject of research.
Ref. 254
Open challenge: calcul ate mixing angles and phases ab initio
Challenge 219 ny
Calculating the mixing angles and phases ab initio, using the statistical distribution of
strand fluctuations, is possible in various ways. In particular, it is interesting to find the
relation between the probability for a tail shift and for a tail braiding. This will allow
checking the statements of this section.
Because the strand model predicts a lack of new physics beyond the standard model
of particle physics, the calculation of neutrino mixing angles is one of the few possible
predictions that are left over in fundamental physics. Since the lepton tangles are still
tentative, a careful investigation is necessary.
One possibility is that only the electron neutrino tangle given above is correct, and
that the other two neutrinos are similar to it, just with more built-in torsion. Figure 114 illustrates this possibility. If this assignment were correct, two of the mixing angles should
be large (and maybe have the zero-order value of 120°/3=40°). In addition, very low mass
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Ref. 253
The strand model predicts that the three neutrinos are massive Dirac particles, not Majorana particles. This has not yet been confirmed by experiment. The strand model thus
predicts that the neutrino mixing matrix has only one CP-violating phase. (It would have
three such phases if neutrinos were Majorana particles.) The value of this phase is predicted to follow from the neutrino tangles and a proper accounting of their fluctuations.
Also this calculation is still a subject of research.
On the one hand, the strand model suggests the appearance of CP violation in neutrinos. On the other hand, it is unclear when the value of the CP-violating phase will
ever be measured with sufficient precision. This is one of the hardest open challenge of
experimental particle physics.
The mechanism of CP violation has important consequences in cosmology, in particular for the matter–antimatter asymmetry. Since the strand model predicts the absence of the see-saw mechanism, the strand model rules out leptogenesis, an idea invented to explain the lack of antimatter in the universe. The strand model is more on the
line with electroweak baryogenesis.
Motion Mountain – The Adventure of Physics
CP viol ation in neu trinos
m ixing a ng les
377
F I G U R E 114 An alternative candidate assignment for the three neutrino tangles that generates large
mixing between neighbouring generations and strong preference for one handedness.
Page 162
The strand model implies that mixing angles for quarks and neutrinos are properties of
their tangle families. The existence of mixing is due to the shape of tangles and their
fluctuations. As a result, strands explain why mixing angles are not free parameters, but
discrete and unique constants of nature. The strand model also predicts that mixing
angles are constant during the evolution of the universe.
We have shown that tangles of strands predict non-zero mixing angles for quarks and
neutrinos, as well as CP-violation in both cases. The strand model also predicts that
the mixing angles of quarks and neutrinos can be calculated from strand fluctuations.
Strands predict that mixing matrices are unitary and that they run with energy. Strands
also predict a specific sequence of magnitudes among matrix elements; the few predictions so far agree with the experimental data. Finally, the strand model rules out leptogenesis.
We have thus partly settled four further items from the millennium list of open issues.
All qualitative aspects and some sequences are reproduced correctly, but no hard quantities were deduced yet. The result is somewhat disappointing, but it is also encouraging.
At present, no other explanation for quark and neutrino mixing is known. Future calculations will allow either improving the checks or refuting the strand model. We leave this
topic unfinished and proceed to the most interesting topic that is left: understanding the
coupling constants.
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Summary on mixing angles and the millennium list
Motion Mountain – The Adventure of Physics
values would arise naturally, in normal ordering. This tangle assignment would also explain the difficulty of observing neutrinos of opposite handedness: neutrinos would have
an extremely high preference for one handedness (belt-trick); the mirror images of the
tangles in Figure 114 would correspond to antineutrinos. Despite these appealing aspects,
this tentative assignment has one unclear issue: explaining the lack of a fourth neutrino
generation is not straightforward.
378
12 pa rticle properties d ed uced from stra nds
1/α i
weaker
60
50
Data
Prediction
by the standard
model
1/α1
40
electromagnetism
1/α2
30
20
stronger
θw
1/α3
√3 α1/5 weak
hypercharge
√α2 weak
interaction
10
10
5
10
10
15
10
19
10
Q / GeV
coupling constant s and unificat ion
Ref. 5
In nature, electric, weak and strong charge are quantized. No experiment has ever found
even the smallest deviation from charge quantization. All charges in nature are integer
multiples of a smallest charge unit. Specifically, the electric charge of every free particle
is observed to be an integer multiple of the positron electric charge. We call the integer
the electric charge quantum number.
In nature, the strength of a gauge interaction for a unit charge is described by its coupling constant. The coupling constant gives the probability with which a unit charge emits
a virtual gauge boson, or, equivalently, the average phase change produced by the absorption of a gauge boson. There are three charge types and three coupling constants: for the
electromagnetic, for the weak and for the strong interaction. All particles with a given
charge type and value share the same coupling constant, even if their masses differ. The
three coupling constants depend on energy. The known data and the change with energy
predicted by the standard model of particle physics are shown in Figure 115.
In nature, the fine structure constant ?, i.e., the electromagnetic coupling constant, at
the lowest possible energy, 0.511 MeV, has the well-known measured value
? = 1/137.035 999 139(31) .
(212)
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
F I G U R E 115 Left: How the three coupling constants (squared) change with energy, as predicted by the
standard model of particle physics; the graph shows the constant ?1 = 53 ?/ cos2 ?W for the weak
hypercharge coupling (related to the electromagnetic fine structure constant ? through the weak
mixing angle ?W and a historical factor 5/3 that is useful in grand unification), the coupling constant
?2 = ?? = ?/ sin2 ?W for the weak interaction, and the coupling constant ?3 = ?s for the strong
interaction. The three black points are measurement points; at lower and slightly higher energies, data
and calculation match within experimental errors. (Courtesy Wim de Boer) Right: The relation between
the coupling constants ? for the electromagnetic U(1)EM , ?2 = ?? for the weak SU(2), ?1 for the weak
hypercharge U(1)Y gauge groups and the weak mixing angle ?W .
Motion Mountain – The Adventure of Physics
0
0
10
√α
coupling consta nts a nd unification
379
Equivalently, the electromagnetic coupling of the positron can also be described by the
equivalent number
√? = 1/11.706 237 6167(13) = 0.085424543114(10) ,
(213)
Interaction strengths and strands
In the strand model, all three gauge interactions are due to shape changes of tangle cores.
We first classify the possible shape changes. Given a tangle core, the following shape
changes can occur:
In the strand model, the fluctuation probabilities for each Reidemeister move – twist,
poke or slide – determine the coupling constants. We thus need to determine these
probability values. We can directly deduce a number of conclusions, without any detailed calculation:
— The coupling constants are not free parameters, but are specified by the geometric,
three-dimensional shape of the particle tangles.
— By relating coupling constants to shape fluctuation probabilities, the strand model
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
— Small changes of core shape do not produce any crossing switch. Small shape changes
thus have no physical significance: for a given observer, they leave all observables
unchanged.
— Twist shape changes of a strand segment in the core produce an electric field, if the
particle is charged. More precisely, the electric field around a particle is the difference
between the average number ?tr of right twists and the average number ?tl of inverse,
left twists that a particle tangle produces per unit time.
— Poke shape changes of a strand segment in the core produce a weak interaction field.
More precisely, the weak field is the asymmetry among the probabilities ?p? , ?p? and
?p? for the three fundamental poke types and their inverses.
— Slide shape changes of a strand segment in the core produce a colour field, if the
particle has colour. More precisely, the colour field is the asymmetry among the probabilities ?s1 to ?s8 for the eight fundamental slide types and their inverses.
— A combination of these moves can also appear.
Motion Mountain – The Adventure of Physics
which is also called the electric charge unit (at low energy). Quantum electrodynamics
predicts the precise change with energy of this charge unit; the experiments performed so
far, up to over 100 GeV, agree with this prediction. Quantum electrodynamics also predicts that the charge unit, when extrapolated right up to the Planck energy, would have a
value of 1/10.2(1). These predictions are shown, in a common, but somewhat scrambled
way, in Figure 115.
Explaining the value of ?, which determines all colours and all material properties
in nature, is the most famous millennium issue. If the strand model cannot reproduce
every observation about ? and other coupling constants, it is wrong. In particular, we
thus need to understand, using the strand model, the quantization of charges on the one
hand, and the mysterious value of the charge unit – either at low energy or at Planck
energy – on the other hand.
380
12 pa rticle properties d ed uced from stra nds
predicts that coupling constants are positive numbers and smaller than 1 for all energies. This is indeed observed.
— A still stricter bound for coupling constants can also be deduced. The sum of all
possible fluctuations for a particular tangle has unit probability. We thus have
8
1 = ?small + ?tr + ?tl + ∑ (?p? + ?p−? ) + ∑ (?s? + ?s−? ) + ?combination . (214)
?=?,?,?
In summary, the strand model implies, like quantum field theory, that coupling constants
are probabilities. The obvious consequences are
⊳
⊳
⊳
⊳
? < 1,
?w < 1,
?s < 1
? + ?w + ?s < 1 and √? + √?w + √?s < 1.
These properties are valid both in quantum field theory and in the strand model. Despite
the agreement with experiment, we have not deduced any new result yet – except one.
Strands imply unification
Page 275
In fact, one new point is made by the strand model. Each gauge interaction is due to a
different Reidemeister move. However, given a specific tangle core deformation, different
observers will classify the deformation as a different Reidemeister move. Indeed, every
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
The strand model thus predicts that the sum of the three charge units must be strictly
smaller than 1, for every energy value. This is easily checked, both with the data
and with the prediction of quantum field theory. In quantum field theory, the three
(modified) coupling constants are given, as a function of energy, in the popular graph
shown in Figure 115. The values are a combination of experimental data – for low energies – and theoretical extrapolations – for high energies. In this popular graph, the
electromagnetic coupling is traditionally multiplied by 5/(3 cos2 ?W ). (This is done
in order to test grand unification; we keep the traditional factor, even though grand
unification is shown by experiment and predicted by the strand model not to apply to
nature.) The graph allows us to confirm that the sum of the three unmodified charge
units is indeed smaller than 1 for all energy values, as predicted by the strand model.
— The strand model also predicts that the three coupling constants are related by small
numbers, as the corresponding fluctuations differ only in the number of involved
strands. This is also observed, as Figure 115 shows – especially if we remember that
the couplings are the square roots of the values shown in the graph, corrected for the
traditional factor.
— The strand model further predicts that the coupling constants are independent of
time and space, and that in particular, they do not depend on the age of the universe.
This is also observed, despite occasional claims to the contrary.
— Finally, strand model predicts that the coupling constants are the same for particles
and antiparticles, as is observed.
Motion Mountain – The Adventure of Physics
Ref. 255
?=1
coupling consta nts a nd unification
Reidemeister move can be realized by the same deformation of a single strand: for each
Reidemeister move, it is sufficient to add a curved section to a straight strand segment.
Such a deformation can look like a type I Reidemeister move for one observer, like a type
II move for another, and like a type III move for a third observer.
Because all interactions follow from the same kind of strand deformation of tangle
cores, the strand model thus provides unification of the interactions. This result is new:
in fact, this kind of strand unification of the interactions differs completely from any
other approach ever proposed. In contrast to other approaches, strand unification does
not require that the three coupling constants have the same value at high energy.
A given shape deformation thus has five probabilities associated to it: the probabilities describe what percentage of observers sees this deformation as a type I move, as a
type II move, as a type III move, as a combination of such moves, or as no move at all,
i.e., as a small move without any crossing switch. On the other hand, at energies measurable in the laboratory, the moves can almost always be distinguished, because for a given
reaction, usually all probabilities but one practically vanish, due to the time averaging
and spatial scales involved.* In short, at energies measurable in the laboratory, the three
gauge interactions almost always differ.
Calcul ating coupling constants
⊳ The strand model predicts that the fine structure constant can be calculated
by determining the probability of twists, i.e., Reidemeister I moves, in the
fluctuating tangle shapes of a given particle with nonzero electrical charge.
In other words, the strand model must show that the probability of the first Reidemeister
move in chiral particle tangles is quantized. This probability must be an integer multiple
of a unit that is common to all tangles; and this coupling unit must be the fine structure
constant. Any check for the existence of a coupling unit requires the calculation of twist
appearance probabilities for each chiral particle tangle. The strand model is only correct
if all particles with the same electric charge yield the same twist emission probability.
Instead of emission, also absorption can be used to calculate the fine structure constant:
* The strand model thus predicts that at extremely high energy, meaning near the Planck energy, for each
gauge interaction, also particles with zero charge can interact. At Planck energy, when horizons form, the
time averaging is not perfect, and interactions become possible even with zero charge.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
The strand model predicts that the calculation of the three coupling constants is a problem of tangle geometry and fluctuation statistics. Thus it can be approached, at each
energy scale, either analytically or with computer calculations. The calculations need to
determine the probabilities of the corresponding Reidemeister moves. If the results do
not agree with the experimental values, the strand model is false. We note that there is
no freedom to tweak the calculations towards the known experimental results.
In particular, in the strand model, one way to proceed is the following. The (square
root of the) fine structure constant is the probability for the emission of twists by a fluctuating chiral tangle.
Motion Mountain – The Adventure of Physics
Challenge 220 e
381
382
12 pa rticle properties d ed uced from stra nds
⊳ The strand model predicts that the fine structure constant can be calculated
from the average angle that a tangle core rotates when absorbing a photon.
Challenge 221 r
First hint: the energy dependence of physical quantities
Page 378
⊳ Coupling constants, masses and mixing angles change with energy because
they are quantities that depend on the average geometrical details, and in
particular, on the scale of the underlying particle tangles.
More precisely, the running quantities depend on the fluctuations of the geometric tangle
shapes, and these fluctuations depend somewhat on the spatial and thus the energy scale
under consideration. We note that the strand model predicts a running only for these
three types of observables; all the other observables – spin, parities or other quantum
numbers – are predicted to depend on the topology of the particle tangles, and thus to be
independent of energy. This prediction agrees with observation. Therefore, we can now
Ref. 227
Page 331
* In the standard model of particle physics, the running of the electromagnetic and weak coupling constants
– the slope in Figure 115 – depends on the number of existing Higgs boson types. The (corrected) strand
model predicts that this number is one. Measuring the running of the constants thus allows checking the
number of Higgs bosons. Unfortunately, the difference is small; for the electromagnetic coupling, the slope
changes by around 2 % if the Higgs number changes by one. But in future, such a measurement accuracy
might be possible.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
In nature, all effective charges, i.e., the coupling constants, change with energy. One also
says that they run with energy. Figure 115 shows the details. Running also occurs for
masses and mixing angles. All other intrinsic particle properties, such as spin, parities
and all other quantum numbers, are found not to change with energy. For the coupling constants, the measured changes between everyday energy and about 100 GeV agree
with the prediction from quantum field theory.*
The strand model predicts
Motion Mountain – The Adventure of Physics
We will pursue this alternative shortly.
So far, there do not seem to exist any analytical tool that permits the calculation of
shape deformation probabilities. Thus, at present, computer calculations seem to be the
only possible choice. Of all existing software programs, the most adapted to calculating
fluctuation probabilities are the programs that simulate the dynamics of tangled polymers; but also the programs that simulate the dynamics of cosmic strings or the dynamics of helium vortices are candidates. The main issue, apart from a large computer time, is
the correct and self-consistent specification of the shape fluctuation distribution at each
energy scale.
In summary, using the strand model we expect to be able to calculate the electromagnetic coupling constant and to understand its validity across all elementary particles. The
same expectation obviously also holds for the two nuclear interactions. If any of the expectations on tangle interactions are found to be incorrect, the strand model is false. The
strand model must yield quantized tangle equivalence classes for the electromagnetic, weak
and colour charge. Even though the calculation issues are still subject of research, there
are encouraging hints that these expectations will be validated.
coupling consta nts a nd unification
383
explore the details of the running.
Second hint: the running of the coupling constants at low
energy
Page 378
Page 334
d?
>0.
d?
Also for the two nuclear interactions, the washing out effect for loose tangle cores
by the vacuum does occur as predicted by quantum field theory. In the weak interaction, the antiscreening of the weak charge appears in this way. In the strong interaction,
both virtual quark–antiquark pairs and virtual gluon pairs can appear from the strands
that make up the vacuum. Virtual quark–antiquark pairs lead to screening, as virtual
electron–antielectron pairs do for the electromagnetic interaction. In addition, however,
we have seen that the strand model of mesons implies that virtual gluon pairs lead to
antiscreening. (In contrast, virtual photon pairs do not lead to such an effect.) Because
the strand model fixes the number of quark and gluons, the strand model is consistent
with the result that the screening of the colour charge by quark pairs is overcompensated
by the antiscreening of the virtual gluon pairs.
In other words, the strand model reproduces the observed signs for the slopes of the
coupling constants in Figure 115, for the same reason that it reproduces the quantum field
theoretic description of the three gauge interactions. The predicted running could also
be checked quantitatively, by taking statistical averages of tangle fluctuations of varying
dimension. This is a challenge for future research.
Third hint: further predictions at low energy
As we just saw, the complete explanation of the running of the couplings depends on
the explicit boson and fermion content of nature and on the fact that the strand model
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
⊳
Motion Mountain – The Adventure of Physics
The strand model proposes a new view on the screening and antiscreening effects that
are part of quantum field theory. In the strand model, screening effects are consequences
of the statistics of shape deformations for loose tangle cores that are embedded into the
strands that form the vacuum. Since these statistical effects can in principle be calculated,
it is expected that such calculations can be compared with the predictions of quantum
field theory shown in Figure 115. This check is in progress. A few results, however, can
be deduced without any calculations at all.
In the strand model, the electromagnetic interaction is due to the first Reidemeister
move, the twist. For a charged particle – thus one with a chiral tangle core – the average
difference in the occurrence of right and left twists determines the effective charge. It is
expected that this difference decreases when the strand core is loose, because the loose
strands are more similar to those of the surrounding vacuum, so that the differences
due to the chirality of the tangle will be washed out. In the language of quantum field
theory, the virtual particle-antiparticle pairs – created by the fluctuations of the vacuum
strands – screen the central, naked charge. The screening is reduced when the energy
is increased, and thus when the scales are reduced. In other words, the strand model
predicts that the electromagnetic coupling increases with energy, as is observed:
384
12 pa rticle properties d ed uced from stra nds
⊳ ?em < ?w < ?s .
The running of the coupling constants up to Pl anck energy
At energies near the Planck energy, quantum field theory is modified: effects due to the
strand diameter start to play a role. Near Planck energy, tangles get tighter and tighter
and fluctuations get weaker, because there is less room for them. In other words, near
Planck energy tangles tend to approach the structure of horizons. Therefore, near the
Planck energy, the strand model predicts deviations from the energy dependence of the
coupling constants that is predicted by quantum field theory. So far, estimating such
deviations has not been possible.
Another calculation might seem more promising: to calculate the coupling constants
near Planck energy. It could be argued that the approach to calculate the low-energy
coupling constants from Planck-energy values seems unsatisfactory, due to the approximations and extrapolations involved. But it is possible if we are convinced that quantum
field theory is correct up to Planck energy. And this is just what the strand model predicts. Such a Planck-scale calculation might then allow us to estimate the low-energy
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 378
The prediction matches observations. Unfortunately, this argument is not reliable. If
the strand number were the only cause of the running, the argument would imply that
the three slopes for the running of the three coupling constants should behave like 3:2:1.
However, the graph of Figure 115 shows otherwise, even if the difference between the
electromagnetic coupling and the weak hypercharge coupling is taken into account. Indeed, the running of the coupling constants is not due to strand number only, but also
to the explicit boson and fermion content of nature, as we just saw.
Motion Mountain – The Adventure of Physics
reproduces quantum field theory. Interestingly, the strand model also proposes a simpler,
though less precise explanation of the running.
At energies much smaller than the Planck energy, such as everyday energies, the
strand model implies that the average size of the tangle core is of the order of the position
uncertainty of a particle. In other words, any thickness of the strands – real or effective
– can be neglected at low energies. Therefore, at low energies, the average strand length
within a particle tangle core is also of the order of the de Broglie wavelength. Low, everyday energy thus implies large, loose and spherical/ellipsoidal tangle cores.
At low energies, shape fluctuations can lead to any of the three Reidemeister moves.
The probabilities of such shape deformations will scale with some power of the average strand length within the tangle core. In other words, coupling constants depend on
energy. But how exactly?
We note directly that higher Reidemeister moves, which involve larger numbers of
strand segments, will scale with larger power values. In particular, the longer the strand
in the core – i.e., the lower the energy – the more the relative probability for the higher
Reidemeister moves will increase.
In summary, the strand model predicts that when a tangle is loose and long, i.e., when
energies are low, the strong nuclear interaction, due to the third Reidemeister move,
is the strongest gauge interaction, followed by the weak nuclear interaction, due to the
second Reidemeister move, in turn followed by the electromagnetic interaction:
coupling consta nts a nd unification
385
coupling constants from their Planck energy values. However, so far, also this approach
has not led to success, despite a number of attempts. The challenge seems to be to understand core deformation for case of tight tangle cores. We keep this option in mind.
L imits for the fine structure constant d o not provide
expl anations
Ref. 260
When searching for ways to determine the fine structure constant, we need to be careful.
Here is an example that explains why.
Numerous observations of nature imply a limit on the fine structure constant. A pretty
one appeared in a post on the internet in 2017. The electrostatic repulsion between two
electrons at a given distance must be larger than the radiation force between to small
neutral black holes at that same distance. In other words,
Vol. V, page 150
Here it is assumed that thermal radiation from one black hole acts on the cross section
of the other black hole by pushing it away. Multiplying both sides by ?2 /ℏ? and inserting
the expressions for the black hole luminosity ? bh and the black hole radius ?bh gives
?>
1
.
15 320 π
(216)
The bound is not tight, but is obviously correct.
Various researchers are looking for observations that give the best possible bound for
?. Such a search can indeed yield much better bounds. However, such a search cannot
explain the value of ?. We can indeed use thermodynamics, gravity or other observed
properties to deduce observational limits on ?. Many formulae of physics contain ? in a
more or less obvious way. Maybe, one day, known physics will be able to yield very tight
upper and lower bounds for ?. Still, the explanation of the value of ? would still lack.
To explain the fine structure constant ?, we need an approach based on the final theory, not one based on known, millennium physics, such as expression (215). Millennium
physics can measure ?, but cannot explain it. To explain the fine structure constant, a
final theory is needed. In our case, we need to check whether we can calculate ? with
strands. Therefore, we now explore tangle topology, tangle shapes and tangle motion
with this aim in mind.
Charge quantization and top ological writhe
In the strand model, electric charge is related to the chirality of a tangle. Only chiral
tangles are electrically charged. The strand model thus implies that a topological quantity for tangles – defined for each tangle in the tangle family corresponding to a specific
elementary particle – must represent electric charge. Which quantity could this be?
The first candidate for charge in the strand model is provided by knot theory:
⊳ The usual topological quantity to determine chirality of knots and tangles is
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 261
(215)
Motion Mountain – The Adventure of Physics
2
? bh π ?bh
?2 1
>
.
4π?0 ?2
? 4π?2
386
12 pa rticle properties d ed uced from stra nds
the topological writhe.
To determine its value, we draw a minimal projection, i.e., a two-dimensional knot or
tangle diagram with the smallest number of crossings possible. We then count the righthanded crossings and subtract the number of left-handed crossings. This difference, an
integer, is the topological writhe. Topological writhe is thus a two-dimensional concept
and does not depend on the shape of a knot or tangle. We note:
Page 327
⊳ The electric charge quantum number behaves similarly to topological writhe
(times one third or times one): it is quantized, has two possible signs, vanishes for achiral tangles, is a topological invariant – and thus is conserved.
Challenge 222 e
Challenge 223 ny
Ref. 5
In short, a topological quantity, namely topological writhe, reproduces the electric charge
quantum number in the strand model. Three issues remain. First, given that every
particle is described by a tangle family with an infinite number of members, how is
the electric charge, i.e., the topological writhe of the other tangle family members accounted for? It is not hard to see that family members do not change topological
writhe. The second issue is more thorny: why is the charge definition different for
leptons? We skip this problem for the time being. The third issue is the central one:
What is the origin of the peculiar value of the charge unit, whose square has the value
? = 1/137.035 999 139(31) at low energy?
Charge quantization and linking number
An alternative conjecture for charge quantization is the following:
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
In other terms, the electric charge quantum number can be reproduced with the help of
topological writhe. And indeed, the electric charge of massless bosons, i.e., photons and
gravitons, vanishes.
Let us sum up. In nature, electric charge is quantized. The strand model describes
charged particles with the help of fluctuating alternating tangles, and charge quantization
is a topological effect that results because all particles are made of strands. In particular,
Motion Mountain – The Adventure of Physics
Page 321
— The topological writhe of the W boson tangles is +3 or −3, depending on which mirror image we look at; the topological writhe of the Z boson and Higgs boson tangles
vanishes. The topological writhe of any unknotted strand also vanishes. In this way, if
we define the electric charge quantum number as one third of the topological writhe,
we recover the correct electric charge quantum number of the weak and all other
gauge bosons. We note that the leather trick does not change this result, so that all
family members of a particle share the same topological writhe.
— The tangles of the quarks show that if we define the electric charge quantum number
as one third of the topological writhe, we recover the correct electric charge quantum
number of all quarks. The leather trick has no effect on this definition.
— The tangles of the leptons show that if we define the electric charge quantum number as the topological writhe of the centre region only, we recover the correct electric
charge quantum number of all leptons. Again, the leather trick does not change this
result.
coupling consta nts a nd unification
387
⊳ Electric charge, i.e., twist emission probability, might be proportional to the
linking number of ribbons formed by strand pairs.
The following arguments speak in favour of this conjecture.
How to calcul ate coupling constants
The strand model suggests that crossing number and linking number somehow define
electric and weak charge. In simple words, the model suggests that quantization of all
charge types is a topological effect; quantization is due to the multiple ways in which
strands cross inside tangles.
Coupling constants describe the probability of interaction with gauge bosons. Experiments show that these quantities are slightly scale-dependent, since they run with energy.
But in the strand model, coupling constants are not really shape-dependent: electrons,
muons and antiprotons have the same electric charge and fine structure constant values
despite being described by different tangles. Coupling constants do not depend on the
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Challenge 224 e
In short, linking number, an integer, might be a better topological quantity to explain
electric charge quantization than topological writhe. On the other hand, it might well be
that linking number, being a quantity that depends on two strands, is related to the weak
charge rather than to the electric charge.
If the conjectured relation between linking number and electric or weak charge is
correct, it might lead to a calculation of the corresponding coupling constant, once the
tangle shape or, better, once the tangle dynamics is included in the proper way. For example, the photon emission probability could depend on the writhe or on the twist of
the (averaged) ribbons. Both these properties might lead to virtual photon emission.
(The sum of writhe and twist of a ribbon is given by the linking number, as explained by
Calugareanu’s theorem.)
In this and any topological definition of electric charge, we face two slight hurdles:
First, we have to watch out for the graviton: it is uncharged. Secondly, we have to explain
why the strand model for the simplest family member of the d quark is not chiral. Both
hurdles can be overcome.
If the linking of two strands is connected to weak charge, it might well be that a similar
quantity defined for three strands is related to colour charge. All these possibilities are
topic of research.
Motion Mountain – The Adventure of Physics
— In knot theory, a ribbon is the strip associated to and limited by two strands.
— The linking number of a ribbon is the number of times that the two edges of a ribbon
wind around each other. The linking number is a topological invariant and an integer.
— In particle tangles, only wound up, i.e., linked ribbons should lead to (net) boson
emission. For tangles made of three strands, we define a total linking number as the
sum of all three possible linking numbers.
— The linking number of the Higgs boson strand pairs is zero; that of the Z boson strand
pairs is the sum of 1, 0 and -1, thus also zero. The linking number for the W boson is
3 or -3, that of the quarks is 1, -1, 2 or -2. We thus conjecture that the charge quantum
number is one third of the total linking number.
— Massless bosons, i.e., photons, gluons and gravitons, have no electric charge.
388
12 pa rticle properties d ed uced from stra nds
kind of tangle. Experiments show that they just depend somewhat on its size. In short,
⊳ We need a definition of each coupling constant that is tangle-independent
and shape-independent, and only depends on a topological invariant of
tangles.
In fact, this conclusion eliminates many speculations, including a number of calculation
approaches that were included in this chapter in previous editions. We are left with just
a few options. To explore them, we start with an overview.
C oupling constants in the strand model
In the strand model, neutral particles are those that cannot receive Reidemeister moves
or that receive them all in equal way:
1. Electromagnetism: Neutral ‘tangles’ are made of one strand (e.g., the photon) or are
achiral (e.g., the Z and the neutrinos).
2. Weak interaction: Neutral tangles are made of one strand (e.g., the photon) or of two
straight or unpokeable strand pairs (e.g., the Z, the right-handed leptons and quarks).
3. Strong interaction: Neutral tangles are made of one strand or of three strands.
In the strand model, charged particles are specific tangles:
1. Electric charge is due to the observability of crossings during photon emission or
absorption, i.e., when twists are applied. Particles with electric charge, i.e., with preferred twist transfer, have a global asymmetry, global twistedness, namely topological
chirality. Locally, electrically charged particles have crossings; electric charge is positive or negative. Charge is 1/3 of the signed crossing number. Examples are the charged
leptons, the quarks and the W boson.
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In the strand model, the gauge interactions are modelled as transfers of Reidemeister
moves:
1. The electromagnetic interaction is twist transfer and the electric charge is preferred
twist transfer to or from a massive particle. Twists can be added abd form a circle:
they form a U(1) Lie group. They change the tangle phase by exchanging one observable crossing.
2. The weak interaction is poke transfer and the weak isospin is preferred poke transfer
to or from a massive particle. Pokes exist in three linearly independent directions
and their generators behave like the belt trick: they generate an SU(2) Lie group.
They change the tangle phase by exchanging two observable crossings.
3. The strong interaction is slide transfer and the colour charge is preferred slide transfer
to or from a massive particle. Slides can be added, its generators have a ?3 symmetry
and they form an SU(3) Lie group. They change the tangle phase by exchanging two
or three? crossings.
Motion Mountain – The Adventure of Physics
In experiments, there are the following gauge interactions with their charges:
1. The electromagnetic interaction with electric charge and U(1) symmetry.
2. The weak interaction with weak isospin and SU(2) symmetry.
3. The strong interaction with colour and SU(3) symmetry.
coupling consta nts a nd unification
389
Coupling strength is the ease of crossing rotation, of poke creation, and of slide induction. These connections allow calculating the coupling strength values.
Deducing ? from precession
Page 226
In nature, magnetic fields rotate charged particles. In the strand model, as shown in
Figure 52, magnetic fields are made of moving twists. In fact, from the strand definition of the electromagnetic interaction and the electric charge and from the drawing in
Figure 49, we deduce:
⊳ Moving twists rotate crossings.
We note that this description differs slightly from a pure twist transfer. But this formulation is the key to calculating ?.
We assume that the typical, average crossing is lying in the paper plane, as in the drawing of the fundamental principle. For an average crossing, the two strands lie along the x
and y axes. When a photon, i.e., a twist, arrives along the diagonal in the first quadrant,
it rotates the crossing completely, by one turn. If the twist arrives at a different angle, its
effect is lowered. We approximate this angle effect with simple trigonometry: we assume
that the angular projection describes the reduction of the effect with the incoming angle
of the twist.
For the incident photon, we call ? the angle from the y-axis and ? the angle out of
the paper plane. The average rotation angle induced by an absorbed photon or twist on
a charged particle with three crossings, corresponding to one elementary charge, can be
calculated. We include sin ? for the volume element in spherical coordinates and average
over the possible angle values ? between the strands at the crossing. Further terms arise
from the trigonometric approximation. In particular, a second power arises from the two
tails, and a further squaring is required to get probabilities. Nevertheless, the expression
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Page 231
Motion Mountain – The Adventure of Physics
2. Weak charge is thus due to the observability of crossings during W or Z emission or
absorption, i.e., when pokes are applied. Particles with weak isospin, i.e., with preferred poke transfer, have a global asymmetry that prevents all pokes to act equally
effectively: For fermions, such an asymmetry arises when tangle twistedness and the
belt trick have the same sign; thus all left-handed fermions and right-handed antifermions have weak isospin. Locally, weakly charged fermions behave like a belt buckle
that rotates in the appropriate direction. Due to their tangle topology, some fermions
have positive, others negative weak isospin. For the W boson, the asymmetry is built
into the tangle; due to the tangle structure, the W and its antiparticle have plus or
minus twice the weak isospin of fermions.
3. Colour, strong charge, is due to the observability of crossings during gluon emission
or absorption, i.e., when slides are applied. Particles with colour charge, i.e., with preferred slide transfer, have a global asymmetry that prevents all slides to act equally effectively: Coloured particles are made of exactly two strands with tails in tetrahedron
skeleton directions. Only two-stranded tangles allow certain slides and prevent others. Therefore only quarks have colour charge. Locally, red, blue and green colours
correspond to three directions in one plane that differ by an angle of 2π/3.
390
12 pa rticle properties d ed uced from stra nds
remains open to dispute:
π/2
3 π ?/2
∫
∫
∫
cos ? (sin ?)2 (cos(?π/?) cos ?)4 d? d? d? = 0.15 .
2π2 ?=0 ?=−?/2 ?=−π/2
(217)
The resulting value of 0.15 is not an acceptable approximation to reality, in which √? =
0.08542454311(1) at low energy and √? = 0.10(1) at Planck energy. Neither is the value
a good approximation to the hypercharge coupling, which changes from √?1 = 0.10(1)
at 100 GeV to √?1 = 0.13(1) at Planck energy. We need a better approximation for the
value of the electromagnetic coupling strength.
√?calc =
Deducing the weak coupling
⊳ A moving poke rotates a pair of strands.
π/2
√?w calc = ∫
?=0
2π
∫
sin ? (cos2 ? cos2 ?)4 d? d? ≈ 0.19 .
(218)
?=0
If we need to average over the different angles between the strands that make up the pair
experiencing the poke, we get a different value.
The calculated value of the weak coupling is not an acceptable approximation to reality, in which √?w = 0.18 at the (low) energy of 100 GeV and √?w = 0.14 at Planck
energy. We need a better approximation.
Deducing the strong coupling
Strong fields deform specific three-strand configurations by adding generalized slides.
The generalized slides are due to gluons. Strong colour is related to the order and orient-
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This process is the key to calculating ?w . We note that there is a certain similarity to the
setting used for calculating the electromagnetic coupling: in both cases, the incoming
boson acts on a target consisting of two strands. This similarity is the reason for electroweak mixing.
We calculate the coupling constant for a single belt buckle, assuming parallel strands.
The average rotation angle induced by one incoming weak (unbroken) boson (out of
the three possible cases) is one full turn when the impact is perpendicular to the two
strands and to the plane defined by them. For a general incidence angle the induced
rotation angle is lower. We again use trigonometrical projection to approximate the induced crossing rotation angle in the general case, with the same issues as in the previous
case. We call ? the angle from ideal incidence, and ? the longitude. The average angle is
then given by
Motion Mountain – The Adventure of Physics
Weak fields deform strand (crossing) pairs by adding or transferring generalized pokes.
Weak fields are collections of pokes; pokes represent virtual weak bosons. The weak
isospin, the weak charge, is related to the orientation of the strand pairs. The weak interaction occurs through an incoming poke that deforms a strand pair:
coupling consta nts a nd unification
391
ation of the strands in these specific three-strand configurations. In short:
⊳ Incoming, moving slides deform three-strand configurations.
This is the key to calculating ?s.
We assume that one of eight possible gluons is incident. In an average triple strand
configuration, the three strands are oriented in a way that in the paper plane they look
like three symmetrically arranged rays. One ray lies along the y axis. When a gluon
arrives, it performs a slide. For an incident gluon, we call ? the angle from the y-axis to
the next strand and ? the angle out of the paper. In the trigonometric approximation,
the average slide angle induced on a coloured particle is given by
π/2
√?s calc = ∫
2π
∫
(219)
?=0
This is not an acceptable approximation to reality, in which √?s = 0.7(1) at the (low)
energy of 1 GeV and √?s = 0.13(1) at Planck energy. We need a better approximation
for the strong coupling.
Open challenge: calcul ate coupling constants with precision
Page 352
Ref. 262
Electric dip ole moments
Experimental physicists are searching for electric dipole moments of elementary
particles. No non-zero value has been detected yet. The idea of electric dipole moment is based on a non-spherical distribution of electric charge in space.
In the strand model, particles are tangles. As a consequence, the electric charge dis-
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Challenge 225 r
The approximations used above for estimating the coupling constants can be dismissed
as mere educated guesses. Despite this objection, these guesses show that a determination of the coupling constants from the strand model is within reach, and that it can be
realized with limited effort. It is sufficient to improve the three approximations; this is
can be realized by using computer simulations for the transfer of Reidemeister moves or
by finding an improved analytical model.
Calculating all three coupling constants ab initio with high precision will allow checking the statements of this section in an independent manner and, above all, will allow
testing the strand model. The calculations should be performed at different energies,
to confirm the energy dependence of the couplings. Also the influence of the effective
strand diameter on the fine structure constant should be explored.
In order to reach highest precision, the effects of the various tangle family members
might have to be taken into account, because in the strand model, each particle is described by a family of tangles. On the other hand, the strand model predicts that family
members have a small effect on the coupling constant, so that the family issue can be
neglected in the beginning.
In the case of the nuclear coupling constants, Arnold’s results on plane curves may
help in the estimations and calculations.
Motion Mountain – The Adventure of Physics
?=0
sin ? (cos3 ? cos3 ?)4 d? d? ≈ 0.11 .
392
12 pa rticle properties d ed uced from stra nds
tribution – the distribution of the crossings in a tangle – is intrinsically a slightly nonspherical quantity, thus a quantity unequally distributed in space. However, it is only
non-local on a scale of the order of a Planck length. In other terms, the electric dipole
moment ? of elementary particles is predicted to be
⊳ ? = ? ? ?Pl ,
Ref. 263
Five key challenges abou t coupling strengths
There are many ways to evaluate candidates for unified models. A concrete evaluation
focuses on four key challenges about coupling constants. These challenges must be resolved by any candidate model in order to be of interest.
⊳ The strand model explains why the fine structure constant, or equivalently,
the electric charge, is the same for electrons and protons.
Deducing this equality is a key challenge for any unified model. In fact, all coupling
constants must be independent of particle type. This is the case in the strand model.
2. The second key challenge was the energy-dependence of the coupling constants.
The strand model predicts that coupling constants run with energy in exactly the way that
is predicted by QED, QCD and electroweak theory. We could also argue that this is not a
real challenge for any unified model that reproduces these theories. In the strand model,
the running of the electromagnetic coupling constant can be seen as a consequence of
the gradual tightening of tangles with energy. For a typical electrically charged particle
at low energy, the tangle is very loose; therefore:
⊳ The Planck scale number of crossings is shielded by an additional cloud of
crossings created by the loose strands of the tangle.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
1. So far, we explained particle charges with topological properties of the tangle models of the particles, and we explained coupling strengths with the transfer of crossings,
pokes and slides. This allowed deducing a rough approximation of coupling constants.
By doing so, we have settled a first key challenge:
Motion Mountain – The Adventure of Physics
where the factor ? arises from averaging the tangle and is of order one. Similar values are
predicted by the standard model in the absence of supersymmetry and grand unification.
However, the sensitivity of measurements has not reached these values yet, by several
orders of magnitude.
We note that the strand model predicts that the dipole moment changes, or ‘runs’,
with energy. This follows from the shape-dependence of the dipole moment. Such a
dependence is also predicted by quantum field theory.
In summary, we expect that up to a region close to a Planck length, the strand model
should not yield dipole moments that differ in order of magnitude from those predicted
by the standard model of particle physics. In the future, more precise calculations and
measurements could allow testing the strand model using dipole moments.
coupling consta nts a nd unification
393
In this way, the strand model explains the running of the fine structure constant in exactly the same way as QED.
3. The third key challenge has only been touched upon very briefly:
⊳ Any unified model needs to clarify the relation between the hypercharge,
the electric charge and the weak isospin (the ‘weak charge’).
4. The fourth key challenge, related to the previous one, still needs to be explored in
more detail:
⊳ Any unified model must explain why the mass ratio of the intermediate
weak vector bosons is related to the coupling ratio of the weak and the electromagnetic interaction as
?? 2 ?
) +
=1.
??
?w
(220)
The strand model strongly suggests that it can explain the relation, but the detailed argument must yet be provided. Using more drastic language, we can repeat what many
have said already in the past: explaining the electroweak mixing expression (220) is the
key challenge for any unified model.
In the strand model, the two electroweak coupling constants are measures for interaction probabilities of crossings with twists and with pokes. In contrast, masses are interaction probabilities of crossings with spatial curvature. Why are they related by expression
(220)? Here is a short brainstorm on the issue.
In the strand model, mass appears by tail braiding. Tail braiding adds crossings, and
in this way adds mass. Added crossings also imply added weak and sometimes electric
charges. The Z boson arises from vacuum by different tail braidings than the W. The W
arises by the braiding of two tail pairs at 90 degrees; the Z arises by braiding one tail pair
at 90 degrees.
In case of the W and the Z bosons, the Z tangle produces a larger disturbance of the
vacuum than the W; therefore it is more massive than the W.
At which angle does a clasp start to form a ‘‘enclosed space in between’’? How does
this space change with scale, given that scale might change the clasp angle? This question might be related to the running of masses, mixing angles or coupling constants. In
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
(
Motion Mountain – The Adventure of Physics
The strand model explains electromagnetism as acting on crossings and the weak interaction as acting on parallel strands. This general statement contains the required explanation; but the details still need to be worked out. It is expected that in electromagnetism,
a single crossing is rotated, mainly by rotating one strand around the other. In contrast,
in the weak interaction, two strands are rotated together, producing ar switching two
crossings. The number of crossings differs between electromagnetism and the weak interaction, but the total number of involved strands is tow in both cases. As a result of this
similarity, the two interactions mix. The final explanation of electroweak mixing might
even allow to deduce a intuitive geometric meaning of ?w , the weak mixing angle or
Weinberg angle.
394
12 pa rticle properties d ed uced from stra nds
particular, we should answer the following question: Which physical observable does
this enclosed space influence? Mass, couplings, or mixings? Is mass more related to
ropelength or more related to the enclosed space?
5. The fifth key challenge is, of course, the precise calculation of the coupling constants.
Summary on coupling constants
Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
The strand model implies that coupling constants are geometric properties of tangle families that correspond to charged particles. As a consequence, strands explain why the
coupling constants are not free parameters in nature, but fixed constants. Strands predict
that coupling constants are the same for particles with the same charge, and that coupling
constants are constant during the macroscopic evolution of the universe. Strands predict
small electric dipole moments for elementary particles, compatible with and lower than
present measurement limits. Strands also predict the correct sequence of the coupling
constants at low energy and the correct sign of their running with energy. Strands thus
reproduce all observed qualitative properties of coupling constants. No other unified
model achieves this yet.
Using tangle shapes, the strand model proposes several ways to calculate coupling
constants ab initio. First estimates of the fine structure constant, based on the knotted particle models of 2010, deviated from experiment by 40 %. However, those particle
models turned out to be mistaken. New estimates based on the new tangled particle
models yield a much better accuracy; nevertheless, the errors due to the approximations
are still larger that the measurement errors. Improved calculations are ongoing and will
allow to confirm or to refute the strand model.
C h a p t e r 13
EXPER IMENTA L PR EDICT IONS OF
T HE ST R AND MODEL
“
Es gibt viele Theorien,
die sich jedem Test entziehen.
Diese aber kann man checken,
elend wird sie drum verrecken.**
A
round the world, numerous researchers are involved in experiments that
re searching for new effects. They are searching for new observations that
re unexplained by the standard model of particle physics or by the conventional view of cosmology. At the same time, all these experiments are testing the strand
model presented here. In fact, most people working on these experiments have not
heard about the strand model, so that there is not even the danger of unconscious bias.
To simplify the check with experiments, the most important predictions of the strand
model that we deduced in our adventure are listed in Table 17.
typeface distinguishes predictions that are unsurprising, that are unconfirmed or unique to the strand
model, and those that are both unconfirmed and unique.
Page 36
Page 329
Experiment
Prediction (most
from 2008/2009)
S tat us ( 2 0 1 7)
Planck units (?, ℏ, ?, ?4 /4?)
are limit values.
Higgs boson
2008/9: does not exist.
2012: does exist.
2008/9: implies no Higgs.
2012: implies one Higgs.
2008/9: show non-local
effects at the Large Hadron
Collider.
2012: show no non-local
effects at the Large Hadron
Collider.
None has been
exceeded, but more
checks are possible.
Falsified.
Verified.
No data yet.
No data yet.
No data yet.
Page 331
Page 382
Running of the coupling constants
Page 382
Page 329
Page 331
Longitudinal W and Z boson scattering
None found yet.
** No adequate translation is possible of this rhyme, inspired by Wilhelm Busch, claiming that any theory
that can be tested is bound to die miserably.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
TA B L E 17 The main predictions of the strand model that follow from the fundamental principle. The
Motion Mountain – The Adventure of Physics
”
Anonymous
396
13 experimental predictions
TA B L E 17 (Continued) The main predictions of the strand model that follow from the fundamental
principle. The typeface distinguishes predictions that are unsurprising, that are unconfirmed or unique to
the strand model, and those that are both unconfirmed and unique.
Page 329
Page 352
Page 314
Page 276, page 318
Page 313
Page 375
Page 341
Page 376
Page 328
Page 276, page 328
Page 344
Page 324, page 340
Page 276, page 328
Page 341
Page 341
Page 338
S tat us ( 2 0 1 7)
Longitudinal W and Z boson scattering
W boson g-factor
Unknown fermions (supersymmetric
particles, magnetic monopoles, dyons,
heavy neutrinos etc.)
Unknown bosons (other gauge bosons,
supersymmetric particles, axions etc.)
Unknown interactions, energy scales and
symmetries (grand unification,
supersymmetry, quantum groups,
technicolour etc.)
Particle masses, mixing angles and
coupling constants
is unitary at the LHC.
is near to 2.
do not exist.
Obvious.
Is observed.
None found yet.
do not exist.
None found yet.
do not exist.
None found yet.
Particle masses, mixing angles and
coupling constants
Particle masses, mixing angles, coupling
constants and g-factors
Mixing matrix for quarks
Mixing matrix for neutrinos
Neutrinos
Neutrinos
Neutrino-less double beta decay
Electric dipole moments of elementary
particles, magnetic dipole moment of
neutrinos
Tetraquarks
Glueballs
Proton decay and other rare decays,
neutron-antineutron oscillations
Neutron decay
Neutron charge
Hadron masses and form factors
are calculable by modifying Most not yet
existing software packages. calculated;
approximations very
encouraging.
are constant in time.
Is observed.
are identical for antimatter.
Is observed.
is unitary.
is unitary.
are Dirac particles.
violate CP symmetry.
does not exist.
have extremely small,
calculable values.
Is observed.
No data yet.
No data yet.
No data yet.
Not yet found.
No data yet.
exist.
probably do not exist; if they
do, the spectrum can be
compared to the strand
model.
occur at extremely small,
standard model rates.
follows the standard model.
vanishes.
can be calculated ab initio.
Likely.
Not yet observed.
Not yet observed.
No deviations found.
None observed.
Not yet calculated;
value sequences and
signs correct.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 372
Prediction (most
from 2008/2009)
Motion Mountain – The Adventure of Physics
Page 313
Experiment
of the stra nd m od el
397
TA B L E 17 (Continued) The main predictions of the strand model that follow from the fundamental
principle. The typeface distinguishes predictions that are unsurprising, that are unconfirmed or unique to
the strand model, and those that are both unconfirmed and unique.
Page 353
Page 314
Page 146
Page 293
Page 293
Page 291
Page 304
Page 307
Page 307
Page 309
Page 307
Page 376
Page 310
Page 354
Dark matter
is conventional matter plus
black holes.
S tat us ( 2 0 1 7)
Partly confirmed by
black hole mergers
and lack of other
results.
Standard model of particle physics
2008/9: is essentially correct, Not yet falsified, but
with deviations for the
deviations not yet
scattering of longitudinal
observed.
vector bosons at LHC energy.
2012: is correct for all
All data agrees.
measurable energies.
Additional dimensions
do not exist.
Not observed.
Non-commutative space-time
does not exist.
Not observed.
General relativity
is correct at all accessible
No deviation found.
energies.
Short-distance deviations from universal do not exist.
All data agrees.
gravitation and modified gravity
Space-time singularities, cosmic strings, do not exist.
None observed.
wormholes, time-like loops, negative
energy regions, domain walls
Quantum gravity effects
will not be found.
None observed yet.
Behind a horizon
nothing exists.
Nothing observed.
Cosmological constant (dark energy)
is small and positive.
Is observed.
Cosmological constant (dark energy)
decreases with time squared. Data are inconclusive.
Cosmic matter density
decreases with time squared. Data are inconclusive.
Cosmic inflation
did not occur.
Data not in contrast.
Leptogenesis
did not occur.
Data are inconclusive.
Cosmic topology
is trivial.
As observed.
Vacuum
is stable and unique.
As observed.
In summary: all motion
results from strands.
Not yet falsified.
In this list, the most interesting predictions of the strand model are the numerical
predictions on the decay of the cosmological constant, the various mass ratios and mass
sequences – including the Z/W and Higgs/W mass ratios – and the relative strength
of the three gauge interactions. There is the clear option to calculate all fundamental
constants in the foreseeable future.
In addition, the strand model reproduces the quark model, gauge theory, wave functions and general relativity; at the same time, the model predicts the lack of measurable
deviations. The strand model solves conceptual problems such as the dark matter problem, inflation, confinement, the strong CP problem and the anomaly issue; by doing so,
the strand model predicts the lack of unknown effects in these domains.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 297, page 277
Prediction (most
from 2008/2009)
Motion Mountain – The Adventure of Physics
Page 146
Experiment
398
13 experimental predictions
The fundamental principle of the strand model
Strand model :
?1
Observation :
Some
deformation,
but no
passing
through
? = ℏ/2
Δ? = ?Pl
Δ? = ?Pl
? = ?/2
?2
F I G U R E 116 The fundamental principle of the strand model: Planck units are defined by a crossing
switch in three spatial dimensions. With this principle, as shown in the previous chapters, the
fundamental principle implies general relativity and the standard model of particle physics.
In our adventure, we have argued that Planck’s natural units should be modelled with the
fundamental principle for strands, which is shown again in Figure 116. As we discovered,
the fundamental principle explains the following measured properties of nature:
— Strands explain the principle of least action and the invariance of ?, ℏ, ? and ?.
— Strands explain the three dimensions of space, the existence of gravitation, curvature
and horizons, the equations of general relativity, the value of black hole entropy and
the observations of modern cosmology.
— Strands explain all the concepts used in the Lagrangian of the standard model of
particle physics, including wave functions, the Dirac equation and the finite, discrete
and small mass of elementary particles.
— Strands explain the existence of electromagnetism and of the two nuclear interactions, with their gauge groups and all their other observed properties.
— Strands describe the observed gauge and Higgs bosons, their charges, their quantum
numbers and their mass mass ranges.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
final summary abou t the millennium issues
Motion Mountain – The Adventure of Physics
The strand model deduces all its experimental predictions from a single and simple
fundamental principle: events and Planck units are due to crossing switches of strands.
Provided there are no errors of reasoning, there is no way to change the predictions
summarized here. The strand model is both simple and unmodifiable.
Naturally, errors of reasoning in the preceding chapters are well possible. A few have
occurred in the past. The exploration was performed at high speed – possibly too high. If
any experiment ever contradicts a prediction of the strand model, the model is doomed.
When the above experimental predictions were first deduced in 2008 and 2009, they were
quite unpopular. Practically all other attempts at unification predicted the existence of
yet undiscovered particles and effects. However, so far, experiment does not confirm
these other attempts; in fact, no prediction of the strand model has been falsified yet.
fina l sum mary a b ou t the m illennium issues
399
All these results translate to specific statements on experimental observations. So far,
there is no contradiction between the strand model and experiments. These results allow
us to sum up our adventure in three statements:
Page 21
We have not yet literally reached the top of Motion Mountain – because certain numerical predictions of the fundamental constants are not yet precise enough – but if no cloud
has played a trick on us, we have seen the top from nearby. In particular, we finally know
the origin of colours.
The last leg, the accurate calculation of the constants of the standard model of particle
physics, is still under way. The drive for simplicity and the spirit of playfulness that we
invoked at the start have been good guides.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 19
1. Strands solve all open issues. With one simple fundamental principle, the strand
model solves or at least proposes a way to solve all issues from the millennium list of
open issues in fundamental physics. All fundamental constants can be calculated.
2. Strands agree with all observations. In particular, the strand model implies that general relativity, quantum theory and the standard model of elementary particles are a
precise description of motion for all practical purposes.
3. Nothing new will be discovered in fundamental physics. Unexpectedly but convincingly, strands predict that general relativity, quantum theory and the standard model
of elementary particles are a complete description of motion for all practical purposes.
Motion Mountain – The Adventure of Physics
— Strands explain the three generations of quarks and leptons, their charges and
quantum numbers, their mixing, their mass sequences, as well as their confinement
properties.
— Strands explain the quark model of hadrons, including CP violation, mass sequences,
signs of quadrupole moments, the lack of unobserved hadrons, common Regge
slopes and the existence of tetraquarks.
— Strands do not allow arbitrary values for masses, coupling constants, mixing angles
and CP violating phases.
— Strands enable calculations of particle masses, their coupling constants, their mixing
angles and the CP violating phases. First rough estimates of these values agree with
the (much more precise) experimental data. Computer calculations will allow us to
improve these checks in the near future.
— Strands predict the lack of unknown dark matter and of unknown inflation mechanisms.
— Finally, strands predict that nature does not hide any unknown elementary particle,
fundamental interaction, fundamental symmetry or additional dimension. In particular, strands predict that no additional mathematical or physical concepts are required for a final theory.
C h a p t e r 14
T HE TOP OF MOT ION MOUNTAIN
”
“
The labour we delight in physics pain.
William Shakespeare, Macbeth.
Thales**
our path to the top
”
Our walk had a simple aim: to talk accurately about all motion. This 2500 year old quest
drove us to the top of this mountain. We can summarize our path in three legs: everyday
life, general relativity plus quantum theory, and unification.
** Thales of Miletus (c. 624 – c. 546 bce) was the first known philosopher, mathematician and scientist.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
ho am I? Where do I come from? What shall I do? Where does the
orld come from? Can the whole world really come to a sudden end? What
ill happen in the future? What is beauty? All these questions have a common
aspect: they are questions about motion. But what is motion? Our search for an answer
led us to study motion in all its details. In this quest, every increase in the precision of
our description of motion was a step towards the peak of Motion Mountain. Now that
we arrived there, we can savour what we have achieved and recall the emotions that we
have experienced.
In our ascent, we have learned how we move, how we experience our environment,
how we grow, what parts we are made of, and how our actions and our convictions about
them can be understood. We have learned a lot about the history and a bit about the
future of matter, of radiation and of space. We have experienced and understood the
many ways in which beauty appears in nature: as colours, as shapes, as rhythms and
most of all: as simplicity.
Savouring our achievement means that first of all, we now can look back to where we
came from. Then we enjoy the view we are offered and look out for what we could not
see before. After that, we search for what is still hidden from our sight. And finally, we
take a different path back down to where we live.
Motion Mountain – The Adventure of Physics
W
“
All things are full of gods.
our path to the top
401
Everyday life: the rule of infinit y
Ref. 1, Ref. 3
“
Ref. 2, Ref. 4
Ref. 264
Vorhin haben wir gesehen, daß in der Wirklichkeit
das Unendliche nirgends zu finden ist, was für
Erfahrungen und Beobachtungen und welcherlei
Wissenschaft wir auch heranziehen.*
David Hilbert
”
The idea that nature offers an infinite range of possibilities is often voiced with deep
personal conviction. However, the results of relativity and quantum theory show the
opposite. In nature, speeds, forces, sizes, ages and actions are limited. No quantity in
nature is infinitely large or infinitely small. No quantity in nature is defined with infinite
precision. There never are infinitely many examples of a situation; the number of possibilities is always finite. The world around us is not infinite; neither its size, nor its age,
nor its content. Nature is not infinite. This is general relativity and quantum theory in
one statement.
Relativity and quantum theory show that the idea of infinity appears only in approximate descriptions of nature; it disappears when talking with precision. Nothing in nature
* ‘Above we have seen that in the real world, the infinite is nowhere to be found, whatever experiences and
observations and whatever knowledge we appeal to.’
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
R el ativit y and quantum theory: the absence of infinit y
Motion Mountain – The Adventure of Physics
Galilean physics is the description of everyday life. We all learned Galilean physics before
secondary school. Galilean physics is the exploration and description of the motion of
stones, water, trees, heat, the weather, electricity and light. To achieve this description of
our environment, our first and main act in life is to partition experience into experiences.
In other words, our first intellectual act is the invention of parts; we invented the plural.
The act of partitioning allows us to define sequences among our experiences, and thus
to define the concept of time. The concept of space arises similarly by our possibility to
distinguish observations that occur at the same time. By comparing parts with other
parts, we define measurement. Using all of this, we become able to define velocity, mass
and electric charge, among others. These allow us to introduce action, the quantity that
quantifies change.
For a simple description of observations, we assume that division is possible without
end: thus we introduce the infinitely small. We also assume that widening our scope
of observation is possible without end. Thus we introduce the infinitely large. Defining
parts thus leads us to introduce infinity.
Using parts and, with them, the infinitely small and the infinitely large, we found,
in volumes I and III, that everyday motion has six main properties: it is continuous,
conserved, relative, reversible, mirror-invariant and lazy. Motion is lazy – or efficient –
because it produces as little change as possible.
Nature minimizes change. This is Galilean physics, the description of everyday motion,
in one statement. It allows us to describe all our everyday experiences with stones, fluids,
stars, electric current, heat and light. The idea of change-minimizing motion is based on
a concept of motion that is continuous and predictable, and a concept of nature that
contains the infinitely small and the infinitely large.
402
Ref. 4
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
is infinite. For example, we found in volume II that the sky is dark at night (also) because
space is not infinite. And we found, in volumes IV and V, that quantum theory contains
probabilities because there is a smallest action value in nature. In fact, the statement that
a quantity is infinitely large or infinitely small cannot be confirmed or reproduced by
any experiment. Worse, such a statement is falsified by every measurement. In short, we
found that infinity is a fantasy of the human mind. In nature, it does not appear. Infinity
about nature is always a lie.
The number of particles, their possible positions, the states they can have, our brain,
our creativity, our possible thoughts: all this is not infinite. Nevertheless, quantum theory
and relativity changed the world: they allowed building ultrasound imaging, magnetic
resonance imaging, lasers, satellite navigation systems, music players and the internet.
Despite the vast progress due to modern physics and the related technologies, one
result remains: nothing in our environment is infinite – neither our life, nor our experiences, nor our memories, not even our dreams or our fantasies. Neither the information
necessary to describe the universe, nor the paper to write down the formulae, nor the
necessary ink, nor the time necessary to understand the formulae is infinite. Nature is
not infinite. On the other hand, we also know that the illusion of the existence of infinity
in nature is one the most persistent prejudices and myths ever conceived. Why did we
use it in the first place?
The habit to use infinity to describe the world has many emotional reasons. For some,
it reflects the deep-rooted experience of smallness that we carry within us as a remnant
our personal history, when the world seemed so large and powerful. For others, the idea
of our smallness allows us to deny somehow the responsibility for our actions or the
existence of death. For others again, the idea of a finite universe often, at a first glance,
produces deception, disbelief and discouragement. The absence of infinity means that
we cannot achieve everything we want, and that our dreams and our possibilities are
limited. Clinging to the idea of infinity is a way to avoid confronting this reality.
However, once we face and accept the absence of infinity, we make a powerful experience. We gain in strength. We are freed from the power of those who use this myth
to put themselves above others. It is an illuminating experience to reread all those sentences on nature, on the world and on the universe containing the term ‘infinite’, knowing that they are incorrect, and then clearly experience the manipulations behind them.
The desire to make others bow to what is called the infinite is a common type of human
violence.
At first, the demise of infinity might also bring panic fear, because it can appear as a
lack of guidance. But at closer inspection, the absence of infinity brings strength. Indeed,
the elimination of infinity takes from people one of the deepest fears: the fear of being
weak and insignificant.
Moreover, once we face the limits of nature, we react like in all those situations in
which we encounter a boundary: the limit becomes a challenge. For example, the experience that all bodies unavoidably fall makes parachuting so thrilling. The recognition
that our life is finite produces the fire to live it to the full. The knowledge of death gives
meaning to our actions. In an infinite life, every act could be postponed without any
consequence. The disappearance of infinity generates creativity. A world without limits is discouraging and depressing. Infinity is empty; limits are a source of strength and
pour passion into our life. Only the limits of the world ensure that every additional step
Motion Mountain – The Adventure of Physics
Challenge 226 e
14 the top of m otion m ountain
our path to the top
403
in life brings us forward. Only in a limited universe is progress possible and sensible.
Who is wiser, the one who denies limits, or the one who accepts them? And who lives
more intensely?
Unification: the absence of finitude
“
Pray be always in motion. Early in the morning go
and see things; and the rest of the day go and see
people. If you stay but a week at a place, and that an
insignificant one, see, however, all that is to be seen
there; know as many people, and get into as many
houses as ever you can.
Philip Stanhope,* Letters to his Son on the Fine Art of
Becoming a Man of the World and a Gentleman.
”
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
* Philip D. Stanhope (b. 1694 London, d. 1773 London) was a statesman and writer.
Motion Mountain – The Adventure of Physics
Page 127
The last part of our adventure, described in this volume, produced an unexpected result.
Not only is nature not infinite; nature is not finite either. None of the quantities which
were supposed to be finite turn out to be so. Finitude turns out to be an approximation,
or better, an illusion, though a subtle one. Nature is not finite. This is the unification of
physics in one statement.
Precise observation shows that nothing in nature can be counted. If nature were finite
it would have to be (described by) a set. However, the exploration of Planck scales shows
that such a description is intrinsically incomplete and inaccurate. Indeed, a description
of nature by a set can never explain the number of its elements, and thus cannot explain
finitude itself. In other words, any approach that tries to describe nature as finite is a
belief, and is never correct. Finitude is a lie.
We thus lost our security of thought a second time. Nature is neither infinite nor finite.
We explored the possibilities left over and found that only one option is left: Nature is
indivisible. In other words, all parts that we experience are approximations. Both finitude
and infinity are approximation of nature. All distinctions are approximate. This central
conclusion solved the remaining open issues about motion. Nature has no parts.
The impossibility to count and the lack of parts imply that nature is not a computer,
not an automaton, nor a physical system. Nature is not discrete.
Recognizing all distinctions as being approximate abolishes the distinction between
the permanent aspects of nature (‘objects’, described by mass, charge, spin, etc.) and the
changing aspects (‘states’, described by position, momentum, energy). Taking all distinctions as approximate introduces extended constituents: fluctuating strands. Looking
even closer, these extended constituents are all the same one. Space, formally only used to
describe states, also acquires changing aspects: it is made from fluctuating strands. Also
properties like mass or charge, which formally were seen as static, become aspects of the
ever changing interplay between these fundamental constituents. Describing nature as
one fluctuating strand allows us to avoid finitude and to answer all questions left open
by quantum theory and general relativity.
In a sense, the merging of objects and states is a resolution of the contrasting views
on motion of the Greek thinkers Parmenides – ‘there is no motion’, i.e., in physical
language, ‘there are no states, there is only permanence’ – and Heraclitus – ‘everything
404
14 the top of m otion m ountain
moves’, i.e., in physical language ‘there is no permanence, there are only states’. Both
turn out to be right.
We can thus sum up the progress during our adventure of physics in the following
table:
TA B L E 18 The progress of physics.
Step 1
Step 2
Galilean Physics
Relativity
Step 3
Quantum field theory
Step 4
Unification
Nature is continuous.
Nature has no infinitely
large.
Nature has no infinitely
small.
Nature is not finite.
Nature has no parts.
We live in Galilean space.
We live in Riemannian
space.
We live in a Hilbert/Fock
space.
We do not live in any space;
we are space.
new sights
sustanze e accidenti e lor costume
quasi conflati insieme, per tal modo
che ciò ch’i’ dico è un semplice lume.
La forma universal di questo nodo
credo ch’i’ vidi, perché più di largo,
dicendo questo, mi sento ch’i’ godo.*
Dante, La (Divina) Commedia, Paradiso,
XXXIII, 85-93.
”
Modelling nature as a complicated web of fluctuating strands allowed us to describe at
the same time empty space, matter, radiation, horizons, kefir, stars, children and all our
other observations. All everyday experiences are consequence of everything in nature
being made of one connected strand. This result literally widens our horizon.
* ‘In its depth I saw gathered, bound with love into one volume, that which unfolds throughout the universe:
substances and accidents and their relations almost joined together, in such a manner that what I say is only
a simple image. The universal form of that knot, I think I saw, because, while I am telling about it, I feel deep
joy.’ This is, in nine lines, Dante’s poetic description of his deepest mystical experience: the vision of god.
For Dante, god, at the depth of the light it emanates, is a knot. That knot spreads throughout the universe,
and substances and accidents – physicists would say: particles and states – are aspects of that knot. Dante
Alighieri (b. 1265 Florence, d. 1321 Ravenna) was one of the founders and the most important poet of the
Italian language. Most of the Divine Comedy, his magnum opus, was written in exile, after 1302, the year
when he had been condemned to death in Florence.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
“
Nel suo profondo vidi che s’interna,
legato con amore in un volume,
ciò che per l’universo si squaderna:
Motion Mountain – The Adventure of Physics
In summary, we are made of space. More precisely, we are made of the same constituents as space. In fact, the fascination of this result goes further than that.
new sig hts
405
The beau t y of strands
“
Someday, surely, we will see the principle underlying
existence itself as so simple, so beautiful, so obvious,
that we will all say to each other, “Oh, how could we
all have been so blind, so long.”
John Wheeler, A Journey Into Gravity And Spacetime.
”
Page 163
“
”
Die Natur kann besser Physik als der beste Physiker.*
Carl Ramsauer
As mentioned above, mathematical physicists are fond of generalizing models. Despite
this fondness, we required that any final, unified description must be unique: any final,
unified description must be impossible to reduce, to modify or to generalize. In par* ‘Nature knows physics better than the best physicist.’ Carl Ramsauer (b. 1879 Oldenburg, d. 1955 Berlin),
influential physicist, discovered that electrons behave as waves.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Can the strand model be generalized?
Motion Mountain – The Adventure of Physics
Describing everything as connected does not come natural to us humans. After all, in
our life, we perform only one act: to partition. We define pluralities. There is no way we
can avoid doing this. To observe, to think, to talk, to take a decision, to move, to suffer,
to love or to enjoy life is impossible without partitioning.
Our walk showed us that there are limits to the ability to distinguish. Any kind of
partitioning is always approximate. In fact, most people can summarize their personal
experience by saying that they learned to make finer and finer distinctions. However,
talking with highest precision about a part of the world inevitably leads to talk about the
whole universe. The situation resembles a person who gets a piece of rope in his hand,
and by following it, discovers a large net. He continues to pull and finally discovers that
everything, including himself, is part of the net.
For the strand model, the term ‘theory of everything’ is therefore not acceptable.
Nature cannot be divided into ‘things’. In nature, things are never separable. There is no
way to speak of ‘every’ thing; there are no sets, no elements and no parts in nature. A
theory describing all of nature cannot be one of ‘everything’, as ‘things’ are only approximate entities: properly speaking, they do not exist. The strand model is not a theory of
everything; it is a final theory.
The strand model shows that nature is not made of related parts. Nature is made of
relations only. Parts only exist approximately. The strand model also shows: being in
motion is intrinsic to being a part. Parts, being approximate, are always in motion. As
soon as we divide, we observe motion. The act of dividing, of partitioning, of defining
parts is the very one which produces order out of chaos. Strands force us to rethink this
habit.
Despite being so tough to grasp, strands yield a precise description of motion that
unifies quantum field theory and general relativity. The strand model for the unification
of motion is both simple and powerful. There are no free parameters. There are no
questions left. Our view from the top of the mountain is thus complete. No uncertainty,
no darkness, no fear and no insecurity are left over. Only wonder remains.
406
Challenge 227 r
What is nature?
Ref. 265
“
”
Nature is what is whole in each of its parts.
Hermes Trismegistos, Book of Twenty-four
Philosophers.
At the end of our long adventure, we discovered that nature is not a set: everything is
connected. Nature is only approximately a set. The universe has no topology, because
space-time is not a manifold. Nevertheless, the approximate topology of the universe is
that of an open Riemannian space. The universe has no definite particle number, because
the universe is not a container; the universe is made of the same stuff of which particles
are made. Nevertheless, the approximate particle density in the universe can be deduced.
In nature, everything is connected. This observation is reflected in the conjecture that
all of nature is described by a single strand.
We thus arrive at the (slightly edited) summary given around the year 1200 by the
author who wrote under the pen name Hermes Trismegistos: Nature is what is whole in
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 155
ticular, a final theory must neither be a generalization of particle physics nor of general
relativity. Let us check this.
The strand model is not a generalization of general relativity: the definitions of
curvature, of gravitons and of horizons differ radically from general relativity’s approach.
The strand model is also not a generalization of particle physics: the definitions of
particle and of interactions differ radically from the concepts of quantum field theory.
Indeed, we have shown that quantum field theory and general relativity are approximations to the strand model; they are neither special cases nor reductions of the strand
model.
But what about the other requirements for a unified theory? Can the strand model
be modified or generalized? We have seen that the model does not work in more spatial dimensions, does not work with more families of quarks, does not work with more
interactions, and does not work with other evolution equations in general relativity or
particle physics. The strand model does not work with other fundamental constituents,
such as bifurcating entities, membranes, bands, or networks. (Though it does work with
the equivalent funnels, as explained earlier on, but that description is equivalent to the
one with strands.) The strand model does not work with any modified fundamental principle. Obviously, exploring all possible variations and modifications remains a challenge
for the years to come. If an actual modification of the strand model can be found, the
strand model instantly loses its value: in that case, it would need to be shelved as a failure.
Only a unique unified model can be correct.
In summary, one of the beautiful aspects of the strand model is its radical departure
from twentieth-century physics in its basic concepts, combined with its almost incredible uniqueness. No generalization, no specialization and no modification of the strand
model seems possible. In short, the strand model qualifies as a unified, final theory.
What is a requirement to one person, is a criticism to another. A number of researchers deeply dislike the strand model precisely because it doesn’t generalize previous theories and because it cannot be generalized. This attitude deserves respect, as it is born
from the admiration for several ancient masters of physics. However, the strand model
points into a different direction.
Motion Mountain – The Adventure of Physics
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407
each of its parts. But in contrast to that author, we now also know how to draw testable
conclusions from the statement.
Q uantum theory and the nature of mat ter and vacuum
”
C osmology
Page 8
”
The strand model also showed us how to deduce general relativity. The strand model
clarified the fabric of horizons and explained the three dimensions of space. Most fascinating is the idea of a universe as the product of a single strand. A single strand implies
that there was nothing before the big bang, and that there is nothing outside the night
sky. For example, the strand model implies that there is no ‘multiverse’ and that there
are no hidden worlds of any kind. And the fluctuating strand explains all observations
of our universe.
The cosmological constant is not constant; it only measures the present age and size of
the universe. Therefore, the constant does not need to appear in Figure 1. In other words,
the cosmological constant simply measures the time from the big bang to the present.
* ‘Modern physics is not far from the question whether everything that exists could possibly be made from
aether. These things are the extreme goals of our science, physics.’ Hertz said this in a well-known speech
he gave in 1889. If we recall that ‘aether’ was the term of the time for ‘vacuum’, the citation is particularly
striking.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
“
Der heutigen Physik liegt die Frage nicht mehr
ferne, ob nicht etwa alles, was ist, aus dem Äther
geschaffen sei. Diese Dinge sind die äußersten
Ziele unserer Wissenschaft, der Physik.*
Heinrich Hertz
The strand model shows that as soon as we separate the universe into space-time and
the rest, i.e., as soon as we introduce the coordinates ? and ?, quantum mechanics appears automatically. More precisely, quantum effects are effects of extension. Quantum
theory appears when we realize that observations are composed of smallest events due
to crossing switches, each with a change given by the quantum of action. All events and
observations appear through the fluctuations of the strand that composes nature.
We found that matter is made of tangled strands. In fact, the correct way would be to
say: matter is made of tangled strand segments. This connection leads to Schrödinger’s
equation and to Dirac’s equation.
Insofar as matter is of the same fabric as the vacuum, we can rightly say that everything
is made of vacuum and that matter is made of nothing. But the most appropriate definition arises when we realize that matter is not made from something, but that matter is
a certain aspect of the whole of nature. Unification showed that every single elementary particle results from an arrangement of strands that involves the whole of nature, or,
if we prefer, the entire universe. In other words, we can equally say: matter is made of
everything.
We can also turn the equivalence of matter and vacuum around. Doing so, we arrive
at the almost absurd statement: vacuum is made of everything.
Motion Mountain – The Adventure of Physics
“
In everything there is something of everything.
Anaxagoras of Clazimenes (500
–428 bce Lampsacus)
408
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The ‘big bang’ is the name for what we observe if we try to make observations approaching the limits of nature. The ‘big bang’ appears automatically from the strand
model whenever we observe nature at the most distant times, the largest distances or at
the largest energies: ‘big bang’ is the name for Planck scale physics.
The universe consists of a single strand. There are many particles in nature, because
the strand is tangled up in complicated ways. What we call the ‘horizon’ of the universe
is the place where new tangles appear.
The belief that the big bang or the horizon are examples of creation is incorrect. What
happened at the big bang still happens at the horizon today. Both the black sky at night
and the big bang are nature’s way to tell us: ‘Galilean physics is approximate! Quantum
theory is approximate! General relativity is approximate!’
Musings abou t unification and strands
”
All is made from one sort of thing: all is one substance. This idea, monism, sounds
a lot like what the influential philosopher Baruch Spinoza (b. 1632 Amsterdam,
d. 1677 The Hague) held as conviction. Monism, though mixed up with the idea of
god, is also the basis of the philosophical ideas that Gottfried Wilhelm Leibniz (b. 1646
Leipzig, d. 1716 Hannover) presents in his text La Monadologie.
Ref. 266
Any complete theory of motion, also the strand model, is built on a single statement
about nature: The many exists only approximately. Nature is approximately multiple.
The etymological meaning of the term ‘multiple’ is ‘it has many folds’; in a very specific
sense, nature thus has many folds.
∗∗
Any precise description of nature is free of arbitrary choices, because the divisions that
we have to make in order to think are all common to everybody, and logically inescapable. Because physics is a consequence of this division, it is also ‘theory-free’ and
‘interpretation-free’. This consequence of the final theory will drive most philosophers
up the wall.
∗∗
For over a century, physics students have been bombarded with the statement:
‘Symmetries are beautiful.’ Every expert on beauty, be it a painter, an architect, a
sculptor, a musician, a photographer or a designer, fully and completely disagrees, and
rightly so. Beauty has no relation to symmetry. Whoever says the contrary is blocking
out his experiences of a beautiful landscape, of a beautiful human figure or of a beautiful
work of art.
* Lao Tse (sixth century bce) was an influential philosopher and sage.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
∗∗
Motion Mountain – The Adventure of Physics
“
Continuing motion masters coldness.
Continuing rest masters heat.
Motion based on rest:
Measure of the all-happening for the single one.
Lao Tse,* Tao Te King, XXXXV.
new sig hts
Ref. 267
409
The correct statement is: ‘Symmetries simplify descriptions.’ Symmetries simplify
physical theories. That is the background for the statement of Werner Heisenberg: ‘In
the beginning there was symmetry.’ On the other hand, the strand model shows that
even this statement is incorrect. In fact, neither the search for beauty nor the search
for symmetry were the right paths to advance towards unification. Such statements have
always been empty marketing phrases. In reality, the progress of fundamental theoretical
physics was always driven by the search for simplicity.
∗∗
Strands unify physics. In particular, strands extend our views on quantum theory and
mathematical physics, on particle physics and field theory, on axiomatic physics and algebraic physics, on polymer physics and gauge theory, on general relativity and cosmology. It will take several years before all these extensions will have been explored.
∗∗
Page 22
Almost all discoveries in physics were made at least 30 years too late. The same is true for
the strand model. If we compare the strand model with what many physicists believed
in the twentieth century, we can see why: researchers had too many wrong ideas about
unification. All these wrong ideas can be summarized in the following statement:
— ‘Unification requires generalization of existing theories.’
This statement is subtle: it was rarely expressed explicitly but widely believed. But the
statement is wrong, and it led many astray. On the other hand, the development of the
strand model also followed a specific guiding idea, namely:
— ‘Unification requires simplification.’
Hopefully this guiding idea will not become a dogma itself: in many domains of life,
simplification means not to pay attention to the details. This attitude does a lot of harm.
∗∗
The strand model shows that achieving unification is not a feat requiring difficult abstraction. Unification was not hidden in some almost inaccessible place that can reached only
by a few select, well-trained research scientists. No, unification is accessible to everyone
who has a basic knowledge of nature and of physics. No Ph.D. in theoretical physics is
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
∗∗
Motion Mountain – The Adventure of Physics
Vol. III, page 279
The description of nature with strands is surprisingly simple, mainly because it uses so
few basic concepts. Is this result astonishing? In our daily life, we describe our experiences with the help of a few thousand words, e.g. taking them from the roughly 350 000
words which make up the English language, or from a similar number from another
language. This set is sufficient to talk about everything, from love to suffering, from
beauty to happiness. And these terms are constructed from no more than about 35 basic
concepts, as we have seen already. We should not be too surprised that we can in fact
talk about the whole universe using only a few basic concepts: the act and the results of
(approximate) distinction, or more specifically, a basic event – the crossing switch – and
its observation.
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14 the top of m otion m ountain
∗∗
Page 84
The strand model settles all questions about determinism. Quantum theory and general
relativity are deterministic. Nevertheless, when both descriptions are combined, time
turns out to be an approximate, low-energy concept. The same applies to determinism.
Even though nature is deterministic for all practical purposes and shows no surprises,
determinism shares the fate of all its conceivable opposites, such as fundamental randomness, indeterminism of all kinds, existence of wonders, creation out of nothing, or
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
needed to understand or to enjoy it. The knowledge presented in the previous volumes
of this series is sufficient.
When Andrew Wiles first proved Fermat’s last theorem after three centuries of attempts by the brightest and the best mathematicians, he explained that his search for a
proof was like the exploration of a dark mansion. And seen the conceptual difficulties he
had to overcome, the analogy was fitting. Recalling how many more people have already
searched for unification without success, the first reaction is to compare the search for
unification to the exploration of something even bigger, such as a complex dark cave
system. But that analogy was not helpful. In contrast to the proof of Fermat’s theorem,
the goal of the quest for unification turned out to be simple and lying out in the open.
Researchers had simply overlooked it, because they were convinced that the goal was
complex, hidden in the dark and hard to reach. It was not.
The adventure of climbing Motion Mountain is thus not comparable to climbing
Cerro Torre, which might be the toughest and most spectacular challenge that nature
offers to mountain climbers. Figure 117 gives an impression of the peak. Motion Mountain does not resemble this peak at all. Neither does Motion Mountain resemble the
Langtang Lirung peak in the Nepalese Himalayas shown on the cover of this volume.
Climbing Motion Mountain is more like walking up a gentle green hill, alone, with a
serene mind, on a sunny day, while enjoying the surrounding beauty of nature.
Motion Mountain – The Adventure of Physics
F I G U R E 117 Motion Mountain does not resemble Cerro Torre, but a gentle hill (© Davide Brighenti,
Myriam70)
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411
divine intervention: determinism is an incorrect description of nature at the Planck scale
– like all its alternatives.
∗∗
Challenge 228 e
The strand model also settles most so-called really big questions that John Wheeler used
to ask: Why the quantum? How come existence? It from bit? A "participatory universe"?
What makes "meaning"? Enjoy the exploration.
∗∗
Any unified model of nature encompasses a lot of ideas, issues and knowledge. Due to
the sheer amount of material, publishing it in a journal will be challenging.
∗∗
Ref. 268
∗∗
Page 331
∗∗
Many researchers believed during all their life that the final theory is something useful,
important and valuable. This common belief about the importance and seriousness of
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 8
Historically, the strand model evolved from an exploration, started in the 1990s, of the
maximum force in nature, the belt trick and the entropy of black holes. After the first six
chapters of the present volume were completed in 2002, meditating on their implications
led to the strand model and its fundamental principle.
Above all, it was the description of general relativity with the help of the maximum
force that triggered the search for a unified description that was purely based on Planck
units. Another essential point was the drive to search for a final theory directly, from its
requirements (‘top down’ in Figure 1), and not from the unification of quantum theory
and general relativity (‘bottom up’). In the years from 2002 to 2007, most of the ideas of
the strand model took shape, mainly in Munich’s underground trains, while commuting
between home and work. In those years, it appeared that strands could explain the Dirac
equation, the entropy of black holes, general relativity and the particle spectrum with the
three particle generations. While walking in the woods and fields around Munich during 2008 and 2009, it appeared that strands explain the three gauge interactions, predict
(with almost complete certainty) the lack of a Higgs boson – a big mistake due to faulty
reasoning, as turned out in 2012 – and of any new physical effects beyond the standard
model, and allow calculating the unexplained constants of particle physics. The model
thus yielded all its main predictions before the accelerator experiments at the Large Hadron Collider at CERN in Geneva were switched on in autumn 2010. Thus much of the
work was done in a haste – future will show what is of lasting value.
In 2012, the discovery of the Higgs boson, and in 2014, the comments by Sergei Fadeev
led to an improvement and simplification of the strand model, eliminating knotted
strands. In 2016 and 2017, the experimental results at the LHC, of dark matter searches,
and of the LIGO observatory confirmed the lack of deviations from the standard model
of particle physics and from general relativity, as predicted by the strand model.
Motion Mountain – The Adventure of Physics
The strand model is so simple that it fits on a tombstone - or on a T-shirt. This would
surely be god’s favourite T-shirt. It is available at www.motionmountain.net/gfts.html.
412
14 the top of m otion m ountain
the quest has led, over the past decades, to an increasingly aggressive atmosphere among
these researchers. This unprofessional atmosphere, combined with the dependence of
researchers on funding, has delayed the discovery of the final theory by several decades.
In fact, the final theory is not useful: it adds nothing of practical relevance to the
combination of the standard model and general relativity. The final theory is also not
important: it has no application in everyday life or in industry and does not substantially
change our view of the world; it just influences teaching – somewhat. Finally, the final
theory is not valuable: it does not help people in their life or make them happier. In
short, the final theory is what all fundamental theoretical research is: entertaining ideas.
Even if the strand model were to be replaced by another model, the conclusion remains: the final theory is not useful, not important and not valuable. But it is enjoyable.
∗∗
“
“
”
”
Anonymous
Cum iam profeceris tantum, ut sit tibi etiam tui
reverentia, licebit dimittas pedagogum.*
Seneca
The final theory of motion has a consequence worth mentioning in detail: its lack of
infinity and its lack of finitude eliminate the necessity of induction. This conclusion is of
* ‘When you have profited so much that you respect yourself you may let go your tutor.’ Seneca, the influential Roman poet and philosopher, writes this in his Epistulae morales ad Lucilium, XXV, 6.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
The elimination of induction
Only boring people get bored.
Motion Mountain – The Adventure of Physics
Page 86
The strand model will take a long time to get accepted. The first reason is obvious: The
strand model contradicts thinking habits in many research fields. Researchers working on
the foundations of quantum theory, on general relativity, on cosmic strings, on mathematical physics, on classical and quantum field theory, on polymer physics, on shape
deformations, on quantum gravity, on strings, on the visualization of quantum mechanics, on knot theory, on higher dimensions, on supersymmetry, on the axiomatization
of physics, on group theory, on the foundation of physics, on quantum optics and on
particle physics have to give up many life-long thinking habits. So do all other physicists. Strands supersede particles and points.
There is also a second reason for the slow acceptance of the model presented here:
The strand model, in its simplicity, is only a small step away from present research. Many
researchers are finding out how close they have been to the ideas of the strand model,
and for how long they were overlooking or ignoring such a simple option. The simplicity
of the fundamental principle contrasts with the expectation of most researchers, namely
that the final theory is complicated, difficult and hard to discover. In fact, the opposite is
true. Strands are based on Planck units and provide a simple, almost algebraic description
of nature.
In summary, for many researchers and for many physicists, there is a mixture of confusion, anger and disappointment. It will take time before these feelings subside and are
replaced by the fascination provided by the strand model.
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413
“
That which eludes curiosity can be grasped in action.
Traditional saying.
”
Where do we come from? Where does the world come from? What will future bring?
What is death? All these questions are questions about motion – and its meaning. To
all such questions, the strand model does not provide answers. We are a collection of
tangled strands. We are everything and nothing. The strand(s) we are made of will con-
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
What is still hidden?
Motion Mountain – The Adventure of Physics
importance for general discussions on man’s grasp of nature.
In physics, as in the other natural sciences, there is a tradition to state that a certain
description of nature – once confusingly called a ‘law’ – is valid in all cases. In these
statements, ‘all’ means ‘for all values of the quantities appearing’. As a concrete example,
the ‘law’ of universal gravitation is always claimed to be the same here and today, as well
as at all other places and times, such as on the other end of the universe and in a few
thousand years. The full list of such all-claims is part of the millennium list of open
issues in twentieth-century physics. For many decades, the habit of claiming general
validity from a limited and finite number of experiences, also called induction, has been
seen, and rightly so, as a logically dubious manoeuvre, tolerated only because it works.
But the developments described in this text show that this method is indeed justified.
First of all, a claim of generality is not that enormous as it may seem, because the
number of events that can be distinguished in nature is finite, not infinite. The preceding
sections showed that the maximal number ? of events that can be distinguished in the
universe is of the order of ? = (?0 /?Pl)4 = 10244±2, ?0 being the age of the universe and
?Pl the Planck time. This is a big, but certainly finite number.
The unified description of nature has thus first reduced the various all-claims from
an apparently infinite to a finite number of cases, though still involving astronomically
large numbers. This reduction results from the recognition that infinities do not appear
in the description of nature. We now know that when talking about nature, ‘all’ cases
never means an infinite number.
A second, important result is achieved by the description of nature with strands. In
any all-claim about fundamental motion, the checking of each of the large number of
possibilities is not necessary any more, because all events result from a single entity, in
which we introduce distinctions with our senses and our brain. And the distinctions
we introduce imply automatically that the symmetries of nature – the ‘all-claims’ or
‘inductions’ – that are used in the description of motion are correct. Nature does not
contain separate parts. Therefore, there is no way that separate parts can behave differently. Induction is a consequence of the unity of nature.
Ultimately, the possibility to verify statements of nature is due to the fact that all the
aspects of our experience are related. Complete separation is impossible in nature. The
verification of all-claims is possible because the strand model achieves the full description of how all ‘parts’ of nature are related.
The strand model shows that we can talk and think about nature because we are a part
of it. The strand model also shows that induction works because everything in nature is
related to everything else: nature is one.
414
Challenge 229 s
Vol. I, page 16
14 the top of m otion m ountain
tinue to fluctuate. Birth, life and death are aspects of tangled strands. The universe is a
folded strand that grows in complexity.
Obviously, abstract statements about tangles do not help in any human quest. Indeed, we aimed at achieving a precise description of moving particles and bending space.
Studying them was a sequence of riddles; but solving these riddles does not provide
meaning, not even at the top of Motion Mountain. From the top we cannot see the
evolution of complicated systems; in particular, we cannot see or describe the evolution
of life, the biological evolution of species, or the growth of a human beings. Nor can we
understand why we are climbing at all.
In short, from the top of Motion Mountain we cannot see the details down in the
valleys of human relations or experience; strands do not provide advice or meaning. Remaining too long on the top is of no use. To find meaning, we have to descend back
down to real life.
“
Ref. 270
”
Enjoying life and giving it meaning requires to descend from the top of Motion Mountain. The return path can take various different directions. From a mountain, the most
beautiful and direct descent might be the use of a paraglider. After our adventure, we
take an equally beautiful way: we leave reality.
The usual trail to study motion, also the one of this text, starts from our ability to talk
about nature to somebody else. From this ability we deduced our description of nature,
starting from Galilean physics and ending with the strand model. The same results can
be found by requiring to be able to talk about nature to ourselves. Talking to oneself
is an example of thinking. We should therefore be able to derive all physics from René
Descartes’ sentence ‘je pense, donc je suis’ – which he translated into Latin as ‘cogito
ergo sum’. Descartes stressed that this is the only statement of which he is completely
sure, in opposition to his observations, of which he is not. He had collected numerous
examples in which the senses provide unreliable information.
However, when talking to ourselves, we can make more mistakes than when asking
for checks from others. Let us approach this issue in a radically different way. We directly
proceed to that situation in which the highest freedom is available and the largest number
of mistakes are possible: the world of dreams. If nature would only be a dream, could we
deduce from it the complete set of physical knowledge? Let us explore the issue.
— Dreaming implies the use of distinctions, of memory and of sight. Dreams contain
parts and motion.
— Independently on whether dreams are due to previous observations or to fantasies,
through memory we can define a sequence among them. The order relation is called
time. The dream aspects being ordered are called events. The set of all (dream) events
forms the (dream) world.
— In a dream we can have several independent experiences at the same time, e.g. about
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 269
I hate reality. But it is the only place where one can
get a good steak.
Woody Allen
Motion Mountain – The Adventure of Physics
a return path: je rêve, donc je suis
a return path: j e rêv e, d onc j e suis
415
thirst and about hunger. Sequences thus do not provide a complete classification of
experiences. We call the necessary additional distinction space. Dream space has
three dimensions.* Dreaming thus means to use space and time.
— We can distinguish between dream contents. Distinguishing means that we can
count items in dreams. Counting means that we have a way to define measurements.
Dreams are thus characterized by something which we can call ‘observables’. Dream
experiences at a given instant of time are characterized by a state.
— Because we can describe dreams, the dream contents exist independently of dream
time. We can also imagine the same dream contents at different places and different
times in the dream space. There is thus an invariance of dream concepts in space and
time. There are thus symmetries in dream space.
— Dream contents can interact. Dreams appear to vary without end. Dreams seem to
be infinite.
Challenge 230 s
— Dreams show that space can warp.
— Dream motion, as you may want to check, shows a maximum speed.
— Dreams show a strange limit in distance. There is a boundary to our field of vision,
even though we do not manage to see it.
— Both the number of items we can dream of at the same time and the memory of
previous dreams is finite.
— Dreams have colours.
— There are pixels in dreams, though we do not experience them directly. But we can
do so indirectly: The existence of a highest number of things we can dream of at the
same time implies that dream space has a smallest scale.
In summary, the world of dreams has something similar to a minimum change. The
world of dreams and that of films is described by a simple form of quantum theory. The
difference with nature is that in dreams and films, space is discrete from the outset. But
there is still more to say about dreams.
— There is no way to say that dream images are made of mathematical points, as there
is nothing smaller than pixels.
— In dreams, we cannot clearly distinguish objects (‘matter’) and environment
(‘space’); they often mix.
* Though a few mathematicians state that they can think in more than three spatial dimensions, all of them
dream in three dimensions.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Pondering these issues shows that there are limits to dreams. In summary, the world of
dreams has a maximum size, a maximum speed and three dimensions that can warp.
The world of dreams and of films is described by a simple form of general relativity.
Motion Mountain – The Adventure of Physics
In other words, a large part of the world of dreams is described by a modified form
of Galilean physics. We note that the biggest difference between dreams and nature is
the lack of conservation. In dreams, observations can appear, disappear, start and stop.
We also note that instead of dreams, we could equally explore cinema films. Films, like
dreams, are described by a modified form of Galilean physics. And films, like dreams,
do not follow conservation laws. But dreams teach us much more.
416
—
—
—
—
Challenge 231 s
14 the top of m otion m ountain
In dreams, fluctuations appear both for images as well as for the background.
In dreams, sharp distinctions are impossible. Dream space-time cannot be a set.
Dream motion appears when approximate conservation (over time) is observed.
In dreams, dimensionality at small distances is not clear; two and three dimensions
are mixed up there.
In summary, the world of dreams seems to behave as if points and point particles do not
exist; and since quantum theory and general relativity hold, the world of dreams seems
to be described by extended constituents! We thus conclude this short exploration of
the physics of dreams with a fascinating conjecture: even if nature would be a dream, an
illusion or a fantasy, we might still get most of the results that we discovered in our ascent
of Motion Mountain. (What differences with modern physics would be left?) Speaking
with tongue in cheek, the fear of our own faults of judgement, so rightly underlined by
Descartes and many others after him, might not apply to fundamental physics.
Ref. 5
⊳ The fine structure constant describes the probability that a fluctuation adds
a twist to the chiral tangles of electrically charged particles.
We have not yet deduced an accurate value for the fine structure constant, but we seem
to have found out how to do so.
In short, we seem to glimpse the origin of all colours – and thus of all beauty around
us. Strands provide a beautiful explanation for beauty.
summary: what is motion?
“
”
Deep rest is motion in itself. Its motion rests in itself.
Lao Tse, Tao Te King, VI, as translated by Walter
Jerven.
We can now answer the question that drove us through our adventure:
⊳ Motion is the observation of crossing switches of the one, unobservable,
tangled and fluctuating strand that describes all of nature.
Nature’s strand forms particles, horizons and space-time: these are the parts of nature.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
All colours around us are determined by the fine structure constant ? – the coupling
constant for the electromagnetic interaction at low energy – with its measured value
1/137.035 999 139(31). The fine structure constant is also essential to describe most
everyday devices and machines, as well as all human thoughts and movements. The
constant is an aspect of every electric charge in nature.
The strand model showed us that electrical charge is a property of tangles of strands.
In particular, the strand model showed:
Motion Mountain – The Adventure of Physics
what is the origin of colours?
sum m ary: w hat is m otion?
“
All the great things that have happened in the world first took place in a
person’s imagination, and how tomorrow’s world will look like will largely
depend on the power of imagination of those who are just learning to read right
now.
Astrid Lindgren*
* Astrid Lindgren (b. 1907 Näs, d. 2002 Stockholm) was a beloved writer of children books.
”
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 271
Particles are tangles of strands; horizons and space-time are weaves of strands. The parts
of nature move. The parts move because their strands fluctuate.
Motion appears because all parts in nature are approximate. Indeed, the observation
of crossing switches and the description of strand segments fluctuating in a background
space result and are possible because we approximate from the one strand that makes up
nature to the many parts inside nature. The one strand (approximately) forms the many
elementary particles inside us. Strand segments and particles (approximately) lead us
to introduce background space, matter and radiation. Introducing background space
implies observing motion. Motion thus appears automatically when approximate parts
of nature, such as humans, animals or machines, describe other approximate parts of
nature, such as other bodies or systems.
The observation of motion is due to our introduction of the plural. Motion results from
of our forced use of many (approximate) parts to describe the unity of nature. The observation of motion results from approximations. All these approximate distinctions are
unavoidable and are due to the limitations of our experience.
Motion appears as soon as we divide the world into parts and then follow these parts.
Dividing nature into parts is not a conscious act; our human nature – our senses and
our brain – force us to perform it. And whenever we experience or talk about parts of
the universe, we find motion. Our senses and our brain are made to distinguish and to
divide – and cannot do otherwise. We need to distinguish in order to survive, to think
and to enjoy life. In a sense, we can say that motion appears as a logical consequence
of our limitations; the fundamental limitation is the one that makes us distinguish and
introduce parts, including points and sets.
Motion is an ‘artefact’ of locality. Locality is an approximation and is due to our human nature. Distinction, localization and motion are inextricably linked.
Motion is low energy concept. Motion does not exist at Planck scales, i.e., at the limits
of nature.
Motion is an artefact due to our limitations. This conclusion resembles what Zeno of
Elea stated 2500 years ago, that motion is an illusion. But in contrast to Zeno’s pessimistic view, we now have a fascinating spectrum of results and tools at our disposition:
they allow us to describe motion and nature with high precision. Most of all, these tools
allow us to change ourselves and our environment for the better.
Motion Mountain – The Adventure of Physics
Vol. I, page 15
417
PO ST FACE
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Plato’s Phaedrus, written around 380 b ce, is available in many pocket editions. Do not waste
your time learning ancient Greek to read it; the translated versions are as beautiful as the original.
Plato’s lifelong avoidance of the natural sciences had two reasons. First of all, he was jealous
of Democritus. Plato never even cites Democritus in his texts. Democritus was the most prolific,
daring, admired and successful philosopher of his time (and maybe of all times). Democritus
was a keen student of nature. His written works did not survive, because his studies were not
congenial to the followers of christianity, and thus they were not copied by the monks in the
Middle Ages. The loss of these texts is related to the second reason that kept Plato away from
the natural sciences: he wanted to save his life. Plato had learned one thing from men in the
town: talking about nature is dangerous. Starting around his lifetime, for over 2000 years people
practising the natural sciences were regularly condemned to exile or to death for impiety. Fortunately, this is only rarely the case today. But such violence still occurs, and we can honour the
dangers that those preceding us had to overcome in order to allow us enjoying this adventure.
Motion Mountain – The Adventure of Physics
Perhaps once you will read Plato’s Phaedrus, one of the beautiful philosophical Greek
texts. In it, Socrates is made to say that he almost never left the city walls because to him,
as a ‘lover of learning, trees and the open country do not teach anything, whereas men
in the town do.’ This is a veiled critique of Democritus, the most important and famous
philosopher in Greece during Plato’s time. Democritus was the natural philosopher par
excellence, and arguably had learned from nature – with its trees and open country –
more than anybody else after him.
After this adventure you can decide for yourself which of these two approaches is more
congenial to you. It might be useful to know that Aristotle refused to choose and cultivated them both. There is no alternative in life to following one’s own mind, and to enjoy
doing so. If you enjoyed this particular trip, show it to your friends. For yourself, after
this walk, sense intensively the pleasure of having accomplished something important.
Many before you did not have the occasion. Enjoy the beauty of the view offered. Enjoy
the vastness of horizon it provides. Enjoy the impressions that it creates inside you. Collect them and rest. You will have a treasure that will be useful in many occasions. Then,
when you feel the desire of going further, get ready for another of the adventures life has
to offer.
Appendix A
KNOT AND TANGLE GEOMET RY
The following table provides a terse summary of the mathematics of knot shapes.
TA B L E 19 Important properties of knot, links and tangles.
Defining property
Normal vector or
curvature vector
local vector normal to the curve, in
direction of the centre of the
‘touching’ circle, with length given by
the curvature
Binormal vector
local unit vector normal to the
tangent and to the normal/curvature
vector
local speed of rotation of the
binormal vector; positive (negative)
for right-handed (left-handed) helix
Torsion
Frenet frame at a
curve point
ropelength is integral of
arclength; ropelength is
shape-dependent.
at present, all non-trivial ideal
shapes are only known
approximately; most ideal knots
(almost surely) have kinks.
measures departure from
straightness, i.e., local bending of
a curve.
is given by the second and first
derivatives of the curve.
measures departure from
flatness, i.e., local twisting or
local handedness of a curve;
essentially a third derivative of
the curve.
‘natural’ local orthogonal frame of
the Frenet frame differs at each
reference defined by unit tangent, unit curve point, the Frenet frame is
normal/curvature and binormal
not uniquely defined if the curve
vector
is locally straight.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Knot / link / tangle one closed / several closed / one or
several open curves, all in 3d and
without intersections
Ideal knot, link,
tightest possible knot, link or tangle
tangle (shape)
(shape) assuming a rope of constant
diameter that is infinitely flexible and
infinitely slippery
Ribbon or framing short perpendicular (or non-tangent)
vector attached at each point of a
curve
Curvature of a
inverse curvature radius of ‘touching’
curve
circle
O t h e r p r o pe rt i es
Motion Mountain – The Adventure of Physics
Concept
420
a knot g eom etry
TA B L E 19 (Continued) Important properties of knot, links and tangles.
Concept
Defining property
O t h e r p r o pe rt i es
‘Natural’ framing
or Frenet ribbon
defined by the local normal, i.e., local
curvature vector
for a closed curve, it is always
closed and two-sided, and thus
never a Moebius band.
Linking number
sloppily, number of times that two
topological invariant, i.e.,
between two closed curves wind around each other, or,
shape-independent;
curves
equivalently, half the number of times Lk(?1, ?2) =
? (d? ×d? )
1
that the curves ‘swap’ position
∮ ∮ 12 ?31 2 .
4π ?2 ?1
12
Twist of a ribbon,
open or closed
Tw(?) is the total angle, in units of 2π,
by which the ribbon rotates around
the central axis of the ribbon; sloppily
said, it measures the local helicity; this
type of twist has no relation to the
first Reidemeister move
Tw(?) is the total angle, in units of
2π, by which the Frenet frame rotates
around the tangent direction, or
equivalently, (total) twist of the Frenet
ribbon, also called the total torsion of
the curve; this type of twist has no
relation to the first Reidemeister move
Twist of a curve or
knot
Signed crossing
number
number of times that the edges of the
natural/Frenet ribbon wind around
each other
generalization of the linking number
for knots to open curves
sum of positive minus sum of
negative crossings in a given oriented
2d projection of a curve or knot
(sometimes called ‘2d-writhe’)
2d-writhe of a knot, signed crossing number for a minimal
or topological
crossing number diagram/projection
(sometimes the term ‘2d-writhe’ is
writhe, or Tait
number
used for the signed crossing number
of any configuration)
topological invariant, i.e.,
shape-independent; always an
integer.
not a topological invariant,
because of existence of inflection
points.
usually not an integer.
vanishes for ribbons that are
everywhere flat.
not an integer even in case of
knots; depends on curve/knot
shape; is different from zero for
chiral curves/knots; is zero for
achiral curves/knots that have a
rigid reflective symmetry; twist
and torsion are only equal if the
twist is defined with the Frenet
ribbon – with other framings
they differ.
always an integer; depends on
shape.
is shape-invariant; is always an
integer; differs from 0 for all
chiral knots; has the value 3 for
the trefoil, 0 for the figure-eight
knot, 5 for the 51 and 52 knots, 2
for the 61 knot, 7 for the 71 and
72 knots, 4 for the 81 knot, and 9
for the 92 knot.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
number of times that the edges wind
around each other
Motion Mountain – The Adventure of Physics
Linking number
for a closed
two-sided ribbon
Self-linking number
or ‘natural’ linking
number for a knot
Link integral for an
open curve
knot g eom etry
421
TA B L E 19 (Continued) Important properties of knot, links and tangles.
Concept
Defining property
Writhing number or Wr(?) is the average, over all
3d-writhe of a knot projection directions, of the signed
crossing number; sloppily said, it
measures how wrapped, coiled and
chiral a knot is, i.e., it measures the
global helicity
Writhe of a ribbon
the value is quasi-quantized for
alternating knots with small crossing
numbers (< 11) in values that differ
from ?4/7 by only a few per cent
the value is quasi-quantized for
alternating links with small crossing
numbers (< 11) in values that differ
from 2/7 + ?4/7 by only a few per
cent
sloppily said, measures how wrapped,
coiled and chiral a ribbon is, i.e.,
measures its global helicity
vanishes for plane curves.
for any knot ? and any ribbon ?
attached to it,
Lk(?, ?) = Tw(?, ?) + Wr(?)
for applying the theorem to open
curves, a (standardized) closing
of curves is required.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Writhe of an open
curve
Calugareanu’s
theorem
depends on knot shape; usually
is not an integer; is different
from zero for chiral knots; is
zero for achiral knots that have a
rigid reflective symmetry;
1
∮? ∮? ?12 (d??31 ×d?2) ;
Wr(?) = 4π
12
uses no ribbon and thus is
independent of the ribbon shape
attached to the knot.
is additive under knot addition
for knots with small crossing
numbers (< 11) within less than
1 %.
Motion Mountain – The Adventure of Physics
Writhe of ideal,
alternating knots
and of
odd-component
links
Writhe of ideal,
alternating
even-component
links
O t h e r p r o pe rt i es
CHALLENGE HINT S AND SOLUT IONS
Challenge 2, page 28: Take Δ? Δ? ⩾ 1 and substitute Δ? = ?/Δ? and Δ? = ?/Δ?.
Challenge 16, page 43: Yes. But we can also argue its opposite, namely that matter appears when
space is compressed too much. Both viewpoints are correct.
Challenge 22, page 46: The strictest upper limits are those with the smallest exponent for length,
Challenge 24, page 48: To my knowledge, no such limits have been published. Do it yourself!
Challenge 25, page 48: The system limits cannot be chosen in other ways; after the limits have
been corrected, the limits given here should still apply.
Challenge 28, page 49: Just insert numbers to check this.
Challenge 30, page 50: No.
Challenge 31, page 51: This is a trick question due to two issues. First, is the cosmological con-
Challenge 33, page 52: If you ever write such a table, publish it and send me a copy. I will include
it in the text.
Challenge 36, page 65: Sloppily speaking, such a clock is not able to move its hands in a way that
guarantees precise time reading.
Challenge 40, page 82: The final energy ? produced by a proton accelerator increases with its
radius ? roughly as ? ∼ ?1.2 ; as an example, CERN’s LHC achieved about 13 TeV for a radius of
4.3 km. Thus we would get a radius of more than 100 light years for a Planck energy accelerator.
Building an accelerator achieving Planck energy is impossible.
Nature has no accelerator of this power, but gets near it. The maximum measured value of
cosmic rays, 1022 eV, is about one millionth of the Planck energy. The mechanism of acceleration
is still obscure. Neither black holes nor the cosmic horizon seem to be sources, for some yet
unclear reasons. This issue is still a topic of research.
Challenge 41, page 83: The Planck energy is ?Pl = √ℏ?5 /? = 2.0 GJ. Car fuel delivers about
43 MJ/kg. Thus the Planck energy corresponds to the energy of 47 kg of car fuel, about a tankful.
Challenge 42, page 83: Not really, as the mass error is equal to the mass only in the Planck case.
Challenge 43, page 83: It is improbable that such deviations can be found, as they are masked by
Page 279
the appearance of quantum gravity effects. However, if you do think that you have a prediction
for a deviation, publish it, and send the author an email.
Challenge 44, page 83: The minimum measurable distance is the same for single particles and
systems of particles.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
stant the same for all observers in the universe that are, like ourselves, more or less at rest with
respect to the background radiation? Most researchers would agree that this is the case. Secondly,
is the cosmological constant the same for extremely rapid observers, observers that move at extremely high energy with respect to the background radiation? Enjoy finding out.
Motion Mountain – The Adventure of Physics
and the strictest lower limits are those with the largest exponent of length.
challenge hints and solu tions
423
Challenge 45, page 84: There is no gravitation at those energies and there are no particles. There
Vol. I, page 259
Challenge 70, page 104: You will not find one.
Challenge 72, page 105: If you find one, publish it, and send the author an email.
Challenge 74, page 106: For the description of nature this is a contradiction. Nevertheless, the
Vol. III, page 323
Challenge 84, page 110: Plotinus in the Enneads has defined ‘god’ in exactly this way. Later,
Augustine in De Trinitate and in several other texts, and many subsequent theologians have taken
up this view. (See also Thomas Aquinas, Summa contra gentiles, 1, 30.) The idea they propose is
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Vol. II, page 258
term ‘universe’, ‘set of all sets’ and other mathematical terms, as well as many religious concepts
are of this type.
Challenge 75, page 107: No, for the reasons mentioned earlier on: fundamental measurement
errors for horizon measurements, as well as many other effects, prevent this. The speculation is
another example of misguided fantasy about extremal identity.
Challenge 76, page 108: The physical concepts most related to ‘monad’ are ‘strand’ and
‘universe’, as shown in the second half of this text.
Challenge 77, page 108: The macroscopic content of the universe may be observer-dependent.
But to speak about many universes (Many ‘everythings’?) or a ‘multiverse’ (What is more than
everything? Why only one multiverse?) is pure nonsense.
Challenge 80, page 108: True only if it were possible to do this. Because particles and space are
indistinguishable, removing particles means to remove everything. (The strand model visualizes
this connection most clearly.)
Challenge 82, page 108: True. Existence is the ability to interact. If the ability disappears, existence disappears. In other words, ‘existence’ is a low-energy concept.
Challenge 83, page 110: If you find a sensible statement about the universe, publish it! And send
it to the author as well. The next challenge shows one reason why this issue is interesting. In
addition, such a statement would contradict the conclusions on the combined effects of general
relativity and quantum theory.
Motion Mountain – The Adventure of Physics
is thus no paradox.
Challenge 46, page 84: The issue is still being debated; a good candidate for a minimum momentum of a single particle is given by ℏ/?, where ? is the radius of the universe. Is this answer
satisfying?
Challenge 47, page 85: All mentioned options could be valid at the same time. The issue is not
closed and clear thinking about it is not easy.
Challenge 48, page 85: The precise energy scale is not clear. The scale is either the Planck energy or within a few orders of magnitude from it; the lowest possible energy is thus around a
thousandth of the Planck energy.
Challenge 50, page 87: If you can think of an experiment, publish the proposal, and send the
author an email.
Challenge 51, page 90: The table of aggregates shows this clearly.
Challenge 52, page 91: The cosmic background radiation is a clock in the widest sense of the
term.
Challenge 53, page 92: The way to deduce cosmological limits is presented in detail in the section starting on page 45.
Challenge 64, page 100: Also measurement errors at Planck scales prevent the determination of
topology at those scales.
Challenge 66, page 102: The measurement error is as large as the measurement result.
424
Challenge 232 e
simple: it is possible to clearly say what ‘god’ is not, but it is impossible to say what ‘god’ is.
This statement is also part of the official Roman Catholic Catechism: see part one, section one,
chapter one, IV, 43, found at www.vatican.va/archive/ENG0015/__PC.HTM. Similar statements
are found in Judaism, Hinduism and Buddhism.
In other terms, theologians admit that ‘god’ cannot be defined, that the term has no properties or content, and that therefore the term cannot be used in any positive sentence. The aspects
common to ‘universe’ and to ‘god’ suggest the conclusion that both are the same. Indeed, the
analogy between the two concepts can be expanded to a proof: both concepts have the same
content, the same boundary, and the same domain of application. (This is an intriguing and
fascinating exercise.) In fact, this might be the most interesting of all proofs of the existence of
‘god’, as it lacks all the problems that the more common ‘proofs’ have. Despite its interest, this
proof of equivalence is not found in any book on the topic yet. The reason is twofold. First, the
results of modern physics – showing that the concept of universe has all these strange properties
– are not common knowledge yet. Secondly, the result of the proof, the identity of ‘god’ and
‘universe’ – also called pantheism – is a heresy for most religions. It is an irony that the catholic
catechism, together with modern physics, can be used to show that pantheism is correct, because
any catholic who defends pantheism (or other heresies following from modern physics) incurs
automatic excommunication, latae sententiae, without any need for a formal procedure.
If one is ready to explore the identity of universe and ‘god’, one finds that a statement like
‘god created the universe’ translates as ‘the universe implies the universe’. The original statement
is thus not a lie any more, but is promoted to a tautology. Similar changes appear for many
other – but not all – statements using the term ‘god’. (The problems with the expression ‘in the
beginning’ remain, though.) In fact, one can argue that statements about ‘god’ are only sensible
and true if they remain sensible and true after the term has been exchanged with ‘universe’.
Enjoy the exploration of such statements.
no such effects exist.
Challenge 88, page 113: In fact, no length below the Planck length itself plays any role in nature.
Challenge 90, page 114: You need quantum humour, because the result obviously contradicts a
previous one given on page 93 that includes general relativity.
Challenge 93, page 122: The number of spatial dimensions must be given first, in order to talk
about spheres.
Challenge 94, page 126: This is a challenge to you to find out. It is fun, it may yield a result in
contradiction with the arguments given so far (publish it in this case), or it may yield an independent check of the results of the section.
Challenge 96, page 130: This issue is open and still a subject of research. The conjecture of the
author is that the answer is negative. If you find an alternative, publish it, and send the author
an email.
Challenge 98, page 135: The lid of a box must obey the indeterminacy relation. It cannot be at
perfect rest with respect to the rest of the box.
Challenge 100, page 136: No, because the cosmic background is not a Planck scale effect, but an
effect of much lower energy.
Challenge 101, page 136: Yes, at Planck scales all interactions are strand deformations; therefore
collisions and gravity are indistinguishable there.
Challenge 102, page 136: No. Time is continuous only if either quantum theory and point
particles or general relativity and point masses are assumed. The argument shows that only the
combination of both theories with continuity is impossible.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Challenge 86, page 112: If you find one, publish it and send it also to me. The conjecture is that
Motion Mountain – The Adventure of Physics
Challenge 233 e
challenge hints and solu tions
challenge hints and solu tions
425
Challenge 103, page 136: You should, because at Planck scales nature’s inherent measurement
errors cannot clearly distinguish between different measurement results.
Challenge 104, page 136: We still have the chance to find the best approximate concepts pos-
sible. There is no reason to give up.
Challenge 105, page 136: Here are a few thoughts. A beginning of the big bang does not exist;
something similar is given by that piece of continuous entity which is encountered when going
backwards in time as much as possible. This has several implications.
In summary, no starting point of the big bang exists, because time does not exist there. For the
same reason, no initial conditions for particles or space-time exist. In addition, this shows that
the big bang involved no creation, because without time and without possibility of choice, the
term ‘creation’ makes no sense.
Challenge 106, page 136: The equivalence follows from the fact that all these processes require
Planck energy, Planck measurement precision, Planck curvature, and Planck shutter time.
Page 366
Challenge 107, page 136: No, as explained later on in the text.
Challenge 108, page 137: Probably there is nothing wrong with the argument. For example, in
Challenge 109, page 137: If not, force yourself. Brainstorming is important in life, as is the sub-
sequent step: the checking of the speculations.
Challenge 114, page 149: The author would like to receive a mail on your reasons for disagree-
ment.
Challenge 115, page 151: Let the author know if you succeed. And publish the results.
Challenge 116, page 151: Energy is action per time. Now, the Planck constant is the unit of ac-
tion, and is defined by a crossing switch. A system that continuously produces a crossing switch
for every Planck time running by thus has Planck energy. An example would be a tangle that is
rotating extremely rapidly, once per Planck time, producing a crossing switch for every turn.
Momentum is action per length. A system that continuously produces a crossing switch
whenever it advances by a Planck length has Planck momentum. An example would be a tangle
configuration that lets a switch hop from one strand to the next under tight strand packing.
Force is action per length and time. A system that continuously produces a crossing switch
for every Planck time that passes by and for every Planck length it advances through exerts a
Planck force. A tangle with the structure of a screw that rotates and advances with sufficient
speed would be an example.
Challenge 120, page 160: Yes; the appearance of a crossing does not depend on distance or on
the number of strands in between.
Challenge 121, page 160: No; more than three dimensions do not allow us to define a crossing
switch.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
the strand model, all observables are composed of fundamental events, and so, in some way, all
observables are fundamentally indistinguishable.
Motion Mountain – The Adventure of Physics
— Going backwards in time as far as possible – towards the ‘beginning’ of time – is the same as
zooming to smallest distances: we find a single strand of the amoeba.
— In other words, we speculate that the whole world is one single piece, fluctuating, and possibly
tangled, knotted or branched.
— Going far away into space – to the border of the universe – is like taking a snapshot with a
short shutter time: strands everywhere.
— Whenever we sloppily say that extended entities are ‘infinite’ in size, we only mean that they
reach the horizon of the universe.
426
challenge hints and solu tions
Challenge 122, page 160: If so, let the author know. If the generalization is genuine, the strand
model is not correct.
Challenge 134, page 188: The magnitude at a point should be related to the vectorial sum of all
inverse shortest crossing distances at that point.
Challenge 140, page 199: This algebraic transformation is shown in all textbooks that treat the
Pauli equation. It can also be checked by writing the two equations out component by component.
Challenge 143, page 218: Yes, as can easily be checked by rereading the definitions with the
spinor tangle description in mind.
Challenge 146, page 219: No contradiction is known.
Challenge 147, page 219: In the relativistic case, local space curvature is also taken into account.
Challenge 149, page 219: Find out, publish the result, and let the author know.
Challenge 150, page 219: If the strand interpenetration is allowed generally, quantum theory is
Challenge 152, page 219: The belt trick would imply that a wheel rolls over its own blood supply
at every second rotation.
Challenge 161, page 242: If you find one, publish it!
Challenge 172, page 272: No slide is possible, thus no crossing change appears; thus the situ-
ation has no observable effects. If we deform one slide before the slide – which is possible – we
get back the situation already discussed above.
proof is still missing. The same is expected for the Haag–Kastler axioms.
Challenge 187, page 301: A black hole has at least one crossing, thus at least a Planck mass.
Challenge 190, page 308: The present consensus is no.
Challenge 196, page 325: These tangles are not rational. In the renewed strand model of 2015,
they cannot form; they are not allowed and do not represent any particle.
Challenge 198, page 340: Such a tangle is composed of several gravitons.
Challenge 199, page 346: Tail braiding leads to tangledness, which in turn is the basis for core
rotation. And core rotation is kinetic energy, not rest mass.
Challenge 201, page 347: The issue is topic of research; for symmetry reasons it seems that a
state in which each of the six quarks has the same bound to the other five quarks cannot exist.
Challenge 209, page 369: If you find such an estimate, publish it and send it to the author. A
really good estimate also answers the following question: why does particle mass increase with
core complexity? A tangle with a complex core, i.e., with a core of large ropelength, has a large
mass value. Any correct estimate of the mass must yield this property. But a more complex knot
will have a smaller probability for the belt trick. We seem to be forced to conclude that particle
mass is not due to the belt trick alone.
Challenge 213, page 370: Probably not.
Challenge 214, page 370: Probably not.
Challenge 215, page 370: Probably not.
Challenge 216, page 370: Find out – and let the author know.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Challenge 177, page 273: For the Wightman axioms, this seems to be the case; however, a formal
Motion Mountain – The Adventure of Physics
impossible to derive, as the spinor behaviour would not be possible. If strand interpenetration
were allowed only under certain conditions (such as only for a strand with itself, but not among
two different strands), quantum theory might still possible. A similar process lies at the basis of
mass generation, as shown in the section on the weak interaction.
challenge hints and solu tions
427
Challenge 218, page 374: This would be an interesting result worth a publication.
Challenge 221, page 382: If you plan such a calculation, the author would be delighted to help.
Challenge 225, page 391: Take up the challenge!
Challenge 227, page 406: There is a good chance, however, that such alternatives can be elimin-
ated rather quickly. If you cannot do so, do publish the argument, and let the author know.
Challenge 229, page 414: Nobody can really answer ‘why’-questions about human actions.
Climbing, like every other passion, is also a symbolic activity. Climbing can be a search for
adventure, for meaning, for our mother or father, for ourselves, for happiness, or for peace.
Challenge 230, page 415: Also in dreams, speeds can be compared; and also in dreams, a kind
of causality holds (though not a trivial one). Thus there is an invariant and therefore a maximum
speed.
Challenge 231, page 416: Probably none. The answer depends on whether the existence of
Motion Mountain – The Adventure of Physics
strands can be deduced from dreams. If strands can be deduced from dreams, all of physics
follows. The conjecture is that this deduction is possible. If you find an argument against or in
favour of this conjecture, let the author know.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
BIBLIO GR APHY
“
1
”
See the first volume of the Motion Mountain series, Fall, Flow and Heat, available as free
download at www.motionmountain.net. Cited on pages 17 and 401.
See the second volume of the Motion Mountain series, Relativity, available as free download
at www.motionmountain.net. Cited on pages 17, 18, 401, and 430.
See the third volume of the Motion Mountain series, Light, Charges and Brains, available
as free download at www.motionmountain.net, as well as the mentioned fourth and fifth
volumes. Cited on pages 17 and 401.
4
See the fourth and fifth volumes of the Motion Mountain series, The Quantum of
Change and Pleasure, Technology and the Stars, available as free download at www.
motionmountain.net. Cited on pages 18, 401, 402, and 429.
5
The most precise value of the fine structure constant is determined from a weighted
world average of high-precision measurements by a special international scientific
committee called CODATA. Its website is www.codata.org/committees-and-groups/
fundamental-physical-constants. The site also provides the latest official publication
with the values of the fundamental constants. The most recent value of the fine structure
constant is published at physics.nist.gov/cgi-bin/cuu/Value?alphinv and physics.nist.gov/
cgi-bin/cuu/Value?alph. Cited on pages 18, 226, 378, 386, and 416.
6
See for example, the book by Rob ert L aug hlin, A Different Universe: Reinventing Physics from the Botton Down Basic Books, 2005. Of the numerous books that discuss the idea of
a final theory, this is the only one worth reading, and the only one cited in this bibliography.
The opinions of Laughlin are worth pondering. Cited on page 21.
7
Many physicists, including Steven Weinberg, regularly – and incorrectly – claim in interviews that the measurement problem is not solved yet. Cited on page 21.
8
Undocumented sentences to this effect are regularly attributed to Albert Einstein. Because
Einstein was a pantheist, as he often explained, his statements on the ‘mind of god’ are not
really to be taken seriously. They were all made – if at all – in a humorous tone. Cited on
page 21.
9
For an example for the inappropriate fear of unification, see the theatre play Die Physiker
by the Swiss author Friedrich D ürrenmatt. Several other plays and novels took over
this type of disinformation. Cited on page 21.
* This is a statement from the brilliant essay by the influential writer Samuel Johnson, Review of Soame
Jenyns’ “A Free Enquiry Into the Nature and Origin of Evil”, 1757. See www.samueljohnson.com.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
3
Motion Mountain – The Adventure of Physics
2
The only end of writing is to enable the readers
better to enjoy life, or better to endure it.
Samuel Johnson*
b ib liography
11
See e.g. the 1922 lectures by Lorentz at Caltech, published as H. A . L orentz, Problems of
Modern Physics, edited by H. Bateman, Ginn and Company, 1927, page 99. Cited on page
27.
12
Bohr explained the indivisibilty of the quantum of action in his famous Como lecture, printed in N . B ohr, Atomtheorie und Naturbeschreibung, Springer, 1931. It was translated into
English language as N . B ohr, Atomic Theory and the Description of Nature, Cambridge
University Press, 1934. More statements about the indivisibility of the quantum of action
can be found in N . B ohr, Atomic Physics and Human Knowledge, Science Editions, New
York, 1961. For summaries of Bohr’s ideas by others see Max Ja m mer, The Philosophy
of Quantum Mechanics, Wiley, first edition, 1974, pp. 90–91, and John Honner, The Description of Nature – Niels Bohr and the Philosophy of Quantum Physics, Clarendon Press,
1987, p. 104. Cited on page 28.
13
For an overview of the quantum of action as a basis of quantum theory, see the first chapter
of the fourth volume of the Motion Mountain series, Ref. 4. Cited on page 29.
14
An overview of EBK quantization can be found in the volume on quantum theory. Cited
on page 29.
15
Minimal entropy is discussed by L. Sz il ard, Über die Entropieverminderung in einem
thermodynamischen System bei Eingriffen intelligenter Wesen, Zeitschrift für Physik 53,
pp. 840–856, 1929. This classic paper can also be found in English translation in his collected works. Cited on page 30.
16
See for example A . E . Sha lyt-Margolin & A . Ya . Treg ubovich, Generalized
uncertainty relation in thermodynamics, preprint at arxiv.org/abs/gr-qc/0307018, or
J. Uffink & J. va n L ith- van D is, Thermodynamic uncertainty relations, Foundations
of Physics 29, pp. 655–692, 1999. Cited on page 30.
17
See also the fundamental paper by A . D iSessa, Momentum flow as an alternative perspective in elementary mechanics, 48, p. 365, 1980, and A . D iSessa, Erratum: “Momentum
flow as an alternative perspective in elementary mechanics” [Am. J. Phys. 48, 365 (1980)], 48,
p. 784, 1980. Cited on page 31.
18
The observations of black holes at the centre of galaxies and elsewhere are summarised by
R . B l a ndford & N . G ehrels, Revisiting the black hole, Physics Today 52, June 1999.
Their existence is now well established. Cited on page 31.
19
It seems that the first published statement of the maximum force as a fundamental principle was around the year 2000, in this text, in the chapter on gravitation and relativity. The
author discovered the maximum force principle, not knowing the work of others, when
searching for a way to derive the results of the last part of this adventure that would be so
simple that it would convince even a secondary-school student. In the year 2000, the author
told his friends in Berlin about his didactic approach for general relativity.
The concept of a maximum force was first proposed, most probably, by Venzo de Sabbata
and C. Sivaram in 1993. Also this physics discovery was thus made much too late. In 1995,
Corrado Massa took up the idea. Independently, Ludwik Kostro in 1999, Christoph Schiller
just before 2000 and Gary Gibbons in the years before 2002 arrived at the same concept.
Gary Gibbons was inspired by a book by Oliver Lodge; he explains that the maximum force
value follows from general relativity; he does not make a statement about the converse, nor
do the other authors. The statement of maximum force as a fundamental principle seems
original to Christoph Schiller.
Page 56
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Exploring the spirit of play is the subject of research of the famous National Institute for
Play, founded by Stuart Brown, and found at www.nifplay.org. Cited on page 21.
Motion Mountain – The Adventure of Physics
10
Vol. IV, page 15
Vol. IV, page 182
429
430
b ib lio graphy
Vol. II, page 107
Maximal luminosity is often mentioned in connection with gravitational wave detection;
nevertheless, the general power maximum has never been mentioned before. See for
example L. Ju, D. G . B l a ir & C . Z hao, Detection of gravitational waves, Reports on
Progress in Physics 63, pp. 1317–1427, 2000. See also C . W. Misner, K. S. Thorne &
J. A . Wheeler, Gravitation, Freeman, 1973, page 980. Cited on page 32.
21
See for example Wolf g ang R indler, Relativity – Special, General and Cosmological,
Oxford University Press, 2001, p. 70 ss, or R ay d ’ Inverno, Introducing Einstein’s Relativity, Clarendon Press, 1992, p. 36 ss. Cited on page 33.
22
T. Jacob son, Thermodynamics of spacetime: the Einstein equation of state, Physical Review Letters 75, pp. 1260–1263, 1995, preprint at arxiv.org/abs/gr-qc/9504004; this deep
article remains fascinating to this day. Even the author was scared to draw all the possible
conclusions. The general concepts are explained, almost without formulae, in L. Sm olin,
On the nature of quantum fluctuations and their relation to gravitation and the principle of
inertia, Classical and Quantum Gravity 3, pp. 347–359, 1986. Cited on pages 33 and 293.
23
This relation was pointed out by Achim Kempf. The story is told in A . D. Sa kha rov,
General Relativity and Gravitation 32, pp. 365–367, 2000, a reprint of his paper Doklady
Akademii Nauk SSSR 177, pp. 70–71, 1967. Cited on pages 34 and 43.
24
Indeterminacy relations in general relativity are discussed in C . A . Mea d, Possible connection between gravitation and fundamental length, Physical Review B 135, pp. 849–862, 1964.
The generalized indeterminacy relation is implicit on page 852, but the issue is explained
rather unclearly. Probably the author considered the result too simple to be mentioned explicitly. (That paper took 5 years to get published; comments on the story, written 37 years
later, are found at C . A . Mea d, Walking the Planck length through history, Physics Today
54, p. 15 and p. 81, 2001, with a reply by Frank Wilczek.) See also P. K. Tow nsend, Smallscale structure of space-time as the origin of the gravitational constant, Physical Review D 15,
pp. 2795–2801, 1977, or the paper by M. -T. Ja ekel & S. R enaud, Gravitational quantum
limit for length measurement, Physics Letters A 185, pp. 143–148, 1994. Cited on pages 35,
66, 67, 68, 71, and 118.
25
M. Kramer & al., Tests of general relativity from timing the double pulsar, preprint at arxiv.
org/abs/astro-ph/060941. Cited on page 35.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
20
Motion Mountain – The Adventure of Physics
The temporal order of the first papers on maximum force seems to be V. d e Sa b bata
& C . Sivaram, On limiting field strengths in gravitation, Foundations of Physics Letters 6,
pp. 561–570, 1993, followed by C . Massa, Does the gravitational constant increase?, Astrophysics and Space Science 232, pp. 143–148, 1995, and by L. Kostro & B. L a ng e, Is ?4 /?
the greatest possible force in nature?, Physics Essays 12, pp. 182–189, 1999. The next references
are the paper by G . W. G ibb ons, The maximum tension principle in general relativity,
Foundations of Physics 32, pp. 1891–1901, 2002, preprint at arxiv.org/abs/hep-th/0210109 –
though he developed the ideas before that date – and the older versions of the present text,
i.e., C hristoph S chiller, Motion Mountain – The Adventure of Physics, a free pdf available at www.motionmountain.net. Then came C . S chiller, Maximum force and minimum distance: physics in limit statements, preprint at arxiv.org/abs/physics/0309118, and
C . S chiller, General relativity and cosmology derived from principle of maximum power
or force, International Journal of Theoretical Physics 44, pp. 1629–1647, 2005, preprint
at arxiv.org/abs/physics/0607090. See also R . B eig, G . W. G ib b ons & R . M. S choen,
Gravitating opposites attract, Classical and Quantum Gravity 26, p. 225013, 2009. preprint
at arxiv.org/abs/09071103.
A detailed discussion of maximum force and power is given in the volume on general
relativity, Ref. 2. Cited on pages 32, 42, 294, and 440.
b ib liography
431
27
Maximal curvature, as well as area and volume quantization, are discussed in
A . A shtekar, Quantum geometry and gravity: recent advances, preprint at arxiv.org/
abs/gr-qc/0112038 and in A . A shtekar, Quantum geometry in action: big bang and black
holes, preprint at arxiv.org/abs/math-ph/0202008. Cited on pages 37, 75, and 439.
28
Maximons, elementary particles of Planck mass, are discussed by A . D. Sa kha rov, Vacuum quantum fluctuations in curved space and the theory of gravitation, Soviet Physics –
Doklady 12, pp. 1040–1041, 1968. Cited on pages 39, 78, and 121.
29
Wolf g a ng R indler, Relativity – Special, General and Cosmological, Oxford University
Press, 2001, p. 230. Cited on page 41.
30
Several incorrect counterclaims to the entropy limit were made in R . B ousso, The holographic principle, Review of Modern Physics 74, pp. 825–874, 2002, preprint at arxiv.org/
abs/hep-th/0203101. However, this otherwise good review has some errors in its arguments,
as explained on page 146 in volume V. Bousso has changed his position in the meantime;
he now accepts the entropy limit. Cited on pages 43, 47, 437, and 439.
31
Gamma-ray bursts are discussed by G . P reparata, R . Ruffini & S. - S. Xue, The dyadosphere of black holes and gamma-ray bursts, Astronomy and Astrophysics 338, pp. L87–
L90, 1998, and C . L. B ia nco, R . Ruffini & S. - S. Xue, The elementary spike produced
by a pure e+ e− pair-electromagnetic pulse from a black hole: the PEM pulse, Astronomy and
Astrophysics 368, pp. 377–390, 2001. Cited on page 44.
32
See for example the review in C . W.J. B eena kker & al., Quantum transport in semiconductor nanostructures, pp. 1–228, in H. E hrenreich & D. Turnbull editors, Solid
State Physics, volume 44, Academic Press, 1991. Cited on page 44.
33
A discussion of a different electrical indeterminacy relation, between current and charge,
can be found in Y- Q. L i & B. C hen, Quantum theory for mesoscopic electronic circuits and
its applications, preprint at arxiv.org/abs/cond-mat/9907171. Cited on page 44.
34
Ha ns C . Oha nian & R em o Ruffini, Gravitation and Spacetime, W.W. Norton & Co.,
1994. Cited on pages 45 and 444.
35
The entropy limit for black holes is discussed by J. D. B ekenstein, Entropy
bounds and black hole remnants, Physical Review D 49, pp. 1912–1921, 1994. See also
J. D. B ekenstein, Universal upper bound on the entropy-to-energy ratio for bounded
systems, Physical Review D 23, pp. 287–298, 1981. Cited on pages 47 and 131.
36
The statement is also called the Kovtun-Son-Starinets conjecture. It was published as
P. Kov tun, D. T. S on & A . O. Sta rinets, A viscosity bound conjecture, preprint at
arxiv.org/abs/hep-th/0405231. See also P. Kov tun, D. T. S on & A . O. Sta rinets, Viscosity in strongly interacting quantum field theories from black hole physics, Physical Review
Letters 44, p. 111601, 2005. For an experimental verification, see U. Hohm, On the ratio
of the shear viscosity to the density of entropy of the rare gases and H2, N2, CH4, and CF4,
Chemical Physics 444, pp. 39–42, 2014. Cited on page 48.
37
B ria n G reene, The Elegant Universe – Superstrings, Hidden Dimensions, and the Quest
for the Ultimate Theory, Jonathan Cape 1999. Cited on page 52.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Minimal length and minimal time intervals are discussed, for example, by G . A m elinoC a m elia, Limits on the measurability of space-time distances in (the semiclassical approximation of) quantum gravity, Modern Physics Letters A 9, pp. 3415–3422, 1994, preprint at
arxiv.org/abs/gr-qc/9603014, and by Y. J. Ng & H. va n Da m, Limit to space-time measurement, Modern Physics Letters A 9, pp. 335–340, 1994. Many other authors have explored
the topic. Cited on pages 37 and 67.
Motion Mountain – The Adventure of Physics
26
432
38
39
40
41
42
43
b ib lio graphy
J. L. Friedman & R . D. S orkin, Spin 1/2 from gravity, Physical Review Letters 44,
pp. 1100–1103, 1980. Cited on page 58.
45
A . P. Ba l achandran, G . B imonte, G . Ma rm o & A . Simoni, Topology change and
quantum physics, Nuclear Physics B 446, pp. 299–314, 1995, preprint at arxiv.org/abs/
hep-th/9503046. Cited on page 58.
46
J. E hlers, Introduction – Survey of Problems, pp. 1–10, in J. E hlers, editor, Sistemi
gravitazionali isolati in relatività generale, Rendiconti della scuola internazionale di fisica
“Enrico Fermi”, LXVIIo corso, Società Italiana di Fisica/North Holland, 1979. Cited on
page 58.
See C . S chiller, Le vide diffère-t-il de la matière? in E . G unz ig & S. D iner editors,
Le Vide – Univers du tout et du rien – Des physiciens et des philosophes s’interrogent, Les
Éditions de l’Université de Bruxelles, 1998. An older, English-language version is available
as C . S chiller, Does matter differ from vacuum? preprint at arxiv.org/abs/gr-qc/9610066.
Cited on pages 58, 118, 120, 121, 122, 133, and 134.
See for example R ichard P. Fey nman, Rob ert B. L eig hton & Matthew Sa nd s, The Feynman Lectures on Physics, Addison Wesley, 1977. Cited on page
59.
47
48
49
Stev en Weinb erg, Gravitation and Cosmology, Wiley, 1972. Cited on pages 59, 65,
and 67.
50
The argument is given e.g. in E . P. Wig ner, Relativistic invariance and quantum phenomena, Reviews of Modern Physics 29, pp. 255–258, 1957. Cited on page 64.
The starting point for the following arguments is taken from M. S chön, Operative time
definition and principal indeterminacy, preprint at arxiv.org/abs/gr-qc/9304024, and from
T. Pa d manabhan, Limitations on the operational definition of space-time events and
quantum gravity, Classical and Quantum Gravity 4, pp. L107–L113, 1987; see also Padmanabhan’s earlier papers referenced there. Cited on page 64.
W. Heisenberg, Über den anschaulichen Inhalt der quantentheoretischen Kinematik und
Mechanik, Zeitschrift für Physik 43, pp. 172–198, 1927. Cited on page 64.
E . H. Kenna rd, Zur Quantenmechanik einfacher Bewegungstypen, Zeitschrift für Physik
44, pp. 326–352, 1927. Cited on page 64.
51
52
53
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
44
Motion Mountain – The Adventure of Physics
S. Weinberg, The cosmological constant problem, Reviews of Modern Physics 61, pp. 1–
23, 1989. Cited on page 57.
Stev en Weinb erg, The Quantum Theory of Fields, Cambridge University Press, volumes
I, 1995, and II, 1996. Cited on page 57.
See the excellent presentation on the cosmological constant in general relativity by
E . B ia nchi & C . Rov elli, Why all these prejudices against a constant?, preprint at
arxiv.org/abs/1002.3966 Cited on page 57.
The difficulties are summarised by B. S. D eWit t, Quantum field theory in curved spacetime, Physics Reports 19, pp. 295–357, 1975. Cited on page 57.
C . W. Misner, K. S. Thorne & J. A . Wheeler, Gravitation, Freeman, 1973. Cited on
pages 58, 59, and 67.
J. A . Wheeler, in Relativity, Groups and Topology, edited by C . D eWit t &
B. S. D eWit t, Gordon and Breach, 1994. See also J. A . Wheeler, Physics at the Planck
length, International Journal of Modern Physics A 8, pp. 4013–4018, 1993. However, his
claim that spin 1/2 requires topology change is refuted by the strand model of the vacuum.
Cited on page 58.
b ib liography
433
55
H. Sa lecker, & E . P. Wig ner, Quantum limitations of the measurement of space-time
distances, Physical Review 109, pp. 571–577, 1958. Cited on pages 65, 93, and 114.
56
E . J. Z im merman, The macroscopic nature of space-time, American Journal of Physics 30,
pp. 97–105, 1962. Cited on pages 65, 93, and 114.
57
J. D. B ekenstein, Black holes and entropy, Physical Review D 7, pp. 2333–2346, 1973.
Cited on pages 65, 131, and 290.
58
S. W. Hawking, Particle creation by black holes, Communications in Mathematical Physics 43, pp. 199–220, 1975; see also S. W. Haw king, Black hole thermodynamics, Physical
Review D 13, pp. 191–197, 1976. Cited on pages 65, 131, and 290.
59
P. G ib bs, The small scale structure of space-time: a bibliographical review, preprint at arxiv.
org/abs/hep-th/9506171. Cited on pages 65 and 84.
60
The impossibility of determining temporal ordering in quantum theory is discussed by
J. Oppenheimer, B. R ez nik & W. G . Unruh, Temporal ordering in quantum mechanics, Journal of Physics A 35, pp. 7641–7652, 2001, preprint at arxiv.org/abs/quant-ph/
0003130. Cited on page 66.
61
M. -T. Jaekel & S. R enaud, Gravitational quantum limit for length measurement, Physics Letters A 185, pp. 143–148, 1994. Cited on page 67.
62
D. V. A hluwalia, Quantum measurement, gravitation and locality, Physics Letters B 339,
pp. 301–303, 1994, preprint at arxiv.org/abs/gr-qc/9308007. Cited on page 67.
63
L. G a ray, Quantum gravity and minimum length, International Journal of Modern Physics A 10, pp. 145–165, 1995, preprint at arxiv.org/abs/gr-qc/9403008. This paper also includes an extensive bibliography. See also R . J. A d ler & D. I. Sa ntiago, On gravity and
the uncertainty principle, Modern Physics Letters A 14, pp. 1371–1381, 1999, preprint at arxiv.
org/abs/gr-qc/9904026. Cited on page 67.
64
C . Rov elli & L. Sm olin, Discreteness of area and volume in quantum gravity, Nuclear
Physics B 442, pp. 593–619, 1995. R . L oll, The volume operator in discretized quantum
gravity, preprint at arxiv.org/abs/gr-qc/9506014. See also C . Rov elli, Notes for a brief history of quantum gravity, preprint at arxiv.org/abs/gr-qc/0006061. Cited on page 68.
65
D. A m ati, M . C ia fa loni & G . Venez ia no, Superstring collisions at Planckian energies, Physics Letters B 197, pp. 81–88, 1987. D. J. G ross & P. F. Mend e, The high energy behavior of string scattering amplitudes, Physics Letters B 197, pp. 129–134, 1987.
K. Konishi, G . Pa ffuti & P. P rov ero, Minimum physical length and the generalized
uncertainty principle, Physics Letters B 234, pp. 276–284, 1990. P. A spinwa ll, Minimum
distances in non-trivial string target spaces, Nuclear Physics B 431, pp. 78–96, 1994, preprint
at arxiv.org/abs/hep-th/9404060. Cited on page 68.
66
M. Mag giore, A generalised uncertainty principle in quantum mechanics, Physics Letters
B 304, pp. 65–69, 1993. Cited on page 68.
67
A simple approach is S. D oplicher, K. Fred enhagen & J. E . Rob erts, Space-time
quantization induced by classical gravity, Physics Letters B 331, pp. 39–44, 1994. Cited on
pages 68 and 83.
68
A . Kem pf, Uncertainty relation in quantum mechanics with quantum group symmetry,
Journal of Mathematical Physics 35, pp. 4483–4496, 1994. A . Kem pf, Quantum groups
and quantum field theory with nonzero minimal uncertainties in positions and momenta,
Czechoslovak Journal of Physics 44, pp. 1041–1048, 1994. Cited on page 68.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
M. G. R ay mer, Uncertainty principle for joint measurement of noncommuting variables,
American Journal of Physics 62, pp. 986–993, 1994. Cited on page 64.
Motion Mountain – The Adventure of Physics
54
434
69
70
71
72
74
75
77
78
79
80
N . F. R a m sey & A . Weis, Suche nach permanenten elektrischen Dipolmomenten: ein
Test der Zeitumkehrinvarianz, Physikalische Blätter 52, pp. 859–863, 1996. See also
W. B ernreuther & M. Suz uki, The electric dipole moment of the electron, Reviews of
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
76
E . J. Hellund & K. Ta na ka, Quantized space-time, Physical Review 94, pp. 192–195,
1954. Cited on page 69.
This intriguing extract from a letter by Einstein was made widely known by
John J. Stachel, in his paper The other Einstein: Einstein contra field theory, that is
best found in his book Einstein from ‘B’ to ‘Z’, Birkhäuser, 2002. The German original
of the letter is found in Rob ert S chulmann, A . J. Knox, Michel Ja nssen &
Józ sef Illy, The Collected Papers of Albert Einstein, Volume 8A – The Berlin Years: Correspondence, 1914–1917, letter 299, Princeton University Press, 1998. Barbara Wolff helped
in clarifying several details in the German original. The letter is now available online, at
einsteinpapers.press.princeton.edu/vol8a-doc/463. Cited on page 69.
A . P eres & N . Rosen, Quantum limitations on the measurement of gravitational fields,
Physical Review 118, pp. 335–336, 1960. Cited on page 71.
It is the first definition in Euclid’s Elements, c. 300 b ce. For an English translation see
T. Heath, The Thirteen Books of the Elements, Dover, 1969. Cited on page 72.
A beautiful description of the Banach–Tarski paradox is the one by Ia n Stewart, Paradox of the spheres, New Scientist, 14 January 1995, pp. 28–31. Cited on page 72.
H. S. Sny der, Quantized space-time, Physical Review 71, pp. 38–41, 1947. H. S. Sny der,
The electromagnetic field in quantized space-time, Physical Review 72, pp. 68–74, 1947.
A . S child, Discrete space-time and integral Lorentz transformations, Physical Review 73,
pp. 414–415, 1948. E . L. Hill, Relativistic theory of discrete momentum space and discrete space-time, Physical Review 100, pp. 1780–1783, 1950. H. T. Flint, The quantization of space-time, Physical Review 74, pp. 209–210, 1948. A . Das, Cellular space-time and
quantum field theory, Il Nuovo Cimento 18, pp. 482–504, 1960. Cited on page 74.
D. Finkelstein, ‘Superconducting’ causal nets, International Journal of Theoretical
Physics 27, pp. 473–519, 1985. Cited on page 74.
N . H. C hrist, R . Friedberg & T. D. L ee, Random lattice field theory: general formulation, Nuclear Physics B 202, pp. 89–125, 1982. G . ’ t Ho oft, Quantum field theory for
elementary particles – is quantum field theory a theory?, Physics Reports 104, pp. 129–142,
1984. Cited on page 74.
For a discussion, see R . S ora bji, Time, Creation and the Continuum: Theories in Antiquity
and the Early Middle Ages, Duckworth, 1983. Cited on page 74.
See, for example, L. B om b elli, J. L ee, D. Mey er & R . D. S orkin, Space-time
as a causal set, Physical Review Letters 59, pp. 521–524, 1987. G . B right well &
R . G reg ory, Structure of random space-time, Physical Review Letters 66, pp. 260–
263, 1991. Cited on page 74.
The false belief that particles like quarks or electrons are composite is slow to die out. See for
example: S. Fredriksson, Preon prophecies by the standard model, preprint at arxiv.org/
abs/hep-ph/0309213. Preon models gained popularity in the 1970s and 1980s, in particular
through the papers by J. C . Pati & A. Sa l a m, Lepton number as the fourth “color”, Physical Review D 10, pp. 275–289, 1974, H. Ha ra ri, A schematic model of quarks and leptons,
Physics Letters B 86, pp. 83–86, 1979, M. A . Shupe, A composite model of leptons and
quarks, Physics Letters B 86, pp. 87–92, 1979, and H. Fritzsch & G . Ma nd elbaum,
Weak interactions as manifestations of the substructure of leptons and quarks, Physics Letters B 102, pp. 319–322, 1981. Cited on page 76.
Motion Mountain – The Adventure of Physics
73
b ib lio graphy
b ib liography
81
82
83
84
85
435
?=
?2 ??
1+
and
?=
???
1+
?2 ??
?Pl
.
(221)
Another, similar approach of recent years, with a different proposal, is called ‘doubly
special relativity’. A recent summary is G . A m elino-Camelia, Doubly-special relativity: first results and key open problems, International Journal of Modern Physics 11,
pp. 1643–1669, 2002, preprint at arxiv.org/abs/gr-qc/0210063. The paper shows how conceptual problems hinder the advance of the field. Another such discussion R . A loisio,
A . G al a nte, A . F. G ri llo, E . Luz i o & F. Ménd ez, Approaching space-time through
velocity in doubly special relativity, preprint at arxiv.org/abs/gr-qc/0410020. The lesson from
these attempts is simple: special relativity cannot be modified to include a limit energy
without also including general relativity and quantum theory. Cited on pages 83 and 280.
W. Jauch, Heisenberg’s uncertainty relation and thermal vibrations in crystals, American
Journal of Physics 61, pp. 929–932, 1993. Cited on page 83.
87
H. D. Z eh, On the interpretation of measurement in quantum theory, Foundations of Physics 1, pp. 69–76, 1970. Cited on page 84.
88
See Y. J. Ng, W. A . C hristiansen & H. va n Da m, Probing Planck-scale physics with
extragalactic sources?, Astrophysical Journal 591, pp. L87–L90, 2003, preprint at arxiv.org/
abs/astro-ph/0302372; D. H. C oule, Planck scale still safe from stellar images, Classical
and Quantum Gravity 20, pp. 3107–3112, 2003, preprint at arxiv.org/abs/astro-ph/0302333.
Negative experimental results (and not always correct calculations) are found in R . L ieu
& L. Hillman, The phase coherence of light from extragalactic sources – direct evidence
against first order Planck scale fluctuations in time and space, Astrophysical Journal 585,
pp. L77–L80, 2003, and R . R ag azz oni, M. Turat to & W. G a essler, The lack of observational evidence for the quantum structure of spacetime at Planck scales, Astrophysical
Journal 587, pp. L1–L4, 2003. Cited on page 87.
B. E . S cha efer, Severe limits on variations of the speed of light with frequency, Physical
Review Letters 82, pp. 4964–4966, 21 June 1999. Cited on page 87.
89
90
A . A . A b d o & al., (Fermi GBM/LAT collaborations) Testing Einstein’s special relativity
with Fermi’s short hard gamma-ray burst GRB090510, preprint at arxiv.org/0908.1832. Cited
on page 87.
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86
?2 ??
?Pl
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Modern Physics 63, pp. 313–340, 1991, and the musings in Ha ns D ehm elt, Is the electron
a composite particle?, Hyperfine Interactions 81, pp. 1–3, 1993. Cited on page 77.
K. A ka m a, T. Hat tori & K. Katsuura, Naturalness bounds on dipole moments from
new physics, preprint at arxiv.org/abs/hep-ph/0111238. Cited on page 77.
The paper by J. Ba ron & al., Order of magnitude smaller limit on the electric dipole moment
of the electron, preprint at arxiv.org/abs/1310.7534 gives an upper experimental limit to the
dipole moment of the electron of 8.7 ⋅ 10−31 e m. Cited on page 77.
C . Wolf, Upper limit for the mass of an elementary particle due to discrete time quantum
mechanics, Il Nuovo Cimento B 109, pp. 213–218, 1994. Cited on page 79.
W. G . Unruh, Notes on black hole evaporation, Physical Review D 14, pp. 870–875, 1976.
W. G . Unruh & R. M. Wa ld, What happens when an accelerating observer detects a Rindler particle, Physical Review D 29, pp. 1047–1056, 1984. Cited on page 81.
The first example was J. Mag ueijo & L. Sm olin, Lorentz invariance with an invariant
energy scale, Physical Review Letters 88, p. 190403, 2002, preprint at arxiv.org/abs/hep-th/
0112090. They propose a modification of the mass energy relation of the kind
436
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92
G . A m elino-C amelia, Phenomenological description of space-time foam, preprint at
arxiv.org/abs/gr-qc/0104005. The paper includes a clearly written overview of present
experimental approaches to detecting quantum gravity effects. See also his update
G . A m elino-C amelia, Quantum-gravity phenomenology: status and prospects, preprint at arxiv.org/abs/gr-qc/0204051. Cited on pages 87 and 88.
93
G . A m elino-C amelia, An interferometric gravitational wave detector as a quantum
gravity apparatus, Nature 398, pp. 216–218, 1999, preprint at arxiv.org/abs/gr-qc/9808029.
Cited on page 87.
94
F. Ka rolyhazy, Gravitation and quantum mechanics of macroscopic objects, Il Nuovo Cimento A42, pp. 390–402, 1966. Y. J. Ng & H. va n Da m, Limit to space-time measurement,
Modern Physics Letters A 9, pp. 335–340, 1994. Y. J. Ng & H. va n Da m, Modern Physics
Letters A Remarks on gravitational sources, 10, pp. 2801–2808, 1995. The discussion is neatly
summarised in Y. J. Ng & H. va n Da m,Comment on ‘Uncertainty in measurements of distance’, preprint at arxiv.org/abs/gr-qc/0209021. See also Y. J. Ng, Spacetime foam, preprint
at arxiv.org/abs/gr-qc/0201022. Cited on pages 87 and 93.
95
L. J. G a ray, Spacetime foam as a quantum thermal bath, Physics Review Letters 80,
pp. 2508–2511, 1998, preprint at arxiv.org/abs/gr-qc/9801024. Cited on page 88.
96
G . A m elino-C amelia & T. P ira n, Planck-scale deformation of Lorentz symmetry as
a solution to the UHECR and the TeV-? paradoxes, preprint at arxiv.org/astro-ph/0008107,
2000. Cited on page 88.
97
R . P. Wo odard, How far are we from the quantum theory of gravity?, preprint at arxiv.org/
abs/0907.4238. For a different point of view, see L. Sm olin, Generic predictions of quantum
theories of gravity, preprint at arxiv.org/abs/hep-th/0605052. Cited on pages 88 and 304.
98
A similar point of view, often called monism, was proposed by Ba ruch Spinoz a, Ethics Demonstrated in Geometrical Order, 1677, originally in Latin; an affordable French edition is Ba ruch Spinoza, L’Ethique, Folio-Gallimard, 1954. For a discussion of his ideas,
especially his monism, see D on G a rret editor, The Cambridge Companion to Spinoza,
Cambridge University Press, 1996, or any general text on the history of philosophy. Cited
on page 88.
99
See the lucid discussion by G . F. R. E llis & T. Rothm an, Lost horizons, American
Journal of Physics 61, pp. 883–893, 1993. Cited on pages 93, 97, and 98.
100 See, for example, the Hollywood film Contact by Robert Zemeckis, based on the book by
C a rl Sag a n, Contact, Simon & Schuster, 1985. Cited on page 98.
101 See, for example, the international bestseller by Stephen Haw king, A Brief History of
Time – From the Big Bang to Black Holes, 1988. Cited on page 101.
102 L. Rosenfeld, Quantentheorie und Gravitation, in H. - J. Tred er, editor, Entstehung,
Entwicklung und Perspektiven der Einsteinschen Gravitationstheorie, Springer Verlag, 1966.
Cited on page 103.
103 Holography in high-energy physics is connected with the work of ’t Hooft and Susskind.
See for example G . ’ t Ho oft, Dimensional reduction in quantum gravity, pp. 284–296,
in A . A li, J. E llis & S . R a ndjbar-Daemi, Salaamfeest, 1993, or the much-cited paper
by L. Susskind, The world as a hologram, Journal of Mathematical Physics 36, pp. 6377–
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
G . A m elino-C amelia, J. E llis, N . E . Mav romatos, D. V. Na nop oulos &
S. Sa ka r, Potential sensitivity of gamma-ray-burster observations to wave dispersion in
vacuo, Nature 393, pp. 763–765, 1998, preprint at arxiv.org/abs/astro-ph/9712103. Cited on
page 87.
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437
6396, 1995, preprint at arxiv.org/abs/hep-th/9409089. A good modern overview is Ref. 30.
Cited on pages 105 and 113.
104 D. B ohm & B. J. Hiley, On the intuitive understanding of nonlocality as implied by
quantum theory, Foundations of Physics 5, pp. 93–109, 1975. Cited on page 106.
105 S. L loy d, Computational capacity of the universe, Physical Review Letters 88, p. 237901,
2002. Cited on page 107.
106 G ot tfried Wilhelm L eib niz, La Monadologie, 1714. Written in French, it is avail-
able freely at www.uqac.uquebec.ca/zone30/Classiques_des_sciences_sociales and in various other languages on other websites. Cited on page 108.
107 See, for example, H. Wussing & P. S. A lexa ndrov editors, Die Hilbertschen Probleme,
Akademische Verlagsgesellschaft Geest & Portig, 1983, or B en H. Ya ndell, The Honours
Class: Hilbert’s Problems and their Solvers, A.K. Peters, 2002. Cited on page 108.
108 A large part of the study of dualities in string and M theory can be seen as investiga-
109 See L. Susskind & J. Ug lum, Black holes, interactions, and strings, preprint at arxiv.
110 M. P l anck, Über irreversible Strahlungsvorgänge, Sitzungsberichte der Kaiserlichen
Akademie der Wissenschaften zu Berlin pp. 440–480, 1899. Today it is commonplace to
use Dirac’s ℏ = ℎ/2π instead of Planck’s ℎ, which Planck originally called ?. Cited on page
118.
111 P. Facchi & S. Pasca zio, Quantum Zeno and inverse quantum Zeno effects, pp. 147–217,
in E . Wolf editor, Progress in Optics, 42, 2001. Cited on page 121.
112 A ristotle, Of Generation and Corruption, book I, part 2. See Jea n-Paul D umont,
Les écoles présocratiques, Folio Essais, Gallimard, p. 427, 1991. Cited on page 121.
113 See for example the speculative model of vacuum as composed of Planck-size spheres pro-
posed by F. Winterberg, Zeitschrift für Naturforschung 52a, p. 183, 1997. Cited on page
122.
114 The Greek salt-and-water argument and the fish argument are given by Lucrece, in full Titus
Lucretius Carus, De natura rerum, c. 60 b ce. Cited on pages 123 and 138.
115 J. H. S chwarz, The second superstring revolution, Colloquium-level lecture presented at
the Sakharov Conference in Moscow, May 1996, preprint at arxiv.org/abs/hep-th/9607067.
Cited on pages 124 and 125.
116 Sim plicius, Commentary on the Physics of Aristotle, 140, 34. This text is cited in Jea n-
Paul D umont, Les écoles présocratiques, Folio Essais, Gallimard, p. 379, 1991. Cited on
page 124.
117 D. Oliv e & C . Montonen, Magnetic monopoles as gauge particles, Physics Letters 72B,
pp. 117–120, 1977. Cited on page 125.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
org/abs/hep-th/9410074, or L. Susskind, String theory and the principle of black hole
complementarity, Physical Review Letters 71, pp. 2367–2368, 1993, and M. Ka rliner,
I. Klebanov & L. Susskind, Size and shape of strings, International Journal of Modern Physics A 3, pp. 1981–1996, 1988, as well as L. Susskind, Structure of hadrons implied
by duality, Physical Review D 1, pp. 1182–1186, 1970. Cited on pages 117 and 132.
Motion Mountain – The Adventure of Physics
tions into the detailed consequences of extremal identity. For a review of dualities, see
P. C . A rg yres, Dualities in supersymmetric field theories, Nuclear Physics Proceedings
Supplement 61, pp. 149–157, 1998, preprint at arxiv.org/abs/hep-th/9705076. A classical
version of duality is discussed by M. C . B. A b dall a, A . L. G a delka & I. V. Va ncea,
Duality between coordinates and the Dirac field, preprint at arxiv.org/abs/hep-th/0002217.
Cited on page 113.
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118 A famous fragment from D io g enes L a ërtius (IX 72) quotes Democritus as follows:
‘By convention hot, by convention cold, but in reality, atoms and void; and also in reality
we know nothing, since truth is at the bottom.’ Cited on page 127.
119 This famous statement is found at the beginning of chapter XI, ‘The Physical Universe’,
in A rthur E d dington, The Philosophy of Physical Science, Cambridge, 1939. Cited on
page 128.
120 P l ato, Parmenides, c. 370 b ce. It has been translated into most languages. Reading it
aloud, like a song, is a beautiful experience. A pale reflection of these ideas is Bohm’s
concept of ‘unbroken wholeness’. Cited on page 128.
121 P. G ib bs, Event-symmetric physics, preprint at arxiv.org/abs/hep-th/9505089; see also his
website www.weburbia.com/pg/contents.htm. Cited on page 128.
122 J. B. Hartle, & S. W. Hawking, Path integral derivation of black hole radiance, Phys-
123 J. Ma d d ox, When entropy does not seem extensive, Nature 365, p. 103, 1993. The issue is
now explored in all textbooks discussing black holes. John Maddox (b. 1925 Penllergaer,
d. 1999 Abergavenny) was famous for being one of the few people who was knowledgeable
in most natural sciences. Cited on page 131.
124 L. B om b elli, R . K. Koul, J. L ee & R . D. S orkin, Quantum source of entropy of black
holes, Physical Review D 34, pp. 373–383, 1986. Cited on page 131.
boundaries, Nature 365, p. 792, 1993. Cited on page 131.
126 See the classic text by P ierre-Gilles d e G ennes, Scaling Concepts in Polymer Physics,
Cornell University Press, 1979. Cited on page 132.
127 See for example S. Maj id, Introduction to braided geometry and ?-Minkowski space, pre-
print at arxiv.org/abs/hep-th/9410241, or S. Maj id, Duality principle and braided geometry, preprint at arxiv.org/abs/hep-th/9409057. Cited on pages 133 and 134.
128 The relation between spin and statistics has been studied recently by M. V. B erry &
J. M. Rob b ins, Quantum indistinguishability: spin–statistics without relativity or field theory?, in R . C . Hilb orn & G . M. Tino editors, Spin–Statistics Connection and Commutation Relations, American Institute of Physics, 2000. Cited on page 135.
129 A . G reg ori, Entropy, string theory, and our world, preprint at arxiv.org/abs/hep-th/
0207195. Cited on pages 136 and 137.
130 String cosmology is a pastime for many. Examples include N . E . Mav romatos, String
cosmology, preprint at arxiv.org/abs/hep-th/0111275, and N . G . Sa nchez, New developments in string gravity and string cosmology – a summary report, preprint at arxiv.org/abs/
hep-th/0209016. Cited on page 137.
131 On the present record, see en.wkipedia.org/wiki/Ultra-high-energy_cosmic_ray and fr.
wkipedia.org/wiki/Zetta-particule. Cited on page 138.
132 P. F. Mende, String theory at short distance and the principle of equivalence, preprint at
arxiv.org/abs/hep-th/9210001. Cited on page 138.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
125 The analogy between polymers and black holes is due to G . Web er, Thermodynamics at
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ical Review D 13, pp. 2188–2203, 1976. See also A . Strom inger & C . Va fa, Microscopic origin of Bekenstein–Hawking entropy, Physics Letters B 379, pp. 99–104, 1996,
preprint at arxiv.org/abs/hep-th/9601029. For another derivation of black hole entropy,
see G . T. Horow itz & J. Polchinski, A correspondence principle for black holes and
strings, Physical Review D 55, pp. 6189–6197, 1997, preprint at arxiv.org/abs/hep-th/
9612146. Cited on pages 131 and 140.
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133 An example is given by A . A . Sl avnov, Fermi–Bose duality via extra dimension, preprint
134
135
136
137
139
140
142
143
144
145
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141
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138
at arxiv.org/abs/hep-th/9512101. See also the standard work by Michael Stone editor,
Bosonization, World Scientific, 1994. Cited on page 138.
A weave model of space-time appears in certain approaches to quantum gravity, such as
Ref. 27. On a slightly different topic, see also S. A . Maj or, A spin network primer, preprint
at arxiv.org/abs/gr-qc/9905020. Cited on page 138.
L. Sm olin & Y. Wa n, Propagation and interaction of chiral states in quantum gravity, preprint at arxiv.org/abs/0710.1548, and references therein. Cited on page 138.
A good introduction into his work is the paper D. Kreimer, New mathematical structures
in renormalisable quantum field theories, Annals of Physics 303, pp. 179–202, 2003, erratum
ibid. 305, p. 79, 2003, preprint at arxiv.org/abs/hep-th/0211136. Cited on page 139.
Introductions to holography include E . A lva rez, J. C onde & L. Herna ndez, Rudiments of holography, preprint at arxiv.org/abs/hep-th/0205075, and Ref. 30. The importance of holography in theoretical high-energy physics was underlined by the discovery of
J. Ma ldacena, The large N limit of superconformal field theories and supergravity, preprint at arxiv.org/abs/hep-th/9711200. Cited on page 139.
X. - G . Wen, From new states of matter to a unification of light and electrons, preprint at
arxiv.org/abs/0508020. Cited on page 139.
J. S. Av rin, A visualizable representation of the elementary particles, Journal of Knot Theory and Its Ramifications 14, pp. 131–176, 2005. Cited on pages 139 and 347.
The well-known ribbon model is presented in S. B ilson-Thompson, A topological
model of composite preons, preprint at arxiv.org/hep-ph/0503213; S. B ilson-Thompson,
F. Ma rkop oulou & L. Sm olin, Quantum gravity and the standard model, preprint
at arxiv.org/hep-th/0603022; S. B ilson-Thompson, J. Hackett, L. Kauffman
& L. Sm olin, Particle identifications from symmetries of braided ribbon network invariants, preprint at arxiv.org/abs/0804.0037; S. B ilson-Thompson, J. Hackett &
L. Kauffman, Particle topology, braids, and braided belts, preprint at arxiv.org/abs/0903.
1376. Cited on pages 139, 165, and 347.
R . J. Finkelstein, A field theory of knotted solitons, preprint at arxiv.org/abs/hep-th/
0701124. See also R . J. Finkelstein, Trefoil solitons, elementary fermions, and SUq (2),
preprint at arxiv.org/abs/hep-th/0602098, R. J. Finkelstein & A . C. C a david, Masses
and interactions of q-fermionic knots, preprint at arxiv.org/abs/hep-th/0507022, and
R . J. Finkelstein, A knot model suggested by the standard electroweak theory, preprint
at arxiv.org/abs/hep-th/0408218. Cited on pages 139 and 347.
L ouis H. Kauffman, Knotsand Physics, World Scientific, 1991. A wonderful book. Cited
on pages 139 and 275.
S. K. Ng, On a knot model of the π+ meson, preprint at arxiv.org/abs/hep-th/0210024, and
S. K. Ng, On a classification of mesons, preprint at arxiv.org/abs/hep-ph/0212334. Cited on
pages 139 and 347.
For a good introduction to superstrings, see the lectures by B. Zw iebach, String theory for
pedestrians, agenda.cern.ch/fullAgenda.php?ida=a063319. For an old introduction to superstrings, see the famous text by M. B. G reen, J. H. S chwarz & E . Wit ten, Superstring Theory, Cambridge University Press, volumes 1 and 2, 1987. Like all the other books
on superstrings, they contain no statement that is applicable to or agrees with the strand
model. Cited on pages 139 and 348.
See A . Sen, An introduction to duality symmetries in string theory, in Les Houches Summer
School: Unity from Duality: Gravity, Gauge Theory and Strings (Les Houches, France, 2001),
440
146
147
148
149
150
152
154
155
156
157
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
153
Springer Verlag, 76, pp. 241–322, 2002. Cited on page 139.
Brian Greene regularly uses the name string conjecture. For example, he did so in a podium
discussion at TED in 2009; the video of the podium discussion can be downloaded at www.
ted.org. Cited on page 140.
L. Susskind, Some speculations about black hole entropy in string theory, preprint at arxiv.
org/abs/hep-th/9309145. G . T. Horow itz & J. Polchinski, A correspondence principle
for black holes and strings, Physical Review D 55, pp. 6189–6197, 1997, preprint at arxiv.org/
abs/hep-th/9612146. Cited on pages 140 and 444.
F. Wilcz ek, Getting its from bits, Nature 397, pp. 303–306, 1999. Cited on page 141.
M. R . D ougl as, Understanding the landscape, preprint at arxiv.org/abs/hep-th/0602266;
his earlier papers also make the point. For the larger estimate, see W. Tay lor & Y. N . Wa ng, The F-theory geometry with most flux vacua, preprint at arxiv.org/abs/1511.
03209. Cited on page 141.
The difficulties of the string conjecture are discussed in the well-known internet blog
by P eter Woit, Not even wrong, at www.math.columbia.edu/~woit/blog. Several Nobel Prize winners for particle physics dismiss the string conjecture: Martin Veltman, Sheldon Glashow, Burton Richter, Richard Feynman and since 2009 also Steven Weinberg are
among those who did so publicly. Cited on pages 142 and 166.
The present volume was originally started with the aim to clarify the basic principles of
string theory and to simplify it as much as possible. In particular, the first six chapters and
the last chapter were conceived, structured and written with that aim. They are older than
the strand model. Later on, the project took an unexpected direction, as explained in Ref. 19.
Cited on page 142.
Searches for background-free approaches are described by E . Wit ten, Quantum background independence in string theory, preprint at arxiv.org/abs/hep-th/9306122 and
E . Wit ten, On background-independent open string theory, preprint at arxiv.org/abs/
hep-th/9208027. Cited on page 143.
In fact, no other candidate model that fulfils all requirements for the final theory is available
in the literature so far. This might change in the future, though. Cited on page 149.
S. C a rlip, The small scale structure of spacetime, preprint at arxiv.org/abs/1009.1136. This
paper deduces the existence of fluctuating lines in vacuum from a number of arguments
that are completely independent of the strand model. Steven Carlip has dedicated much of
his research to the exploration of this topic. One summary is S. C a rlip, Spontaneous dimensional reduction in quantum gravity, preprint at arxiv.org/abs/1605.05694; its is also instructive to read his review S. C a rlip, Dimension and dimensional reduction in quantum
gravity, Classical and Quantum Gravity 34, p. 193001, 2017, preprint at arxiv.org/abs/1705.
05417. With the strand model in the back of one’s mind, these results are even more fascinating. Cited on pages 161 and 300.
David Deutsch states that any good explanation must be ‘hard to vary’. This must also apply
to a unified model, as it claims to explain everything that is observed. See D. D eu tsch,
A new way to explain explanation, video talk at www.ted.org. Cited on pages 164 and 406.
L. B om b elli, J. L ee, D. Meyer & R . D. S orkin, Space-time as a causal set, Physical
Review Letters 59, pp. 521–524, 1987. See also the review by J. Henson, The causal set
approach to quantum gravity, preprint at arxiv.org/abs/gr-qc/0601121. Cited on pages 165
and 299.
D. Finkelstein, Homotopy approach to quantum gravity, International Journal of Theoretical Physics 47, pp. 534–552, 2008. Cited on pages 165 and 299.
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158 L. H. Kauffman & S. J. L om onaco, Quantum knots, preprint at arxiv.org/abs/
quant-ph/0403228. See also S. J. L om onaco & L. H. Kauffman, Quantum knots
and mosaics, preprint at arxiv.org/abs/quant-ph/0805.0339. Cited on page 165.
159 Im m anuel Ka nt, Critik der reinen Vernunft, 1781, is a famous but long book that every
philospher pretends to have read. In his book, Kant introduced the ‘a priori’ existence of
space and time. Cited on page 168.
160 The literature on circularity is rare. For two interesting exceptions, see L. H. Kauffman,
Knot logic, downloadable from www2.math.uic.edu/~kauffman, and L. H. Kauffman,
Reflexivity and eigenform, Constructivist Foundations 4, pp. 121–137, 2009. Cited on page
169.
161 Information on the belt trick is scattered across many books and few papers. The best source
162 There is an interesting exploration behind this analogy between a non-dissipative system
Challenge 234 e
163 D. B ohm, R . S chiller & J. Tiom no, A causal interpretation of the Pauli equation (A),
Supplementi al Nuovo Cimento 1, pp. 48 – 66, 1955, and D. B ohm & R . S chiller, A
causal interpretation of the Pauli equation (B), Supplementi al Nuovo Cimento 1, pp. 67–91,
1955. The authors explore an unusual way to interpret the wavefunction, which is of little
interest here; but doing so, they give and explore the description of Pauli spinors in terms
of Euler angles. Cited on page 198.
164 R icha rd P. Fey nman, QED – The Strange Theory of Light and Matter, Princeton Uni-
versity Press 1988. This is one of the best summaries of quantum theory ever written. Every
physicist should read it. Cited on pages 199, 204, 216, 222, and 441.
165 S. Ko chen & E . P. Specker, The problem of hidden variables in quantum mechanics, 17,
pp. 59–87, 1967. This is a classic paper. Cited on page 202.
166 A . A spect, J. Da li bard & G . Ro g er, Experimental tests of Bell’s inequalities using
time-varying analyzers, Physical Review Letters 49, pp. 1804–1807, 1982, Cited on page
206.
167 L. Kauffman, New invariants of knot theory, American Mathematical Monthly 95,
pp. 195–242, 1987. See also the image at the start of chapter 6 of L ouis H. Kauffman,
On Knots, Princeton University Press, 1987. Cited on page 207.
168 The details on the speed of photons are explained in any textbook on quantum electrody-
namics. The issue is also explained by Feynman in Ref. 164 on page 89. Cited on page 210.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
– a free quantum particle moving in vacuum – and a dissipative system – a macroscopic
body drawn through a viscous liquid, say honey. The first question is to discover why this
analogy is possible at all. (A careful distinction between the cases with spin 0, spin 1 and
spin 1/2 are necessary.) The second question is the exploration of the motion of bodies of
general shape in viscous fluids at low Reynolds numbers and under constant force. For the
best overview of this question, see the beautiful article by O. G onz alez, A . B. A. G ra f
& J. H. Ma d d o cks, Dynamics of a rigid body in a Stokes fluid, Journal of Fluid Mechanics
519, pp. 133–160, 2004. Cited on pages 195 and 360.
Motion Mountain – The Adventure of Physics
of information on this topic are websites. For belt trick visualizations see www.evl.uic.edu/
hypercomplex/html/dirac.html, www.evl.uic.edu/hypercomplex/html/handshake.html, or
www.gregegan.net/APPLETS/21/21.html. For an excellent literature summary and more
movies, see www.math.utah.edu/~palais/links.html. None of these sites or the cited references seem to mention that there are many ways to perform the belt trick; this seems to
be hidden knowledge. In September 2009, Greg Egan took up my suggestion and changed
his applet to show an additional version of the belt trick. Cited on pages 176 and 178.
442
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169 J. - M. L év y -L eblond, Nonrelativitic particles and wave equations, Communic-
ations in Mathematical Physics 6, pp. 286–311, 1967. See also A . G a lind o &
C . Sá nchez d el R ío, Intrinsic magnetic moment as a nonrelativistic phenomenon,
American Journal of Physics 29, pp. 582–584, 1961, and V. I. Fushchich, A . G. Nikitin
& V. A . Sa lo g ub, On the non-relativistic motion equations in the Hamiltonian form, Reports on Mathematical Physics 13, pp. 175–185, 1978. Cited on page 213.
170 L. L erner, Derivation of the Dirac equation from a relativistic representation of spin,
European Journal of Phyics 17, pp. 172–175, 1996. Cited on pages 213 and 214.
171 E . P. Bat tey-P ratt & T. J. R acey, Geometric model for fundamental particles, Inter-
national Journal of Theoretical Physics 19, pp. 437–475, 1980. Without knowing this work,
C. Schiller had deduced the same results in 2008. Cited on pages 213, 214, and 453.
172 A . A b ra ham, Prinzipien der Dynamik des Elektrons, Annalen der Physik 10, pp. 105–179,
Bewegung in der relativistischen Quantenmechanik, Berliner Berichte pp. 418–428, 1930,
and Zur Quantendynamik des Elektrons, Berliner Berichte pp. 63–72, 1931. Numerous subsequent papers discuss these publications. Cited on page 216.
174 See for example the book by Ma rtin R ivas, Kinematic Theory of Spinning Particles,
Springer, 2001. Cited on page 216.
175 The basic papers in the field of stochastic quantization are W. Weiz el, Ableitung der
Quantentheorie aus einem klassischen, kausal determinierten Modell, Zeitschrift für Physik
A 134, pp. 264–285, 1953, W. Weiz el, Ableitung der Quantentheorie aus einem klassischen
Modell – II, Zeitschrift für Physik A 135, pp. 270–273, 1954, W. Weiz el, Ableitung der
quantenmechanischen Wellengleichung des Mehrteilchensystems aus einem klassischen Modell, Zeitschrift für Physik A 136, pp. 582–604, 1954. This work was taken up by E . Nelson,
Derivation of the Schrödinger equation from Newtonian mechanics, Physical Review 150,
pp. 1079–1085, 1969, and in E dward Nelson, Quantum Fluctuations, Princeton Univer-
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
173 The concept of Zitterbewegung was formulated in E . S chrödinger, Über die kräftefreie
Motion Mountain – The Adventure of Physics
1903, J. Frenkel, Die Elektrodynamik des rotierenden Elektrons, Zeitschrift für Physik
37, pp. 243–262, 1926, L. H. Thom as, The motion of a spinning electron, Nature April 10,
p. 514, 1926, and L. H. Thom as, The kinematics of an electron with an axis, Philosophical
Magazine 3, pp. 1–22, 1927. See also W. E . Bay lis, Surprising symmetries in relativistic charge dynamics, preprint at arxiv.org/abs/physics/0410197. See also W. E . Bay lis,
Quantum/classical interface: a geometric approach from the classical side, pp. 127–154
and W. E . Bay lis, Geometry of paravector space with applications to relativistic physics,
pp. 363–387 in Computational Noncommutative Algebra and Applications, Proceedings of
the NATO Advanced Study Institute, NATO Science Series II, vol. 136, ed. J. B y rnes,
Kluwer Academic 2004. W. E . Bay lis, R . C a b rera & D. Keselica, Quantum/
classical interface: fermion spin, preprint at arxiv.org/abs/0710.3144. D. Hestenes, Zitterbewegung Modelling, Foundations of Physics 23, pp. 365–386, 1993. D. Hestenes,
Zitterbewegung in quantum mechanics – a research program, preprint at arxiv.org/abs/
0802.2728. See also D. Hestenes, Reading the electron clock, preprint at arxiv.org/abs/
0802.3227 and his webpage modelingnts.la.asu.edu/html/GAinQM.html. A . L oing er &
A. Sparza ni, Dirac equation without Dirac matrices, Il Nuovo Cimento 39, pp. 1140–
1145, 1965. D. B ohm, P. Hillion, T. Takabayasi & J. - P. Vig ier, Relativistic rotators
and bilocal theory, Progress of Theoretical Physics 23, pp. 496–511, 1960. A . C ha llinor,
A. L asenb y, S. Gill & C . D oran, A relativistic, causal account of a spin measurement,
Physics Letters A 218, pp. 128–138, 1996. E . Sa nta mato, The role of Dirac equation in
the classical mechanics of the relativistic top, preprint at arxiv.org/abs/0808.3237. Cited on
pages 215 and 216.
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copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
181
sity Press 1985, also downloadable at www.math.princeton.edu/~nelson/books.html, and
the book E dwa rd Nelson, Stochastic Quantization, Princeton University Press 1985. See
also L. Fritsche & M. Haug k, A new look at the derivation of the Schrödinger equation from Newtonian mechanics, Annalen der Physik 12, pp. 371–402, 2003. A summary
of Nelson’s approach is also given in F. Ma rkop oulou & L. Sm olin, Quantum theory from quantum gravity, Physical Review D 70, p. 124029, 2004, preprint at arxiv.org/
abs/gr-qc/0311059. See also the important criticism by T. C . Wa llstrom, Inequivalence
between the Schrödinger equation and the Madelung hydrodynamic equation, Physical Review A 49, pp. 1613–1617, 1994, and T. C . Wa llstrom, The stochastic mechanics of the
Pauli equation, Transactions of the American Mathematical Society 318, pp. 749–762, 1990.
A proposed answer is L. Sm olin, Could quantum mechanics be an approximation to another theory?, preprint at arxiv.org/quant-ph/abs/0609109. See also S. K. Srinivasan &
E . C . G. Sudarshan, A direct derivation of the Dirac equation via quaternion measures,
Journal of Physics A 29, pp. 5181–5186, 1996. Cited on page 216.
Julia n S chw inger, Quantum Mechanics – Symbolism of Atomic Measurements,
Springer, 2001. Cited on page 219.
H. Nikolić,How (not) to teach Lorentz covariance of the Dirac equation, European Journal
of Physics 35, p. 035003, 2014, preprint at arxiv.org/abs/1309.7070. Cited on page 219.
For such an attempt, see the proposal by M. R a iner, Resolution of simple singularities
yielding particle symmetries in space-time, Journal of Mathematical Physics 35, pp. 646–
655, 1994. Cited on page 222.
C . S chiller, Deducing the three gauge interactions from the three Reidemeister moves, preprint at arxiv.org/abs/0905.3905. Cited on pages 222 and 224.
G . T. Horow itz & J. Polchinski, Gauge/gravity duality, preprint at arxiv.org/
abs/gr-qc/0602037. Note also the statement in the introduction that a graviton might
be a composite of two spin-1 bosons, which is somewhat reproduced by the strand
model of the graviton. A more concrete approach to gauge–gravity duality is made by
M. va n R aamsd onk, Building up spacetime with quantum entanglement, preprint at
arxiv.org/1005.3035. This approach to gauge–gravity duality is close to that of the strand
model. Cited on page 224.
K. R eidemeister, Elementare Begründung der Knotentheorie, Abhandlungen aus dem
Mathematischen Seminar der Universität Hamburg 5, pp. 24–32, 1926. Cited on pages 224
and 275.
For an attempt to reconcile braided particle models and SU(5) GUT, see D. C a rtin, Braids
as a representation space of SU(5), preprint at arxiv.org/pdf/1506.08067. Cited on page 242.
Sheld on G l ashow, confirmed this to the author in an email; R icha rd Fey nman,
makes the point in Ja mes G leick, Genius: The Life and Science of Richard Feynman, Vintage Books, 1991, page 288 and also in Rob ert C rease & C ha rles Ma nn, The Second
Creation: Makers of the Revolution in Twentieth-Century Physics, Macmillan Publishing,
page 418; Ma rtin Veltm an, writes this in his Nobel Prize Lecture, available on www.
nobel.org. Cited on page 255.
For some of the background on this topic, see F. Wilcz ek & A . Z ee Appearance of gauge
structures in simple dynamical systems, Physical Review Letters 52, pp. 2111–2114, 1984,
A . S ha pere & F. Wilcz ek, Self-propulsion at low Reynolds number, Physical Review
Letters 58, pp. 2051–2054, 1987, and A . Sha pere & F. Wilcz ek, Gauge kinematics of
deformable bodies, American Journal of Physics 57, pp. 514–518, 1989. Cited on page 272.
R . B rit to, F. C achazo, B. Feng & E . Wit ten, Direct proof of tree-level recursion relation in Yang–Mills theory, preprint at arxiv.org/abs/hep-th/0501052. Cited on page 273.
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444
b ib lio graphy
186 D. V. A hluwalia-Kha lilova, Operational indistinguishability of double special re-
187
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lativity from special relativity, Classical and Quantum Gravity 22, pp. 1433–1450, 2005, preprint at arxiv.org/abs/gr-qc/0212128; see also N . Ja fa ri & A . Sha riati, Doubly special
relativity: a new relativity or not?, preprint at arxiv.org/abs/gr-qc/0602075. Cited on page
280.
E . Verlinde, On the origin of gravity and the laws of Newton, preprint at arxiv.org/abs/
1001.0785. Cited on page 281.
G . - L. L esag e, Lucrèce Newtonien, Nouveaux mémoires de l’Académie Royale des
Sciences et Belles Lettres pp. 404–431, 1747, or www3.bbaw.de/bibliothek/digital/
struktur/03-nouv/1782/jpg-0600/00000495.htm. See
also
en.wikipedia.org/wiki/
Le_Sage’s_theory_of_gravitation. In fact, the first to propose the idea of gravitation as
a result of small particles pushing masses was Nicolas Fatio de Duillier in 1688. Cited on
page 283.
G . ’ t Ho oft, Dimensional reduction in quantum gravity, preprint at arxiv.org/abs/gr-qc/
9310026. Many of the ideas of this paper become easier to understand and to argue when
the strand model is used. Cited on page 288.
S. C a rlip, Logarithmic corrections to black hole entropy from the Cardy formula, Classical
and Quantum Gravity 17, pp. 4175–4186, 2000, preprint at arxiv.org/abs/gr-qc/0005017.
Cited on page 289.
D. N . Pag e, The Bekenstein Bound, preprint at arxiv.org/abs/1804.10623. Cited on page
291.
On the limit for angular momentum of black holes, see Ref. 34. Cited on page 291.
F. Ta m burini, C. C uofano, M. D ell a Va lle & R. G ilmozzi, No quantum gravity
signature from the farthest quasars, preprint at arxiv.org/abs/1108.6005. Cited on page 296.
B. P. A b b ot t & al., (LIGO Scientific Collaboration and Virgo Collaboration) Observation of gravitational waves from a binary black hole merger, Physical Review Letters
116, p. 061102, 2016, also available for free download at journals.aps.org/prl/pdf/10.1103/
PhysRevLett.116.061102. See also the website www.ligo.caltech.edu. More about this discovery and its implications is told in volume II of the Motion Mountain series. Cited on page
297.
On torsion, see the excellent review by R . T. Ha m mond, New fields in general relativity,
Contemporary Physics 36, pp. 103–114, 1995. Cited on page 299.
H. Kleinert, & J. Z a a nen, World nematic crystal model of gravity explaining the absence of torsion, Physics Letters A 324, pp. 361–365, 2004. Cited on page 299.
The analogy between the situation around line defects and general relativity is explained in E kkeha rt Kröner, Kontinuumstheorie der Versetzungen und Eigenspannungen, Springer, 1958, These ideas have been taken up and pursued by J. D. E shelby,
B. A . B ilb y, and many others after them. Cited on page 299.
Loop quantum gravity is a vast research field. The complete literature is available at arxiv.
org/archive/gr-qc. Cited on page 299.
G . ’ t Ho oft, Crystalline Gravity, International Journal of Modern Physics A 24,
pp. 3243–3255, 2009, and also G . ’ t Ho oft, A locally finite model of gravity, preprint
at arxiv.org/abs/0804.0328. Cited on page 299.
L. Susskind, New concepts for old black holes, preprint at arxiv.org/abs/1311.3335, and also
reference Ref. 147. Cited on page 300.
M. B ot ta C antcheff, Spacetime geometry as statistic ensemble of strings, preprint at
arxiv.org/abs/1105.3658. Cited on page 300.
b ib liography
445
202 N . A rka ni-Hamed, L. Motl, A . Nicolis & C . Va fa, The string landscape, black
holes and gravity as the weakest force, preprint at arxiv.org/abs/hep-th/0601001. The paper
contradicts the strand model in multiple ways. Cited on page 301.
203 M. va n R aamsd onk, Comments on quantum gravity and entanglement, preprint at
arxiv.org/abs/0907.2939. Cited on page 301.
204 W. H. Zurek & K. S. Thorne, Statistical mechanical origin of the entropy of a rotating,
charged black hole, Physical Review Letters 54, pp. 2171–2175, 1985. Cited on page 301.
205 M. Sha p oshnikov & C . Wet terich, Asymptotic safety of gravity and the Higgs boson
mass, preprint at arxiv.org/abs/0912.0208. Cited on page 302.
206 M. M. Anb er & J. F. D ono ghue, On the running of the gravitational constant, preprint
at arxiv.org/abs/1111.2875. Cited on page 302.
207 The 2016 data about modified Newtonian dynamics is found in S. McG augh, F. L elli
208 C . H. L ineweaver & T. M. Dav is, Misconceptions about the big bang, Scientific Amer-
ican pp. 36–45, March 2005. Cited on page 305.
209 Supernova Sea rch Tea m C oll aboration, A . G . R iess & al., Observational
evidence from supernovae for an accelerating universe and a cosmological constant, Astronomical Journal 116, pp. 1009–1038, 1998, preprint at arxiv.org/abs/astro-ph/9805201.
Cited on page 305.
210 Stephen Haw king & Roger P enrose, The Nature of Space and Time, Princeton Uni211 C . Ba l á zs & I. Sz a pudi, Naturalness of the vacuum energy in holographic theories, pre-
print at arxiv.org/abs/hep-th/0603133. See also C . Ba m bi & F. R . Urban, Natural extension of the generalised uncertainty principle, preprint at arxiv.org/abs/0709.1965. The same
point is made by D. A . E asson, P. H. Fra mpton & G . F. Sm o ot, Entropic accelerating
universe, preprint at arxiv.org/abs/1002.4278. Cited on page 307.
212 W. Fischler & L. Susskind, Holography and Cosmology, preprint at arxiv.org/abs/
hep-th/9806039. Cited on page 307.
213 For a review of recent cosmological data, see D. N . Spergel, R . B ea n, O. D oré,
M. R . Nolta, C . L. B ennett, G . Hinshaw, N . Ja rosik, E . Kom atsu,
L. Pag e, H. V. P eiris, L. Verde, C . Ba rnes, M. Ha lpern, R . S. Hill,
A. Ko g u t, M. L im on, S. S. Meyer, N. Od egard, G. S. Tucker, J. L. Weil and,
E . Woll ack & E . L. Wrig ht, Wilkinson Microwave Anisotropy Probe (WMAP) three
year results: implications for cosmology, preprint at arxiv.org/abs/astro-ph/0603449. Cited
on pages 308 and 309.
214 There is a large body of literature that has explored a time-varying cosmological constant,
especially in relation to holography. An example with many references is L. Xu, J. Lu &
W. L i, Time variable cosmological constants from the age of the universe, preprint at arxiv.
org/abs/0905.4773. Cited on page 308.
215 D. Wiltshire, Gravitational energy and cosmic acceleration, preprint at arxiv.org/abs/
0712.3982 and D. Wiltshire, Dark energy without dark energy, preprint at arxiv.org/abs/
0712.3984. Cited on page 308.
216 The attribution to Voltaire could not be confirmed. Cited on page 313.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
versity Press, 1996. Cited on page 306.
Motion Mountain – The Adventure of Physics
& J. S chombert, The radial acceleration relation in rotationally supported galaxies, preprint at arxiv.org/abs/1609.05917, and in F. L elli, S. S. McG augh, J. M. S chombert
& M. S. Pawlowski, One law to rule them all: the radial acceleration relation of galaxies,
preprint at arxiv.org/abs/1610.08981. Cited on page 303.
446
b ib lio graphy
217 V. C red e & C . A . Mey er, The experimental status of glueballs, Progress in Particle and
218
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222
224
225
226 J. W. Moffat & V. T. Thot, A finite electroweak model without a Higgs particle, preprint
at arxiv.org/abs/0812.1991. The ideas go back to D. Ev ens, J. W. Moffat, G . Kleppe
& R . P. Wo oda rd, Nonlocal regularizations of gauge theories, Physical Review D 43,
pp. 499–519, 1991. For more details on how to introduce non-locality while maintaining
current conservation and unitarity, see G . Kleppe & R . P. Wo odard, Non-local YangMills, Nuclear Physics B 388, pp. 81–112, 1992, preprint at arxiv.org/abs/hep-th/9203016.
For a different approach that postulates no specific origin for the W and Z masses, see
J. W. Moffat, Ultraviolet complete electroweak model without a Higgs particle, preprint
at arxiv.org/abs/1006.1859. Cited on page 331.
227 H. B. Nielsen & P. Olesen, A vortex line model for dual strings, Nuclear Physics B 61,
pp. 45–61, 1973. Cited on pages 334 and 382.
228 B. A nd ersson, G . G ustaf son, G . Ing elman & T. Sj östra nd, Parton fragmentation and string dynamics, Physics Reports 97, pp. 31–145, 1983. Cited on page 334.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
223
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Nuclear Physics 63, pp. 74–116, 2009. Cited on page 324.
E . Klem p t & A . Z a itsev, Glueballs, hybrids, multiquarks. Experimental facts versus
QCD inspired concepts, Physics Reports 454, 2007, preprint at arxiv.org/abs/0708.4016.
Cited on pages 324 and 344.
R . V. B uniy & T. W. Kepha rt, A model of glueballs, preprint at arxiv.org/pdf/
hep-ph/0209339; R . V. B uniy & T. W. Kepha rt, Universal energy spectrum of tight
knots and links in physics, preprint at arxiv.org/pdf/hep-ph/0408025; R . V. B uniy &
T. W. Kepha rt, Glueballs and the universal energy spectrum of tight knots and links,
preprint at arxiv.org/pdf/hep-ph/0408027. See also J. P. R a lston, The Bohr atom of
glueballs, preprint at arxiv.org/pdf/hep-ph/0301089. Cited on page 324.
A . J. Niem i, Are glueballs knotted closed strings?, pp. 127–129, in H. Sug a numa,
N . Ishii, M. Oka, H. E nyo, T. Hatsuda, T. Kunihiro & K. Ya z a ki editors,
Color confinement and hadrons in quantum chromodynamics, World Scientific, 2003, preprint at arxiv.org/pdf/hep-th/0312133. See also Y. M. C ho, B. S. Pa rk & P. M. Z ha ng,
New interpretation of Skyrme theory, preprint at arxiv.org/pdf/hep-th/0404181; K. Kond o,
A. Ono, A. Shibata, T. Shinohara & T. Mura kami, Glueball mass from quantized
knot solitons and gauge-invariant gluon mass, Jornal of Physics A 39, pp. 13767–13782, 2006,
preprint at arxiv.org/abs/hep-th/0604006. Cited on pages 324 and 345.
See the one million dollar prize described at www.claymath.org/millennium/
Yang-Mills_Theory. Cited on page 325.
For a clear review on the topic and the planned experiments, see E . Fiorini, Measurement
of neutrino mass in double beta decay, Europhysics News 38, pp. 30–34, 2007, downloadable
at www.europhysicsnews.org. Cited on page 328.
For example, see the detailed discussion of neutrino properties at pdg.web.cern.ch or, in
print, in Ref. 231. Cited on page 329.
For a possible third approach, see A . F. Nicholson & D. C. Kennedy, Electroweak theory without Higgs bosons, International Journal of Modern Physics A 15, pp. 1497–1519,
2000, preprint at arxiv.org/abs/hep-ph/9706471. Cited on page 330.
M. Veltman, The Higgs system, lecture slides at www.nikhef.nl/pub/theory/
academiclectures/Higgs.pdf . See also his CERN Yellow Report 97-05, Reflections on the
Higgs system, 1997, and the paper H. Veltm an & M. Veltm an, On the possibility of
resonances in longitudinally polarized vector boson scattering, Acta Physics Polonica B 22,
pp. 669–695, 1991. Cited on page 330.
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447
229 C . B. Thorn, Subcritical string and large N QCD, preprint at arxiv.org/abs/0809.1085.
230
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236
237
238
239
240
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Cited on page 334.
A . J. B uchmann & E . M. Henley, Intrinsic quadrupole moment of the nucleon, Physical Review C 63, p. 015202, 2000. Alfons Buchmann also predicts that the quadrupole moment of the other, strange ? = 1/2 octet baryons is positive, and predicts a prolate structure for all of them (private communication). For the decuplet baryons, with ? = 3/2,
the quadrupole moment can often be measured spectroscopically, and is always negative. The four Δ baryons are thus predicted to have a negative intrinsic quadrupole moment and thus an oblate shape. This explained in A . J. B uchmann & E . M. Henley,
Quadrupole moments of baryons, Physical Review D 65, p. 073017, 2002. For recent updates, see A . J. B uchm ann, Charge form factors and nucleon shape, pp. 110–125, in
C . N . Pa pa nicolas & A ron B ernstein editors, Shape of Hadrons Workshop Conference, Athens, Greece, 27-29 April 2006, AIP Conference Proceedings 904. Cited on pages
338 and 343.
C . Patrignani & al., (Particle Data Group), Chinese Physics C 40, p. 100001, 2016, or
pdg.web.cern.ch. Cited on pages 338, 339, 340, 359, 361, 372, 373, 374, 375, 446, and 448.
A review on Regge trajectories and Chew-Frautschi plots is W. D rechsler, Das ReggePol-Modell, Naturwissenschaften 59, pp. 325–336, 1972. See also the short lecture on
courses.washington.edu/phys55x/Physics557_lec11.htm. Cited on page 338.
Kurt G ot tfried & Victor F. Weisskopf, Concepts of Particle Physics, Clarendon
Press, Oxford, 1984. Cited on page 339.
G . ’ t Ho oft, G. Isid ori, L. Ma ia ni, A. D. Polosa & V. R iq uer, A theory of scalar
mesons, Physics Letters B 662, pp. 424–430, 2008, preprint at arxiv.org/abs/0801.2288. However, other researchers, such as arxiv.org/abs/1404.5673, argue against the tetraquark interpretation. The issue is not closed. Cited on page 344.
M. Karliner, Doubly heavy tetraquarks and baryons, preprint at arxiv.org/abs/1401.4058.
Cited on page 347.
J. Viro & O. Viro, Configurations of skew lines, Leningrad Mathematical Journal 1,
pp. 1027–1050, 1990, and updated preprint at arxiv.org/abs/math.GT/0611374. Cited on page
347.
W. Thom son, On vortex motion, Transactions of the Royal Society in Edinburgh pp. 217–
260, 1868. This famous paper stimulated much work on knot theory. Cited on page 347.
H. Jehle, Flux quantization and particle physics, Physical Review D 6, pp. 441–457, 1972,
and H. Jehle, Flux quantization and fractional charge of quarks, Physical Review D 6,
pp. 2147–2177, 1975. Cited on page 347.
T. R . Mong an, A holographic charged preon model, preprint at arxiv.org/abs/0801.3670.
Cited on page 347.
The arguments can be found in A . H. C ha mseddine, A . C onnes & V. Mukha nov,
Geometry and the quantum: basics, preprint at arxiv.org/abs/1411.0977 and in
A . H . C ha m seddine & A . C onnes, Why the standard model, Journal of Geometry
and Physics 58, pp. 38–47, 2008, preprint at arxiv.org/abs/0706.3688. Cited on page 348.
Jacob’s rings are shown, for example, in the animation on www.prestidigitascience.fr/index.
php?page=anneaux-de-jacob. They are already published in the book by Tom Tit, La science amusante, 1870, and the images were reprinted the popular science books by Edi Lammers, and, almost a century later on, even in the mathematics column and in one of the
books by Martin Gardner. See also www.lhup.edu/~dsimanek/scenario/toytrick.htm. Cited
on page 351.
448
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242 R . B oug hez al, J. B. Tausk & J. J. va n d er B ij, Three-loop electroweak corrections to
243
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the W-boson mass and sin2 ?eff in the large Higgs mass limit, Nuclear Physics B 725, pp. 3–
14, 2005, preprint at arxiv.org/abs/hep-ph/0504092. Cited on page 359.
The topic of the g-factor of the W boson and of charged fermions is covered in the delightful
paper by Ba rry R . Holstein, How large is the ‘‘natural’’ magnetic moment?, American
Journal of Physics 74, pp. 1104–1111, 2006, preprint at arxiv.org/abs/hep-ph/0607187. Cited
on page 361.
The calculations have been performed in August 2016 by Eric Rawdon. Cited on pages 359
and 361.
The calculations have been performed by Eric Rawdon and Maria Fisher. Cited on page
363.
The quark masses at Planck energy are due to a private communication by Xing Zhizhong and Zhou Shun. They are calculated following the method presented in Quark
mass hierarchy and flavor mixing puzzles, preprint at arxiv.org/abs/1411.2713 and Z hiz hong X ing, He Z ha ng & Shun Z hou, Updated values of running quark and lepton
masses, preprint at arxiv.org/abs/0712.1419. Cited on page 363.
See H. Fritzsch, A . D. Öz er, A scaling law for quark masses, preprint at arxiv.org/abs/
hep-ph/0407308. Cited on page 363.
K. A . Meissner & H. Nicol ai, Neutrinos, axions and conformal symmetry, preprint at
arxiv.org/abs/0803.2814. Cited on pages 366 and 367.
M. Sha p oshnikov, Is there a new physics between electroweak and Planck scale?, preprint
at arxiv.org/abs/0708.3550. Cited on page 367.
Y. D iao, C . E rnst, A . Por & U. Z ieg ler, The roplength of knots are almost linear in
terms of their crossing numbers, preprint at arxiv.org/abs/0912.3282. No citations.
H. Fritzsch & Z. - Z. X ing, Lepton mass hierarchy and neutrino mixing, preprint at
arxiv.org/abs/hep-ph/0601104 Cited on page 374.
The effects of neutrino mixing, i.e., neutrino oscillations, were measured in numerous
experiments from the 1960s onwards; most important were the experiments at SuperKamiokande in Japan and at the Sudbury Neutrino Observatory in Canada. See Ref. 231.
Cited on page 375.
M. Fukugita & T. Yanagida, Baryogenesis without grand unification, Physics Letters
B 174, pp. 45–47, 1986. Cited on page 376.
J. M. C line, Baryogenesis, preprint at arxiv.org/abs/hep-ph/0609145 or the review by
L. C a netti, M. D rew es & M. Sha p oshnikov, Matter and Antimatter in the Universe, preprint at arxiv.org/abs/1204.4186. They explain the arguments that the standard
model with its CKM-CP violation is not sufficient to explain baryogenesis. The opposite view, by the same authors, is found in L. C a net ti, M. D rewes, T. Frossard
& M. Sha p oshnikov, Dark matter, baryogenesis and neutrino oscillations from right
handed neutrinos, preprint at arxiv.org/abs/1208.4607; another opposing view is found in
T. B rauner, CP violation and electroweak baryogenesis in the Standard Model, EPJ Web
of Conferences 0̌ 70/ p. 00078, 2014. Cited on page 376.
Several claims that the coupling constants changed with the age of the universe have
appeared in the literature. The first claim was by J. K. Web b, V. V. Fl a m baum,
C . W. C hurchill, M. J. D rinkwater & J. D. Ba rrow, A search for time variation
of the fine structure constant, Physical Review Letters 82, pp. 884–887, 1999, preprint at
arxiv.org/abs/astro-ph/9803165. None of these claims has been confirmed by subsequent
measurements. Cited on page 380.
b ib liography
449
256 P. P iera nski, S. P rz y byl & A . Stasia k, Tight open knots, European Physical Journal
E 6, pp. 123–128, 2001, preprint at arxiv.org/abs/physics/0103016. No citations.
257 That tight tangles correlate with random tangles was first shown by V. Katritch,
J. B ed nar, D. Michoud, R . G . Sha rein, J. D ub o chet & A . Stasia k, Geometry and physics of knots, Nature 384, pp. 142–145, 1996. It was confirmed by
E . J. Ja nse va n R ensburg, E . Orl a ndini, D. W. Sum ners, M. C . Tesi &
S. G. Whit tington, The writhe of knots in the cubic lattice, Journal of Knot Theory
and its Ramifications 6, pp. 31–44, 1997. No citations.
258 The quasi-quantization of writhe was discovered by V. Katritch, J. B ed na r,
259 The probabilities for random orientations have been calculated with the help of Tyler Spaeth
Wr i t h e
2
Crossing number
3
0
4
0.00046
5
0
6
0
7
0
8
0
9
0
10
0
11
0
12
0
13
0
3
4
5
0.485
0
0.045
0
0.00022
0
0.000007
0
0
0
0
0
0.392
0
0.076
0
0.00011
0
0.000002
0
0
0
0
0
0
0
0
0
0
0
0.000004
0
0.000002
These are the probabilities of knot orientations with a given writhe and crossing number for
the tight open trefoil knot 31 . The smaller numbers are expected to be calculation artefacts.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
and Ronan Lamy, using ideal knot shapes provided by Jason Cantarella. The calculations
yield the following relative sizes of regions with different crossing numbers and writhes for
the open trefoil and the open figure-eight knot.
Motion Mountain – The Adventure of Physics
D. Michoud, R . G . Sha rein, J. D ub o chet & A . Stasiak, Geometry and physics of knots, Nature 384, pp. 142–145, 1996. See also P. P iera nski, In search of ideal
knots, pp. 20–41, and A . Stasiak, J. D ub o chet, V. Katritch & P. P iera nski,
Ideal knots and their relation to the physics of knots, pp. 1–19, both found in A . Stasiak,
V. Katritch & L. H. Kauffman, editors, Ideal Knots, World Scientific, 1998. Most
pedagogic is P. P iera nski & S. P rz y byl, Quasi-quantization of writhe in ideal knots,
European Physical Journal E 6, pp. 117–121, 2001, preprint at arxiv.org/abs/physics/0102067.
See also C . C erf & A . Stasiak, Linear behavior of the writhe versus the number of crossings in rational knots and links, pp. 111–126, in M. I. Monast yrsky editor, Topology
in Molecular Biology, 2007. The quasi-quantization, and in particular the lack of precise
quantization, was confirmed in 2010 by T. A shton, J. C a ntarella, M. P iatek &
E . R aw d on, private communication. No citations.
450
b ib lio graphy
Wr i t h e
-1
1
0.561
0
0.296
0
0.083
0
0.043
0
0.013
0
0.00008
0
0.000005
0
0.00083
0
0.00065
0
0.00038
0
0.0001
0
0.00004
0
0
0
260 The poster on www.physicsoverflow.org referred to J. P. L estone, Physics based calcula-
tion of the fine structure constant, preprint at arxiv.org/abs/physics/0703151. The preprint
has never been published. Cited on page 385.
261 For a highly questionable, but still intriguing argument based on black hole thermodynam-
ics that claims to deduce the limit ? > ln 3/48π ≈ 1/137.26, see S. Hod, Gravitation, thermodynamics, and the fine-structure constant, International Journal of Modern Physics D
19, pp. 2319–2323, 2010. It might well be that similar or other arguments based on textbook
physics will yield more convincing or even better limits in the future. Cited on page 385.
262 V. A rnold, Topological Invariants of Plane Curves and Caustics, American Mathematical
Society, 1994. Cited on page 391.
263 See M. Pospelov & A . R itz, Electric dipole moments as probes of new physics, preprint
at arxiv.org/abs/hep-ph/0504231. Cited on page 392.
264 D. Hilbert, Über das Unendliche, Mathematische Annalen 95, pp. 161–190, 1925. Cited
on page 401.
265 The Book of Twenty-four Philosophers, c. 1200, is attributed to the god Hermes Trismegistos,
Page 423
but was actually written in the middle ages. The text can be found in F. Hud ry, ed., Liber
viginti quattuor philosophorum, Turnholt, 1997, in the series Corpus Christianorum, Continuatio Mediaevalis, CXLIII a, tome III, part 1, of the Hermes Latinus edition project headed
by P. Lucentini. There is a Spinozian cheat in the quote: instead of ‘nature’, the original says
‘god’. The reason why this substitution is applicable is given above. Cited on page 406.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
These are the probabilities of knot orientations with a given writhe and crossing number for
the tight open figure-eight knot 41 . The limits to the precision of the calculation are clearly
noticeable.
The lack of precision is due to the lack of precision of the available tight knot shapes. The
probabilities for random changes in orientation are then deduced from the values in these
tables. Because the knots are tight, it is a good approximation to assume that Reidemeister
I and Reidemeister II moves can be distinguished without ambiguity – in contrast to the
loose knot case, where this is not possible – from the writhe and crossing numbers of the
start and end orientation. No citations.
Motion Mountain – The Adventure of Physics
Crossing number
4
0
5
0.001
6
0
7
0.0004
8
0
9
0.00038
10
0
11
0.000087
12
0
13
0
14
0
15
0
16
0
0
b ib liography
451
266 As a disappointing example, see G illes D eleuze, Le Pli – Leibniz et le baroque, Les Edi-
tions de Minuit, 1988. In this unintelligible, completely crazy book, the author pretends to
investigate the implications of the idea that the fold (in French ‘le pli’) is the basic entity of
matter and ‘soul’. Cited on page 408.
267 Werner Heisenberg, Der Teil und das Ganze, Piper, 1969. The text shows well how
boring the personal philosophy of an important physicist can be. Cited on page 409.
268 John Barrow wrote to the author saying that he might indeed have been the first to have
Motion Mountain – The Adventure of Physics
used the T-shirt image, in his 1988 Gifford Lectures at Glasgow that were a precursor to his
book John D. Ba rrow, Theories of Everything: The Quest for Ultimate Explanation, 1991.
He added that one can never be sure, though. Cited on page 411.
269 R ené D esca rtes, Discours de la méthode, 1637. He used and discussed the sentence
again in his Méditations métaphysiques 1641, and in his Les principes de la philosophie 1644.
These books influenced many thinkers in the subsequent centuries. Cited on page 414.
270 D. D. Kelly, Sleep and dreaming, in Principles of Neural Science, Elsevier, New York, 1991.
The paper summarises experiments made on numerous humans and shows that even during dreams, people’s estimate of time duration corresponds to that measured by clocks.
Cited on page 414.
271 Astrid Lindgren said this in 1977, in her speech at the fiftieth anniversary of Oetinger Verlag,
her German publisher. The German original is: ‘Alles was an Großem in der Welt geschah,
vollzog sich zuerst in der Phantasie eines Menschen, und wie die Welt von morgen aussehen
wird, hängt in großem Maß von der Einbildungskraft jener ab, die gerade jetzt lesen lernen.’
The statement is found in A strid L ind gren, Deshalb brauchen Kinder Bücher, Oetinger
Almanach Nr. 15, p. 14, 1977. Cited on page 417.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
CR EDIT S
Acknowled gments
Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
This volume was first published in 2009. No other person helped developing or exploring the
strand model, until in 2014, Sergei Fadeev suggested to rethink the strand models for the W
and Z bosons. His suggestion triggered many improvements, including a much clearer relation
between the three Reidemeister moves and the intermediate gauge bosons of the three gauge
interactions. The results were first included in 2015, in edition 28. The calculation of the fine
structure constant became more involved, but is still possible.
A few other people helped to achieve progress on specific issues or provided encouragement.
In the first half of the text, stimulating discussions in the 1990s with Luca Bombelli helped structuring the chapter on the contradictions between general relativity and quantum theory, as well
as the chapter on the difference between vacuum and matter. In the years up to 2005, stimulating
discussions with Saverio Pascazio, Corrado Massa and especially Steven Carlip helped shaping
the chapter on limit values.
The second half of the text, on the strand model, owes much to Louis Kauffman. The ideas
found in his books and in his papers inspired the ideas of this text long before we met and exchanged mails. His papers – available on www2.math.uic.edu/~kauffman – and his books are all
worth reading; among them is the fascinating paper Knot Logic and the wonderful book Knots
and Physics, World Scientific, 1991. His ideas on knots, on quantum theory, on measurement, on
particle physics, on set theory and on foundational issues convinced me that strands are a promising direction in the search for a final theory. The breadth of Louis’s knowledge and interests,
the depth of his passion and his warm humanity are exemplary.
I thank Eric Rawdon and Maria Fisher for their ropelength calculations. I also thank Claus
Ernst, Andrzej Stasiak, Ralf Metzler and Jason Cantarella for their input and the fruitful discussions we had.
Hans Aschauer, Roland Netz, Gerrit Bauer, Stephan Schiller, Richard Hoffmann, Axel Schenzle, Reinhard Winterhoff, Alden Mead, Franca Jones-Clerici, Damoon Saghian, Frank Sweetser,
Franz Aichinger, Marcus Platzer, Miles Mutka, and a few people who want to remain anonymous
provided valuable help. My parents, Isabella and Peter Schiller, strongly supported the project.
I thank my mathematics and physics teachers in secondary school, Helmut Wunderling, for the
fire he has nurtured inside me.
The typesetting and book design is due to the professional consulting of Ulrich Dirr. The
typography was much improved with the help of Johannes Küster and his Minion Math font.
The design of the book and its website also owe much to the suggestions and support of my wife
Britta.
From 2007 to 2011, the electronic edition and distribution of the Motion Mountain text was
generously supported by the Klaus Tschira Foundation. I also thank the lawmakers and the taxpayers in Germany, who, in contrast to most other countries in the world, allow residents to use
the local university libraries.
cred its
453
Film credits
Image credits
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
The photograph of the east side of the Langtang Lirung peak in the Nepalese Himalayas, shown
on the front cover, is courtesy and copyright by Kevin Hite and found on his blog thegettingthere.
com. The photograph of the Ultra Deep Field project on page 16 is courtesy of NASA. The drawing by Maurits Escher on page 62 is copyright by the M.C. Escher Heirs, c/o Cordon Art, Baarn,
The Netherlands, who kindly gave permission for its use. The passport photograph of Andrei Sakharov on page 78 is copyright by himself. The photograph of an apheresis machine on
page 179 is courtesy of Wikimedia. The drawing by Peter Battey-Pratt and Thomas Racey of the
belt trick on page 213, taken from Ref. 171, is courtesy and copyright by Springer Verlag. The
graph of the running coupling constants on page 378 is courtesy and copyright by Wim de Boer
and taken from his home page at www-ekp.physik.uni-karlsruhe.de/~deboer. On page 410, the
photograph of Cerro Torre is copyright and courtesy of Davide Brighenti, and found on Wikimedia; the photograph of the green hill on the same page is copyright and courtesy of Myriam70,
and found on her site www.flickr.com/photos/myriam70. The photograph on the back cover, of
a basilisk running over water, is courtesy and copyright by the Belgian group TERRA vzw and
found on their website www.terravzw.org. All drawings are copyright by Christoph Schiller.
Motion Mountain – The Adventure of Physics
The animations of the belt trick on page 176 are copyright and courtesy of Greg Egan; they can be
found on his website www.gregegan.net/APPLETS/21/21.html. I am also grateful to Greg Egan
for expanding his applet so as to show a second option out of the many possible ones for the belt
trick.
The beautiful animation of the belt trick on page 177 and the wonderful and so-far unique
animation of the fermion exchange on page 182 are copyright and courtesy of Antonio Martos.
He made them for this text. They can be found at vimeo.com/62228139 and vimeo.com/62143283.
The animation of the rotating blue ball attached to a sheet on page 178 was made for this book
and is part of the software found at www.ariwatch.com/VS/Algorithms/DiracStringTrick.htm.
The colourful animations of the belt trick with many tails on page 179 and page 180 are copyright and courtesy of Jason Hise at www.entropygames.net, and were made for this text and for
the Wikimedia Commons website.
The film of the chain ring trick on page 350 is copyright and courtesy of Franz Aichinger.
The animations and the film of the falling chain ring trick were included into the pdf file with
the help of a copy of the iShowU software sponsored by Neil Clayton from www.shinywhitebox.
com.
NAME INDEX
A
Abdall a
on particles as Moebius
bands 347
C
Cabibbo, Nicola 372
Cabrera, R. 442
Cachazo, F. 443
Cadavid, A.C. 439
Canetti, L. 448
Cantarella, Jason 449, 452
Carlip, Steven 440, 444, 452
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
B
Balachandran, A.P. 432
Balázs, C. 445
Bambi, C. 445
Barnes, C. 445
Baron, J. 435
Barrow, J.D. 448
Barrow, John D. 451
Bateman, H. 429
Battey-Pratt, Peter 442, 453
Bauer, Gerrit 452
Baylis, W.E. 442
Bean, R. 445
Bednar, J. 449
Beenakker, C.W.J. 431
Beig, R. 430
Bekenstein, Jacob 431, 433
on the entropy bound 131
Bennett, C.L. 445
Berlin, Isaiah 138
Bernreuther, W. 434
Bernstein, Aron 447
Berry, M.V. 438
Besso, Michele 68
Bianchi, E. 432
Bianco, C.L. 431
Bij, J.J. van der 448
Bilby, B.A. 444
Bilson-Thompson, Sundance
439
on particles as triple
ribbons 347
Bimonte, G. 432
Blair, D.G. 430
Blandford, R. 429
Boer, Wim de 378, 453
Bohm, David 437, 438, 441,
442
on entanglement 205
on wholeness 106
Bohr, Niels 429
on minimum action 28
on thermodynamic
indeterminacy 30
Bombelli, Luca 300, 434, 438,
440, 452
Bonner, Yelena 78
Botta Cantcheff, Marcelo 444
on fluctuating strings 300
Boughezal, R. 448
Bousso, R. 431
Brauner, T. 448
Brighenti, Davide 410, 453
Brightwell, G. 434
Britto, R. 443
Bronshtein, Matvei 8
Brown, Stuart 429
Buchmann, A.J. 447
Buchmann, Alfons 447
Buniy, R.V. 446
Busch, Wilhelm 395
Byrnes, J. 442
Motion Mountain – The Adventure of Physics
A
Abbott, B.P. 444
Abdalla, M.C.B. 437
Abdo, A.A. 435
Abraham, A. 442
Adams, Douglas 106
Adler, R.J. 433
Ahluwalia, D.V. 433, 444
Aichinger, Franz 350, 452, 453
Akama, K. 435
Alexandrov, P.S. 437
Ali, A. 436
Allen, Woody 414
Aloisio, R. 435
Alvarez, E. 439
Amati, D. 433
Amelino-Camelia, G. 431, 435,
436
Anaxagoras of Clazimenes
on unification 407
Anber, M.M. 445
Andersson, B. 446
Antonio Martos 182
Argyres, P.C. 437
Aristotle 166, 168, 437
on learning 418
on points 121
on vacuum 83
Arkani-Hamed, N. 445
Arnold, V. 450
Aschauer, Hans 452
Ashtekar, A. 431
Ashton, T. 449
Aspect, A. 441
Aspinwall, P. 433
Augustine of Hippo 423
Avrin, Jack 439
na m e ind ex
C
Cartin
E
Easson, D.A. 445
Eddington, Arthur 438
on particle number 127
Egan, Greg 176, 441, 453
Ehlers, Jürgen 432
on point particles 58
Ehrenfest, Paul
on spinors 198
Ehrenreich, H. 431
Einstein, Albert
last published words 86
on continuity 86
on dropping the
continuum 68, 69, 86
on gods 428
on his deathbed 38
on mathematics 107
on modifying general
relativity 163
on thinking 56
on ultimate entities 69
on unification 22
Ellis, G.F.R. 436
Ellis, J. 436
Enyo, H. 446
Ernst, Claus 230, 448, 452
Escher, Maurits 453
Heirs 62
illustrating circularity 61
Eshelby, J.D. 444
Euclid
on points 72
Evens, D. 446
F
Facchi, P. 437
Fadeev, Sergei 251, 411, 452
Fatio de Duillier, Nicolas 444
Faust 110
Feng, B. 443
Feynman, Richard 255, 432,
440, 441, 443
on many-particle wave
functions 204
Finkelstein, David 74, 299,
434, 440
Finkelstein, Robert 439
on fermions as knots 347
Fiorini, E. 446
Fischler, W. 445
Fisher, Maria 363, 448, 452
Flambaum, V.V. 448
Flint, H.T. 434
Frampton, P.H. 445
Fredenhagen, K. 433
Fredriksson, S. 434
Frenkel, J. 442
Friedberg, R. 434
Friedman, J.L. 432
Fritsche, L. 443
Fritzsch, H. 434, 448
Frossard, T. 448
Fukugita, M. 448
Fushchich, V.I. 442
G
Gadelka, A.L. 437
Gaessler, W. 435
Galante, A. 435
Galindo, A. 442
Garay, L. 433, 436
Gardner, Martin 447
Garret, Don 436
Gehrels, N. 429
Gell-Mann, Murray
on strings 142
Gennes, Pierre-Gilles de 438
Gibbons, Gary 429, 430
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
D
Dalibard, J. 441
Dam, H. van 93, 431, 435, 436
Dante Alighieri
on the basic knot 404
Das, A. 434
Davis, T.M. 445
Dehmelt, Hans 435
Deleuze, Gilles 451
Della Valle, M. 444
Democritus
on learning 418
on particles and vacuum
127
Descartes, René 414, 451
Deutsch, David 161, 440
on explanations 164
DeWitt, B.S. 432
DeWitt, C. 432
Diao, Y. 448
Diner, S. 432
Dirac, Paul 214
on unification 22
Dirr, Ulrich 452
Dis, J. van Lith-van 429
DiSessa, A. 429
Donoghue, J.F. 445
Doplicher, S. 433
Doran, C. 442
Doré, O. 445
Douglas, M.R. 440
Drechsler, W. 447
Drewes, M. 448
Drinkwater, M.J. 448
Dubochet, J. 449
Dumont, Jean-Paul 437
Dyson, Freeman 352
Dällenbach, Werner 69
Dürrenmatt, Friedrich 428
Motion Mountain – The Adventure of Physics
on fluctuating lines 161,
300
Cartin, D. 443
Cerf, C. 449
Challinor, A. 442
Chamseddine, A.H. 447
Chen, B. 431
Cho, Y.M. 446
Christ, N.H. 434
Christiansen, W.A. 435
Churchill, C.W. 448
Ciafaloni, M. 433
Cicero, Marcus Tullius
life 105
Clay Mathematics Institute
325
Clayton, Neil 453
Cline, J.M. 448
CODATA 428
Conde, J. 439
Connes, A. 447
Connes, Alain 348
Cordon Art 453
Coule, D.H. 435
Crease, Robert 443
Crede, V. 446
Cuofano, C. 444
455
456
G
Gibbs
I
Illy, József 434
Ingelman, G. 446
Inverno, Ray d’ 430
Ishii, N. 446
Isidori, G. 447
J
Jacobson, T. 33, 430
Jaekel, M.-T. 430, 433
Jafari, N. 444
Jammer, Max 429
Janssen, Michel 434
Jarlskog, Cecilia 374
Jarosik, N. 445
Jauch, W. 83, 435
Jehle, Herbert 447
on particles as knots 347
Jerven, Walter 416
Johnson, Samuel 428
Jones-Clerici, Franca 452
Ju, L. 430
K
Kaluza, Theodor
on unification 22
Kant, Immanuel 168, 441
Karliner, M. 437, 447
Karliner, Marek 347
Karolyhazy, F. 436
Katritch, V. 449
Katsuura, K. 435
Kauffman, Lou 214
Kauffman, Louis 439, 441,
449, 452
on commutation relations
207
Kelly, D.D. 451
Kempf, Achim 430, 433
Kennard, E.H. 432
Kennedy, D.C. 446
Kephart, T.W. 446
Keselica, D. 442
Klaus Tschira Foundation 452
Klebanov, I. 437
Kleinert, Hagen 299, 444
Klempt, E. 446
Kleppe, G. 446
Knox, A.J. 434
Kochen, S. 441
Kogut, A. 445
Komatsu, E. 445
Kondo, K. 446
Konishi, K. 433
Kostro, L. 430
Kostro, Ludwik 429
Koul, R.K. 438
Kovtun, P. 431
Kramer, M. 430
Kreimer, Dirk 439
on knots in QED 139
Kronecker, Leopold
life 107
Kröner, Ekkehart 299, 444
Kunihiro, T. 446
Küster, Johannes 452
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
H
Hackett, J. 439
Halpern, M. 445
Hammond, R.T. 444
Harari, H. 434
Hartle, J.B. 438
Hatsuda, T. 446
Hattori, T. 435
Haugk, M. 443
Hawking, Stephen 433, 436,
438, 445
Heath, T. 434
Hegel, Friedrich 142
Heisenberg, Werner 432, 451
on symmetry 409
on thermodynamic
indeterminacy 30
on unification 22
Hellund, E.J. 434
Henley, E.M. 447
Henson, J. 440
Hermes Trismegistos 406, 450
Hernandez, L. 439
Hertz, Heinrich
on everything 407
Hestenes, D. 442
Higgs, Peter 331
Hilbert, David 450
famous mathematical
problems 108
his credo 143
on infinity 401
on unification 22
Hilborn, R.C. 438
Hildebrandt, Dieter
life 218
Hiley, B.J. 437
Hill, E.L. 434
Hill, R.S. 445
Hillion, P. 442
Hillman, L. 435
Hinshaw, G. 445
Hise, Jason 179, 180, 453
Hite, Kevin 453
Hod, S. 450
Hoffer, Eric 66
Hoffmann, Richard 452
Hohm, U. 431
Holstein, Barry R. 448
Honner, John 429
’t Hooft, Gerard 74, 299, 434,
436, 444, 447
Hooke, Robert 282
Horowitz, G.T. 438, 440, 443
Hudry, F. 450
Motion Mountain – The Adventure of Physics
Gibbs, Phil 433, 438
on event symmetry 84, 128
Gill, S. 442
Gilmozzi, R. 444
Glashow, Sheldon 255, 440,
443
Gleick, James 443
Goethe, Johann Wolfgang von
on searching 110
Gonzalez, O. 441
Gottfried, Kurt 447
Graf, A.B.A. 441
Green, M.B. 439
Greene, Brian 431, 440
on popularization 52
on superstrings 140
Gregori, Andrea 438
on particle mass 136
Gregory, R. 434
Grillo, A.F. 435
Gross, D.J. 433
Gunzig, E. 432
Gustafson, G. 446
na m e ind ex
na m e ind ex
L
L ammers
N
Nanopoulos, D.V. 436
NASA 17, 453
Nelson, Edward 442, 443
Netz, Roland 452
Newton, Isaac 282
Ng Sze Kui 347, 439
Ng, Y.J. 93, 431, 435, 436
Nicholson, A.F. 446
Nicolai, H. 448
Nicolis, A. 445
Nielsen, H.B. 446
Niemi, A.J. 446
Nietzsche, Friedrich
on walking 149
Nikitin, A.G. 442
Nikolić, H. 443
Nikolić, Hrvoje 219
Nolta, M.R. 445
O
Occam, William of 127
Odegard, N. 445
Ohanian, Hans 431
Oka, M. 446
Olesen, P. 446
Olive, D. 437
Ono, A. 446
Oppenheimer, J. 433
Orlandini, E. 449
Özer, A.D. 448
P
Padmanabhan, T. 432
Paffuti, G. 433
Page, D.N. 444
Page, L. 445
Papanicolas, C.N. 447
Park, B.S. 446
Parmenides 128
Pascazio, Saverio 437, 452
Pati, J.C. 434
Patrignani, C. 447
Pauli, Wolfgang
on gauge theory 248
Pawlowski, M.S. 445
Peiris, H.V. 445
Penrose, Roger 445
Peres, A. 434
Phaedrus 418
Piatek, M. 449
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
M
Maddocks, J.H. 441
Maddox, John 438
life 438
Maggiore, M. 433
Magueijo, J. 435
Maiani, L. 447
Maimonides 64
Majid, S. 438
Major, S.A. 439
Maldacena, J. 439
Mandelbaum, G. 434
Mann, Charles 443
Markopoulou, F. 439, 443
Marmo, G. 432
Marsden, Jerry 272
Martos, Antonio 177, 453
Marx, Groucho
on principles 38
Massa, Corrado 429, 430, 452
Mavromatos, N.E. 436, 438
McGaugh, S. 445
McGaugh, S.S. 445
Mead, Alden 272, 430, 452
Meissner, K.A. 448
Mende, P.F. 433, 438
Mende, Paul
on extension checks 138
Metzler, Ralf 452
Meyer, C.A. 446
Meyer, D. 434, 440
Meyer, S.S. 445
Michoud, D. 449
Mills, Robert 248
Misner, C.W. 430, 432
Moffat, J.W. 446
Monastyrsky, M.I. 449
Mongan, Tom 447
on particles as tangles 347
Montonen, C. 437
Motl, L. 445
Mukhanov, V. 447
Murakami, T. 446
Mutka, Miles 452
Myriam70 410, 453
Méndez, F. 435
Motion Mountain – The Adventure of Physics
L
Lammers, Edi 447
Lamy, Ronan 449
Lange, B. 430
Lao Tse
on motion 408, 416
Lasenby, A. 442
Laughlin, Robert 428
Laërtius, Diogenes 438
Lee Tsung Dao 248, 434
Lee, J. 434, 438, 440
Leibniz, Gottfried Wilhelm
408, 437
on parts 108
on relations in nature 306
Leighton, Robert B. 432
Lelli, F. 445
Lenin (Vladimir Ilyich
Ulyanov) 82
Lerner, L. 442
Lesage, Georges-Louis 444
on universal gravitation
283
Lestone, J.P. 450
Li, W. 445
Li, Y-Q. 431
Lichtenberg, Georg Christoph
on truth 25
Lieu, R. 435
Limon, M. 445
Lindgren, Astrid 417, 451
Lineweaver, C.H. 445
Lloyd, Seth 437
on information 107
Loinger, A. 442
Loll, R. 433
Lomonaco, S.J. 441
Loren, Sophia
on everything 355
Lorentz, Hendrik Antoon 429
on the speed limit 27
Lu, J. 445
Lucentini, P. 450
Lucrece, in full Titus
Lucretius Carus 437
Luzio, E. 435
Lévy-Leblond, J.-M. 441
457
458
P
Pieranski
S
Sabbata, V. de 430
Sabbata, Venzo de 429
Sagan, Carl 436
Saghian, Damoon 452
Sakar, S. 436
Sakharov, Andrei 430, 431
life 78
on matter constituents 121
on maximum particle
mass 39
on minimum length 43
portrait 78
Salam, Abdus 434
on unification 22
Salecker, H. 114, 433
Salogub, V.A. 442
Sanchez, N.G. 438
Sands, Matthew 432
Santamato, E. 442
Santiago, D.I. 433
Schaefer, B.E. 435
Schenzle, Axel 452
Schild, A. 434
Schiller, Britta 452
Schiller, Christoph 429, 430,
432, 443, 453
Schiller, Isabella 452
Schiller, Peter 452
Schiller, R. 441
Schiller, Stephan 452
Schoen, R.M. 430
Schombert, J. 445
Schombert, J.M. 445
Schrödinger, Erwin 442
on thinking 116
Schulmann, Robert 434
Schulz, Charles 135
Schwarz, J.H. 437, 439
Schwinger, Julian 238, 443
Schön, M. 432
Sen, A. 439
Seneca, Lucius Annaeus 412
Shakespeare, William 124, 139,
400
Shalyt-Margolin, A.E. 429
Shapere, Alfred 272, 443
Shaposhnikov, M. 445, 448
Shaposhnikov, Mikhail 302
Sharein, R.G. 449
Shariati, A. 444
Shibata, A. 446
Shinohara, T. 446
Shupe, M.A. 434
Simoni, A. 432
Simplicius 437
Simplicius of Cilicia 124
Sivaram, C. 429, 430
Sjöstrand, T. 446
Slavnov, A.A. 439
Smolin, L. 430, 433, 435, 436,
439, 443
Smoot, G.F. 445
Snyder, H.S. 434
Socrates 418
Son, D.T. 431
Sorabji, R. 434
Sorkin, R.D. 432, 434, 438,
440
Spaeth, Tyler 449
Sparzani, A. 442
Specker, E.P. 441
Spergel, D.N. 445
Spinoza, Baruch 408, 436
Springer Verlag 213, 453
Srinivasan, S.K. 443
Stachel, John J. 434
Stanhope, Philip 403
Starinets, A.O. 431
Stasiak, Andrzej 449, 452
Stewart, Ian 434
Stone, Michael 439
Strominger, A. 438
Sudarshan, E.C.G. 443
Suganuma, H. 446
Sumners, D.W. 449
Supernova Search Team
Collaboration 445
Susskind, L. 444
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
R
Raamsdonk, Mark van 301,
443, 445
Racey, Thomas 442, 453
Ragazzoni, R. 435
Rainer, M. 443
Ralston, J.P. 446
Ramsauer, Carl
life 405
Ramsey, N.F. 434
Randjbar-Daemi, S. 436
Rawdon, Eric 363, 448, 449,
452
Raymer, Michael 433
on the indeterminacy
relation 64
Reidemeister, Kurt 443
on knot deformations 224,
275
Renaud, S. 430, 433
Rensburg, E.J. Janse van 449
Reznik, B. 433
Richter, Burton 440
Riemann, Bernhard
on geometry 38
Riess, A.G. 445
Rindler, Wolfgang 430, 431
Riquer, V. 447
Ritz, A. 450
Rivas, Martin 442
Robbins, J.M. 438
Roberts, J.E. 433
Roger, G. 441
Rosen, N. 434
Rosenfeld, L. 103, 436
Rothman, T. 436
Rovelli, C. 432, 433
Ruffini, Remo 431
Rutherford, Ernest 146
Motion Mountain – The Adventure of Physics
Pieranski, Piotr 449
Piran, T. 436
Pittacus 118
Planck, M. 437
Plato 94, 438
on love 418
on nature’s unity 128
Platzer, Marcus 452
Plotinus 423
Polchinski, J. 438, 440, 443
Polosa, A.D. 447
Pontecorvo, Bruno 375
Por, A. 448
Pospelov, M. 450
Preparata, G. 431
Provero, P. 433
Przybyl, S. 449
na m e ind ex
na m e ind ex
Susskind, Leonard 300, 436,
437, 440, 445
on Planck scale scattering
117
Suzuki, M. 434
Sweetser, Frank 452
Szapudi, I. 445
Szilard, Leo 429
on minimum entropy 30
Sánchez del Río, C. 442
S
Susskind
Wiltshire, D. 308, 445
Winterberg, F. 437
Winterhoff, Reinhard 452
Witten, Edward 439, 440, 443
on duality 137
on infinities 140
on unification 22
Woit, Peter 440
Wolf, C. 435
Wolf, E. 437
Wolff, Barbara 434
Wollack, E. 445
Woodard, R.P. 436, 446
Wright, E.L. 445
Wunderling, Helmut 452
Wussing, H. 437
W
Wald, R.M. 435
Wallstrom, T.C. 443
Wan, Y. 439
Wang, Y.-N. 440
Webb, J.K. 448
Weber, G. 438
Weiland, J.L. 445
Weinberg, Steven 103, 428,
432, 440
on unification 22
Weis, A. 434
Weisskopf, Victor F. 447
Weizel, W. 442
Wen, X.-G. 439
Wetterich, C. 445
Wetterich, Christof 302
Wheeler, John A. 157, 159, 160,
296, 430, 432
life 58
on nature’s principles 405
on topology change 58
on unification 22
really big questions 411
Whittington, S.G. 449
Wigner, Eugene 114, 432, 433
Wikimedia 179, 453
Wilczek, Frank 272, 430, 440,
443
Wilde, Oscar 92, 93
life 91
Wiles, Andrew
on research 410
William of Occam 127
X
Xing Zhi-zhong 448
Xing, Z.-Z. 448
Xing, Zhi-zhong 448
Xu, L. 445
Xue, S.-S. 431
Y
Yanagida, T. 448
Yandell, Ben H. 437
Yang Chen Ning 248
Yazaki, K. 446
Z
Zaanen, J. 444
Zaitsev, A. 446
Zee, A. 443
Zeh, H.D. 435
Zemeckis, Robert 436
Zeno of Elea 70, 119, 121, 128,
218
on motion as an illusion
417
on size 124
Zhang, He 448
Zhang, P.M. 446
Zhao, C. 430
Zhou Shun 448
Zhou, Shun 448
Ziegler, U. 448
Zimmerman, E.J. 114, 433
Zurek, W.H. 445
Zwiebach, B. 439
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
U
Uffink, J. 429
Uglum, J. 437
Unruh, W.G. 433, 435
Urban, F.R. 445
V
Vafa, C. 438, 445
Vancea, I.V. 437
Veltman, H. 446
Veltman, Martin 255, 440,
443, 446
Veneziano, G. 433
Verde, L. 445
Verlinde, Erik 444
on universal gravitation
281
Vigier, J.-P. 442
Viro, Julia 347, 447
Viro, Oleg 347, 447
Voltaire
life 313
Motion Mountain – The Adventure of Physics
T
Takabayasi, T. 442
Tamburini, F. 444
Tanaka, K. 434
Tausk, J.B. 448
Taylor, Gareth 177
Taylor, W. 440
Terence, in full Publius
Terentius Afer
life 115
Tesi, M.C. 449
Thales of Miletus 400
Thomas Aquinas 423
Thomas, L.H. 442
Thomson–Kelvin, William
447
on atoms as knotted
vortices 347
Thorn, C.B. 447
Thorne, K.S. 430, 432, 445
Thot, V.T. 446
Tillich, Paul 90
Tino, G.M. 438
Tiomno, J. 441
Tit, Tom 447
Townsend, P.K. 430
Treder, H.-J. 436
Tregubovich, A.Ya. 429
Tschira, Klaus 452
Tucker, G.S. 445
Turatto, M. 435
Turnbull, D. 431
459
na m e ind ex
460
Z
Zwiebach
Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
SUBJECT INDEX
averaging of strands 187
Avrin, Jack
on particles as Moebius
bands 347
axioms
in a final theory 148
in physics 108, 166–169
axion 273
B
background
continuous 154
dependence 148
differs from physical space
283, 285
illustration of 168
independence 169
space 206, 285
space-time 166
Balinese candle dance 176
ball
tethered 368
Banach–Tarski
paradox/theorem 70, 72
band models 139, 165
Barbero–Immirzi parameter
288
bare quantity 237
baryogenesis 376
baryon
density in universe 309
form factor 343
masses 338
number 315
number conservation 251,
267, 315
number limit 310
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
amoeba and nature 129–130,
170
Anaxagoras of Clazimenes
on unification 407
angle
A
weak mixing 360
acceleration
angular frequency
indeterminacy 28
upper limit, or Planck 36
upper limit, or Planck 36
angular momentum
accelerator, Planck 82
limit for black holes 291
accuracy, maximum 94
lower limit and spin 40
action
upper limit 45–46
as fundamental quantity
anomaly issue 140, 348
401
anti-twister mechanism 180
defined with strands 158,
antimatter
208, 209
belt trick and 179
is change 26
indistinguishability from
lower limit 28–29
matter 53, 77–78, 104
no lower limit for virtual
antiparticle see antimatter
particles 40
apheresis machine 180
principle of least 208
illustration of 179
upper limit 45
aphorism, a physical 114
action, quantum of, ℏ
apple counting 107
as lower limit 36
area
from strands 150
lower limit, or Planck 37
ignored by relativity 57–58 argument
lower limit 28–29
encouraging 22
physics and 8
Aristotle
Planck scales and 60–61
on learning 418
quantum theory implied
on points 121
by 18, 28
on vacuum 83
addition of states 189
arrow
Feynman’s rotating 216
ADM mass 103
Zeno’s flying 121, 218
AdS/CFT correspondence 224
Aspect experiment 205
aether
asymptotic safety 302
useless 70, 157
atoms 116
vortices in 347
Motion Mountain – The Adventure of Physics
Numbers
3
omnipresence in the
standard model 353
462
B
bath
41, 42
microstates 288
no microscopic 301, 367
radiation 290
Schwarzschild 31
shape of 292
size limit 35
sphericity of 292
strand definition 285
universe as inverted 305
universe lifetime and 51
upper power limit 42
blood platelets 180
blurring of tangle 185, 194
Bohm, David
on entanglement 205
on wholeness 106
Bohr, Niels
on minimum action 28
on thermodynamic
indeterminacy 30
Bohr–Einstein discussion 29
Boltzmann constant ? 30, 150
physics and 8
book
perfect, on physics 109
boost see Lorentz boost
border of space 153
Bose–Einstein condensates 49
boson
tangle structure quiz 351
bosonization 138
bosons
as radiation particles 18
definition 174
gauge 317
illustration of 183
illustration of exchange 183
in millennium description
18
masses of W and Z 358
none at Planck scales 77
strand model 183
weak gauge 248
weak intermediate 248
Botta Cantcheff, Marcelo
on fluctuating strings 300
bound see limit
boundary
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
spin 1/2 and 174–180
spin and 134
SU(2) and 245–248
torsion and 299
two options 178
wheels and 219
with 96 tails 180
beta decay, neutrinoless
double 328
big bang
creation and 408
distance in time 93
initial conditions and 136
lack of 148
no creation in 425
not a singularity 101
not an event 92
precision and 53
size of 99
strand illustration of 306
strands and the 306–307
Bilson-Thompson, Sundance
on particles as triple
ribbons 347
biology 169
bit 160
black hole
see also Schwarzschild
radius
as size limit 35
as smallest systems 39
cannot have Planck mass
301, 367
charge limit 291
charged 41
clock limits and 64
definition 31
entropy 131, 287–289
strand illustration of 288
entropy limit 47–48
evaporation 42, 46, 290
falling into a 301
information loss 290
limits 290
lower power limit 46
lower temperature limit 49
magnetic field limit and 44
mass 286
maximum force and 31, 32,
Motion Mountain – The Adventure of Physics
number of leptons 328
number of quarks 320
observed number of 19,
101, 162
quadrupole moment 447
Regge trajectories 339
shape 343
spin 335
strand illustration of
342–344
strand model of 299,
341–344
bath
gluon 269
measurement and 200
perfect 84
photon 228
vacuum as a 155
weak boson 246
beauty
in physics 56
of strands 405
symmetry is not 408
beginning of time 92
Bekenstein’s entropy bound
43, 47–48, 291
Bekenstein, Jacob
on the entropy bound 131
beliefs
about unification 22
in finitude 403
Occam and 127
belt trick 264
and Dirac equation 214
antimatter and 179
Dirac equation and
212–217
illustration of 213
fermions and 331
illustration of 175
parity violation and 179,
247
particle mass and 195,
363–365
phase and 196
quantum theory and 220
saving lives 180
space-time symmetries
and 224
sub j ect ind ex
sub j ect ind ex
B
boxes
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
lower limit 67
mass and vacuum 124
particle rotation and 282
quantum effects and
60–61, 106, 216
table with examples 60
upper limit 51
vacuum and 80
computer
nature is not a 403
conditions
initial, of universe 101, 306
confinement 322
conformal invariance 140
conjecture
no avail 304
conjecture, no avail 301
consciousness
and final theory 21
constant
cosmological see
cosmological constant
coupling see coupling
constant
constants
values of fundamental 150
constituents
common, of particles and
space 86
extended 299
fundamental 147
continuity 70, 158
as time average 170
discreteness and 157
lack of 39, 69, 70, 86, 125
of motion 401
of space and time 68
continuum
see continuity
contradictions between
relativity and quantum
theory 57–63
coordinates
fermionic 133
Grassmann 133
core, tangle
deformation of 224
rotation 222
rotation of 224
Motion Mountain – The Adventure of Physics
electric 228
of objects 117
electric, from strands 152
of point particles 118
elementary ?, physics and
of space 100
8
boxes, limits to 116
fractional 44
braid 133
limit for black holes 291
braid symmetry 134
magnetic, no 230
braiding 351
and mass 352
quantization 386
of tails 373, 374
unit, electric 379
of tails, and mass 366
weak 246–248
brain, and circularity 169
illustration of 247
breaking of SU(2) 250
chirality 229, 385
Broglie, de, wavelength
circularity
lower limit 67
contradictions and 111
Bronshtein cube 8, 301
fundamental 166–169, 187
bucket experiment, resolution
in classical physics 59
218
resolution 218
in modern physics 61
C
in physics 109
Cabibbo angle 372
resolution 128
calculations
classical gravitation 35, 280
non-perturbative 236
classicality
perturbative 236
of measurement apparatus
Calugareanu’s theorem 387,
200
421
climbing
capacitors 44
a green hill 410
capacity
Motion Mountain 410
indeterminacy of 44
symbolism of 427
Carlip, Steven
clocks
on fluctuating lines 161,
limits and Planck time
300
64–66
Casimir effect 57
CODATA 428
catechism, catholic 424
cogito ergo sum 414
categories 106
Coleman–Mandula theorem
centre of group 260
273
Cerro Torre 410
collapse of wave function 201
chain
colour charge 269, 270
film of falling ring 350
strand illustration of 322
illustration of falling ring
three types 270
349
colours in nature, origin of 18,
chain ring trick 352
416
challenge
combination, linear 188
classification 9
complex numbers 272
change
compositeness and strand
is action 26
number 345
nature minimizes 26, 401
Compton wavelength
charge
as displacement limit 29
see also electromagnetism
elementarity and 76, 78
conjugation 242
final theory and 147
463
464
C
corpuscules
D
D-branes 170
dance 176
dangers of a final theory 21
Dante Alighieri
on the basic knot 404
dark energy see cosmological
constant, 19, 162, 304, 307
dark matter 18, 304, 309, 353
challenge 309
is conventional matter 353
death 413
decay
neutrino-less double-beta
353
decoherence 202
defects in vacuum 284, 297
definition
circular 167
deformation
gauge groups and 272
of core 224
of tails 224
degrees of freedom
and boundary 140
and entropy 120
and quantum theory 349
and system surface 49
and volume 84
entropy limit and 41
fundamental 149
in universe 49
of space-time 48
delocalization of W and Z
bosons 331, 332
Democritus
on learning 418
on particles and vacuum
127
denseness 70
density limit for black holes
291
desert
high-energy 350, 354
determinism 84, 85, 410
Deutsch, David
on explanations 164
devils 21
diffeomorphism invariance
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
in neutrinos 376
CP-violating phase 373
CPT invariance 351
CPT symmetry 74, 88, 112
not valid 104
creation 106, 107, 425
is impossible 101
cross sections at Planck scales
117
crossing
as simplest tangle 320
density 188
illustration of 185
illustration of definition
152
in knot theory 153
number, signed 420
of strands 152
orientation average 188
position density 187
switch 150
switch as event 152
switch in space-time 151
switch is observable 346
switch, definition 153, 184
switch, illustration of 151
crystal, nematic world 299
crystals and vacuum 36
cube
Bronshtein 8
physics 8
curiosity 7
current
indeterminacy 44
curvature
see also space-time
around spherical masses
292
of curve 419
space, from strands 283,
292
strand illustration of 283
total 328, 370
upper limit, or Planck 37
curve
rotation, of stars in
galaxies 303
unknotted 316
cutting matter 120
Motion Mountain – The Adventure of Physics
thethered 368
corpuscules ultra-mondains
283
cosmic background radiation
patterns in 107
cosmic string 297, 303
not possible 298
strand illustration of 297
cosmological constant
and strands 307
cosmological constant Λ
307–308
as millennium issue 19, 162
cosmological limits and 51
cosmology and 304
cosmology implied by 18
for flat vacuum 210
from thermodynamics 34
general relativity 51
implies cosmology 50
is dark energy 307
minimum length and 43
problem 143
time variation 311
vacuum density and 51
cosmological limit
see also system-dependent
limit
lowest force 51
to observables 45, 51–52
cosmological scales 90
cosmology 304–312
in one statement 50
Coulomb’s inverse square
relation deduced from
strands 230
counting objects 107
coupling constant
calculation of 381–391
comparison of 384
definition 378
electromagnetic, and
Planck limits 43
illustration of running 378
running and the Higgs
boson 382
covering, topological 71
CP problem, strong 267, 273
CP violation 339, 373, 374, 376
sub j ect ind ex
sub j ect ind ex
D
differences
experiment and gravity
298
with strands 199, 200
doubly special relativity 279,
435
dreams 414–416
duality 167
as an argument for
extension 126
between large and small
124, 125
gravity/gauge 224
space-time see space-time
duality
strings and 437
superstrings and 139
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
E
Eddington, Arthur
on particle number 127
Eddington–Finkelstein
coordinates 71
efficiency
of nature 27
Ehlers, Jürgen
on point particles 58
Ehrenfest, Paul
on spinors 198
Einstein’s field equations
see field equations
Einstein’s hole argument 58
Einstein, Albert
last published words 86
on continuity 86
on dropping the
continuum 68, 69, 86
on gods 428
on his deathbed 38
on mathematics 107
on modifying general
relativity 163
on thinking 56
on ultimate entities 69
on unification 22
Einstein–Bohr discussion 29
Einstein–Brillouin–Keller
quantization 29
electric charge quantum
number 378, 386
electric dipole moment 76,
276, 391–392
electric field
illustration of 231
lower limit 50
upper limit 43, 231
electric potential 196
electrodynamics
from strands 226–243
electromagnetic coupling
constant
see fine structure constant
electromagnetic energy
from strands 230
electromagnetism
cosmological limits 50
from strands 226–243
illustration of 229
Planck limits 43–44, 231
electron
?-factor 199, 236, 242
tangle of 327
electroweak interaction 250
elementary charge see fine
structure constant
elementary particle
see also particle
action limit and 28
cannot be point particle 75
cannot have Planck mass
367
definition 39
properties 313, 355
shape of 116, 119, 120
size limit and 28, 29, 35, 39
size of 117
speed limit and 27
upper energy limit 39
upper mass limit 39
upper momentum limit 39
virtual particles and 40
elements of sets
none in nature 53, 105
elongation 132
emotions
beautiful 7
end
of strand 170
energy
Motion Mountain – The Adventure of Physics
71, 74, 125, 130
differences are approximate
127
dimensions
higher 139, 160
higher, and the final
theory 144
no higher 144, 276, 277,
348
none and superstrings 85
none at Planck scales 53, 71
of space undefined 148
orgin of three 206
origin of spatial 206
dinner parties, physics for 26
dipole moment, electric 76
Dirac equation 73, 212
and belt trick 214
explanation 217, 220
from strands 172
from tangles 212–217
ingredients 216
visualizing the 215
Dirac, Paul
on unification 22
discreteness
continuity and 157
none in nature 111, 403
displacement
indeterminacy 29
limit, quantum 28
distance
defined with strands 158
lower limit, or Planck 37
distinction
none in nature 111
divergence
none in the strand model
276
of QED 237
of vacuum energy 210
dogmas about unification 22
domain walls 297, 303
donate
for this free pdf 9
to this book 9
double beta decay,
neutrinoless 328
double-slit
465
466
E
F
Faust 110
featureless
strands are 153
Fermat’s theorem 410
fermion
as matter particle 18
definition 173, 174, 181
exchange and extension
133–134
from strands 174
illustration of 158
illustration of exchange 181
illustration of wave
function 173
in general relativity 58, 302
in millennium description
18
none at Planck scales 77
spin and extension 134–135
fermionic coordinates 133
Feynman diagram
QED, illustration of 235–237
and braiding 352
high-order QED 139
mechanism for 222–278
strands and 235
strong, strand illustration
of 270
weak 253
weak, illustration of 253
weak, strand illustration of
254
Feynman’s rotating arrow 216
Feynman, Richard
on many-particle wave
functions 204
fiction, no science 354
field
electric 230
magnetic 230
without field 159
field equations
deduced from a drawing
294
from maximum force
32–35
from strands 293–295
films
dreams and 415
Hollywood 98
final theory 405, 428
arguments against 20–21
candidates 22, 139, 165,
299, 347
dangers 21
disinformation 428
extension and 143
higher dimensions and
144, 348
how to find it 144
list of testable predictions
24, 395
modification 147
of motion 20
requirements of 146
steps of the search for a 24
supersymmetry and 144
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
definition 59, 153
from strands 152
fundamental 150
fundamental, illustration
of 147
horizon 305
in dreams 414
symmetry 84, 128
exchange
extension and 133, 134
existence and Planck scales
108
exotic manifold 170
experiment
hard challenge 376
explanation
definition of 164
extension
essential for spin 1/2 177
exchange and 133, 134
final theory and 143
importance of 149
in superstrings 139
of constituents 299
spin and 134
tests of 137
unification and 146
extinction 197
extremal identity 112–113
Motion Mountain – The Adventure of Physics
dark see cosmological
constant
density, critical 308
electromagnetic, from
strands 230
from strands 192, 208, 284
indeterminacy 30, 35, 64
kinetic 208
no region with negative
298
of horizons 286
potential 208
quanton definition 192
speed 27
energy–momentumsystem-dependent limits
and 45
upper limit for elementary
particles 39, 82
energy–momentum tensor 34
ensembles 106
entangled state 202–206
entanglement 202–206, 219
quantum gravity and 301
entropy
at Planck scales 85
Bekenstein’s bound 47–48,
131, 291, 307
Bekenstein–Hawking 289
black hole 47–48, 303
defined with strands 158
lower limit, or Boltzmann
constant ? 30, 36
of black holes 287–289
of gravity 281
of horizons 287–289
of vacuum 47
particle shape and 120
upper limit 291
upper limit to 47–48
equations
non-existence of evolution
148
Escher, Maurits
illustrating circularity 61
essence of universe 110
Euclid
on points 72
Euler angles 198
event
sub j ect ind ex
sub j ect ind ex
F
fine
black hole and 41, 42
electric charge and 41
general relativity implied
by 17, 30–36
principle 34
quantum effects and 36
size limit and 39
why gravitational 41
form factor
of baryons 343
of mesons 338
framing of tangle 419
freedom
asymptotic 272
Frenet frame 419
Frenet ribbon 420
Fulling–Davies–Unruh effect
81, 281, 303
fundamental principle 150
illustration of 398
funnels 169, 406
illustration of 169
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
G
?-factor 199, 236, 242
g-factor
of W boson 361
Galilean physics 26, 59, 401,
415
circular reasoning
resolution 218
gamma-ray bursts 44, 87, 231
gasoline 83
gauge
choice 232–234
covariant derivative 235
freedom, illustration of 232
group, no other 318
interaction 234
interaction antiscreening
383
interaction screening 383
interactions 222–278
interactions, summary 275
symmetry 19, 162, 222,
232–234
symmetry and cores 224
symmetry, not valid 73, 104
theory, non-Abelian 248,
267, 317, 325
transformation 232
U(1) freedom, illustration
of 233
gauge boson
no other types 318
role of 317
strand illustration of 316
weak, illustration of 249
weak, illustration of
incorrect 252
Gedanken experiment see
thought experiment
Gell-Mann matrices 258
Gell-Mann, Murray
on strings 142
gender and physics 137
general relativity
see also field equations
contradicts quantum
theory 57–63
deviations from 303
from maximum force
32–35
from strands 279
horizons and 32
in one statement 30
indeterminacy relation 35
millennium issues and
19–20, 162–163
minimum force 51
non-locality 75
size limit 35
generalizations of the strand
model 163, 405
generalized indeterminacy
principle see
indeterminacy relation,
generalized
generalized uncertainty
principle see
indeterminacy relation,
generalized
generations of quarks 323
generators
of SU(3) and slides 262
Gibbs, Phil
on event symmetry 84, 128
Glashow, Sheldon
Motion Mountain – The Adventure of Physics
table of requirements 147
unmodifiable 163
fine structure constant 18, 152
see also coupling constant,
electromagnetic
charge unit and 50, 378
dead end 385
electrodynamics and 226
estimation of 381
how not to calculate it 385
Planck limits and 43
fine-tuning
none 370–371
finitude
absence of 403
Finkelstein, Robert
on fermions as knots 347
fish in water 123, 138
flavour quantum numbers 315
flavour-changing charged
currents 322
fluctuating lines 300
fluctuations of strands 155, 159
fluid
tangle motion in 195, 360
foam
quantum see foam,
space-time
space-time 311
foam, space-time 157, 296
folds 130, 408, 451
fool
making one of oneself 347
foolishness 21, 347
force
is momentum flow 17, 31
lower limit 46, 51
maximum see force limit,
maximum
no fifth 275
Planck see force limit,
maximum
surface and 31
unlimited for virtual
particles 40
upper limit 42
force limit, maximum ?4 /4?
36
4-force and 42
467
468
G
global
H
Haag’s theorem 273
Haag–Kastler axioms 273
hadron
see meson, baryon
heat and horizons 33
Heisenberg picture 207
Heisenberg, Werner
on symmetry 409
on thermodynamic
indeterminacy 30
on unification 22
heresy, religious 424
Hertz, Heinrich
on everything 407
hidden variables 187
hierarchy
of particle masses 195, 357,
366
Higgs boson
2012 update 331
mass 361
mass prediction 331
predictions about 329
strand illustration of 329,
332
Higgs mechanism 366
Hilbert action 295–296
definition 295
Hilbert space 192
Hilbert’s problems 108
Hilbert’s sixth problem 54,
108–109
Hilbert, David
famous mathematical
problems 108
his credo 143
on infinity 401
on unification 22
hill, gentle green 410
hole argument 58
Einstein’s 284
Hollywood films 98
holography 105, 113, 139, 140,
306–308, 431, 436, 439, 445,
447
’t Hooft, Gerard 299
hoop conjecture 31, 35, 286,
303
hopping
from strand to strand 352
horizon 100, 284
see also black hole
and Planck scales 112
behind a 285
cosmic 50, 305
cosmic, diameter of 100,
102–104
cosmic, distance 97–100
cosmic, none 104–105
cosmic, shape of 100–101
electric charge 41
energy 286
entropy 287–289
entropy limit 47–48
heat flow 33
maximum force 31–35,
41–42
maximum power 31–35,
41–42
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
universal 35, 280–283
gravitational constant ? 30
see also force limit,
maximum, see also power,
upper limit
absence from quantum
theory 58
as conversion constant 59
physics and 8
gravitational wave 87
detectors 87
emitted from atoms 302
graviton 296, 320, 324
illustration of 324
strand illustration of 296
gravity
see gravitation
weak, conjecture 301
Greene, Brian
on popularization 52
on superstrings 140
Gregori, Andrea
on particle mass 136
group
centre of 260
slide 262
GUT see grand unification
Motion Mountain – The Adventure of Physics
on fake unification 255
global coordinate systems 68
glueballs 324–325, 340, 345
gluon
as slide 267
Lagrangian 268–269
self-interaction
illustration of 268
waves 318
gluonic waves 318
gluons 257, 266
Gödel’s incompleteness
theorem 20, 106, 167
gods
and Dante 404
and Einstein 70, 428
and integers 107
and Kronecker 107
and Leibniz 408
and Thales 400
and Trismegistos 450
definition of 423
existence proof 424
favorite T-shirt 411
final theory and 21
intervention of 411
things and 400
Goethe, Johann Wolfgang von
on searching 110
grand unification 380, 396
does not exist 242, 255, 273,
276, 319
not final 164
Grassmann coordinates 133
gravitation 281
see also general relativity,
quantum gravity
and double-slit experiment
298
classical 35, 280
entropic 281–283
entropy and 281
from strands 284
maximum force and 30–35
of superposition 298
quantum 297
strand illustration of 280
strands and 280, 287
surface 31, 33, 47
sub j ect ind ex
sub j ect ind ex
H
Hubble
J
Jarlskog invariant 374
Jehle, Herbert
on particles as knots 347
Jerven, Walter 416
K
Kaluza, Theodor
on unification 22
kaons and quantum gravity 88
Kauffman, Louis
on commutation relations
207
key issues
of unification 392
Killing vector field 34
Klein–Gordon equation 212
knife, limitations of 116
knot
closed 318
definition 318
long 316, 318
models of mesons 345
models of nature 165
models of particles 139
open 316, 318
table of properties 419
topological invariant 229
writhe probabilities 449
knots
dimensionality and 71
in strands 170
Kochen–Specker theorem 202
Kovtun-Son-Starinets
conjecture 431
Kreimer, Dirk
on knots in QED 139
Kronecker, Leopold 107
Kruskal–Szekeres coordinates
71
L
Lagrangian
description of physics 56
Dirac 220
electromagnetic field 231
electroweak 252
from strands 209
gluon 268–269
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
I
idea, Platonic
of observables 94
identity, extremal 112–113, 137
illusion
of motion 417
impenetrability of strands 151,
219
incompleteness theorem,
Gödel’s 20, 106, 167
indeterminacy principle
see indeterminacy relation
indeterminacy relation 194
all together 36
equivalence to Planck
limits 36
for capacitors 44
for current 44
for temperature 30
generalized 68
Heisenberg’s 29, 64, 68, 83
in general relativity 35
in quantum theory 29
in special relativity 28
in thermodynamics 30
indivisibility of nature 403
induction: not a problem 412
infinity
absence of 20, 401
as a lie 402
inflation
and strands 307
lack of 307, 310, 311, 397
information
in the universe 106
no loss 290
initial conditions
of the universe 101, 306
integers 107
interaction
definition 223
electroweak 250
from tangles 224
gauge 222
inversion and 113
mixing 250
no fifth 275
interference
from strands 196, 199
illustration of 197, 199, 200
visualized with strands 199
invariant
see also action, quantum
of, see also force limit, see
also Lorentz invariance,
see also Planck units, see
also speed of light
knot 229
maximum force as 17
Planck unit as 24, 26–44
quantum of action as 18
speed of light as 17
topological 229
inversion 113
irreducibility
computational 20
isotropy of the vacuum 68
issue
key, of unification 392
open: ending funnels 170
open: funnel diameter
behaviour under boosts
170
open: lepton tangles 329
open: W and Z tangles 251
Motion Mountain – The Adventure of Physics
nature of 305
no space beyond 71
nothing behind 397
nothing behind a 71
puzzle 286
quantum effects at 91
radius 32
relation 33
singularities and 58
strand illustration of 285
symmetries at the 104
temperature 290
temperature limit 41
thermodynamic properties
293
types 305
Hubble radius 98
Hubble time 95
hydrogen atom 217
illustration of 217
hypercharge, weak 242, 378
469
470
L
L agrangian
360
list
millennium 18, 161
of experimental
predictions 395
of requirements for a final
theory 147
three important kinds 24
Lloyd, Seth
on information 107
locality
lack of 148
need to abandon 75
none at Planck scales 117
quantum theory vs.
general relativity 58
long knots 316
loop quantum gravity 138
loop, twisted
electromagnetism and 226
loops
time-like 297
Loren, Sophia
on everything 355
Lorentz boosts
maximum force and 31
quantum theory and 215
Lorentz invariance
fluctuations and 75
none at Planck scales 71,
83, 87, 88
of strand model 157
of the vacuum 210
quantum gravity and 88
quantum theory and 214
Lorentz symmetry
see Lorentz invariance
Lorentz transformations
from invariance of ? 211
lattices and 74
minimum length and 71
temperature 49
Lorentz, Hendrik Antoon
on the speed limit 27
M
machine, braiding 351
magnetic
monopole, none 230
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
number 315
strand illustration of 327,
365
tangle of 327
Lesage, Georges-Louis
on universal gravitation
283
LHC see Large Hadron
Collider
Lichtenberg, Georg Christoph
on truth 25
lie, infinity as a 402
life
meaning 413
origin of 21
saving with belt trick 180
light
see also speed of light
deflection by the Sun 59
onion 97
propagation and quantum
gravity 87
scattering of 68
Lilliput
no kingdom 349
lily, beauty of 17
limit
cosmological see
cosmological limit
Planck see Planck limits
limits
in nature, summary 54
our human 417
physics in 26
Planck units as 26
size-dependent 45
system dependent, to all
observables 45–52
system-dependent 45
to cutting 117
to measurements 67
to motion 26
to observables, additional
46
to precision see precision
linear combination 188
lines, skew 347
linking number 387, 420
liquid, tangle motion in 195,
Motion Mountain – The Adventure of Physics
none for strands 348
of QED 234
of general relativity 311
of the standard model 277
of the strong interaction
271
of the weak interaction 249
properties 236
strands and 156, 208–209
superstrings and 141, 142
Lagrangian density see
Lagrangian
Lamb shift, gravitational 88
Landau pole 237
Langtang Lirung 410
Lao Tse
on motion 408, 416
Large Hadron Collider
no discoveries 354
no Higgs boson 331
strand model and 411
W and Z scattering 331,
395, 396
large number hypothesis 103
lattice space-time 74
laziness of nature 26
least action principle
from strands 172, 208–209
in nature 17, 26, 56
valid for strands 277, 295
leather trick 321, 323, 363
illustration of 323
Leibniz, Gottfried Wilhelm
on parts 108
on relations in nature 306
length
defined with strands 158
definition 59
indeterminacy 28
intrinsic 122
lower limit, or Planck 37
maximum 98
minimum 158
Lenin (Vladimir Ilyich
Ulyanov) 82
leptogenesis
none 376, 377
lepton
mass ratios 364
sub j ect ind ex
sub j ect ind ex
M
magnetic
form factor 338
from tangles 334
heavy, illustration of 337
illustration of 334
knotted 345
mass sequences 338
pseudoscalar 334
Regge trajectories 339
shape 338
vector 334
metre rule
cosmic horizon and 98
Planck scales and 66
metric
Planck scales and 71
space 72
microstates of a black hole
288
millennium description of
physics 17–20
millennium list
final summary 398, 399
not solved by superstring
conjecture 142
of open issues 18–20,
161–163
millennium problems
from the Clay
Mathematics Institute 325
minimal coupling 196, 199,
234
minimal crossing number 229
minimization of change
see least action
minimum length 158
Minion Math font 452
mixed state 206
mixing
angle, weak 360
angles 372
matrices 372
quark 373
model
non-commutative 348
tangle, of particles 159
model, topological particle
347
modification of final theory
147
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
sequences of mesons 338
upper limit for elementary
particles 39, 78–79
W and Z bosons 358
without mass 159
matchboxes and universe 99
mathematics
of nature, simplicity of 38,
54
matter
density in universe 309
difference from vacuum 64
extension of 85, 115–144
indistinguishable from
vacuum 52–53, 81–82
made of everything 407
made of nothing 407
mixes with vacuum 80, 81
mattress analogy of vacuum
36
maximons 78
maximum force
see force limit, maximum
maximum speed
see speed of light ?
Maxwell’s field equations 226,
232, 240
meaning in life 413
measurement
always electromagnetic 159
averaging in 85
classicality of 200
definition 401
from strands 201
none at Planck scales 73
precision see precision
problem in quantum
theory 21
problem, quantum 21
mechanism for Feynman
diagrams 222–278
membranes 139
Mende, Paul
on extension checks 138
mesons
charmed, illustration of
336
CP violation 339
excited 338
Motion Mountain – The Adventure of Physics
magnetic charge 230
magnetic field
lower limit 50
upper limit 43, 231
magnetic moment
anomalous 238–240
neutrino 329
magnetic vector potential 196
man-years of work in
superstrings 142
manifolds
see also space, space-time
definition 72
exotic 170
lack of 71
none at Planck scales 39
many-particle state 203
Marx, Groucho
on principles 38
mass
absolute value for particles
366
ADM 103
and braiding of tails 366
black hole 286
calculation for neutrinos
370
calculation of 356
crossing switch rate as 367
eigenstate of quark 373
elementary particle 311, 357
flow, upper limit 40
from strands 194, 284, 330,
346
gap 325
generation 250
gravitational 79, 80, 103,
356, 359
hierarchy 195, 357, 366–369
in universe 102
inertial 80, 356, 360
inverse of 125
maximum density 37
measurement 78–82
negative 83
of bosons 358
rate limit 291
ratios of leptons 364
ratios of quarks 362
471
472
M
modified
rare decays 329
tangle of 327
O
object
motion and 59
observables
basic 157
defined with crossing
switches 156
none at Planck scales 73
system-dependent limits
45
unexplained, millennium
issues 18
value definition 170
observer
definition 82
Occam’s razor 127, 166
octonions 272
Olbers’ paradox 42
operator
Hermitean 207
unitary 207
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
N
National Institute for Play 429
natural units see also Planck
limits, see also Planck
units, 37, 150
naturalness
none 370–371
of standard model 152
nature
and description, table of
difference 167
efficiency of 27
event symmetry 84
has no meaning 111
is indivisible 403
is not finite 403
laziness of 26
limits motion 36
made of one strand 407
multiplicity of 128
no infinity in 401
no sets nor elements 53–54
non-locality of 75
not a computer 403
not a set 128
not discrete 403
table of properties 112
unity of 128
vs. people 418
whether deterministic 84,
410
whole in each of its parts
406
nematic world crystal 299
neurobiology 169
neutrino
magnetic moment 329
mass calculation 370
mixing 375
illustration of 375
tangle of 327
neutrinoless double beta
decay 328
neutron
decay and strand model
255
form factor 343
neutron–antineutron
oscillations 276
Newton’s bucket 218
Newtonian physics
see Galilean physics
Nietzsche, Friedrich
on walking 149
night sky 17
meaning of 408
nightmare scenario 354
no avail conjecture 301, 304
no-hair theorem 132, 286
non-commutative model 348
non-locality 148
natural 169
solves contradictions 75
non-perturbative calculations
236
non-zero action 28
norm of quantum state 191
nothing
difference from universe
90
NSA dream 306
number of stars 102
numbers, no real 52, 73
Motion Mountain – The Adventure of Physics
modified Newtonian
dynamics 303
momentum 194
flow is force 17, 31
indeterminacy 29
quanton definition 194
upper limit for elementary
particles 39
momentum indeterminacy 67
monad 108
Mongan, Tom
on particles as tangles 347
monism 408, 436
monopoles, magnetic, none
230
motion 173
as an illusion 417
continuity of 401
essence of 416
fast 8, 22
helical 360
limited in nature 26, 36
limits to 26
none at Planck scales 86
of particles through
vacuum 352
powerful 8, 22
predictability of 401
quantum 174
strand illustration of 350
tiny 8, 22
translational 352
ultimate questions and
400
uniform 8, 22
Motion Mountain 21
climbing 410
nature of 410
supporting the project 9
top of 400
move see Reidemeister move
multi-particle state
see many-particle state
multiplicity
approximate 130
multiverse nonsense 108, 111,
348, 407, 423
muon
?-factor 199, 236, 242
sub j ect ind ex
sub j ect ind ex
order out of chaos 405
origin
human 413
oscillator, harmonic 273
overcrossing 250, 251
O
order
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
disappearance of 228
entangled
illustration of 205
model of 227
speed of 210
stability of 228
physical space differs from
background space 283
physical system 27, 106
see also system-dependent
limit
illustration of 154
physicists
conservative 55, 412
physics 167
approximations and the
sky 408
beauty in 56
book, perfect 109
definition 17
Galilean 26, 59, 401, 415
gender and 137
golden age 22
in four steps 404
in limit statements 26
in the year 2000 18
map of 8
motion limits in 26, 36
progress of 404
simplicity in 409
simplicity of 26–36, 38
table of progress 404
the science of motion 17
unification in one
statement 403
physics cube 8
Planck acceleration 36
Planck accelerator 82
Planck action ℏ
see action, quantum of
Planck angular frequency 36
Planck area 37
Planck constant ℏ
see action, quantum of
Planck curvature 37
Planck density 37, 79
Planck distance 37
Planck energy 41, 57, 82
see also Planck scales
Motion Mountain – The Adventure of Physics
vacuum 123
of four or more strands 341
pair creation 58, 60, 100
properties 313, 355
quantum 159, 175
spectrum, explanation 352,
P
353
pantheism 424
spectrum, prediction 353
parity 242
spin 1 314–319
none at Planck scales 78
stable 174
violation 246–248
table of correspondence
belt trick and 179
with tangles 346
part of nature see parts
tangle as 160
particle
translational motion 352
see also elementary
virtual 40, 153, 157, 172
particle, see also matter, see parts
also virtual particle
are approximate 127
circular definition of
in nature 127, 401
space-time and 59
in nature, lack of 127
common constituents with
none in nature 403
vacuum 88
pastime, unification as 21
definition 59, 173, 313
path
electrically charged 228
helical 360
exchange 74, 77, 133
integral 199
in the millennium
integral formulation 186
description of physics 18
Pauli equation 199, 212
internal structure 346
from tangles 198
intrinsic property list 315
Pauli matrices 198
lower speed limit 51
Pauli, Wolfgang
made of one strand
on gauge theory 248
314–319
Penrose conjecture 286, 303
made of three strands 326
pentaquarks 347
spin of 340
permutation symmetry
made of two strands 319
not valid 74, 104, 133
mass 356
origin of 184
mass, absolute value 366
perturbation theory
motion 59
convergence of 240
motion through vacuum
failure of 331
352
validity of 237
no exchange at horizon
phase 173
scales 104
average 188
no point 32, 39, 41, 58,
CP-violating 373
75–78
quantum 188
none at Planck scales
tangle 226
77–82, 88, 160
Philippine wine dance 176
photography, limits of 118
number in the universe
photon 227
101–104, 107, 110, 112,
affected by quantum
127–129
gravity 87
number that fits in
473
474
P
Pl anck
incompatible with
unification 24, 63
shape of 115
size of 121, 123
poke see also Reidemeister
move
basic
illustration of 247
gauge group
illustration of 245
transfer
illustration of 244
posets 106
position
from strands 173
indeterminacy 66
positron charge 341
potential
and strands 196
electric 230
indeterminacy 44
electromagnetic 232
magnetic 230
power
lower limit to 46
misuse of 21
surface and 31
upper limit 42
upper limit ?5 /4? 30–36
precession 178
precision
does not increase with
energy 126
fun and 114
lack at Planck scales 71
lack of at Planck scales 68
limited by quantum theory
58
limits 77
limits to 52–54
maximum 94
of age measurements 93
of clocks 66, 93–98
of final theory 147
of length measurements 66
predictability
of motion 401
predictions
about axions 273
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
surprising behaviour at
64–89
vacuum and matter at
52–53, 81–82
Planck speed ? see speed of
light ?
Planck time 150
age measurement and 93
as measurement limit
64–66, 93, 97, 114
shutters and 118
value of 60
Zeno effect and 121
Planck units 36, 148, 150, 158
as invariants 26, 37
as key to unification 143
as limits 26, 37, 67
as natural units 67
corrected, definition 37
definition 24
key to unification 54
Planck value 37
see natural units, see
Planck units
Planck volume 37, 44, 67
number in the universe 49
plate trick 176
platelets 180
Plato
on love 418
on nature’s unity 128
Platonic idea
of time 94
play 21
plural 401
and motion 417
Poincaré symmetry
of vacuum 212
point particles
do not exist 32, 39, 41, 58,
75–78
points
as clouds 119
as tubes 123
cross section of 123
do not exist 24, 32, 40, 53,
66–73
exchange 133
in vacuum 121
Motion Mountain – The Adventure of Physics
definition 39
Planck entropy 150
Planck force ?4 /4?
see force limit, maximum
Planck length 37, 150
see also Planck scales,
Planck energy
antimatter and 53
as measurement limit
66–73
duality and 113
extremal identity and 113
mass limit and 78
shutters and 118
space-time lattices and 74
value of 60
Planck limits
see also Planck units,
natural units
curiosities and challenges
40–44
definition 36
electromagnetic 43–44
Planck mass 78, 367
definition 39
does not exist as black hole
301, 367
does not exist as
elementary particles 367
stand model 367
strand model of 367
Planck mass density 37
Planck momentum
definition 39
Planck scales
as domain of
contradictions 59
definition 37
general relativity and
quantum theory at 59–63
large symmetry at 125
nature at 64–89
no dimensions at 71
no events at 69
no measurements at 73–74
no observables at 73–74
no space-time at 72
no supersymmetry at 78
no symmetries at 73–74
sub j ect ind ex
sub j ect ind ex
P
preon
QED
271–274
226–243
flavour 315
lepton number 315
parity 315
spin 315
quantum particle
properties 313
quantum state 185
quantum theory
and space-time curvature
81
contradicts general
relativity 57–63
displacement limit 28, 29
implied by quantum of
action 18
in one statement 28
measurement problem 21
millennium issues and 19,
162
no infinity in 401
non-zero action 28
of matter 172
space-time curvature and
57
vacuum and 79
quark
flavour change
illustration of 323
mixing
illustration of 372
model acceptance 335
Planck energy table 364
ropelength table 364
quarks
are elementary 75
fractional charge 44
generations of 323
mass ratios 362
mesons and 334
mixing 372, 373
strand illustration of 321,
362
tangles 320–324
quasars 231
quaternions 272
qubit 161
R
race, in quantum gravity 87
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Q
QCD
convergence of 240
quantities
bare 237
quantum action principle
209, 219
quantum effects
are due to extension 407
quantum field theory 217, 383
as approximation of the
strand model 275
strand hopping and 352
quantum fluctuations 106
quantum foam see foam,
space-time
quantum geometry 52, 75, 113
quantum gravity 297
QED and 243
definition of 296
does not exist 304
effects on photons 87
entanglement and 301
entropy and 288
experiments in 87–88
extension and 138, 288
finite entropy 131
from strands 311
gravity waves and 87
is unobservable 304
loop 299
Lorentz symmetry and 88
minimum distance and 37
minimum power and 47
no such theory 23
non-locality 75
Planck scales and 73
predictions 303, 397
predictions about 277
strands and 293, 298
topology and 310
quantum groups 396
quantum lattices 106
quantum measurement
from tangles 199
quantum mechanics see also
quantum theory, 217
quantum numbers
all 315
baryon number 315
charge(s) 315
Motion Mountain – The Adventure of Physics
about cosmology 310
about dark matter 353
about general relativity 302
about grand unification
276
about mesons 338
about supersymmetry 276
about the number of
interactions 275
about the strong
interaction 273
of the strand model 395
on charge quantization 378
on coupling constants 381
on the weak interaction
255
preon models 434
pride 21
principle
fundamental 150
fundamental, illustration
of 147
of least action 17, 26, 56,
208, 277
of least change 17, 26, 56
of maximum force see
force limit, maximum
of non-zero action 28
quantum action 219
probability density 191
process
fundamental 152
projection, minimal 386
propagator 216
properties
intrinsic 85, 313
unexplained, as
millennium issues 18
proton
charge 341
decay 276
form factor 343
mass 343
tangle of 341
puzzle about strands 325
475
476
R
resolution in measurements
118
Reynolds number 360
ribbon
framing 419
models 139, 165, 347
ribbon, mathematical 387
Ricci scalar 295
Ricci tensor 34
Riemann, Bernhard
on geometry 38
ring chain trick 352
rope braiding 351
ropelength 359
measured in diameters 359
rotation
of tangle cores 222
tethered 180
rotation curve
of stars in galaxies 303
rule, superselection 190
running
of coupling 238, 253, 272
of coupling constants 383
of coupling, data 382
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
S
S-duality 124
safety, asymptotic 302
Sakharov, Andrei 78
on matter constituents 121
on maximum particle
mass 39
on minimum length 43
portrait 78
Salam, Abdus
on unification 22
scalar multiplication 189
scalar product 191
scale
extremal, nature at 111
scales, cosmological 90
scattering
by vacuum 82
of longitudinal W and Z
bosons 330, 332
to determine mass 80
to determine size 76, 117
Schrödinger equation 73, 192,
212
Schrödinger picture 186, 207
Schrödinger, Erwin
on thinking 116
Schwarzschild black hole 285
see black hole
Schwarzschild radius
see also black hole
as limit of general
relativity 59
as measurement limit 67,
79, 80
definition 31
entropy and 131
extension and 115
lack of sets and 106
mass and vacuum 124
table with examples 60
science fiction, no 354
scissor trick 176
see-saw mechanism 366, 376
self-linking number 420
sets
not useful to describe
nature 128
not useful to describe
universe 53–54, 105–106
shape
of points 115
touching 119
sheet
and belt trick 178
shivering 151
and divergences 276
short-time average 185
shutter
limits of a 118
time table 118
SI units 150
simplicity
of physics 38
simplification
as guiding idea 409
single atom 83
singularities
horizon and 58
none in nature 84, 148
none in the strand model
291
Motion Mountain – The Adventure of Physics
Raychaudhuri equation 34,
293
Raymer, Michael
on the indeterminacy
relation 64
real numbers, no 52, 73
reductionism
and the final theory 21
and the strand model 160
Regge slope
illustration of 339
Regge trajectories 338
region
of negative energy 303
Raychaudhuri region of negative energy
not possible 298
regions of negative energy 297
regularization
non-local 331
weak bosons and 331
Reidemeister move
and gauge bosons 317
first or type I or twist
225–243
illustration of types 225
second or type II or poke
225, 244–256
third or type III or slide
225, 257–274, 317
Reidemeister’s theorem 275
Reidemeister, Kurt
on knot deformations 224,
275
relativity
as approximation of the
strand model 311
doubly special 435
general see general
relativity
no infinity in 401
special see special relativity
summary on 311
renormalization
at Planck scales 74
of QCD 272
of QED 236
of gravity 303
requirements
for a final theory 146
sub j ect ind ex
sub j ect ind ex
S
size
speed
lower limit 46, 51
of energy 27
of light ?
from strands 209–212
physics and 8
strands and 210
tangles and 351
special relativity implied
by maximum 17, 27
unlimited for virtual
particles 40
upper limit 27, 36
spin
and strand number 340
at Planck scales 77
belt trick and 134, 299
entangled
illustration of 204
extension and 134
foam 138, 299
from strands 174
from tangles 174
general relativity and 58
importance of 134
many-particle
illustration of 203
minimal action and 40
operator 198
orientation 173
superposition
illustration of 201
three-dimensionality and
207
without topology change
432
spin–statistics theorem 174
spinor 198, 215, 218
visualization 215
sponsor
this book 9
this free pdf 9
standard model
and the final theory 23
standard model of particle
physics
millennium description of
physics 18, 146
strands and 152
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
background space 285
physical, from strands 166
shivering 151, 154, 170, 351
topological 72
topology change, not
needed 182
space-time see also curvature
as statistical average 75
as thermodynamic limit 75
circular definition of
particles and 59
continuity 39, 69, 70, 86,
125
curvature 34
curvature and quantum
theory 57, 81
discrete and continuous
157
duality 113, 124, 224
elasticity 36
entropy of 47
foam 157, 296, 311
motion of 48
must be fluctuating 75
no dimensionality at
Planck scales 71
non-commutative 276
not a lattice 74, 106
not a manifold 72
results from upper energy
speed 27
shivering 151, 154, 170, 351
symmetries 224
topology change, not
needed 182
spatial order 68
special relativity
double or deformed 302
doubly special 279
falsified by minimum
length 71
implied by maximum
speed 17
in one statement 27–28
massive tangles and 352
strands and 279
spectrum
of particles, explanation
353
Motion Mountain – The Adventure of Physics
none inside black holes
286
none predicted in nature
303
size
indeterminacy 35
system-dependent limits
and 45
size limit
due to cosmology 50
due to general relativity 35
due to quantum theory 29
due to special relativity 28
skew lines 347
sky, at night 17
slide see Reidemeister move
group 262
illustration of observable
263
illustration of
unobservable 262
slide transfer
illustration of 257
slit
double, and gravity 298
double, and strands 199,
200
Sokolov–Ternov effect 303,
304
space see also background, see
also vacuum
airless, breathing in 86
background 154, 166
border of 153
constituents of 121
curved 284
definition 401
in dreams 415
isotropy and strands 156,
157, 187, 210, 212
mathematical 72
metric 72
no points in 121
none at Planck scales 88
not a lattice 74
not a manifold 72
physical 154, 285
physical, definition of 157
physical, differs from
477
478
S
star
principle 73
superselection rules 190
superstrings
basic principles of 142
black hole entropy and 289
conjecture 140
dimensionality and 85
joke 141
Lagrangian 141
not an explanation 165
overview 139–144
status 142
summary 142
vs. strands 348
supersymmetry
final theory and 144
not correct 74, 78, 133, 164,
242, 255, 276, 319, 353, 396
required 22
strings and 139
support
this book 9
this free pdf 9
surface
force and 31
gravity 31, 33, 47, 290
physical 31, 43
surprises in nature 85, 410
Susskind, Leonard
on Planck scale scattering
117
switch of crossing 150
symmetry
at the horizon 104
beauty and 408
between large and small
124
breaking 250
event 84
no higher 277
none at Planck scales 74,
148
space-time 224
total 125
system, physical
table of correspondence
with strands 155
table of correspondence
with tangles 299
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
extension and 146
foundations 150
fundamental principle 150
generalizations of 163, 405
history of 411
illustration of fundamental
event 147
illustration of fundamental
principle 147
is natural 152
list of predictions 395
of Planck mass 367
of the cosmological
constant 307
other research and the 412
simplicity of 165
slow acceptance 412
table of predictions 395
vs. superstrings 348
stress-energy tensor 42
string nets 139
string trick see also belt trick,
176
quantum theory and 217,
220
strings see superstrings
strong interaction
conservation properties
271
strong nuclear interaction
257–274
SU(2) 245, 378
SU(2) breaking 250
SU(2) field
classical waves 317
SU(3)
and slides 261–268
multiplication table 260
properties 258–261
SU(3) field
classical waves 318
supergravity
not correct 242, 255, 276
supermembranes 139
superparticles 276
superposition
from strands 188
gravitational field of 298
of clocks 58
Motion Mountain – The Adventure of Physics
star
rotation curve in galaxies
303
Stark effect, gravitational 88
stars
in the universe 101
number of 102
state
in dreams 415
Stern–Gerlach experiment
199
stones 174
strand
as cloud 153
averaging 187
braiding and mass 352
definition 153
density 157, 170
diameter 150
diameter behaviour under
boosts 154
diameter, issues 153
ends 170
evolution equation 187
fluctuations, conditions on
186, 294
hopping 352
illustration of fluctuations
186
impenetrability 151, 300
impenetrability of 219
knotted 170
linear combination
illustration of 191
puzzle 325
scalar multiplication
illustration of 190
spacing 157
substance of a 159
translation in other
languages 149
vs. superstring 348
strand model
anomalous magnetic
moment and 238–240
basis of 146
beauty of 405
checking the 171
confirmation of 355
sub j ect ind ex
sub j ect ind ex
Szilard, Leo
on minimum entropy 30
S
Szil ard
of everything 129
of everything does not
exist 405
physical, definition 21
thermodynamics in one
statement 30
thinking
extreme 21
mechanism of 18
Thomson–Kelvin, William
on atoms as knotted
vortices 347
thought experiment
constituents and 121
on extension 137
on force and power 32
on shape 120
on vacuum 80, 123
time 166
beginning of 92
coordinate 66
defined with strands 158
definition 59, 401
does not exist 94
in dreams 414
indeterminacy 44, 64
issue of 151
lower limit, or Planck 36
maximum 91
measurement 58
Platonic idea 94
proper, end of 66
time-like loops 297
tombstone 411
topological models 347
topological space 72
topological writhe 229
definition 386
topology change
of space-time, not needed
182
topology of the universe 310
toroidal black holes 297
torsion
in general relativity 299,
303
of curve 419
total, of a curve 420
translation invariance 68
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
rational, of high
complexity 324
table of properties 419
tails 173, 175
three-stranded 341
tight 358
tight, illustration of 358
topological invariant 229
trivial 319
two-stranded 325
tangle classes
one strand, illustration of
314
three strands, illustration
of 326
two strands, illustration of
319
tangle model
of particles 159
tau
tangle of 327
technicolour 396
temperature
indeterminacy 30
lower limit 49
upper limit 41
vacuum 281
tetraquark
strand illustration of 345
tetraquarks 344
theorem
Banach–Tarski see
Banach–Tarski
paradox/theorem
Calugareanu 421
Coleman–Mandula 273
Fermat’s last 410
Gödel’s see
incompleteness theorem,
Gödel’s
Kochen–Specker 202
no-hair 132, 286
Reidemeister’s 275
spin–statistics 174
Weinberg–Witten 273
theory
final 428
final, not of everything 405
freedom of 408
Motion Mountain – The Adventure of Physics
T
T-duality 124
T-shirt 411
first use 451
tachyon 27
tail
braiding 250
deformations 224
essential for spin 1/2 177
model for particle 134
shifting 374
Tait number 420
tangle 156
alternating 386
as particle 159
blurred 185, 194
braided 341
chirality 385
coloured, illustration of
269
core 173
core deformation
illustration of 223
core rotation 222
illustration of 223
family 335
family of leptons 364
four-stranded 345
framing of 419
function 184, 185
functions are wave
functions 187–194
ideal 358, 419
lepton 327
locally knotted 320
moving
illustration of 193, 211
of four or more strands 341
one-stranded 318–319
phase 226
prime 320
quark 320
rational 320, 341
rational, as coloured
fermion 270
rational, definition 270
479
480
T
trick
unmodifiability of final
theory 163
Unruh effect see
Fulling–Davies–Unruh
effect
Unruh radiation see
Fulling–Davies–Unruh
effect
V
vacuum see also space
as a bath 155
breathing in 86
common constituents with
particles 55, 61, 63, 81, 86,
88, 132
defects in 284, 297
difference from matter 64
elasticity of 36
energy density 19, 57, 162,
210, 307
entropy bound 47–48
entropy of 47
illustration of schematic
156
indistinguishable from
matter 52–53, 81–82
Lorentz invariance 210
made of everything 407
mass error 79–81
mixes with matter 80
none at Planck scales 88
Poincaré symmetry 212
relativity vs. quantum
theory 57–63
strand model 157, 172
tangle function 187
temperature 281
uniqueness of 157, 172, 210,
277, 284, 354
variables, hidden 202
vector
binormal 419
normal 419
Verlinde, Erik
on universal gravitation
281
violence and infinity 402
virtual particles 40, 153, 157,
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
steps of the search for 24
three key issues 392
unified models
assumptions of 166
complexity of 165
requirements of 146
uniqueness 406
uniqueness
of the unified model 406
unitarity
violation in W and Z
boson scattering 330
units
Planck’s natural see Planck
units, natural units
universal gravitation 35, 280
universe
age error of 93, 94
age of 91
U
as inverted black hole 305
U(1) 226–235, 242, 378
definition of 111, 113
U(1) invariance 196
difference from nothing 90
U-duality 124
essence of 110
uncertainty see indeterminacy
fate of 98
uncertainty principle see
finiteness of 304
indeterminacy relation
has no meaning 111
uncertainty relation see
horizon of 305
indeterminacy relation
information and 107
unification see also final
initial conditions 101, 306
theory
luminosity limit 42
arguments against 20–21
mass of 102–104
as lack of finitude 403
matter density 309
as pastime 21
no boundary 104–105
as riddle 21, 24
no information in 107
beliefs and dogmas 22
not a container 129
difficulty of 300
not a physical system 106
disinformation 428
not a set 53–54, 105–106
grand see grand
oscillating 91
unification
power limit 47
is possible 125
radius 100
key to 54
sense of 110
millennium issues and 18
size of 98
of interactions 275, 276, 381
strand illustration of 305
quantum theory and
strand model 304
relativity 56
system dependent limits
reason for failure 255
and 45–52
table of properties 110
requiring extreme
topology of 310
thinking 21
volume of 100
simplicity of 22, 38
Motion Mountain – The Adventure of Physics
trick
belt 176
dirty Higgs 331
leather 363
plate 176
scissor 176
tubes in space 123
tuning, fine
none 370–371
Turing machines 106
twist see Reidemeister move,
317, 387, 420
and gauge
illustration of 233
generalized 234
twist transfer
illustration of 226
twisted loop 226
sub j ect ind ex
sub j ect ind ex
172
at Planck scales 77
viscous fluids
tangles and 195, 360
volume 72
lower limit, or Planck 37
of the universe 100
vortices in the aether 347
V
viscous
crystal, nematic 299
of dreams 414
origin of 413
wormholes 170, 296, 297, 310
do not exist 298, 303
writhe 387
2d 420
3d 421
topological 229, 420
topological, definition 386
writhing number 421
Y
Yang–Mills theory see gauge
theory, non-Abelian
Yukawa coupling 366
Yukawa mechanism 366
Z
Z boson 251
mass 361
mass of 358
strand model 248
two mass values 331
Zeno effect, quantum 28, 121
Zeno of Elea
on motion as an illusion
417
on size 124
Zeno’s argument against
motion 70, 119, 121
resolution 218
zero-point energy 57, 72
Zitterbewegung 216
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
weak bosons 248
weak charge 246–248
weak current
absence of
flavour-changing 347
weak gravity conjecture 301
weak hypercharge 242, 378
weak interaction 244–256
weak mixing angle 360
Weinberg, Steven
on unification 22
Weinberg–Witten theorem
273
Wheeler, John A.
on ‘it from bit’ 160
on mass without mass 159
on nature’s principles 405
on space-time foam 157,
296
on topology change 58
on unification 22
really big questions 411
Wheeler–DeWitt equation 73
wheels
none in nature 219
wholeness, Bohm’s unbroken
106, 438
Wightman axioms 273
Wiles, Andrew
on research 410
Witten, Edward
on duality 137
on infinities 140
on unification 22
words and physics 409
world
Motion Mountain – The Adventure of Physics
W
W boson
g-factor of 361
mass 361
mass of 358
strand model 248
two mass values 331
W polarization
illustration of 330
walls
limitations to 117
water flow
upper limit 40
wave function 173, 185
and crossing size 188
as blurred tangles 217
as rotating cloud 187
collapse 201
collapse from tangles 199
definition 187
is a tangle function
187–194
visualizing the 215
wave, gravitational
emitted from atoms 302
waves
gluonic 318
481
MOTION MOUNTAIN
The Adventure of Physics – Vol. VI
The Strand Model –
A Speculation on Unification
What is the origin of colours?
Which problems in physics are unsolved since the year 2000
and what might be their solution?
At what distance between two points does it become
impossible to find room for a third one in between?
Why do change and motion exist?
What is the most fantastic voyage possible?
Answering these and other questions, this book
gives an entertaining and mind-twisting introduction
to the search for the final theory of physics. The
search leads to the strand model: Based on a
simple principle, strands reproduce quantum theory,
the standard model of particle physics and general
relativity. Strands leave no room for alternative
theories, agree with all experimental data and
allow estimating the fine structure constant.
Christoph Schiller, PhD Université Libre de Bruxelles,
is a physicist and physics popularizer. This entertaining
book is for students, teachers and anybody interested
in modern research about fundamental physics.
Pdf file available free of charge at
www.motionmountain.net
MOTION MOUNTAIN
the adventure of physics – vol.vi
the strand model –
a speculation on unification
www.motionmountain.net
Christoph Schiller
Motion Mountain
The Adventure of Physics
Volume VI
The Strand Model –
A Speculation on Unification
Edition 30, available as free pdf
with films at www.motionmountain.net
Editio trigesima.
Proprietas scriptoris © Chrestophori Schiller
tertio anno Olympiadis trigesimae primae.
Omnia proprietatis iura reservantur et vindicantur.
Imitatio prohibita sine auctoris permissione.
Non licet pecuniam expetere pro aliqua, quae
partem horum verborum continet; liber
pro omnibus semper gratuitus erat et manet.
Thirtieth edition.
Copyright © 1990–2018 by Christoph Schiller,
from the third year of the 24th Olympiad
to the third year of the 31st Olympiad.
This pdf file is licensed under the Creative Commons
Attribution-Noncommercial-No Derivative Works 3.0 Germany
Licence, whose full text can be found on the website
creativecommons.org/licenses/by-nc-nd/3.0/de,
with the additional restriction that reproduction, distribution and use,
in whole or in part, in any product or service, be it
commercial or not, is not allowed without the written consent of
the copyright owner. The pdf file was and remains free for everybody
to read, store and print for personal use, and to distribute
electronically, but only in unmodified form and only at no charge.
To Britta, Esther and Justus Aaron
τῷ ἐμοὶ δαὶμονι
Die Menschen stärken, die Sachen klären.
PR EFACE
T
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Christoph Schiller
Motion Mountain – The Adventure of Physics
his book is for anybody who is intensely curious about motion. Why and how do
hings, people, trees, stars, images or empty space move? The answer leads
o many adventures, and this book presents one of the best of them: the search
for a precise, unified and final description of all motion.
The aim to describe all motion – everyday, quantum and relativistic – implies a large
project. This project can be structured using the diagram shown in Figure 1, the so-called
Bronshtein cube. The previous volumes have covered all points in the cube – all domains
of motion – except the highest one. This remaining point contains the final and unified
description of all motion. The present volume briefly summarizes the history of this old
quest and then presents an intriguing, though speculative solution to the riddle.
The search for the final, unified description of motion is a story of many surprises.
First, twentieth-century research has shown that there is a smallest measurable distance
in nature, the Planck length. Then it appeared that matter cannot be distinguished from
empty space at those small distances. A last surprise dates from this century: particles and
space appear to be made of strands, instead of little spheres or points. The present text
explains how to reach these surprising conclusions. In particular, quantum field theory,
the standard model of particle physics, general relativity and cosmology are shown to
follow from strands. The three gauge interactions, the three particle generations and the
three dimensions of space turn out to be due to strands. In fact, all the open questions
of twentieth-century physics about the foundations of motion, including the origin of
colours and of the parameters of the standard model, appear to be answerable.
The strand model, as presented in this text, is an unexpected result from a threefold
aim that the author has pursued since 1990, in the five previous volumes of this series:
to present the basics of motion in a way that is up to date, captivating and simple. While
the previous volumes introduced the established parts of physics, this volume presents,
in the same captivating and playful way, a speculation about unification. Nothing in this
volume is established knowledge – yet. The text is the original presentation of the topic.
The aim for maximum simplicity has been central in deducing this speculation.
The search for a final theory is one of the great adventures of life: it leads to the limits
of thought. The search overthrows several of our thinking habits about nature. This can
produce fear, but by overcoming it we gain strength and serenity. Changing thinking
habits requires courage, but it produces intense and beautiful emotions. Enjoy them.
8
preface
Final, unified description of motion
Adventures: describing precisely all motion, understanding
the origin of colours, space -time and particles, enjoying
extreme thinking, calculating masses and couplings,
catching a further, tiny glimpse of bliss (vol. VI).
PHYSICS:
Describing motion with precision,
i.e., using the least action principle.
Special relativity
Adventures: light,
magnetism, length
c contraction, time
limits dilation and
G
h, e, k
fast E0 = mc2
limits
limit
motion (vol. II).
uniform
tiny
motion
motion
Quantum field theory
(the ‘standard model’)
Adventures: building
accelerators, understanding quarks, stars,
bombs and the basis of
life, matter & radiation
(vol. V).
Quantum theory
Adventures: biology,
birth, love, death,
chemistry, evolution,
enjoying colours, art,
paradoxes, medicine
and high-tech business
(vol. IV and vol. V).
Galilean physics, heat and electricity
The world of everyday motion: human scale, slow and weak.
Adventures: sport, music, sailing, cooking, describing
beauty and understanding its origin (vol. I);
using electricity, light and computers,
understanding the brain and people (vol. III).
F I G U R E 1 A complete map of physics, the science of motion, as first proposed by Matvei Bronshtein
(b. 1907 Vinnytsia, d. 1938 Leningrad). The map is of central importance in the present volume. The
Bronshtein cube starts at the bottom with everyday motion, and shows the connections to the fields of
modern physics. Each connection increases the precision of the description and is due to a limit to
motion that is taken into account. The limits are given for uniform motion by the gravitational constant
G, for fast motion by the speed of light c, and for tiny motion by the Planck constant h, the elementary
charge e and the Boltzmann constant k.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Classical gravity
Adventures:
climbing, skiing,
space travel,
the wonders of
astronomy and
geology (vol. I).
Quantum theory
with classical gravity
Adventures: bouncing
neutrons, understanding tree
growth (vol. V).
Motion Mountain – The Adventure of Physics
General relativity
Adventures: the
night sky, measuring curved and
wobbling space,
exploring black
holes and the
universe, space
and time (vol. II).
An arrow indicates an
increase in precision by
adding a motion limit.
preface
9
Using this bo ok
To get a quick overview, read the first chapter and continue with the summary sections
only. There are summaries at the end of each chapter. In addition, throughout the text,
⊳ Important ideas are marked with a triangle.
Feedback
Supp ort
Your donation to the minuscule, charitable, tax-exempt non-profit organisation that produces, translates and publishes this book series is welcome. For details, see the web page
www.motionmountain.net/donation.html. The German tax office checks the proper use
of your donation. If you want, your name will be included in the sponsor list. Thank you
in advance for your help, on behalf of all readers across the world. And now, enjoy the
reading.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Receiving an email from you at fb@motionmountain.net, either on how to improve the
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net/prizes.html, would be delightful. All feedback will be used to improve the next edition. For a particularly useful contribution you will be mentioned – if you want – in the
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Motion Mountain – The Adventure of Physics
Marginal notes refer to bibliographic references, to other pages or to challenge solutions. In the colour edition, such notes and also the pointers to footnotes and to other
websites are typeset in green. In the free pdf edition of this book, available at www.
motionmountain.net, all green pointers and links are clickable. The pdf edition also contains all films; they can be watched directly in Adobe Reader. Over time, links on the internet tend to disappear. Most links can be recovered via www.archive.org, which keeps
a copy of old internet pages.
Challenges are included regularly. Solutions and hints are given in the appendix. Challenges are classified as easy (e), standard student level (s), difficult (d) and research level
(r). Challenges for which no solution has yet been included in the book are marked (ny).
A paper edition of this book is available, either in colour or in black and white, from
www.amazon.com or www.createspace.com. So is a Kindle edition.
CONTENTS
17
1
From m illennium physics to unification
Against a final theory 20 • What went wrong in the past 22 • An encouraging
argument 22
Summary: how to find the final theory of motion
2
P hysics in lim it statements
Simplifying physics as much as possible
Everyday, or Galilean, physics in one statement 26 • Special relativity in one statement 27 • Quantum theory in one statement 28 • Thermodynamics in one statement 30 • General relativity in one statement 30 • Deducing general relativity 32
• Deducing universal gravitation 35 • The size of physical systems in general relativity 35 • A mechanical analogy for the maximum force 35
Planck limits for all physical observables
Physics, mathematics and simplicity 38 • Limits to space, time and size 38 • Mass
and energy limits 39 • Virtual particles – a new definition 40 • Curiosities and
fun challenges about Planck limits 40
Cosmological limits for all physical observables
Size and energy dependence 45 • Angular momentum and action 45 • Speed 46
• Force, power and luminosity 46 • The strange charm of the entropy bound 47
• Curiosities and fun challenges about system-dependent limits to observables 48
• Cosmology in one statement 50 • The cosmological limits to observables 51
• Minimum force 51 • Limits to measurement precision and their challenge to
thought 52 • No real numbers 52 • Vacuum and mass: two sides of the same
coin 52 • Measurement precision and the existence of sets 53
Summary on limits in nature
3
G eneral rel ativity v ersus q ua ntum theory
The contradictions 57 • The origin of the contradictions 58 • The domain of contradictions: Planck scales 59 • Resolving the contradictions 61 • The origin of
points 62
Summary on the clash between the two theories
4
D oes m at ter d iffer from vacuum?
Farewell to instants of time 64 • Farewell to points in space 66 • The generalized indeterminacy relation 68 • Farewell to space-time continuity 68
•
Farewell to dimensionality 71 • Farewell to the space-time manifold 72 • Farewell
to observables, symmetries and measurements 73 • Can space or space-time be a
lattice? 74 • A glimpse of quantum geometry 75 • Farewell to point particles 75
• Farewell to particle properties 77 • A mass limit for elementary particles 78 •
Farewell to massive particles – and to massless vacuum 79 • Matter and vacuum
are indistinguishable 81 • Curiosities and fun challenges on Planck scales 82 •
Common constituents 86
• Experimental predictions 87
Summary on particles and vacuum
5
What is the d ifference b et w een the univ erse a nd nothing?
Cosmological scales 90 • Maximum time 91 • Does the universe have a definite
24
26
26
36
45
54
56
63
64
88
90
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
P reface
Using this book 9 • Feedback 9 • Support 9
Motion Mountain – The Adventure of Physics
7
contents
age? 91 • How precise can age measurements be? 92 • Does time exist? 93 •
What is the error in the measurement of the age of the universe? 94 • Maximum
length 98 • Is the universe really a big place? 98 • The boundary of space – is
the sky a surface? 100 • Does the universe have initial conditions? 101 • Does the
universe contain particles and stars? 101 • Does the universe have mass? 102 • Do
symmetries exist in nature? 104 • Does the universe have a boundary? 104 • Is
the universe a set? 105 • Curiosities and fun challenges about the universe 107
• Hilbert’s sixth problem settled 108 • The perfect physics book 109 • Does the
universe make sense? 110
• Abandoning sets and discreteness eliminates contradictions 111 • Extremal scales and open questions in physics 111 • Is extremal
identity a principle of nature? 112
Summary on the universe
A physical aphorism 114
113
115
6
116
The sha pe of p oints – extension in nature
124
131
133
134
135
137
143
7
The basis of the stra nd m od el
Requirements for a final theory 146
• Introducing strands 150 • Events, processes, interactions and colours 152 • From strands to modern physics 152 • Vacuum 156 • Observable values and limits 157 • Particles and fields 159 • Curiosities and fun challenges about strands 159 • Do strands unify? – The millennium list
of open issues 161 • Are strands final? – On generalizations and modifications 163
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
127
Motion Mountain – The Adventure of Physics
The size and shape of elementary particles
Do boxes exist? 116 • Can the Greeks help? – The limitations of knives 116 • Are
cross sections finite? 117 • Can we take a photograph of a point? 118 • What is
the shape of an electron? 119 • Is the shape of an electron fixed? 120 • Summary
of the first argument for extension 121
The shape of points in vacuum
Measuring the void 123 • What is the maximum number of particles that fit inside
a piece of vacuum? 123 • Summary of the second argument for extension 123
The large, the small and their connection
Is small large? 124 • Unification and total symmetry 125 • Summary of the third
argument for extension 126
Does nature have parts?
Does the universe contain anything? 129 • An amoeba 129 • Summary of the
fourth argument for extension 130
The entropy of black holes
Summary of the fifth argument for extension 132
Exchanging space points or particles at Planck scales
Summary of the sixth argument for extension 134
The meaning of spin
Summary of the seventh argument for extension 135
Curiosities and fun challenges about extension
Gender preferences in physics 137
Checks of extension
Current research based on extended constituents 138 • Superstrings – extension
plus a web of dualities 139 • Why superstrings and supermembranes are so appealing 140 • Why the mathematics of superstrings is difficult 141 • Testing superstrings: couplings and masses 141 • The status of the superstring conjecture 142
Summary on extension in nature
121
146
11
12
contents
• Why strands? – Simplicity 165 • Why strands? – The fundamental circularity
of physics 166 • Funnels – an equivalent alternative to strands 169 • Knots and
the ends of strands 170
Summary on the fundamental principle – and on continuity
170
172
8
244
257
G auge interactions d ed uced from stra nds
Interactions and phase change 222 • Tail deformations versus core deformations 224
Electrodynamics and the first Reidemeister move
Strands and the twist, the first Reidemeister move 226 • Can photons decay, disappear or break up? 228 • Electric charge 228 • Challenge: What topological invariant is electric charge? 229 • Electric and magnetic fields and potentials 230 •
The Lagrangian of the electromagnetic field 231 • U(1) gauge invariance induced
by twists 232 • U(1) gauge interactions induced by twists 234 • The Lagrangian of
QED 234 • Feynman diagrams and renormalization 235 • The anomalous magnetic moment 238 • Maxwell’s equations 240 • Curiosities and fun challenges
about QED 242 • Summary on QED and experimental predictions 242
The weak nuclear interaction and the second Reidemeister move
Strands, pokes and SU(2) 245 • Weak charge and parity violation 246 • Weak bosons 248 • The Lagrangian of the unbroken SU(2) gauge interaction 249 • SU(2)
breaking 250 • Open issue: are the W and Z tangles correct? 251 • The electroweak
Lagrangian 252 • The weak Feynman diagrams 253 • Fun challenges and curiosities about the weak interaction 255 • Summary on the weak interaction and
experimental predictions 255
The strong nuclear interaction and the third Reidemeister move
Strands and the slide, the third Reidemeister move 257 • An introduction to
SU(3) 258 • From slides to SU(3) 261 • The strand model for gluons 266 • The
gluon Lagrangian 268 • Colour charge 269 • Properties of the strong interaction 271 • The Lagrangian of QCD 271 • Renormalization of the strong interaction 272 • Curiosities and fun challenges about SU(3) 272 • Summary on the
strong interaction and experimental predictions 273
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
226
9
Motion Mountain – The Adventure of Physics
Strands, vacuum and particles 172 • Rotation, spin 1/2 and the belt trick 174 • The
belt trick is not unique 178 • An aside: the belt trick saves lives 180 • Fermions
and spin 181 • Bosons and spin 183 • Spin and statistics 184 • Tangle functions:
blurred tangles 184 • Details on fluctuations and averages 186 • Tangle functions
are wave functions 187 • Deducing the Schrödinger equation from tangles 192 •
Mass from tangles 194 • Potentials 196 • Quantum interference from tangles 196
• Deducing the Pauli equation from tangles 198 • Rotating arrows, path integrals
and interference 199 • Measurements and wave function collapse 199 • Hidden
variables and the Kochen–Specker theorem 202 • Many-particle states and entanglement 202 • Mixed states 206 • The dimensionality of space-time 206 •
Operators and the Heisenberg picture 207 • Lagrangians and the principle of least
action 208 • Special relativity: the vacuum 209 • Special relativity: the invariant
limit speed 210 • Dirac’s equation deduced from tangles 212 • Visualizing spinors
and Dirac’s equation using tangles 215 • Quantum mechanics vs. quantum field
theory 217 • A flashback: settling three paradoxes of Galilean physics 218 • Fun
challenges about quantum theory 218
Summary on quantum theory of matter: experimental predictions
220
222
Q ua ntum theory of m at ter d ed uced from stra nd s
contents
13
279
311
10 G eneral rel ativity d ed uced from stra nds
Flat space, special relativity and its limitations 279 • Classical gravitation 280 •
Deducing universal gravitation from black hole properties 281 • Summary on universal gravitation from strands 283 • Curved space 283 • The structure of horizons and black holes 284 • Is there something behind a horizon? 285 • Energy of
black hole horizons 286 • The nature of black holes 286 • Entropy of vacuum and
matter 287 • Entropy of black holes deduced from the strand model 287 • Temperature, radiation and evaporation of black holes 290 • Black hole limits 290 •
Curvature around black holes 292 • The shape of non-rotating black holes 292 •
The field equations of general relativity 293 • Equations from no equation 294 •
The Hilbert action of general relativity 295 • Space-time foam 296 • Gravitons,
gravitational waves and their detection 296 • Open challenge: Improve the argument for the graviton tangle 297 • Other defects in vacuum 297 • The gravity
of superpositions 298 • Torsion, curiosities and challenges about quantum gravity 299 • Predictions of the strand model about gravity 302
Cosmology
The finiteness of the universe 304
• The big bang – without inflation 306 •
The cosmological constant 307 • The value of the matter density 309 • Open challenge: What are the effects of dark matter? 309 • The topology of the universe 310
• Predictions of the strand model about cosmology 310
Summary on millennium issues about relativity and cosmology
313
11 The pa rticle spectrum d ed uced from stra nd s
304
314
319
326
341
346
Particles and quantum numbers from tangles 313
Particles made of one strand
Unknotted curves 316 • Gauge bosons – and Reidemeister moves 317 • Open or
long knots 318 • Closed tangles: knots 318 • Summary on tangles made of one
strand 318
Particles made of two strands
Quarks 320 • Quark generations 323 • The graviton 324 • Glueballs 324 • The
mass gap problem and the Clay Mathematics Institute 325 • A puzzle 325 • Summary on two-stranded tangles 325
Particles made of three strands
Leptons 327 • Open issue: are the lepton tangles correct? 329 • The Higgs boson –
the mistaken section from 2009 329 • The Higgs boson – the corrected section of
2012 331 • 2012 predictions about the Higgs 333 • Quark-antiquark mesons 334
• Meson form factors 338 • Meson masses, excited mesons and quark confinement 338 • CP violation in mesons 339 • Other three-stranded tangles and glueballs 340 • Spin and three-stranded particles 340 • Summary on three-stranded
tangles 341
Tangles of four and more strands
Baryons 341 • Tetraquarks and exotic mesons 344 • Other tangles made of four
or more strands 345 • Summary on tangles made of four or more strands 345
Fun challenges and curiosities about particle tangles
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Summary and predictions about gauge interactions
Predicting the number of interactions in nature 275 • Unification of interactions 275 • No divergences 276 • Grand unification, supersymmetry and other
dimensions 276 • No new observable gravity effects in particle physics 277 • The
status of our quest 277
Motion Mountain – The Adventure of Physics
275
14
352
355
356
372
CPT invariance 351 • Motion through the vacuum – and the speed of light 351
Summary on millennium issues about particles and the vacuum
The omnipresent number 3 353 • Predictions about dark matter, the LHC and the
vacuum 353
395
398
13 E xperimental predictions of the stra nd m od el
Final summary about the millennium issues
400
400
414
416
416
14 The top of Motion Mounta in
Our path to the top
Everyday life: the rule of infinity 401 • Relativity and quantum theory: the absence
of infinity 401
• Unification: the absence of finitude 403
New sights
The beauty of strands 405 • Can the strand model be generalized? 405
• What
is nature? 406 • Quantum theory and the nature of matter and vacuum 407 •
Cosmology 407 • Musings about unification and strands 408 • The elimination
of induction 412
• What is still hidden? 413
A return path: je rêve, donc je suis
What is the origin of colours?
Summary: what is motion?
418
Postface
419
a Knot a nd ta ng le g eom etry
404
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
12 Pa rticle properties d ed uced from stra nds
The masses of the elementary particles
General properties of particle mass values 357 • Boson masses 358 • W/Z boson
mass ratio and mixing angle (in the 2016 tangle model) 359 • The g-factor of the
W boson 361 • The Higgs/Z boson mass ratio 361 • A first approximation for absolute boson mass values 361 • Quark mass ratios 362 • Lepton mass ratios 364
• On the absolute values of particle masses 366 • Analytical estimates for particle
masses 369 • Open issues about mass calculations 370 • On fine-tuning and naturalness 370 • Summary on elementary particle masses and millennium issues 371
Mixing angles
Quark mixing – the experimental data 372 • Quark mixing – explanations 373 •
A challenge 374 • CP violation in quarks 374 • Neutrino mixing 375 • CP violation in neutrinos 376 • Open challenge: calculate mixing angles and phases ab
initio 376 • Summary on mixing angles and the millennium list 377
Coupling constants and unification
Interaction strengths and strands 379 • Strands imply unification 380 • Calculating coupling constants 381 • First hint: the energy dependence of physical quantities 382 • Second hint: the running of the coupling constants at low energy 383
• Third hint: further predictions at low energy 383 • The running of the coupling constants up to Planck energy 384 • Limits for the fine structure constant do
not provide explanations 385 • Charge quantization and topological writhe 385
• Charge quantization and linking number 386 • How to calculate coupling constants 387 • Coupling constants in the strand model 388 • Deducing alpha from
precession 389 • Deducing the weak coupling 390 • Deducing the strong coupling 390 • Open challenge: calculate coupling constants with precision 391 • Electric dipole moments 391 • Five key challenges about coupling strengths 392 •
Summary on coupling constants 394
Motion Mountain – The Adventure of Physics
378
contents
422
C hallenge hints and solu tions
428
B ib liography
452
C redits
Acknowledgments 452 • Film credits 453 • Image credits 453
454
Na m e ind ex
461
Sub ject ind ex
The Strand Model –
A Speculation on
Unification
Where, through the combination of
quantum mechanics and general relativity,
the top of Motion Mountain is reached,
and it is discovered
that vacuum is indistinguishable from matter,
that there is little difference between the large and the small,
that nature can be described by strands,
that particles can be modelled as tangles,
that gauge interactions appear naturally,
that colours are due to strand twisting,
and that a complete description of motion is possible.
Chapter 1
FR OM MILLENNI UM PHYSICS TO
UNIFICAT ION
L
Ref. 2
2. In nature, there is an invariant maximum energy speed, the speed of light ?. This
invariant maximum implies special relativity. Among others, it implies that mass and
energy are equivalent, as is observed.
Page 30
3. In nature, there is an invariant highest momentum flow, the Planck force ?4 /4?. This
invariant maximum implies general relativity, as we will recall below. Among others,
** The photograph on page 16 shows an extremely distant, thus extremely young, part of the universe, with
its large number of galaxies in front of the black night sky (courtesy NASA).
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 1, Ref. 3
1. In nature, motion takes place in three dimensions of space and is described by
the least action principle. Action is a physical quantity that describes how much
change occurs in a process. The least action principle states: motion minimizes change.
Among others, the least change principle implies that motion is predictable, that energy is conserved and that growth and evolution are natural processes, as is observed.
Motion Mountain – The Adventure of Physics
ook at what happens around us. A child that smiles, a nightingale that sings, a
ily that opens: all move. Every shadow, even an immobile one, is due to moving
ight. Every mountain is kept in place by moving electrons. Every star owes its formation and its shine to motion of matter and radiation. Also the darkness of the night
sky** is due to motion: it results from the expansion of space. Finally, human creativity
is due to the motion of molecules, ions and electrons in the brain. Is there a common
language for these and all other observations of nature?
Is there a unified and precise way to describe all motion? How? Is everything that
moves, from people to planets, from light to empty space, made of the same constituents?
What is the origin of motion? Answering these questions is the topic of the present text.
Answering questions about motion with precision defines the subject of physics. Over
the centuries, researchers collected a huge number of precise observations about motion.
We now know how electric signals move in the brain, how insects fly, why colours vary,
how the stars formed, how life evolved, and much more. We use our knowledge about
motion to look into the human body and heal illnesses; we use our knowledge about
motion to build electronics, communicate over large distances, and work for peace; we
use our knowledge about motion to secure life against many of nature’s dangers, including droughts and storms. Physics, the science of motion, has shown time after time that
knowledge about motion is both useful and fascinating.
At the end of the last millennium, humans were able to describe all observed motion
with high precision. This description can be summarized in the following six statements.
18
1 from m illennium physics to unification
Ref. 2
general relativity implies that things fall and that empty space curves and moves, as
is observed.
Ref. 2
4. The evolution of the universe is described by the cosmological constant Λ. It determines the largest distance and the largest age that can presently be observed.
Ref. 4
5. In nature, there is a non-zero, invariant smallest change value, the quantum of action
ℏ. This invariant value implies quantum theory. Among others, it explains what life
and death are, why they exist and how we enjoy the world.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 5
These six statements, the millennium description of physics, describe everything known
about motion in the year 2000. (Actually, 2012 is a more precise, though less striking
date.) These statements describe the motion of people, animals, plants, objects, light,
radiation, stars, empty space and the universe. The six statements describe motion so
precisely that even today there is no difference between calculation and observation,
between theory and practice. This is an almost incredible result, the summary of the
efforts of tens of thousands of researchers during the past centuries.
However, a small set of observations does not yet follow from the six statements. A
famous example is the origin of colours. In nature, colours are consequences of the socalled fine structure constant, a mysterious constant of nature, abbreviated ?, whose value
is measured to be ? = 1/137.035 999 139(31). If ? had another value, all colours would
differ. And why are there three gauge interactions, twelve elementary fermions, thirteen
elementary bosons and three dimensions? What is the origin of particle masses? Why is
the standard model, the sixth statement above, so complicated? How is it related to the
five preceding statements?
A further unexplained observation is the nature of dark matter found around galaxies. We do not know yet what it is. Another unexplained process is the way thinking
forms in our brain. We do not know yet in detail how thinking follows from the above
six statements, though we do know that thinking is not in contrast with them. For this
reason, we will not explore the issue in the following. In the case of dark matter this is
not so clear: dark matter could even be in contrast with the millennium description of
motion.
Finally, why is there motion anyway? In short, even though the millennium description of physics is precise and successful, it is not complete. The list of all those fundamental issues about motion that are unexplained since the year 2000 make up only a
short table. We call them the millennium issues.
Motion Mountain – The Adventure of Physics
Ref. 4
6. In nature, matter and radiation consist of quantum particles. Matter consists of fermions: six quarks, three charged leptons, three neutrinos and their antiparticles. Radiation consists of bosons: the photon, three intermediate weak vector bosons and eight
gluons. In addition, the year 2012 finally brought the discovery of the Higgs boson,
which was already predicted in 1964. Fermions and bosons move and can transform
into each other. The transformations are described by the electromagnetic interaction, the weak nuclear interaction and the strong nuclear interaction. Together with
the masses, quantum numbers, mixing angles and couplings, these transformation
rules form the so-called standard model of particle physics. Among others, the standard model explains how lightning forms, why colours vary, and how the atoms in our
bodies came to be.
from m illennium physics to unification
19
TA B L E 1 The millennium list: everything the standard model and general relativity cannot explain; thus,
also the list of the only experimental data available to test the final, unified description of motion.
O b s e rva b l e P r o p e rt y u n e x p l a i n e d s i n c e t h e y e a r 2 0 0 0
Local quantities unexplained by the standard model: particle properties
Concepts unexplained by the standard model
SU(2)
SU(3)
Renorm. group
?? = 0
? = ∫? SM d?
the origin of the invariant Planck units of quantum field theory
the number of dimensions of physical space and time
the origin of Poincaré symmetry, i.e., of spin, position, energy, momentum
the origin and nature of wave functions
the origin of particle identity, i.e., of permutation symmetry
the origin of the gauge groups, in particular:
the origin of the electromagnetic gauge group, i.e., of the quantization of electric charge, of the vanishing of magnetic charge, and of minimal coupling
the origin of weak interaction gauge group, its breaking and P violation
the origin of strong interaction gauge group and its CP conservation
the origin of renormalization properties
the origin of the least action principle in quantum theory
the origin of the Lagrangian of the standard model of particle physics
Global quantities unexplained by general relativity and cosmology
0
1.2(1) ⋅ 1026 m
?de = Λ?4 /(8π?)
≈ 0.5 nJ/m3
(5 ± 4) ⋅ 1079
?dm
the observed flatness, i.e., vanishing curvature, of the universe
the distance of the horizon, i.e., the ‘size’ of the universe (if it makes sense)
the value and nature of the observed vacuum energy density, dark energy or
cosmological constant
the number of baryons in the universe (if it makes sense), i.e., the average
visible matter density in the universe
the density and nature of dark matter
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
?, ℏ, ?
3+1
SO(3,1)
Ψ
?(?)
Gauge symmetry
U(1)
Motion Mountain – The Adventure of Physics
? = 1/137.036(1) the low energy value of the electromagnetic coupling or fine structure constant
?w or ?w
the low energy value of the weak coupling constant or the value of the weak
mixing angle
?s
the value of the strong coupling constant at one specific energy value
?q
the values of the 6 quark masses
?l
the values of 6 lepton masses
?W
the value of the mass of the ? vector boson
?H
the value of the mass of the scalar Higgs boson
?12 , ?13 , ?23
the value of the three quark mixing angles
?
the value of the CP violating phase for quarks
?
?
?
?12
, ?13
, ?23
the value of the three neutrino mixing angles
?
? , ?1 , ?2
the value of the three CP violating phases for neutrinos
3⋅4
the number of fermion generations and of particles in each generation
J, P, C, etc.
the origin of all quantum numbers of each fermion and each boson
20
1 from m illennium physics to unification
TA B L E 1 (Continued) The millennium list: everything the standard model and general relativity cannot
explain; also the only experimental data available to test the final, unified description of motion.
O b s e rva b l e P r o p e rt y u n e x p l a i n e d s i n c e t h e y e a r 2 0 0 0
?0 (1, ..., c. 1090 )
the initial conditions for c. 1090 particle fields in the universe (if or as long as
they make sense), including the homogeneity and isotropy of matter distribution, and the density fluctuations at the origin of galaxies
Concepts unexplained by general relativity and cosmology
?, ?
R × S3
???
?? = 0
? = ∫? GR d?
the origin of the invariant Planck units of general relativity
the observed topology of the universe
the origin and nature of curvature, the metric and horizons
the origin of the least action principle in general relativity
the origin of the Lagrangian of general relativity
Against a final theory
— It is regularly claimed that a final theory cannot exist because nature is infinite and
mysteries will always remain. But this statement is wrong. First, nature is not infinite.
Second, even if it were infinite, knowing and describing everything would still be
possible. Third, even if knowing and describing everything would be impossible, and
if mysteries would remain, a final theory remains possible. A final theory is not useful
for every issue of everyday life, such as choosing your dish on a menu or your future
profession. A final theory is simply a full description of the foundations of motion:
the final theory just combines and explains particle physics and general relativity.
Page 168
— It is sometimes argued that a final theory cannot exist due to Gödel’s incompleteness
theorem or due to computational irreducibility. However, in such arguments, both
theorems are applied to domains were they are not valid. The reasoning is thus wrong.
— Some state that it is not clear whether a final theory exists at all. We all know from experience that this is wrong. The reason is simple: We are able to talk about everything.
In other words, all of us already have a ‘theory of everything’, or a final theory of
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
We know that a final theory exists: it is the theory that describes how to calculate the
fine structure constant ? = 1/137.036(1). The theory does the same for about two dozen
other constants, but ? is the most famous one. In other terms, the final theory is the
theory that explains all colours found in nature.
A fixed list of arguments are repeated regularly against the search for a final, unified
theory of motion. Reaching the final theory and enjoying the adventure is only possible
if these arguments are known – and then put gently aside.
Motion Mountain – The Adventure of Physics
The millennium list contains everything that particle physics and general relativity
cannot explain. In other words, the list contains every issue that was unexplained in the
domain of fundamental motion in the year 2000. The list is short, but it is not empty.
Every line in the millennium list asks for an explanation. The quest for unification – and
the topic of this text – is the quest for these explanations. We can thus say that a final
theory of motion is a theory that eliminates the millennium list of open issues.
from m illennium physics to unification
—
Ref. 6
Page 160
—
—
Ref. 8
—
Ref. 9
—
Ref. 10
These arguments show us that we can reach the final unified theory – which we symbolically place at the top of Motion Mountain – only if we are not burdened with ideological or emotional baggage. (We get rid of all baggage in the first six chapters of this
volume.) The goal we have set requires extreme thinking, i.e., thinking up to the limits.
After all, unification is the precise description of all motion. Therefore, unification is a
riddle. The search for unification is a pastime. Any riddle is best approached with the
light-heartedness of playing. Life is short: we should play whenever we can.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
—
nature. Also a physical theory is a way to talk about nature, and for the final theory
we only have to search for those concepts that enable us to talk about all of motion
with full precision. Because we are just looking for a way to talk, we know that the
final theory exists. And searching for it is fascinating and exciting, as everybody busy
with this adventure will confirm.
Some claim that the search for a final theory is a reductionist endeavour and cannot
lead to success, because reductionism is flawed. This claim is wrong on three counts.
First, it is not clear whether the search is a reductionist endeavour, as will become
clear later on. Second, there is no evidence that reductionism is flawed. Third, even
if it were, no reason not to pursue the quest would follow. The claim in fact invites
to search with a larger scope than was done in the past decades – an advice that will
turn out to be spot on.
Some argue that searching for a final theory makes no sense as long as the measurement problem of quantum theory is not solved, or consciousness is not understood,
or the origin of life is not understood. Now, the measurement problem is solved by
decoherence, and in order to combine particle physics with general relativity, understanding the details of consciousness or of the origin of life is not required. Neither
is understanding child education required – though this can help.
Some people claim that searching for a final theory is a sign of foolishness or a sin
of pride. Such defeatist or envious comments should simply be ignored. After all, the
quest is just the search for the solution to a riddle.
Some believe that understanding the final theory means to read the mind of god, or
to think like god, or to be like god. This is false, as any expert on god will confirm.
In fact, solving a riddle or reading a physics textbook does not transform people into
gods. This is unfortunate, as such an effect would provide excellent advertising.
Some fear that knowing the final theory yields immense power that harbours huge
dangers of misuse, in short, that knowing the final theory might change people into
devils. However, this fear is purely imaginary; it only describes the fantasies of the
person that is talking. Indeed, the millennium description of physics is already quite
near to the final theory, and nothing to be afraid of has happened. Sadly, another great
advertising opportunity is eliminated.
Some people object that various researchers in the past have thought to have found
the final theory, but were mistaken, and that many great minds tried to find a final
theory, but had no success. That is true. Some failed because they lacked the necessary
tools for a successful search, others because they lost contact with reality, and still
others because they were led astray by prejudices that limited their progress. We just
have to avoid these mistakes.
Motion Mountain – The Adventure of Physics
Ref. 7
Vol. IV, page 143
21
22
1 from m illennium physics to unification
What went wrong in the past
Vol. V, page 268
The twentieth century was the golden age of physics. Scholars searching for the final
theory explored candidates such as grand unified theories, supersymmetry and numerous other options. These candidates will be discussed later on; all were falsified by experiment. In other words, despite a large number of physicists working on the problem,
despite the availability of extensive experimental data, and despite several decades of research, no final theory was found. Why?
During the twentieth century, many successful descriptions of nature were deformed
into dogmatic beliefs about unification. Here are the main examples, with some of their
best known proponents:
An encouraging argument
Page 8
The Bronshtein cube in Figure 1 shows that physics started from the description of motion in everyday life. At the next level of precision, physics introduced the observed limits to motion and added the description of powerful, i.e., as uniform as possible motion (classical gravity), as fast as possible motion (special relativity), and as tiny as possible motion (quantum theory). At the following level of precision, physics achieved all
possible combinations of two of these motion types, by taking care of two motion lim-
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
All these beliefs appeared in the same way: first, some famous scholar – in fact, many
more than those mentioned – explained the idea that guided his discovery; then, he and
most other researchers started to believe the guiding idea more than the discovery itself. The most explored belief were those propagated by Salam and Weinberg: they –
unknowingly – set thousands of researchers on the wrong path for dozens of years. The
most detrimental has been the belief that unification is complicated and difficult: it kept
the smartest physicists from producing progress. In fact, all the mentioned beliefs can
be seen as special cases of the first one. And like the first belief, they are all, as we will
discover in the following, wrong.
Motion Mountain – The Adventure of Physics
— ‘Unification requires generalization of existing theories.’
— ‘Unification requires finding higher symmetries.’ (Werner Heisenberg)
— ‘Unification requires generalizing electroweak mixing to include the strong interaction.’ (Abdus Salam)
— ‘Unification requires extending the standard model of particle physics with supersymmetry.’ (Steven Weinberg)
— ‘Unification requires axiomatization.’ (David Hilbert)
— ‘Unification requires searching for beauty.’ (Paul Dirac)
— ‘Unification requires new quantum evolution equations.’ (Werner Heisenberg)
— ‘Unification requires new field equations of gravitation.’ (Albert Einstein)
— ‘Unification requires more dimensions of space.’ (Theodor Kaluza)
— ‘Unification requires topology change.’ (John Wheeler)
— ‘Unification is independent of Planck’s natural units.’
— ‘Unification requires using complicated mathematics and solving huge conceptual
difficulties.’ (Edward Witten)
— ‘Unification is only for a selected few.’
— ‘Unification is extremely useful, important and valuable.’
from m illennium physics to unification
23
its at the same time: fast and uniform motion (general relativity), fast and tiny motion
(quantum field theory), and tiny and uniform motion (quantum theory with gravity).
The only domain left over is the domain where motion is fast, tiny and as uniform as
possible at the same time. When this last domain is reached, the precise description of
all motion is completed.
But Figure 1 suggests even stronger statements. First of all, no domain of motion is
left: the figure covers all motion. Secondly, the final description appears when general
relativity, quantum field theory and quantum theory with gravity are combined. In other
words, the final theory appears when relativity and quantum theory and interactions are
all described together. But a third conclusion is especially important. Each of these three
fields can be deduced from the unified final theory by eliminating a limitation: either
that of tiny motion, that of straight motion, or that of fast motion. In other words:
⊳ Intermediate steps or theories do not exist before the final theory.
This is a strong statement. In the foundations of motion, apart from the final theory, no
further theory is missing. For example, the figure implies that there is no separate theory
of relativistic quantum gravity or no doubly special relativity.
In particular, Figure 1 implies that, conceptually, we are already close to the final theory. The figure suggests that there is no need for overly elaborate hypotheses or concepts
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Speaking even more bluntly, and against a common conviction of researchers in the field,
the figure suggests: The standard model follows from the final theory by eliminating gravity.
These connections eliminate many candidates for the unified final theory that were proposed in the research literature in the twentieth and twenty-first century. But more importantly, the connections leave open a range of possibilities – and interestingly enough,
this range is very narrow.
The figure allows stronger statements still. Progress towards the final theory is
achieved by taking limitations to motion into account. Whatever path we take from
everyday physics to the final theory, we must take into consideration all limits to motion.
The order can differ, but all limits have to be taken into account. Now, if any intermediate steps – due to additional motion limitations – between quantum field theory and the
final theory existed in the upper part of the figure, corresponding steps would have to
appear also in the lower part of the figure, between everyday physics and classical gravity. In the same way, if any intermediate limits between general relativity and the final
theory really existed, these limits would also have to appear between everyday motion
and quantum theory.
Experiments show that no intermediate steps or limits exist between everyday motion
and the next level of precision. Using the top-down symmetry of Figure 1, this implies:
Motion Mountain – The Adventure of Physics
⊳ General relativity follows from the final theory by eliminating the quantum
of action ℏ, i.e., taking the limit ℏ → 0.
⊳ Quantum field theory, including quantum electrodynamics, follows from
the final theory by eliminating ?, i.e., taking the limit ? → 0.
⊳ Quantum theory with gravity follows from the final theory by eliminating
the speed limit ?, i.e., taking the limit 1/? → 0.
24
1 from m illennium physics to unification
to reach the final theory:
⊳ We just have to add ? to the standard model or ℏ and ? to general relativity.
In short, the final, unified theory of motion cannot be far.
summary: how to find the final theory of motion
Page 147
At this point, after the first half of our adventure, we obtain a detailed requirement list
for the final theory. This list allows us to proceed rapidly towards our goal, without being
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
1. We first simplify modern physics. Twentieth century physics deduced several invariant properties of motion. These invariants, such as the speed of light or the quantum
of action, are called Planck units. The invariant Planck units allow motion to be measured. Above all, these invariants are also found to be limit values, valid for every example of motion.
2. Combining quantum theory and general relativity, we discover that at the Planck limits, the universe, space and particles are not described by points. We find that as long as
we use points to describe particles and space, and as long as we use sets and elements
to describe nature, a unified description of motion is impossible.
3. The combination of quantum theory and general relativity teaches us that space and
particles have common constituents.
4. By exploring black holes, spin, and the limits of quantum theory and gravity, we discover that the common constituents of space and particles are extended, without ends,
one-dimensional and fluctuating: the common constituents of space and particles are
fluctuating strands.
5. We discover that we cannot think or talk without continuity. We need a background
to describe nature. We conclude that to talk about motion, we have to combine continuity and non-continuity in an appropriate way. This is achieved by imagining that
fluctuating strands move in a continuous three-dimensional background.
Motion Mountain – The Adventure of Physics
We have a riddle to solve: we want to describe precisely all motion and discover its origin.
In order to achieve this, we need to find a final theory that solves and explains each open
issue given in the millennium list. This is our starting point.
We proceed in steps. We first simplify quantum theory and gravitation as much as
possible, we explore what happens when the two are combined, and we deduce the requirement list that any final theory must fulfil. Then we deduce the simplest possible
model that fulfils the requirements; we check the properties of the model against every
experiment performed so far and against every open issue from the millennium list. Discovering that there are no disagreements, no points left open and no possible alternatives,
we know that we have found the final theory. We thus end our adventure with a list of
testable predictions for the proposed model.
In short, three lists structure our quest for a final theory: the millennium list of open
issues, the list of requirements for the final theory, and the list of testable predictions. To
get from one list to the next, we proceed along the following legs.
sum m ary: how to find the fina l theory of m otion
25
led astray:
“
Es ist fast unmöglich, die Fackel der Wahrheit
durch ein Gedränge zu tragen, ohne jemandem
den Bart zu sengen.*
Georg Christoph Lichtenberg
”
* ‘It is almost impossible to carry the torch of truth through a crowd without scorching somebody’s beard.’
Georg Christoph Lichtenberg (b. 1742 Ober-Ramstadt, d. 1799 Göttingen) was a famous physicist and essayist.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
At the end of this path, we will thus have unravelled the mystery of motion. It is a truly
special adventure. But be warned: almost all of the story presented here is still speculative, and thus open to question. Everything presented in the following agrees with experiment. Nevertheless, with almost every sentence you will find at least one physicist or
philosopher who disagrees. That makes the adventure even more fascinating.
Motion Mountain – The Adventure of Physics
Page 395
6. We discover a simple fundamental principle that explains how the maximum speed
?, the minimum action ℏ, the maximum force ?4 /4? and the cosmological constant Λ
follow from strands. We also discover how to deduce quantum theory, relativity and
cosmology from strands.
7. We discover that strands naturally yield the existence of three spatial dimensions,
flat and curved space, black holes, the cosmological horizon, fermions and bosons.
We find that all known physical systems are made from strands. Also the process of
measurement and all properties of the background result from strands.
8. We discover that fermions emit and absorb bosons and that they do so with exactly
those properties that are observed for the electromagnetic, the weak and the strong
nuclear interaction. In short, the three known gauge interactions – and their parity
conservation or violation – follow from strands in a unique way. In addition, we discover that other interactions do not exist.
9. We discover that strands naturally yield the known elementary fermions and bosons,
grouped in three generations, and all their observed properties. Other elementary
particles do not exist. We thus recover the standard model of elementary particles.
10. We discover that the fundamental principle allows us to solve all the issues in the
millennium list, and that all properties deduced from strands agree with experiment.
In particular, the strand model allows us to calculate the fine structure constant and
the other gauge coupling strengths. An extensive list of testable predictions can be
given. These predictions will all be tested – by experiment or by calculation – in the
coming years.
11. We discover that motion is due to crossing switches of strands. Motion is an inescapable consequence of observation: motion is an experience that we make because we
are, like every observer, a small, approximate part of a large whole.
Chapter 2
PHYSICS IN LIMIT STAT EMENT S
T
At dinner parties, physicists are regularly asked to summarize physics in a few sentences.
It is useful to have a few simple statements ready to answer such a request. Such statements are not only useful to make other people think; they are also useful in our quest
for the final theory. Here they are.
Everyday, or Galilean, physics in one statement
Everyday motion is described by Galilean physics. It consists of only one statement:
⊳ Motion minimizes change.
In nature, change is measured by physical action ?. More precisely, change is measured
by the time-averaged difference between kinetic energy ? and potential energy ?. In
other words, motion obeys the so-called least action principle, written as
?? = 0 , where
? = ∫(? − ?) d? .
(1)
This statement determines the effort we need to move or throw stones, and explains why
cars need petrol and people need food. In other terms, nature is as lazy as possible. Or:
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
simplifying physics as much as possible
Motion Mountain – The Adventure of Physics
wentieth century physics deduced several invariant properties of motion.
hese invariants, such as the speed of light or the quantum of action, define
he so-called Planck units. The invariant Planck units are important for two reasons: first, they allow motion to be measured; second, the invariants are limit values. In
fact, the Planck units provide bounds for all observables.
The main lesson of modern physics is thus the following: When we simplify physics
as much as possible, we discover that nature limits the possibilities of motion. Such limits
lie at the origin of special relativity, of general relativity and of quantum theory. In fact,
we will see that nature limits every aspect of motion. Exploring the limits of motion will
allow us to deduce several astonishing conclusions. These conclusions contradict all that
we learned about nature so far.
sim plifying physics as m uch as p ossible
27
⊳ Nature is maximally efficient.
Vol. I, page 29
The efficiency or laziness of nature implies that motion is conserved, relative and predictable. In fact, the laziness of motion and nature is valid throughout modern physics,
for all observations, provided a few limit statements are added.
Special rel ativit y in one statement
Ref. 11
The step from everyday, or Galilean, physics to special relativity can be summarized in a
single limit statement on motion. It was popularized by Hendrik Antoon Lorentz:
⊳ There is a maximum energy speed value ? in nature.
Challenge 1 e
Vol. II, page 27
* A physical system is a region of space-time containing mass–energy, the location of which can be followed over time and which interacts incoherently with its environment. The speed of a physical system is
thus an energy speed. The definition of physical system excludes images, geometrical points or incomplete,
entangled situations.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Vol. II, page 100
All results peculiar to special relativity follow from this principle. A few well-known facts
set the framework for the discussion that follows. The speed ? is less than or equal to the
speed of light ? for all physical systems;* in particular, this speed limit is valid both for
composite systems and for elementary particles. No exception has ever been found. (Try
it.)
The energy speed limit is an invariant: the local energy speed limit is valid for all observers. In this context it is essential to note that any observer must be a physical system,
and must be close to the moving energy.
The speed limit ? is realized by massless particles and systems; in particular, it is realized by electromagnetic waves. For matter systems, the speed is always below ?.
Only a maximum energy speed ensures that cause and effect can be distinguished in
nature, or that sequences of observations can be defined. The opposite hypothesis, that
energy speeds greater than ? are possible, which implies the existence of so-called (real)
tachyons, has been explored and tested in great detail; it leads to numerous conflicts with
observations. Tachyons do not exist.
The maximum energy speed forces us to use the concept of space-time to describe
nature, because the existence of a maximum energy speed implies that space and time
mix. It also implies observer-dependent time and space coordinates, length contraction,
time dilation, mass–energy equivalence, horizons for accelerated observers, and all the
other effects that characterize special relativity. Only a maximum speed leads to the principle of maximum ageing that governs special relativity; and only this principle leads to
the principle of least action at low speeds. In addition, only with a finite speed limit is
it possible to define a unit of speed that is valid at all places and at all times. If there
were no global speed limit, there could be no natural measurement standard for speed,
independent of all interactions; speed would not then be a measurable quantity.
Motion Mountain – The Adventure of Physics
For all physical systems and all observers, the local energy speed ? is limited by the speed
of light ?:
? ⩽ ? = 3.0 ⋅ 108 m/s .
(2)
28
2 physics in lim it statem ents
Special relativity also limits the size of systems – whether composite or elementary.
Indeed, the limit speed implies that acceleration ? and size ? cannot be increased independently without bounds, because the two ends of a system must not interpenetrate.
The most important case concerns massive systems, for which we have
?⩽
?2
.
?
(3)
This size limit is induced by the speed of light ?; it is also valid for the displacement ? of
a system, if the acceleration measured by an external observer is used. Finally, the speed
limit implies a relativistic ‘indeterminacy relation’
Δ? Δ? ⩽ ?2
for the length and acceleration indeterminacies. You may wish to take a minute to deduce
this relation from the time–frequency indeterminacy. All this is standard knowledge.
Q uantum theory in one statement
Ref. 12
The difference between Galilean physics and quantum theory can be summarized in a
single statement on motion, due to Niels Bohr:
⊳ There is a minimum action value ℏ in nature.
For all physical systems and all observers, the action ? obeys
? ⩾ ℏ = 1.1 ⋅ 10−34 Js .
Challenge 3 e
Challenge 4 e
(5)
The Planck constant ℏ is the smallest observable action value, and the smallest observable change of angular momentum. The action limit is valid for all systems, thus both
for composite and elementary systems. No exception has ever been found. (Try it.) The
principle contains all of quantum theory. We call it the principle of non-zero action, in
order to avoid confusion with the principle of least action.
The non-zero action limit ℏ is an invariant: it is valid with the same numerical value
for all observers. Again, any such observer must be a physical system.
The action limit is realized by many physical processes, from the absorption of light
to the flip of a spin 1/2 particle. More precisely, the action limit is realized by microscopic
systems where the process involves a single particle.
The non-zero action limit is stated less frequently than the speed limit. It starts from
the usual definition of the action, ? = ∫(? − ?) d?, and states that between two observations performed at times ? and ? + Δ?, even if the evolution of a system is not known, the
measured action is at least ℏ. Since physical action measures the change in the state of a
physical system, there is always a minimum change of state between two different observations of a system.* The non-zero action limit expresses the fundamental fuzziness of
* For systems that seem constant in time, such as a spinning particle or a system showing the quantum Zeno
effect, finding this minimum change is tricky. Enjoy the challenge.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Vol. IV, page 15
Motion Mountain – The Adventure of Physics
Challenge 2 s
(4)
sim plifying physics as m uch as p ossible
Ref. 13
?⩾
Challenge 5 e
Vol. IV, page 24
ℏ
.
??
(6)
In other words, the (reduced) Compton wavelength of quantum theory appears as the
lower limit on the displacement of a system, whenever gravity plays no role. Since this
quantum displacement limit also applies to elementary systems, it also applies to the size
of a composite system. However, for the same reason, this size limit is not valid for the
sizes of elementary particles.
The limit on action also implies Heisenberg’s well-known indeterminacy relation for
the displacement ? and momentum ? of physical systems:
Δ? Δ? ⩾
ℏ
.
2
(7)
This relation is valid for both massless and massive systems. All this is textbook knowledge.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
nature at a microscopic scale.
It can easily be checked that no observation – whether of photons, electrons or macroscopic systems – gives a smaller action than the value ℏ. The non-zero action limit has
been verified for fermions, bosons, laser beams, matter systems, and for any combination
of these. The opposite hypothesis, implying the existence of arbitrary small change, has
been explored in detail: Einstein’s long discussion with Bohr, for example, can be seen as
a repeated attempt by Einstein to find experiments that would make it possible to measure arbitrarily small changes or action values in nature. In every case, Bohr found that
this could not be achieved. All subsequent attempts were equally unsuccessful.
The principle of non-zero action can be used to deduce the indeterminacy relation, the
tunnelling effect, entanglement, permutation symmetry, the appearance of probabilities
in quantum theory, the information-theoretic formulation of quantum theory, and the
existence of elementary particle reactions. Whenever we try to overcome the smallest
action value, the experimental outcome is probabilistic. The minimum action value also
implies that in quantum theory, the three concepts of state, measurement operation, and
measurement result need to be distinguished from each other; this is done by means
of a so-called Hilbert space. Finally, the non-zero action limit is also the foundation of
Einstein–Brillouin–Keller quantization.
The existence of a non-zero action limit has been known from the very beginning
of quantum theory. It is at the basis of – and completely equivalent to – all the usual
formulations of quantum theory, including the many-path and the information-theoretic
formulations.
We also note that only a non-zero action limit makes it possible to define a unit of
action. If there were no action limit, there could be no natural measurement standard
for action: action would not then be a measurable quantity.
The upper bounds for speed and for action for any physical system, ? ⩽ ? and
? ⩽ ?? ⩽ ???, when combined with the quantum of action, imply a limit on the
displacement ? of a system between any two observations:
Motion Mountain – The Adventure of Physics
Ref. 14
29
30
2 physics in lim it statem ents
Thermodynamics in one statement
Thermodynamics can also be summarized in a single statement about motion:
⊳ There is a smallest entropy value ? in nature.
Written symbolically,
Challenge 6 e
Ref. 15
? ⩾ ? = 1.3 ⋅ 10−23 J/K .
1
?
Δ? ⩾ .
?
2
(9)
General rel ativit y in one statement
Page 54
This text can be enjoyed most when a compact and unconventional description of general
relativity is used; it is presented in the following. However, the conclusions do not depend
on this description; the results are also valid if the usual approach to general relativity is
used; this will be shown later on.
The most compact description summarizes the step from universal gravity to general
relativity in a single statement on motion:
⊳ There are maximum force and power values in nature.
For all physical systems and all observers, force ? and power ? are limited by
?⩽
Challenge 7 e
?4
= 3.0 ⋅ 1043 N
4?
and ? ⩽
?5
= 9.1 ⋅ 1051 W .
4?
(10)
No exception has ever been found. (Try it.) These limit statements contain both the speed
of light ? and the gravitational constant ?; they thus qualify as statements about relativistic gravitation. Before we deduce general relativity, let us explore these limits.
The numerical values of the limits are huge. The maximum power corresponds to
converting 50 solar masses into massless radiation within 1 millisecond. And applying
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
This relation was first given by Bohr and then discussed by Heisenberg and many others.
Motion Mountain – The Adventure of Physics
The entropy ? is limited by the Boltzmann constant ?. No exception has ever been found.
(Try it.) This result is almost 100 years old; it was stated most clearly by Leo Szilard. All
of thermodynamics can be deduced from this relation, together with the quantum of
action.
The entropy limit is an invariant: it is valid for all observers. Again, any observer must
be a physical system.
The entropy limit is realized only by physical systems made of a single particle. In
other words, the entropy limit is again realized only by microscopic systems. Therefore
the entropy limit provides the same length limit for physical systems as the action limit.
Like the other limit statements we have examined, the entropy limit can also be
phrased as a indeterminacy relation between temperature ? and energy ?:
Δ
Ref. 16
(8)
sim plifying physics as m uch as p ossible
Ref. 17
Vol. I, page 230
31
the maximum force value along a distance ? costs as much energy as a black hole of
diameter ?.
Force is change of momentum; power is change of energy. Since momentum and energy are conserved, force and power are the flow of momentum and energy through a
surface. Force and power, like electric current, describe the change in time of conserved
quantity. For electric current, the conserved quantity is charge, for force, it is momentum,
for power, it is energy. In other words, like current, also force is a flow across a surface.
This is a simple consequence of the continuity equation. Therefore, every discussion of
maximum force implies a clarification of the underlying surface.
Both the force and the power limits state that the flow of momentum or of energy
through any physical surface – a surface to which an observed can be attached at every
one of its points – of any size, for any observer, in any coordinate system, never exceeds
the limit value. In particular:
Vol. II, page 110
Challenge 8 e
Challenge 9 e
Vol. II, page 107
Vol. II, page 83
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 18
In all other situations, the observed values are strictly smaller than the maximum values.
The force and power limit values are invariants: they are valid for all observers and
for all interactions. Again, any observer must be a physical system and it must be located
on or near the surface used to define the flow of momentum or energy.
The value of the force limit is the energy of a Schwarzschild black hole divided by its
diameter; here the ‘diameter’ is defined as the circumference divided by π. The power
limit is realized when such a black hole is radiated away in the time that light takes to
travel along a length corresponding to the diameter.
An object of mass ? that has the size of its own Schwarzschild radius 2??/?2 is
called a black hole, because according to general relativity, no signals and no light from
inside the Schwarzschild radius can reach the outside world. In this text, black holes
are usually non-rotating and usually uncharged; in this case, the terms ‘black hole’ and
‘Schwarzschild black hole’ are synonymous.
The value of the maximum force, as well as being the mass–energy of a black hole
divided by its diameter, is also the surface gravity of a black hole times its mass. Thus the
force limit means that no physical system of a given mass can be concentrated in a region
of space-time smaller than a (non-rotating) black hole of that mass. (This is the so-called
hoop conjecture.) In fact, the mass–energy concentration limit can easily be transformed
algebraically into the force limit: they are equivalent.
It is easily checked that the maximum force limit is valid for all systems observed
in nature, whether they are microscopic, macroscopic or astrophysical. Neither the
‘gravitational force’ (as long as it is operationally defined) nor the electromagnetic or
nuclear interactions are ever found to exceed this limit.
But is it possible to imagine a system that exceeds the force limit? An extensive discussion shows that this is impossible. For example, the force limit cannot be overcome with
Lorentz boosts. We might think that a boost can be chosen in such a way that a 3-force
value ? in one frame is transformed into any desired value ?? in another, boosted frame.
This thought turns out to be wrong. In relativity, 3-force cannot be increased beyond all
bounds using boosts. In all reference frames, the measured 3-force can never exceed the
Motion Mountain – The Adventure of Physics
⊳ The force limit is only realized at horizons. The power limit is only realized
with horizons.
32
Vol. II, page 107
proper force, i.e., the 3-force value measured in the comoving frame.
Also changing to an accelerated frame does not help to overcome the force limit, because for high accelerations ?, horizons appear at distance ?2 /?, and a mass ? has a
minimum diameter given by ? ⩾ 4??/?2 .
In fact, the force and power limits cannot be exceeded in any thought experiment,
as long as the sizes of observers or of test masses are taken into account. All apparent
exceptions or paradoxes assume the existence of point particles or point-like observers;
these, however, are not physical: they do not exist in general relativity.
Fortunately for us, nearby black holes or horizons are rare. Unfortunately, this means
that neither the force limit nor the power limit are realized in any physical system at
hand, neither at everyday length scales, nor in the microscopic world, nor in astrophysical systems. Even though the force and power limits have never been exceeded, a direct
experimental confirmation of the limits will take some time.
The formulation of general relativity as a consequence of a maximum force is not
common; in fact, it seems that it was only discovered 80 years after the theory of general
relativity had first been proposed.
Deducing general rel ativit y*
Ref. 20
?=
?
.
?
(11)
Since we are at a horizon, we need to insert the maximum possible values. In terms of
* This section can be skipped at first reading.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
In order to elevate the force or power limit to a principle of nature, we have to show that,
just as special relativity follows from the maximum speed, so general relativity follows
from the maximum force.
The maximum force and the maximum power are only realized at horizons. Horizons
are regions of space-time where the curvature is so high that it limits the possibility of
observation. The name ‘horizon’ is due to an analogy with the usual horizon of everyday
life, which also limits the distance to which we can see. However, in general relativity
horizons are surfaces, not lines. In fact, we can define the concept of horizon in general
relativity as a region of maximum force; it is then easy to prove that a horizon is always
a two-dimensional surface, and that it is essentially black (except for quantum effects).
The connection between horizons and the maximum force or power allows us to deduce the field equations in a simple way. First, there is always a flow of energy at a horizon. Horizons cannot be planes, since an infinitely extended plane would imply an infinite energy flow. To characterize the finite extension of a given horizon, we use its radius
? and its total area ?.
The energy flow across a horizon is characterized by an energy ? and a proper length
? of the energy pulse. When such an energy pulse flows perpendicularly across a horizon,
the momentum change d?/d? = ? is given by
Motion Mountain – The Adventure of Physics
Ref. 19
2 physics in lim it statem ents
sim plifying physics as m uch as p ossible
33
the horizon area ? and radius ?, we can rewrite the limit case as
?4
1
?
= 4π?2
4? ?
?
(12)
where we have introduced the maximum force and the maximum possible area 4π?2 of
a horizon of (maximum local) radius ?. The ratio ?/? is the energy per unit area flowing
across the horizon.
Horizons are often characterized by the so-called surface gravity ? instead of the radius
?. In the limit case, two are related by ? = ?2 /2?. This leads to
?=
Ref. 21
1 2
? ?? .
4π?
(13)
?? =
Ref. 22
?2
? ?? .
8π?
(15)
In this form, the relation between energy and area can be applied to general horizons,
including those that are irregularly curved or time-dependent.*
In a well-known paper, Jacobson has given a beautiful proof of a simple connection:
if energy flow is proportional to horizon area for all observers and all horizons, and if
the proportionality constant is the correct one, then general relativity follows. To see
the connection to general relativity, we generalize the horizon relation (15) to general
coordinate systems and general directions of energy flow.
* The horizon relation (15) is well known, though with different names for the observables. Since no communication is possible across a horizon, the detailed fate of energy flowing across a horizon is also unknown.
Energy whose detailed fate is unknown is often called heat, and abbreviated ?. The horizon relation (15)
therefore states that the heat flowing through a horizon is proportional to the horizon area. When quantum
theory is introduced into the discussion, the area of a horizon can be called ‘entropy’ ? and its surface
gravity can be called ‘temperature’ ?; relation (15) can then be rewritten as ?? = ???. However, this translation of relation (15), which requires the quantum of action, is unnecessary here. We only cite it to show
the relation between horizon behaviour and quantum aspects of gravity.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
This horizon relation makes three points. First, the energy flowing across a horizon is limited. Secondly, this energy is proportional to the area of the horizon. Thirdly, the energy
flow is proportional to the surface gravity. These three points are fundamental, and characteristic, statements of general relativity. (We also note that due to the limit property
of horizons, the energy flow towards the horizon just outside it, the energy flow across a
horizon, and the energy inside a horizon are all the same.)
Taking differentials, the horizon relation can be rewritten as
Motion Mountain – The Adventure of Physics
Special relativity shows that at horizons the product ?? of proper length and acceleration is limited by the value ?2 /2. This leads to the central relation for the energy flow at
horizons:
?2
?=
?? .
(14)
8π?
34
2 physics in lim it statem ents
The proof uses tensor notation. We introduce the general surface element dΣ and the
local boost Killing vector field ? that generates the horizon (with suitable norm). We then
rewrite the left-hand side of relation (15) as
?? = ∫ ??? ?? dΣ? ,
(16)
where ??? is the energy–momentum tensor. This is valid in arbitrary coordinate systems and for arbitrary energy flow directions. Jacobson’s main result is that the righthand side of the horizon relation (15) can be rewritten, using the (purely geometric)
Raychaudhuri equation, as
? ?? = ?2 ∫ ??? ?? dΣ? ,
(17)
(19)
Motion Mountain – The Adventure of Physics
where Λ is a constant of integration whose value is not determined by the problem. These
are the full field equations of general relativity, including the cosmological constant Λ.
This value of this constant remains undetermined, though.
The field equations are thus shown to be valid at horizons. Now, it is possible, by
choosing a suitable coordinate transformation, to position a horizon at any desired
space-time event. To achieve this, simply change to the frame of an observer accelerating away from that point at the correct distance, as explained in the volume on relativity.
Therefore, because a horizon can be positioned anywhere at any time, the field equations
must be valid over the whole of space-time.
Since it is possible to have a horizon at every event in space-time, there is the same
maximum possible force (or power) at every event in nature. This maximum force (or
power) is thus a constant of nature.
In other words, the field equations of general relativity are a direct consequence of
the limited energy flow at horizons, which in turn is due to the existence of a maximum
force or power. We can thus speak of the maximum force principle. Conversely, the field
equations imply maximum force and power. Maximum force and general relativity are
thus equivalent.
By the way, modern scholars often state that general relativity and gravity follow from
the existence of a minimum measurable length. The connection was already stated by
Sakharov in 1969. This connection is correct, but unnecessarily restrictive. The maximum
force, which is implicit in the minimal length, is sufficient to imply gravity. Quantum
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
where ??? is the Ricci tensor describing space-time curvature.
Combining these two steps, we find that the energy–area horizon relation (15) can be
rewritten as
?4
∫ ??? ?? dΣ? =
(18)
∫ ??? ?? dΣ? .
8π?
Jacobson shows that this equation, together with local conservation of energy (i.e., vanishing divergence of the energy–momentum tensor), can only be satisfied if
??? =
Vol. II, page 98
Ref. 23
?4
1
(? − ( ? + Λ) ??? ) ,
8π? ??
2
sim plifying physics as m uch as p ossible
35
theory – or ℏ – is (obviously) not necessary to deduce gravity.
Deducing universal gravitation
Challenge 10 e
Challenge 11 e
The size of physical systems in general rel ativit y
General relativity, like the other theories of modern physics, implies a limit on the size ?
of systems. There is a limit to the amount of matter that can be concentrated into a small
volume:
4??
?⩾ 2 .
(20)
?
Ref. 24
Ref. 25
Experimental data are available only for composite systems; all known systems
comply with it. For example, the latest measurements for the Sun give ??⊙ /?3 =
4.925 490 947(1) μs; the error in ? is thus much smaller than the (scaled) error in its
radius, which is known with much smaller precision. The ‘indeterminacy relation’ (21)
is not as well known as that from quantum theory. In fact, tests of it – for example with
binary pulsars – may distinguish general relativity from competing theories. We cannot
yet say whether this inequality also holds for elementary particles.
A mechanical analo gy for the maximum force
The maximum force is central to the theory of general relativity. Indeed, its value (adorned with a factor 2π) appears in the field equations. The importance of the maximum
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 300
The size limit is only realized for black holes, those well-known systems which swallow
everything that is thrown into them. The size limit is fully equivalent to the force limit.
(Also the hoop conjecture is understood to be true.) All composite systems in nature
comply with the lower size limit. Whether elementary particles fulfil or even match this
limit remains open at this point. More about this issue below.
General relativity also implies an ‘indeterminacy relation’ for the measurement errors
of size ? and energy ? of systems:
Δ?
?4
⩽
.
(21)
Δ?
4?
Motion Mountain – The Adventure of Physics
Page 281
Universal gravitation follows from the force limit in the case where both forces and
speeds are much smaller than the maximum values. The first condition implies
√4??? ≪ ?2 , the second ? ≪ ? and ?? ≪ ?2 . Let us apply this to a specific case.
Consider a satellite circling a central mass ? at distance ? with acceleration ?. This
system, with length ? = 2?, has only one characteristic speed. Whenever this speed ? is
much smaller than ?, ?2 must be proportional both to the squared speed calculated by
?? = 2?? and to the squared speed calculated from √4??? . Taken together, these two
conditions imply that ? = ???/?2 , where ? is a numerical factor. A quick check, for
example using the observed escape velocity values, shows that ? = 1.
Forces and speeds much smaller than the limit values thus imply that gravity changes
with the inverse square of distance. In other words, nature’s limit on force implies universal gravitation. Other deductions of universal gravity from limit quantities are given
later.
36
2 physics in lim it statem ents
pl anck limits for all physical observables
The existence of a maximum force in nature is equivalent to general relativity. As a result,
a large part of modern physics can be summarized in four simple and fundamental limit
statements on motion:
? ⩾ℏ
? ⩾?
? ⩽?
General relativity follows from the force limit:
Challenge 12 e
? ⩽
?4
.
4?
(22)
These (corrected) Planck limits are valid for all physical systems, whether composite or
elementary, and for all observers. Note that the limit quantities of quantum theory, thermodynamics, special and general relativity can also be seen as the right-hand sides of the
respective indeterminacy relations. Indeed, the set (4, 7, 9, 21) of indeterminacy relations
is fully equivalent to the four limit statements (22).
We note that the different dimensions of the four fundamental limits (22) in nature
mean that the four limits are independent. For example, quantum effects cannot be used
to overcome the force limit; similarly, the power limit cannot be used to overcome the
speed limit. There are thus four independent limits on motion in nature.
By combining the four fundamental limits, we can obtain limits on a number of physical observables. The following limits are valid generally, for both composite and elementary systems:
time interval:
?⩾√
4?ℏ
?5
=
1.1 ⋅ 10−43 s
(23)
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Quantum theory follows from the action limit:
Thermodynamics follows from the entropy limit:
Special relativity follows from the speed limit:
Motion Mountain – The Adventure of Physics
force becomes clearer when we return to our old image of space-time as a deformable
mattress. Like any material body, a mattress is described by a material constant that
relates the deformation values to the values of applied energy. Similarly, a mattress, like
any material, is described by the maximum stress it can bear before it breaks. These
two values describe all materials, from crystals to mattresses. In fact, for perfect crystals (without dislocations), these two material constants are the same.
Empty space somehow behaves like a perfect crystal, or a perfect mattress: it has a
deformation-energy constant that is equal to the maximum force that can be applied
to it. The maximum force describes the elasticity of space-time. The high value of the
maximum force tells us that it is difficult to bend space.
Now, materials are not homogeneous: crystals are made up of atoms, and mattresses
are made up of foam bubbles. What is the corresponding structure of space-time? This is
a central question in the rest of our adventure. One thing is sure: unlike crystals, vacuum
has no preferred directions. We now take a first step towards answering the question of
the structure of space-time and particles by putting together all the limits found so far.
pl a nck lim its for a ll physical ob servables
time-distance product:
?? ⩾
4?ℏ
?4
?7
4?ℏ
acceleration:
?⩽√
angular frequency:
? ⩽ 2π√
37
=
3.5 ⋅ 10−78 ms
(24)
=
2.8 ⋅ 1051 m/s2
(25)
5.8 ⋅ 1043 /s .
(26)
?5
=
2?ℏ
Adding the knowledge that space and time can mix, we get
distance:
area:
curvature:
mass density:
3.2 ⋅ 10−35 m
(27)
1.0 ⋅ 10−69 m2
(28)
3.4 ⋅ 10−104 m3
(29)
1.0 ⋅ 1069 /m2
(30)
3.2 ⋅ 1095 kg/m3 .
(31)
⊳ Every natural unit or (corrected) Planck unit is the limit value of the corresponding physical observable.
Page 58
Ref. 26
Ref. 27
Most of these limit statements are found scattered throughout the research literature,
though the numerical factors often differ. Each limit has attracted a string of publications.
The existence of a smallest measurable distance and time interval of the order of the
Planck values is discussed in all approaches to quantum gravity. The maximum curvature
has been studied in quantum gravity; it has important consequences for the ‘beginning’
of the universe, where it excludes any infinitely large or small observable. The maximum
mass density appears regularly in discussions on the energy of the vacuum.
In the following, we often call the collection of Planck limits the Planck scales. We will
discover shortly that at Planck scales, nature differs in many ways from what we are used
to at everyday scales.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Of course, speed, action, angular momentum, entropy, power and force are also limited,
as already stated. The limit values are deduced from the commonly used Planck values
simply by substituting 4? for ?. These limit values are the true natural units of nature. In
fact, the ideal case would be to redefine the usual Planck values for all observables to these
extremal values, by absorbing the numerical factor 4 into the respective definitions. In
the following, we call the limit values the corrected Planck units or corrected Planck limits
and assume that the numerical factor 4 has been properly included. In other words:
Motion Mountain – The Adventure of Physics
volume:
4?ℏ
=
?3
4?ℏ
?⩾ 3
=
?
3/2
4?ℏ
?⩾( 3 ) =
?
3
?
?⩽
=
4?ℏ
?5
?⩽
=
16?2 ℏ
?⩾√
38
2 physics in lim it statem ents
“
Die Frage über die Gültigkeit der Voraussetzungen der Geometrie im
Unendlichkleinen hängt zusammen mit der Frage nach dem innern Grunde der
Massverhältnisse des Raumes. Bei dieser Frage, welche wohl noch zur Lehre
vom Raume gerechnet werden darf, kommt die obige Bemerkung zur
Anwendung, dass bei einer discreten Mannigfaltigkeit das Princip der
Massverhältnisse schon in dem Begriffe dieser Mannigfaltigkeit enthalten ist,
bei einer stetigen aber anders woher hinzukommen muss. Es muss also
entweder das dem Raume zu Grunde liegende Wirkliche eine discrete
Mannigfaltigkeit bilden, oder der Grund der Massverhältnisse ausserhalb, in
darauf wirkenden bindenden Kräften, gesucht werden.*
Bernhard Riemann, 1854, Über die Hypothesen, welche der Geometrie zu
Grunde liegen.
Physics, mathematics and simplicit y
”
“
Those are my principles, and if you don’t like
them ... well, I have others.
Groucho Marx***
”
* ‘The question of the validity of the hypotheses of geometry in the infinitely small is connected to the
question of the foundation of the metric relations of space. To this question, which may still be regarded as
belonging to the study of space, applies the remark made above; that in a discrete manifold the principles
of its metric relations are given in the notion of this manifold, while in a continuous manifold, they must
come from outside. Either therefore the reality which underlies space must form a discrete manifold, or the
principles of its metric relations must be sought outside it, in binding forces which act upon it.’
Bernhard Riemann is one of the most important mathematicians. 45 years after this statement, Max
Planck confirmed that natural units are due to gravitation, and thus to ‘binding forces’.
** Interestingly, he also regularly wrote the opposite, as shown on page 86.
*** Groucho Marx (b. 1890 New York City, d. 1977 Los Angeles), well-known comedian.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
L imits to space, time and size
Motion Mountain – The Adventure of Physics
The four limits of nature of equation (22) – on action, entropy, speed and force – are astonishing. Above all, the four limits are simple. For many decades, a silent assumption has
guided many physicists: physics requires difficult mathematics, and unification requires
even more difficult mathematics.
For example, for over thirty years, Albert Einstein searched with his legendary intensity for the final theory by exploring more and more complex equations. He did so even
on his deathbed!** Also most theoretical physicists in the year 2000 held the prejudice
that unification requires difficult mathematics. This prejudice is a consequence of over a
century of flawed teaching of physics. Flawed teaching is thus one of the reasons that the
search for a final theory was not successful for so long.
The summary of physics with limit statements shows that nature and physics are
simple. In fact, the essence of the important physical theories is extremely simple: special relativity, general relativity, thermodynamics and quantum theory are each based on
a simple inequality.
The summary of a large part of physics with inequalities is suggestive. The summary
makes us dream that the description of the remaining parts of physics – gauge fields,
elementary particles and the final theory – might be equally simple. Let us continue to
explore where the dream of simplicity leads us to.
pl a nck lim its for a ll physical ob servables
We have seen that the four fundamental limits of nature (22) result in a minimum distance and a minimum time interval. As the expressions for the limits shows, these minimum intervals arise directly from the unification of quantum theory and relativity: they
do not appear if the theories are kept separate. In other terms, unification implies that
there is a smallest length in nature. This result is important: the formulation of physics as a set of limit statements shows that the continuum model of space and time is not
completely correct. Continuity and manifolds are only approximations, valid for large actions, low speeds and small forces. Formulating general relativity and quantum theory
with limit statements makes this especially clear.
The existence of a force limit in nature implies that no physical system can be smaller
than a Schwarzschild black hole of the same mass. In particular, point particles do not
exist. The density limit makes the same point. In addition, elementary particles are predicted to be larger than the corrected Planck length. So far, this prediction has not been
tested by observations, as the scales in question are so small that they are beyond experimental reach. Detecting the sizes of elementary particles – for example, with electric
dipole measurements – would make it possible to check all limits directly.
Mass and energy limits
for elementary particles: ? ⩽
ℏ
.
??
(32)
Using this limit, we find the well-known mass, energy and momentum limits that are
valid only for elementary particles:
Ref. 28
ℏ?
= 1.1 ⋅ 10−8 kg = 0.60 ⋅ 1019 GeV/c2
4?
for (real) elementary particles:
?⩽√
for (real) elementary particles:
?⩽√
ℏ?5
= 9.8 ⋅ 108 J = 0.60 ⋅ 1019 GeV
4?
for (real) elementary particles:
?⩽√
ℏ?3
= 3.2 kg m/s = 0.60 ⋅ 1019 GeV/c . (33)
4?
These elementary-particle limits are the (corrected) Planck mass, Planck energy and
Planck momentum. They were discussed in 1968 by Andrei Sakharov, though with different numerical factors. They are regularly cited in elementary particle theory. All known
measurements comply with them.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Mass plays a special role in all these arguments. The four limits (22) do not make it possible to extract a limit statement on the mass of physical systems. To find one, we have
to restrict our aim somewhat.
The Planck limits mentioned so far apply to all physical systems, whether composite
or elementary. Other limits apply only to elementary systems. In quantum theory, the
distance limit is a size limit only for composite systems. A particle is elementary if its size
? is smaller than any measurable dimension. In particular, it must be smaller than the
reduced Compton wavelength:
Motion Mountain – The Adventure of Physics
Page 58
39
40
2 physics in lim it statem ents
Virtual particles – a new definition
Vol. II, page 72
Vol. IV, page 193
In fact, there are elementary particles that exceed all three limits that we have encountered so far. Nature does have particles which move faster than light, which show
actions below the quantum of action, and which experience forces larger than the force
limit.
We know from special relativity that the virtual particles exchanged in collisions move
faster than light. We know from quantum theory that the exchange of a virtual particle
implies actions below the minimum action. Virtual particles also imply an instantaneous
change of momentum; they thus exceed the force limit.
In short, virtual particles exceed all the limits that hold for real elementary particles.
Curiosities and fun challenges abou t Pl anck limits*
∗∗
The minimum action may come as a surprise at first, because angular momentum and
spin have the same unit as action; and nature contains particles with spin 0 or with spin
1/2 ℏ. A minimum action indeed implies a minimum angular momentum. However, the
angular momentum in question is total angular momentum, including the orbital part
with respect to the observer. The measured total angular momentum of a particle is never
smaller than ℏ, even if the spin is smaller.
In terms of mass flows, the power limit implies that flow of water through a tube is limited
in throughput. The resulting limit d?/d? ⩽ ?3 /4? for the change of mass with time seems
to be unrecorded in the research literature of the twentieth century.
∗∗
Vol. II, page 107
A further way to deduce the minimum length using the limit statements which structure
this adventure is the following. General relativity is based on a maximum force in nature,
or alternatively, on a maximum mass change per time, whose value is given by d?/d? =
?3 /4?. Quantum theory is based on a minimum action ? in nature, given by ℏ. Since a
distance ? can be expressed as
?
?2 =
,
(34)
d?/d?
we see directly that a minimum action and a maximum rate of change of mass imply
a minimum distance. In other words, quantum theory and general relativity force us to
conclude that in nature there is a minimum distance. In other words, at Planck scales the
term ‘point in space’ has no theoretical or experimental basis.
* Sections called ‘Curiosities’ can be skipped at first reading.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
∗∗
Motion Mountain – The Adventure of Physics
The (corrected) Planck limits are statements about properties of nature. There is no way
to measure values exceeding these limits, with any kind of experiment. Naturally, such a
claim provokes the search for counter-examples and leads to many paradoxes.
pl a nck lim its for a ll physical ob servables
41
∗∗
With the single-particle limits, the entropy limit leads to an upper limit for temperature:
?⩽√
ℏ?5
= 0.71 ⋅ 1032 K .
4??2
(35)
This corresponds to the temperature at which the energy per degree of freedom is given
by the (corrected) Planck energy √ℏ?5 /4? . A more realistic value would have to take
account of the number of degrees of freedom of a particle at Planck energy. This would
change the numerical factor. However, no system that is even near this temperature value
has been studied yet. Only Planck-size horizons are expected to realize the temperature
limit, but nobody has managed to explore them experimentally, so far.
∗∗
∗∗
∗∗
The gravitational attraction between two masses never yields force values high enough
to exceed the force limit. Why? First of all, masses ? and ? cannot come closer together
than the sum of their horizon radii. Using ? = ???/?2 with the distance ? given by the
(naive) sum of the two black hole radii as ? = 2?(? + ?)/?2 , we get
?⩽
?4
??
,
4? (? + ?)2
(36)
which is never larger than the force limit. Thus even two attracting black holes cannot
exceed the force limit – in the inverse-square approximation of universal gravity. In short,
the minimum size of masses means that the maximum force cannot be exceeded.
∗∗
Ref. 29
It is well known that gravity bends space. Therefore, if they are to be fully convincing,
our calculation for two attracting black holes needs to be repeated taking into account
the curvature of space. The simplest way is to study the force generated by a black hole
on a test mass hanging from a wire that is lowered towards a black hole horizon. For an
unrealistic point mass, the force would diverge at the horizon. Indeed, for a point mass
? lowered towards a black hole of mass ? at (conventionally defined radial) distance ?,
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Challenge 13 e
At first sight, it seems that electric charge can be used in such a way that the acceleration
of a charged body towards a charged black hole is increased to a value, when multiplied with the mass, that exceeds the force limit. However, the changes in the horizon for
charged black holes prevent this.
Motion Mountain – The Adventure of Physics
How can the maximum force be determined by gravity alone, which is the weakest interaction? It turns out that in situations near the maximum force, the other interactions
are usually negligible. This is the reason why gravity must be included in a unified description of nature.
42
2 physics in lim it statem ents
the force would be
?=
???
?2 √1 −
2??
??2
.
(37)
∗∗
Challenge 14 e
An absolute power limit implies a limit on the energy that can be transported per unit
time through any imaginable physical surface. At first sight, it may seem that the combined power emitted by two radiation sources that each emit 3/4 of the maximum value
should give 3/2 times the maximum value. However, the combination forms a black hole,
or at least prevents part of the radiation from being emitted by swallowing it between the
two sources.
Challenge 15 e
One possible system that actually achieves the Planck power limit is the final stage of
black hole evaporation. But even in this case, the power limit is not exceeded.
∗∗
Ref. 19
The maximum force limit states that the stress-energy tensor, when integrated over any
physical surface, does not exceed the limit value. No such integral, over any physical
surface, of any tensor component in any coordinate system, can exceed the force limit,
provided that it is measured by a realistic observer, in particular, by an observer with a
realistic proper size. The maximum force limit thus applies to any component of any force
vector, as well as to its magnitude. It applies to gravitational, electromagnetic, and nuclear
forces; and it applies to all realistic observers. It is not important whether the forces are
real or fictitious; nor whether we are discussing the 3-forces of Galilean physics or the
4-forces of special relativity. Indeed, the force limit applied to the zeroth component of
the 4-force is the power limit.
∗∗
The power limit is of interest if applied to the universe as a whole. Indeed, it can be used to
partly explain Olbers’ paradox: the sky is dark at night because the combined luminosity
of all light sources in the universe cannot be brighter than the maximum value.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
∗∗
Motion Mountain – The Adventure of Physics
This diverges at ? = 0, the location of the horizon. However, even a test mass cannot be
smaller than its own gravitational radius. If we want to reach the horizon with a realistic
test mass, we need to choose a small test mass ?: only a small mass can get near the
horizon. For vanishingly small masses, however, the resulting force tends to zero. Indeed,
letting the distance tend to the smallest possible value by letting ? = 2?(? + ?)/?2 →
2??/?2 requires ? → 0, which makes the force ?(?, ?) vanish. If on the other hand,
we remain away from the horizon and look for the maximum force by using a mass as
large as can possibly fit into the available distance (the calculation is straightforward),
then again the force limit is never exceeded. In other words, for realistic test masses,
expression (37) is never larger than ?4 /4?. Taking into account the minimal size of test
masses, we thus see that the maximum force is never exceeded in gravitational systems.
pl a nck lim its for a ll physical ob servables
43
∗∗
Page 35
Challenge 16 s
The force limit and its solid state analogy might be seen to suggest that the appearance of
matter might be nature’s way of preventing space from ripping apart. Does this analogy
make sense?
∗∗
Ref. 23
In fact, the connection between minimum length and gravity is not new. Already in 1967,
Andrei Sakharov pointed out that a minimum length implies gravity. He showed that
regularizing quantum field theory on curved space with a cut-off at small distances will
induce counter-terms that include to lowest order the cosmological constant and then
the Einstein–Hilbert action of general relativity.
∗∗
⊳ No surface is physical if any part of it requires a localization in space-time to
scales below the minimum length.
Ref. 30
For example, a physical surface must not cross any horizon. Only by insisting on physical surfaces can we eliminate unphysical examples that contravene the force and power
limits. For example, this condition was overlooked in Bousso’s early discussion of Bekenstein’s entropy bound – though not in his more recent ones.
Challenge 17 e
The equation ? = ? ? implies that energy and mass are equivalent. What do the equations ? = (4?/?2 )? = (4?/?4 )? for length and ? = ℏ? for action imply?
∗∗
Our discussion of limits can be extended to include electromagnetism. Using the (lowenergy) electromagnetic coupling constant ?, the fine structure constant, we get the following limits for physical systems interacting electromagnetically:
electric charge:
? ⩾ √4π?0 ??ℏ = ? = 0.16 aC
electric field:
?⩽√
?7
?4
=
= 1.9 ⋅ 1062 V/m
64π?o ?ℏ?2
4??
(39)
magnetic field:
?⩽√
?5
?3
=
= 6.3 ⋅ 1053 T
2
64π?0 ?ℏ?
4??
(40)
voltage:
?⩽√
?4
1 ℏ?5
= √
= 6.1 ⋅ 1027 V
16π?0 ??
? 4?
(41)
inductance:
?⩾
1
4?ℏ
1 √ 4?ℏ3
√
=
= 4.4 ⋅ 10−40 H .
4π?o ? ?7
?2
?5
(38)
(42)
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
∗∗
2
Motion Mountain – The Adventure of Physics
We said above that a surface is physical if an observer can be attached to each of its points.
The existence of a smallest length – and a corresponding shortest time interval – implies
44
2 physics in lim it statem ents
With the additional assumption that in nature at most one particle can occupy one Planck
volume, we get
9
π?o ? ?5
√ ?
= 4.7 ⋅ 1084 C/m3
=
?
16?3 ℏ
64?3 ℏ3
charge density:
?e ⩽ √
capacitance:
? ⩾ 4π?0 ?√
4?ℏ
4?
= ?2 √ 5 = 2.6 ⋅ 10−47 F .
3
?
?ℏ
(43)
(44)
For the case of a single conduction channel, we get
electric resistance:
electric current:
Ref. 31
Ref. 32
?⩽√
π?0 ??6
?5
= ?√
= 1.5 ⋅ 1024 A .
?
4ℏ?
(45)
(46)
(47)
∗∗
The general relation that to every limit value in nature there is a corresponding indeterminacy relation is valid also for electricity. Indeed, there is an indeterminacy relation
for capacitors, of the form
Δ? Δ? ⩾ ? ,
(48)
where ? is the positron charge, ? capacity and ? potential difference. There is also an
indeterminacy relation between electric current ? and time ?
Δ? Δ? ⩾ ? .
Ref. 33
Both these relations may be found in the research literature.
(49)
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
The magnetic field limit is significant in the study of extreme stars and black holes. The
maximum electric field plays a role in the theory of gamma-ray bursters. For current,
conductivity and resistance in single channels, the limits and their effects were studied
extensively in the 1980s and 1990s.
The observation of quarks and of collective excitations in semiconductors with charge
?/3 does not necessarily invalidate the charge limit for physical systems. In neither case
is there is a physical system – defined as localized mass–energy interacting incoherently
with the environment – with charge ?/3.
Motion Mountain – The Adventure of Physics
electric conductivity:
1
ℏ
= 2 = 4.1 kΩ
4π?0 ?? ?
?2
? ⩽ 4π?0 ?? =
= 0.24 mS
ℏ
?⩾
cosm olo gical lim its for a ll physical ob serva bles
45
cosmolo gical limits for all physical observables
Vol. II, page 107
Size and energy dependence
Angul ar momentum and action
Challenge 18 e
Ref. 34
Vol. IV, page 151
It only takes a moment to check that the ratio of angular momentum ? to energy ? times
length ? has the dimensions of inverse speed. Since speeds are limited by the speed of
light, we get
1
?system ⩽ ?? .
(50)
?
Indeed, in nature there do not seem to be any exceptions to this limit on angular momentum. In no known system, from atoms to molecules, from ice skaters to galaxies,
does the angular momentum exceed this value. Even the most violently rotating objects,
the so-called extremal black holes, are limited in angular momentum by ? ⩽ ??/?. (Actually, this limit is correct for black holes only if the energy is taken as the irreducible
* Quantum theory refines this definition: a physical system is a part of nature that in addition interacts
incoherently with its environment. In the following discussion we will assume that this condition is satisfied.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
While looking for additional limits in nature, we note a fundamental fact. Any upper
limit for angular momentum, and any lower limit for power, must be system-dependent.
Such limits will not be absolute, but will depend on properties of the system. Now, a
physical system is a part of nature characterized by a boundary and its content.* Thus the
simplest properties shared by all systems are their size (characterized in the following by
the diameter) ? and their energy ?. With these characteristics we can deduce systemdependent limits for every physical observable. The general method is straightforward:
we take the known inequalities for speed, action, power, charge and entropy, and then
extract a limit for any observable, by inserting the length and energy as required. We
then have to select the strictest of the limits we find.
Motion Mountain – The Adventure of Physics
In our quest to understand motion, we have focused our attention on the four fundamental limitations to which motion is subject. Special relativity posits a limit to speed,
namely the speed of light ?. General relativity limits force and power respectively by
?4 /4? and ?5 /4?, and quantum theory introduces a smallest value ℏ for action. Nature
imposes the lower limit ? on entropy. If we include the limit ? on electric charge changes,
these limits induce extremal values for all physical observables, given by the corresponding (corrected) Planck values.
A question arises: does nature also impose limits on physical observables at the opposite end of the measurement scale? For example, there is a highest force and a highest
power in nature. Is there also a lowest force and a lowest power? Is there also a lowest
speed?
We will show that there are indeed such limits, for all observables. We give the general
method to generate such bounds, and explore several examples. This exploration will take
us on an interesting survey of modern physics; we start by deducing system-dependent
limits and then go on to the cosmological limits.
46
2 physics in lim it statem ents
mass times ?2 ; if the usual mass is used, the limit is too large by a factor of 4.) The limit
deduced from general relativity, given by ? ⩽ ?2 ?3 /4?, is not stricter than the one just
given. By the way, no system-dependent lower limit for angular momentum can be deduced.
The maximum value for angular momentum is also interesting when it is seen as an
action limit. Action is the time integral of the difference between kinetic and potential
energy. Though nature always seeks to minimize the action ?, systems, of size ?, that
maximize action are also interesting. You might check for yourself that the action limit
? ⩽ ??/?
Challenge 19 e
(51)
is not exceeded in any physical process.
Speed
?system ⩾ ℏ?2
Challenge 20 e
Challenge 22 s
(52)
This is not a new result; it is just a form of the indeterminacy relation of quantum theory.
It gives a minimum speed for any system of energy ? and diameter ?. Even the extremely
slow radius change of a black hole by evaporation just realizes this minimal speed.
Continuing with the same method, we also find that the limit deduced from general
relativity, ? ⩽ (?2 /4?)(?/?), gives no new information. Therefore, no system-dependent
upper speed limit exists – just the global limit ?.
Incidentally, the limits are not unique. Other limits can be found in a systematic way.
Upper limits can be multiplied, for example, by factors of (?/?)(?4 /4?) or (??)(2/ℏ?),
yielding less strict upper limits. A similar rule can be given for lower limits.
Force, p ower and luminosity
We have seen that force and power are central to general relativity. The force exerted
by a system is the flow of momentum out of the system; emitted power is the flow of
energy out of the system. Thanks to the connection ? = ??? between action ?, force
?, distance ? and time ?, we can deduce
?system ⩾
ℏ 1
.
2? ?2
(53)
Experiments do not reach this limit. The smallest forces measured in nature are those
in atomic force microscopes, where values as small as 1 aN are observed. But even these
values are above the lower force limit.
The power ? emitted by a system of size ? and mass ? is limited by
?3
?
?
⩾ ?system ⩾ 2ℏ? 3 .
?
?
(54)
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Challenge 21 e
1
.
??
Motion Mountain – The Adventure of Physics
Speed times mass times length is an action. Since action values in nature are limited from
below by ℏ, we get a limit for the speed of a system:
cosm olo gical lim its for a ll physical ob serva bles
47
The limit on the left is the upper limit for any engine or lamp, as deduced from relativity;
not even the universe exceeds it. The limit on the right is the minimum power emitted by any system through quantum gravity effects. Indeed, no physical system is completely tight. Even black holes, the systems with the best ability to keep components inside their enclosure, radiate. The power radiated by black holes should just meet this
limit, provided the length ? is taken to be the circumference of the black hole. Thus the
claim of the quantum gravity limit is that the power emitted by a black hole is the smallest power that is emitted by any composite system of the same surface gravity. (However,
the numerical factors in the black hole power appearing in the research literature are not
yet consistent.)
The strange charm of the entropy bound
Ref. 35
?
?c.Planck
⩽
?
? c.Planck
(55)
which gives
?⩽
??3
?,
4?ℏ
(56)
?
?c.Planck
⩽
? c.Planck
?c.Planck ? c.Planck
?
?
?
(57)
we get
?⩽
Ref. 30
π??
2π??
?? =
?? .
ℏ
ℏ
(58)
This is called Bekenstein’s entropy bound. It states that the entropy of any physical system is finite and limited by its mass ? and size ?. No exception has ever been found or
constructed, despite many attempts. Again, the limit value itself is only realized for black
holes.
We need to explain the strange assumption used above. We are investigating the entropy of a horizon. Horizons are not matter, but limits to empty space. The entropy of
horizons is due to the large number of virtual particles found at them. In order to deduce the maximum entropy of expression (57) we therefore have to use the properties of
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
where ? is the surface of the system. Equality is realized only for black holes. The old
question of the origin of the factor 4 in the entropy of black holes is thus answered here:
it is due to the factor 4 in the force or power bound in nature. Time will tell whether this
explanation will be generally accepted.
We can also derive a more general relation by using a mysterious assumption, which
we will discuss afterwards. We assume that the limits for vacuum are opposite to those
for matter. We can then write ?2 /4? ⩽ ?/? for the vacuum. Using
Motion Mountain – The Adventure of Physics
In 1973, Bekenstein discovered a famous limit that connects the entropy ? of a physical
system with its size and mass. No system has a larger entropy than one bounded by a
horizon. The larger the horizon surface, the larger the entropy. We write
48
Ref. 36
2 physics in lim it statem ents
the vacuum. In other words, either we use a mass-to-length ratio for vacuum above the
Planck limit, or we use the Planck entropy as the maximum value for vacuum.
Other, equivalent limits for entropy can be found if other variables are introduced. For
example, since the ratio of the shear viscosity ? to the volume density of entropy (times
?) has the dimensions of action, we can directly write
?⩽
Challenge 23 e
?
?? .
ℏ
(59)
Curiosities and fun challenges abou t system-dependent limits to
observables
∗∗
Challenge 24 r
The content of a system is characterized not only by its mass and charge, but also by
its strangeness, isospin, colour charge, charge and parity. Can you deduce the limits for
these quantities?
∗∗
Challenge 25 s
In our discussion of black hole limits, we silently assumed that they interact, like any
thermal system, in an incoherent way with the environment. Which of the results of this
section change when this condition is dropped, and how? Which limits can be overcome?
∗∗
Challenge 26 e
Can you find a general method to deduce all limits of observables?
∗∗
Bekenstein’s entropy bound leads to some interesting speculations. Let us speculate that
the universe itself, being surrounded by a horizon, meets the Bekenstein bound. The
entropy bound gives a bound to all degrees of freedom inside a system: it tells us that the
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Also the system-dependent limit values for all physical observables, like the Planck values, yield a plethora of interesting questions. We study a few examples.
Motion Mountain – The Adventure of Physics
Again, equality is only attained in the case of black holes. In time, no doubt, the list of
similar bounds will grow longer.
Is there also a smallest, system-dependent entropy? So far, there does not seem to be a
system-dependent minimum value for entropy: the present approach gives no expression
that is larger than ?.
The establishment of the entropy limit is an important step towards making our description of motion consistent. If space-time can move, as general relativity maintains, it
also has an entropy. How could entropy be finite if space-time were continuous? Clearly,
because of the existence of a minimum distance and minimum time in nature, spacetime cannot be continuous, but must have a finite number of degrees of freedom, and
thus a finite entropy.
cosm olo gical lim its for a ll physical ob serva bles
Challenge 27 e
49
number ?d.o.f . of degrees of freedom in the universe is roughly
?d.o.f . ≈ 10132 .
(60)
Compare this with the number ?Pl. vol. of Planck volumes in the universe
?Pl. vol. ≈ 10183
(61)
and with the number ?part. of particles in the universe
?part. ≈ 1091 .
(62)
∗∗
A lower limit for the temperature of a thermal system can be found using the following
idea: the number of degrees of freedom of a system is limited by its surface, or more
precisely, by the ratio between the surface and the Planck surface. We get the limit
?⩾
Vol. II, page 62
Challenge 29 ny
(63)
This is the smallest temperature that a system of mass ? and size ? can have. Alternatively, using the method given above, we can use the limit on the thermal energy
??/2 ⩾ ℏ?/2π? (the thermal wavelength must be smaller than the size of the system)
together with the limit on mass ?2 /4? ⩾ ?/?, and deduce the same result.
We have met the temperature limit already: when the system is a black hole, the limit
yields the temperature of the emitted radiation. In other words, the temperature of black
holes is the lower limit for all physical systems for which a temperature can be defined,
provided they share the same boundary gravity. The latter condition makes sense: boundary gravity is accessible from the outside and describes the full physical system, since it
depends on both its boundary and its content.
So far, no exception to the claim on the minimum system temperature is known. All
systems from everyday life comply with it, as do all stars. Also the coldest known systems in the universe, namely Bose–Einstein condensates and other cold gases produced
in laboratories, are much hotter than the limit, and thus much hotter than black holes
of the same surface gravity. (We saw earlier that a consistent Lorentz transformation for
temperature is not possible; so the minimum temperature limit is only valid for an observer at the same gravitational potential as the system under consideration and stationary relative to it.)
By the way, there seems to be no consistent way to define an upper limit for a sizedependent temperature. Limits for other thermodynamic quantities can be found, but
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Challenge 28 s
4?ℏ ?
.
π?? ?2
Motion Mountain – The Adventure of Physics
We see that particles are only a tiny fraction of what moves around. Most motion must be
movement of space-time. At the same time, space-time moves far less than might be naively expected. To find out how all this happens is the challenge of the unified description
of motion.
50
2 physics in lim it statem ents
we do not discuss them here.
∗∗
When electromagnetism plays a role in a system, the system also needs to be characterized by a charge ?. Our method then gives the following limit for the electric field ?:
? ⩾ 4??
?2
.
?2 ?2
(64)
We write the field limit in terms of the elementary charge ?, though it might be more appropriate to write it using the fine structure constant via ? = √4π?0 ?ℏ? . In observations,
the electric field limit has never been exceeded. For the magnetic field we get
?⩾
(65)
Again, this limit is satisfied by all known systems in nature.
Similar limits can be found for the other electromagnetic observables. In fact, several
of the limits given earlier are modified when electric charge is included. Does the size
limit change when electric charge is taken into account? In fact, an entire research field
is dedicated to deducing and testing the most general limits valid in nature.
∗∗
C osmology in one statement
We now continue our exploration of limits to the largest systems possible. In order to do
that, we have a simple look at cosmology.
Cosmology results from the equations of general relativity when the cosmological
constant is included. Cosmology can thus be summarized by any sufficiently general
statement that includes the cosmological constant Λ. The simplest statement can be deduced from the observation that the present distance ?0 of the night sky horizon is about
?0 ≈ 1/√Λ . From this we can summarize cosmology by stating
⊳ There is a maximum distance value of the order of 1.4/√Λ in nature.
For all systems and all observers, sizes, distances and lengths are limited by the relation
?≲
1.4
= 1.3 × 1026 m = 1.4 × 1010 al .
√Λ
(66)
This expression contains all of cosmology. The details of the numerical factor 1.4 are not
of importance here and we will often neglect it in the following. This statement on length
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Many cosmological limits have not been discussed here nor anywhere else. The following
could all be worth a publication: What is the limit for momentum? Energy? Pressure?
Acceleration? Mass change? Lifetime?
Motion Mountain – The Adventure of Physics
Challenge 30 s
4?? ?2
.
? ?2 ?2
cosm olo gical lim its for a ll physical ob serva bles
Challenge 31 s
51
should be added to the four fundamental Planck limits as a fifth limit statement in nature.
By the way, can you show that the cosmological constant is observer-invariant?
The cosmological limits to observables
Minimum force
The negative energy volume density −Λ?4 /4π? introduced by the positive cosmological
constant Λ corresponds to a negative pressure (both quantities have the same dimensions). When multiplied by the minimum area it yields a force value
?=
Λℏ?
= 4.8 ⋅ 10−79 N .
2π
(68)
Apart from the numerical factor, this is the cosmological force limit, the smallest possible
force in nature. This is also the gravitational force between two corrected Planck masses
located at the cosmological distance √π/4Λ .
As a note, we are led the fascinating conjecture that the full theory of general relativity,
including the cosmological constant, is defined by the combination of a maximum and
a minimum force in nature.
In summary,
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Challenge 32 e
It has never been reached or approached by any observation.
Many cosmological limits are related to black hole limits. The observed average mass
density of the universe is not far from the corresponding black hole limit. The black
hole lifetime limit might thus provide an upper limit for the full lifetime of the universe.
However, the age of the universe is far from that limit by a large factor. In fact, since the
universe’s size and age are increasing, the lifetime limit is pushed further into the future
with every second that passes. The universe evolves so as to escape its own decay.
Motion Mountain – The Adventure of Physics
From the system-dependent limits for speed, action, force and entropy we can deduce
system-dependent limits for all other physical observables. In addition, we note that the
system-dependent limits can (usually) be applied to the universe as a whole; we only need
to insert the size and energy content of the universe. Usually, we can do this through a
limit process, even though the universe itself is not a physical system. In this way, we get
an absolute limit for every physical observable that contains the cosmological constant Λ
and that is on the opposite end of the Planck limit for that observable. We can call these
limits the cosmological limits.
The simplest cosmological limit is the upper limit to length in the universe. Since the
cosmological length limit also implies a maximum possible Compton wavelength, we
get a minimum particle mass and energy. We also get an cosmological lower limit on
luminosity.
For single particles, we find an absolute lower speed limit, the cosmological speed limit,
given by
?
?particle ⩾ c. Planck ? = √4?ℏ/? √Λ ≈ 7 ⋅ 10−53 m/s .
(67)
? universe
52
2 physics in lim it statem ents
⊳ Nature provides two limits for each observable: a Planck limit and a cosmological limit.
Challenge 33 s
Every observable has a lower and an upper limit. You may want to summarize them into
a table. This has important consequences that we will explore now.
L imits to measurement precision and their challenge to thought
No real numbers
Because of the fundamental limits to measurement precision, the measured values of
physical observables do not require the full set of real numbers. In fact, limited precision
implies that observables cannot be described by the real numbers! This staggering result
appears whenever quantum theory and gravity are brought together. But there is more.
Page 58
Ref. 37
There is a limit to the precision of length measurements in nature. This limit is valid both
for length measurements of empty space and for length measurements of matter (or radiation). Now let us recall what we do when we measure the length of a table with a ruler.
To find the ends of the table, we must be able to distinguish the table from the surrounding air. In more precise terms, we must be able to distinguish matter from vacuum.
Whenever we want high measurement precision, we need to approach Planck scales.
But at Planck scales, the measurement values and the measurement errors are of the
same size. In short, at Planck scales, the intrinsic measurement limitations of nature imply that we cannot say whether we are measuring vacuum or matter. We will check this
conclusion in detail later on.
In fact, we can pick any other observable that distinguishes vacuum from matter –
for example, colour, mass, size, charge, speed or angular momentum – and we have the
same problem: at Planck scales, the limits to observables lead to limits to measurement
precision, and therefore, at Planck scales it is impossible to distinguish between matter
and vacuum. At Planck scales, we cannot tell whether a box is full or empty.
To state the conclusion in the sharpest possible terms: vacuum and matter do not differ
at Planck scales. This counter-intuitive result is one of the charms of the search for a
final, unified theory. It has inspired many researchers in the field and some have written
best-sellers about it. Brian Greene was particularly successful in presenting this side of
quantum geometry to the wider public.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Vacuum and mass: t wo sides of the same coin
Motion Mountain – The Adventure of Physics
We now know that in nature, every physical measurement has a lower and an upper
bound. One of the bounds is cosmological, the other is given by the (corrected) Planck
unit. As a consequence, for every observable, the smallest relative measurement error
that is possible in nature is the ratio between the Planck limit and the cosmological limit.
In particular, we have to conclude that all measurements are limited in precision.
All limits, those to observables and those to measurement precision, only appear
when quantum theory and gravity are brought together. But the existence of these limits,
and in particular the existence of limits to measurement precision, forces us to abandon
some cherished assumptions.
cosm olo gical lim its for a ll physical ob serva bles
Page 81
53
Limited measurement precision also implies that at the Planck energy it is impossible
to speak about points, instants, events or dimensionality. Similarly, at the Planck length
it is impossible to distinguish between positive and negative time values: so particles and
antiparticles are not clearly distinguished at Planck scales. All these conclusions are so
far-reaching that we must check them in more detail. We will do this shortly.
Measurement precision and the existence of sets
Page 58
⊳ Nature has no parts.
Page 58
In summary, at Planck scales, perfect separation is impossible in principle. We cannot
distinguish observations. At Planck scales it is impossible to split nature into separate parts
or entities. In nature, elements of sets cannot be defined. Neither discrete nor continuous
sets can be constructed:
⊳ Nature does not contain sets or elements.
Since sets and elements are only approximations, the concept of a ‘set’, which assumes
separable elements, is too specialized to describe nature. Nature cannot be described at
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 58
Motion Mountain – The Adventure of Physics
Page 58
In physics, it is generally assumed that nature is a set of components or parts. These components, called elements by mathematicians, are assumed to be separable from each other.
This tacit assumption is introduced in three main situations: it is assumed that matter
consists of separable particles, that space-time consists of separable events or points, and
that the set of states consists of separable initial conditions. Until the year 2000, physics
has built the whole of its description of nature on the concept of a set.
The existence of a fundamental limit to measurement precision implies that nature is
not a set of such separable elements. Precision limits imply that physical entities can be
distinguished only approximately. The approximate distinction is only possible at energies much lower than the Planck energy √ℏ?5 /4? . As humans, we do live at such small
energies, and we can safely make the approximation. Indeed, the approximation is excellent in practice; we do not notice any error. But at Planck energy, distinction and separation is impossible in principle. In particular, at the cosmic horizon, at the big bang, and
at Planck scales, any precise distinction between two events, two points or two particles
becomes impossible.
Another way to reach this result is the following. Separation of two entities requires
different measurement results – for example, different positions, different masses or different velocities. Whatever observable is chosen, at the Planck energy the distinction
becomes impossible because of the large measurements errors. Only at everyday energies is a distinction possible. In fact, even at everyday energies, any distinction between
two physical systems – for example, between a toothpick and a mountain – is possible
only approximately. At Planck scales, a boundary can never be drawn.
A third argument is the following. In order to count any entities in nature – a set of
particles, a discrete set of points, or any other discrete set of physical observables – the
entities have to be separable. But the inevitable measurement errors contradict separability. Thus at the Planck energy it is impossible to count physical objects with precision:
54
2 physics in lim it statem ents
Planck scales – i.e., with full precision – if any of the concepts used for its description
presupposes sets. However, all concepts used in the past 25 centuries to describe nature
– particles, space, time, observables, phase space, wave functions, Hilbert space, Fock
space, Riemannian space, particle space, loop space or moduli space – are based on sets.
They must all be abandoned at Planck energy.
⊳ No correct mathematical model of nature can be based on sets.
Page 108
In other terms, nature has no parts: nature is one.
None of the approaches to unification pursued in the twentieth century has abandoned sets. This requirement about the final theory is thus powerful and useful. Indeed,
the requirement to abandon sets will be an efficient guide in our search for the unification of relativity and quantum theory. The requirement will even solve Hilbert’s sixth
problem.
If we exclude gauge interactions, we can summarize the rest of physics in a few limit
statements:
Page 19
The speed limit is equivalent to special relativity.
The force limit is equivalent to general relativity.
The action limit is equivalent to quantum theory.
The entropy limit is equivalent to thermodynamics.
The distance limit is equivalent to cosmology.
All these limits are observer-invariant. The invariance of the limits suggests interesting
thought experiments, none of which leads to their violation.
The invariant limits imply that in nature every physical observable is bound on one
end by the corresponding (corrected) Planck unit and on the other end by a cosmological
limit. Every observable in nature has an upper and lower limit value.
The existence of lower and upper limit values to all observables implies that measurement precision is limited. As a consequence, matter and vacuum are indistinguishable, the description of space-time as a continuous manifold of points is not correct, and
nature can be described by sets and parts only approximately. At Planck scales, nature
does not contain sets or elements.
Nature’s limits imply that Planck units are the key to the final theory. Since the most
precise physical theories known, quantum theory and general relativity, can be reduced
to limit statements, there is a good chance that the final, unified theory of physics will
allow an equally simple description. Nature’s limits thus suggest that the mathematics of
the final, unified theory might be simple.
At this point of our adventure, many questions are still open. Answering any of the
open issues of the millennium list still seems out of reach. But this impression is too
pessimistic. Our discussion implies that we only need to find a description of nature that
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
⊳
⊳
⊳
⊳
⊳
Motion Mountain – The Adventure of Physics
summary on limits in nature
sum m ary on lim its in nature
55
is simple and without sets. And a natural way to avoid the use of sets is a description of
empty space, radiation and matter as being made of common constituents. But before we
explore this option, we check the conclusions of this chapter in another way. In particular, as a help to more conservative physicists, we check all conclusions we found so far
without making use of the maximum force principle.
Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Chapter 3
GENER AL R EL AT IVIT Y VER SUS
QUANT UM T HEORY
** ‘One needs to replace habits of thought by necessities of thought.’
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Vol. I, page 247
”
he two accurate descriptions of motion available in the year 2000, namely
hat of general relativity and that of the standard model, are both useful and
horoughly beautiful. This millennium description of motion is useful because its
consequences are confirmed by all experiments, to the full measurement precision. We
are able to describe and understand all examples of motion that have ever been encountered. We can use this understanding to save lives, provide food and enjoy life. We
have thus reached a considerable height in our mountain ascent. Our quest for the full
description of motion is not far from completion.
The results of twentieth century physics are also beautiful. By this, physicists just mean
that they can be phrased in simple terms. This is a poor definition of beauty, but physicists
are rarely experts on beauty. Nevertheless, if a physicist has some other concept of beauty
in physics, avoid him, because in that case he is really talking nonsense.
The simplicity of twentieth-century physics is well-known: all motion observed in
nature minimizes action. Since in physics, action is a measure of change, we can say that
all motion observed in nature minimizes change. In particular, every example of motion
due to general relativity or to the standard model of particle physics minimizes action:
both theories can be described concisely with the help of a Lagrangian.
On the other hand, some important aspects of any type of motion, the masses of the
involved elementary particles and the strength of their coupling, are unexplained by general relativity and by the standard model of particle physics. The same applies to the origin of all the particles in the universe, their initial conditions, and the dimensionality of
space-time. Obviously, the millennium description of physics is not yet complete.
The remaining part of our adventure will be the most demanding. In the ascent of
any high mountain, the head gets dizzy because of the lack of oxygen. The finite amount
of energy at our disposal requires that we leave behind all unnecessary baggage and
everything that slows us down. In order to determine what is unnecessary, we need to
focus on what we want to achieve. Our aim is the precise description of motion. But even
though general relativity and quantum theory are extremely precise, useful and simple,
we do carry a burden: the two theories and their concepts contradict each other.
Motion Mountain – The Adventure of Physics
T
“
Man muß die Denkgewohnheiten durch
Denknotwendigkeiten ersetzen.**
Albert Einstein
g enera l rel ativit y v ersus q ua ntum theory
57
The contradictions
Ref. 38
Vol. II, page 243
In classical physics and in general relativity, the vacuum, or empty space, is a region with
no mass, no energy and no momentum. If particles or gravitational fields are present, the
energy density is not zero, space is curved and there is no complete vacuum.
In everyday life, vacuum has an energy density that cannot be distinguished from
zero. However, general relativity proposes a way to check this with high precision: we
measure the average curvature of the universe. Nowadays, cosmological measurements
performed with dedicated satellites reveal an average energy density ?/? of the intergalactic ‘vacuum’ with the value of
?
≈ 0.5 nJ/m3 .
?
Vol. V, page 122
Vol. V, page 128
Ref. 41
Ref. 40
The approximation is valid for the case in which the cut-off frequency ?max is much larger
than the rest mass ? of the particles corresponding to the field under consideration. The
limit considerations given above imply that the cut-off energy has to be of the order of
the Planck energy √ℏ?5 /4? , about 0.6 ⋅ 1019 GeV= 1.0 GJ. That would give a vacuum
energy density of
?
(71)
≈ 10111 J/m3 ,
?
which is about 10120 times higher than the experimental measurement. In other words,
something is slightly wrong in the calculation due to quantum field theory.*
General relativity and quantum theory contradict each other in other ways. Gravity
is curved space-time. Extensive research has shown that quantum field theory, which
describes electrodynamics and nuclear forces, fails for situations with strongly curved
space-time. In these cases the concept of ‘particle’ is not precisely defined. Quantum
field theory cannot be extended to include gravity consistently, and thus to include general relativity. Without the concept of the particle as a discrete entity, we also lose the
ability to perform perturbation calculations – and these are the only calculations possible
* It is worthwhile to stress that the ‘slight’ mistake lies in the domain of quantum field theory. There is no
mistake and no mystery, despite the many claims to the contrary found in newspapers and in bad research
articles, in general relativity. This well-known point is made especially clear by Bianchi and Rovelli.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 39
In short, cosmological data show that the energy density of intergalactic space is not
exactly zero; nevertheless, the measured value is extremely small and can be neglected
in all laboratory experiments.
On the other hand, quantum field theory tells a different story on vacuum energy
density. A vacuum is a region with zero-point fluctuations. The energy content of a vacuum is the sum of the zero-point energies of all the fields it contains. Indeed, the Casimir
effect ‘proves’ the reality of these zero-point energies. Following quantum field theory,
the most precise theory known, their energy density is given, within one order of magnitude, by
? 4πℎ ?max 3
πℎ 4
? d? = 3 ?max
.
(70)
≈ 3 ∫
?
? 0
?
Motion Mountain – The Adventure of Physics
Ref. 39
(69)
58
Vol. V, page 294
Vol. V, page 44
Vol. II, page 284
Ref. 44, Ref. 45
Ref. 46
The origin of the contradictions
Ref. 47
All contradictions between general relativity and quantum mechanics have the same
origin. In 20th-century physics, motion is described in terms of objects, made up of
* John Archibald Wheeler (b. 1911, Jacksonville, d. 2008, Hightstown), was a physicist and influential teacher
who worked on general relativity.
** As we will see below, the strand model provides a way to incorporate fermions into an extremely accurate approximation of general relativity, without requiring any topology change. This effectively invalidates
Wheeler’s argument.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 42, Ref. 43
in quantum field theory. In short, quantum theory only works because it assumes that
gravity does not exist. Indeed, the gravitational constant ? does not appear in quantum
field theory.
On the other hand, general relativity neglects the commutation rules between physical
quantities discovered in experiments on a microscopic scale. General relativity assumes
that the classical notions of position and momentum of material objects are meaningful.
It thus ignores Planck’s constant ℏ, and only works by neglecting quantum effects.
The concept of measurement also differs. In general relativity, as in classical physics,
it is assumed that arbitrary precision of measurement is possible – for example, by using
finer and finer ruler marks. In quantum mechanics, on the other hand, the precision of
measurement is limited. The indeterminacy relation yields limits that follow from the
mass ? of the measurement apparatus.
The contradictions also concern the concept of time. According to relativity and classical physics, time is what is read from clocks. But quantum theory says that precise
clocks do not exist, especially if gravitation is taken into account. What does ‘waiting 10
minutes’ mean, if the clock goes into a quantum-mechanical superposition as a result of
its coupling to space-time geometry? It means nothing.
Similarly, general relativity implies that space and time cannot be distinguished,
whereas quantum theory implies that matter does make a distinction between them. A
related difference is the following. Quantum theory is a theory of – admittedly weird –
local observables. In general relativity, there are no local observables, as Einstein’s hole
argument shows.
The contradiction between the two theories is shown most clearly by the failure of
general relativity to describe the pair creation of particles with spin 1/2, a typical and
essential quantum process. John Wheeler* and others have argued that, in such a case, the
topology of space necessarily has to change; in general relativity, however, the topology
of space is fixed. Equivalently, quantum theory says that matter is made of fermions, but
fermions cannot be incorporated into general relativity.**
Another striking contradiction was pointed out by Jürgen Ehlers. Quantum theory is
built on point particles, and point particles move on time-like world lines. But following
general relativity, point particles have a singularity inside their black hole horizon; and
singularities always move on space-like world lines. The two theories thus contradict each
other at smallest distances.
No description of nature that contains contradictions can lead to a unified or to a
completely correct description. To eliminate the contradictions, we need to understand
their origin.
Motion Mountain – The Adventure of Physics
Page 64
3 g eneral rel ativity v ersus q uantum theory
g enera l rel ativit y v ersus q ua ntum theory
Ref. 48
59
The d omain of contradictions: Pl anck scales
?S = 2??/?2 .
Ref. 42, Ref. 49
(72)
The gravitational constant ? and the speed of light ? act as conversion constants. Indeed,
as the Schwarzschild radius of an object is approached, the difference between general
relativity and the classical 1/?2 description of gravity becomes larger and larger. For example, the barely measurable gravitational deflection of light by the Sun is due to the
light approaching the Sun to within 2.4 ⋅ 105 times its Schwarzschild radius. Usually, we
are forced to stay away from objects at a distance that is an even larger multiple of the
Schwarzschild radius, as shown in Table 2. Only for this reason is general relativity unne* In the following, we use the terms ‘vacuum’ and ‘empty space’ interchangeably.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Despite their contradictions and the underlying circular definition, both general relativity and quantum theory are successful theories for the description of nature: they agree
with all data. How can this be?
Each theory of modern physics provides a criterion for determining when it is necessary and when classical Galilean physics is no longer applicable. These criteria are the
basis for many arguments in the following chapters.
General relativity shows that it is necessary to take into account the curvature of empty
space* and space-time whenever we approach an object of mass ? to within a distance
of the order of the Schwarzschild radius ?S , given by
Motion Mountain – The Adventure of Physics
particles, and space-time, made up of events. Let us see how these two concepts are
defined.
A particle – and in general any object – is defined as a conserved entity that has a
position and that can move. In fact, the etymology of the word object is connected to the
latter property. In other words, a particle is a small entity with conserved mass, charge,
spin and so on, whose position can vary with time.
An event is a point in space and time. In every physics text, time is defined with the
help of moving objects, usually called ‘clocks’, or moving particles, such as those emitted by light sources. Similarly, length is defined in terms of objects, either with an oldfashioned ruler or in terms of the motion of light, which is itself motion of particles.
Modern physics has sharpened our definitions of particles and space-time. Quantum
mechanics assumes that space-time is given (as a symmetry of the Hamiltonian), and
studies the properties of particles and their motion, both for matter and for radiation.
Quantum theory has deduced the full list of properties that define a particle. General relativity, and especially cosmology, takes the opposite approach: it assumes that the properties of matter and radiation are given (for example, via their equations of state), and
describes in detail the space-time that follows from them, in particular its curvature.
However, one fact remains unchanged throughout all these advances: in the millennium description of nature, the two concepts of particle and of space-time are each defined
with the help of the other. This circular definition is the origin of the contradictions
between quantum mechanics and general relativity. In order to eliminate the contradictions and to formulate a complete theory, we must eliminate this circular definition.
60
3 g eneral rel ativity v ersus q uantum theory
TA B L E 2 The size, Schwarzschild radius and Compton wavelength of some objects appearing in nature.
The lengths in quotation marks make no physical sense, as explained in the text.
Dia- Mass ?
m eter
?
galaxy
neutron star
Sun
Earth
human
molecule
atom (12 C)
proton p
pion π
up-quark u
electron e
neutrino ?e
≈ 1 Zm
10 km
1.4 Gm
13 Mm
1.8 m
10 nm
0.6 nm
2 fm
2 fm
< 0.1 fm
< 4 am
< 4 am
S c h wa r z - R a t i o
schild
?/?S
r a d i u s ?S
≈ 5 ⋅ 1040 kg
2.8 ⋅ 1030 kg
2.0 ⋅ 1030 kg
6.0 ⋅ 1024 kg
75 kg
0.57 zg
20 yg
1.7 yg
0.24 yg
5 ⋅ 10−30 kg
9.1 ⋅ 10−31 kg
< 3 ⋅ 10−36 kg
≈ 70 Tm
4.2 km
3.0 km
8.9 mm
0.11 ym
‘8.5 ⋅ 10−52 m’
‘3.0 ⋅ 10−53 m’
‘2.5 ⋅ 10−54 m’
‘3.6 ⋅ 10−55 m’
‘7 ⋅ 10−57 m’
‘1.4 ⋅ 10−57 m’
‘< 5 ⋅ 10−63 m’
≈ 107
2.4
4.8 ⋅ 105
1.4 ⋅ 109
1.6 ⋅ 1025
1.2 ⋅ 1043
2.0 ⋅ 1043
8.0 ⋅ 1038
5.6 ⋅ 1039
< 1 ⋅ 1040
< 3 ⋅ 1039
n.a.
C om p ton R atio
wav e ?/? C
l e n g t h ?C
(red.)
‘≈ 10−83 m’
‘1.3 ⋅ 10−73 m’
‘1.0 ⋅ 10−73 m’
‘5.8 ⋅ 10−68 m’
‘4.7 ⋅ 10−45 m’
6.2 ⋅ 10−19 m
1.8 ⋅ 10−17 m
2.0 ⋅ 10−16 m
1.5 ⋅ 10−15 m
7 ⋅ 10−14 m
3.9 ⋅ 10−13 m
> 1 ⋅ 10−7 m
≈ 10104
8.0 ⋅ 1076
8.0 ⋅ 1081
2.2 ⋅ 1074
3.8 ⋅ 1044
1.6 ⋅ 1010
3.2 ⋅ 107
9.6
1.4
< 0.001
< 1 ⋅ 10−5
< 3 ⋅ 10−11
?C =
ℏ
.
??
(73)
In this case, Planck’s constant ℏ and the speed of light ? act as conversion factors to
transform the mass ? into a length scale. Of course, this length is only relevant if the
object is smaller than its own Compton wavelength. At these scales we get relativistic
quantum effects, such as particle–antiparticle pair creation or annihilation. Table 2 shows
that the approach distance is near to or smaller than the Compton wavelength only in
the microscopic world, so that such effects are not observed in everyday life. Only for
this reason we do not need quantum field theory to describe common observations.
Combining concepts of quantum field theory and general relativity is required in situations where both conditions are satisfied simultaneously. The necessary approach distance for such situations is calculated by setting ?S = 2? C (the factor 2 is introduced for
simplicity). We find that this is the case when lengths or times are – within a factor of
order 1 – of the order of
?Pl = √ℏ?/?3
= 1.6 ⋅ 10−35 m, the Planck length,
?Pl = √ℏ?/?5
= 5.4 ⋅ 10−44 s, the Planck time.
(74)
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
cessary in everyday life. We recall that objects whose size is given by their Schwarzschild
radius are black holes; smaller objects cannot exist.
Similarly, quantum mechanics shows that Galilean physics must be abandoned and
quantum effects must be taken into account whenever an object is approached to within
distances of the order of the (reduced) Compton wavelength ? C , given by
Motion Mountain – The Adventure of Physics
Challenge 34 e
Object
g enera l rel ativit y v ersus q ua ntum theory
Challenge 35 e
61
R esolving the contradictions
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
The contradictions between general relativity and quantum theory have little practical
consequences. Therefore, for a long time, the contradictions were accommodated by
keeping the two theories separate. It is often said that quantum mechanics is valid at
small scales and general relativity is valid at large scales. This attitude is acceptable as
long as we remain far from the Planck length. However, this accommodating attitude
also prevents us from resolving the circular definition, the contradictions and therefore,
the millennium issues.
The situation resembles the well-known drawing, Figure 2, by Maurits Escher (b. 1898
Leeuwarden, d. 1972 Hilversum) in which two hands, each holding a pencil, seem to
be drawing each other. If one hand is taken as a symbol of vacuum and the other as a
symbol of particles, with the act of drawing taken as the act of defining, the picture gives
a description of twentieth-century physics. The apparent circular definition is solved by
recognizing that the two concepts (the two hands) both originate from a third, hidden
concept. In the picture, this third entity is the hand of the artist. In physics, the third
concept is the common origin of vacuum and particles.
We thus conclude that the contradictions in physics and the circular definition are
solved by common constituents for vacuum and matter. In order to find out what these
common constituents are and what they are not, we must explore the behaviour of nature
at the Planck scales.
Motion Mountain – The Adventure of Physics
Whenever we approach objects at these scales, both general relativity and quantum
mechanics play a role, and effects of quantum gravity appear. Because the values of the
Planck dimensions are extremely small, this level of sophistication is unnecessary in
everyday life, in astronomy and even in particle physics.
In the millennium description of nature, all the contradictions and also the circular
definition just mentioned are effective only at Planck scales. You can check this yourself.
This is the reason that general relativity and quantum theory work so well in practice.
However, to answer the questions posed at the beginning – why do we live in three dimensions, why are there three interactions, and why is the proton 1836.15 times heavier
than the electron? – we require a precise and complete description of nature. To answer
these questions, we must understand physics at Planck scales.
In summary, general relativity and quantum theory do contradict each other. However, the domains where these contradictions play a role, the Planck scales, are not accessible by experiment. As a consequence, the contradictions and our lack of knowledge
of how nature behaves at the Planck scales have only one effect: we do not see the solutions to the millennium issues.
We note that some researchers argue that the Planck scales specify only one of several domains of nature where quantum mechanics and general relativity apply simultaneously. They mention horizons and the big bang as separate domains. However, it is more
appropriate to argue that horizons and the big bang are situations where Planck scales
are essential.
62
3 g eneral rel ativity v ersus q uantum theory
F I G U R E 2 ‘Tekenen’ by Maurits
The origin of p oints
Page 57
⊳ The use of points in space and of separate, point-like particles are the reasons
for the mistaken vacuum energy calculation (71) that is wrong by 120 orders
of magnitude.
In short, only the circular definition of space and matter allows us to define points
and point particles. This puts us in a strange situation. On the one hand, experiment tells
us that describing nature with space points and with point particles works. On the other
hand, reason tells us that this is a fallacy and cannot be correct at Planck scales. We need
a solution.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
General relativity is built on the assumption that space is a continuum of points. Already
at school we learn that lines, surfaces and areas are made of points. We take this as granted, because we imagine that finer and finer measurements are always possible. And all
experiments so far agree with the assumption. Fact is: in this reasoning, we first idealized measurement rulers – which are made of matter – and then ‘deduced’ that points
in space exist.
Quantum theory is built on the assumption that elementary particles are point-like.
We take this as granted, because we imagine that collisions at higher and higher energy
are possible that allow elementary particles to get as close as possible. And all experiments so far agree with the assumption. Fact is: in this reasoning, we first imagined infinite energy and momentum values – which is a statement on time and space properties
– and then ‘deduce’ that point particles exist.
Motion Mountain – The Adventure of Physics
Escher, 1948 – a metaphor for the
way in which ‘particles’ and
‘space-time’ are defined: each with
the help of the other (© M.C. Escher
Heirs).
sum m ary on the cl ash b et ween the t wo theories
63
summary on the cl ash bet ween the t wo theories
General relativity and quantum theory contradict each other. In practice however, this
happens only at Planck scales. The reason for the contradiction is our insistence on a
circular definition of space and particles. Indeed, we need this circularity: Only such a
circular definition allows us to define points and point particles at all.
In order to solve the contradictions between general relativity and quantum theory
and in order to understand nature at Planck scales, we must introduce common constituents for space and particles. But common constituents have an important consequence:
common constituents force us to stop using points to describe nature. We now explore
this connection.
Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Chapter 4
D OES MAT T ER DIFFER FROM
VACUUM?
Vol. II, page 24
T
Farewell to instants of time
Ref. 50, Ref. 51
Ref. 52, Ref. 53
Ref. 54
Page 74
”
Measurement limits appear most clearly when we investigate the properties of clocks and
metre rules. Is it possible to construct a clock that is able to measure time intervals shorter
than the Planck time? Surprisingly, the answer is no, even though the time–energy indeterminacy relation Δ?Δ? ⩾ ℏ seems to indicate that by making Δ? large enough, we can
make Δ? arbitrary small.
Every clock is a device with some moving parts. The moving parts can be mechanical wheels, or particles of matter in motion, or changing electrodynamic fields (i.e.,
photons), or decaying radioactive particles. For each moving component of a clock the
indeterminacy relation applies. As explained most clearly by Michael Raymer, the indeterminacy relation for two non-commuting variables describes two different, but related,
situations: it makes a statement about standard deviations of separate measurements on
many identical systems; and it describes the measurement precision for a joint measurement on a single system. In what follows, we will consider only the second situation.
For a clock to be useful, we need to know both the time and the energy of each hand.
Otherwise it would not be a recording device. More generally, a clock must be a classical system. We need the combined knowledge of the non-commuting variables for each
moving component of the clock. Let us focus on the component with the largest time indeterminacy Δ?. It is evident that the smallest time interval ?? that can be measured by
** Moses Maimonides (b. 1135 Cordoba, d. 1204 Egypt) was a physician, philosopher and influential theologian. However, there is no evidence for ‘time atoms’ in nature, as explained below.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
“
Time is composed of time atoms ... which in
fact are indivisible.
Maimonides**
Motion Mountain – The Adventure of Physics
he appearance of the quantum of action in the description of motion leads
o limitations for all measurements: Heisenberg’s indeterminacy relations.
hese relations, when combined with the effects of gravitation, imply an almost
unbelievable series of consequences for the behaviour of nature at Planck scales. The
most important ones are the necessity to abandon points, instants and events, and the
equivalence of vacuum and matter. Here we show how these surprising and important
conclusions follow from simple arguments based on the indeterminacy relations, the
Compton wavelength and the Schwarzschild radius.
d oes m at ter d iffer from vacuum?
65
a clock is always larger than the quantum limit, i.e., larger than the time indeterminacy
Δ? for the most ‘uncertain’ component. Thus we have
?? ⩾ Δ? ⩾
ℏ
,
Δ?
(75)
where Δ? is the energy indeterminacy of the moving component. Now, Δ? must be smaller than the total energy ? = ?2 ? of the component itself: Δ? < ?2 ?.* Furthermore, a
clock provides information, so signals have to be able to leave it. Therefore the clock must
not be a black hole: its mass ? must be smaller than a black hole of its size, i.e., ? ⩽ ?2 ?/?,
where ? is the size of the clock (neglecting factors of order unity). Finally, for a sensible
measurement of the time interval ??, the size ? of the clock must be smaller than ? ??,
because otherwise different parts of the clock could not work together to produce the
same time display: ? < ???.** If we combine these three conditions, we get
ℏ?
?5 ??
(76)
or
?? ⩾ √
Vol. II, page 163
Ref. 49
Challenge 36 s
Ref. 55, Ref. 56
Ref. 57, Ref. 58
(77)
In summary, from three simple properties of any clock – namely, that it is only a single
clock, that we can read its dial, and that it gives sensible read-outs – we conclude that
clocks cannot measure time intervals shorter than the Planck time. Note that this argument
is independent of the nature of the clock mechanism. Whether the clock operates by
gravitational, electrical, mechanical or even nuclear means, the limit still applies.***
The same conclusion can be reached in other ways. For example, any clock small
enough to measure small time intervals necessarily has a certain energy indeterminacy
due to the indeterminacy relation. Meanwhile, on the basis of general relativity, any energy density induces a deformation of space-time, and signals from the deformed region
arrive with a certain delay due to that deformation. The energy indeterminacy of the
source leads to an indeterminacy in the deformation, and thus in the delay. The expression from general relativity for the deformation of the time part of the line element due
to a mass ? is ?? = ??/??3 . From the mass–energy relation, we see that an energy spread
* Physically, this condition means being sure that there is only one clock: if Δ? > ?, it would be impossible
to distinguish between a single clock and a clock–anticlock pair created from the vacuum, or a component
together with two such pairs, and so on.
** It is amusing to explore how a clock larger than ? ?? would stop working, as a result of the loss of rigidity
in its components.
*** Gravitation is essential here. The present argument differs from the well-known study on the limitations
of clocks due to their mass and their measuring time which was published by Salecker and Wigner and
summarized in pedagogical form by Zimmerman. In our case, both quantum mechanics and gravity are
included, and therefore a different, lower, and more fundamental limit is found. Also the discovery of black
hole radiation does not change the argument: black hole radiation notwithstanding, measurement devices
cannot exist inside black holes.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 59
ℏ?
= ?Pl .
?5
Motion Mountain – The Adventure of Physics
?? ⩾
66
4 d oes m at ter d iffer from vacuum?
Δ? produces an indeterminacy Δ? in the delay:
Δ? =
Δ? ?
.
? ?5
(78)
This determines the precision of the clock. Furthermore, the energy indeterminacy of the
clock is fixed by the indeterminacy relation for time and energy Δ? ⩾ ℏ/Δ?. Combining
all this, we again find the relation ?? ⩾ ?Pl for the minimum measurable time.
We are forced to conclude that in nature, it is impossible to measure time intervals
shorter than the Planck time. Thus
⊳ In nature there is a minimum time interval.
“
Ref. 24
Our greatest pretenses are built up not to hide
the evil and the ugly in us, but our emptiness.
The hardest thing to hide is something that is
not there.
Eric Hoffer,* The Passionate State of Mind
”
In a similar way, we can deduce that it is impossible to make a metre rule, or any other
length-measuring device, that is able to measure lengths shorter than the Planck length.
Obviously, we can already deduce this from ?Pl = ? ?Pl , but an independent proof is also
possible.
For any length measurement, joint measurements of position and momentum are necessary. The most straightforward way to measure the distance between two points is to
put an object at rest at each position. Now, the minimal length ?? that can be measured
* Eric Hoffer (b. 1902 New York City, d. 1983 San Francisco), philosopher.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Farewell to p oints in space
Motion Mountain – The Adventure of Physics
Ref. 60
In other words, at Planck scales the term ‘instant of time’ has no theoretical or experimental
basis. But let us go on. Special relativity, quantum mechanics and general relativity all rely
on the idea that time can be defined for all points of a given reference frame. However,
two clocks a distance ? apart cannot be synchronized with arbitrary precision. Since the
distance between two clocks cannot be measured with an error smaller than the Planck
length ?Pl , and transmission of signals is necessary for synchronization, it is not possible
to synchronize two clocks with a better precision than ?Pl /? = ?Pl , the Planck time. So
use of a single time coordinate for a whole reference frame is only an approximation.
Reference frames do not have a single time coordinate at Planck scales.
Moreover, since the time difference between events can only be measured within a
Planck time, for two events distant in time by this order of magnitude, it is not possible
to say with complete certainty which of the two precedes the other. But if events cannot
be ordered, then the very concept of time, which was introduced into physics to describe
sequences, makes no sense at Planck scales. In other words, after dropping the idea of a
common time coordinate for a complete frame of reference, we are forced to drop the
idea of time at a single ‘point’ as well. The concept of ‘proper time’ loses its meaning at
Planck scales.
d oes m at ter d iffer from vacuum?
67
must be larger than the position indeterminacy of the two objects. From the indeterminacy relation we know that neither object’s position can be determined with a precision
Δ? better than that given by Δ? Δ? = ℏ, where Δ? is the momentum indeterminacy. The
requirement that there be only one object at each end (avoiding pair production from
the vacuum) means that Δ? < ??: together, these requirements give
?? ⩾ Δ? ⩾
ℏ
.
??
(79)
Ref. 42, Ref. 49
Ref. 24
Ref. 61
Ref. 26
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 62, Ref. 63
Length measurements are limited by the Planck length.
Another way to deduce this limit reverses the roles of general relativity and quantum
theory. To measure the distance between two objects, we have to localize the first object
with respect to the other within a certain interval Δ?. The corresponding energy indeterminacy obeys Δ? = ?(?2 ?2 + (Δ?)2 )1/2 ⩾ ?ℏ/Δ?. However, general relativity shows
that a small volume filled with energy changes the curvature of space-time, and thus
changes the metric of the surrounding space. For the resulting distance change Δ?, compared with empty space, we find the expression Δ? ≈ ?Δ?/?4 . In short, if we localize the
first particle in space with a precision Δ?, the distance to a second particle is known
only with precision Δ?. The minimum length ?? that can be measured is obviously larger
than either of these quantities; inserting the expression for Δ?, we find again that the
minimum measurable length ?? is given by the Planck length.
We note that every length measurement requires a joint measurement of position and
momentum. This is particularly obvious if we approach a metre ruler to an object, but it
is equally true for any other length measurement.
We note that, since the Planck length is the shortest possible length, there can be
no observations of quantum-mechanical effects for a situation where the corresponding de Broglie or Compton wavelength is smaller than the Planck length. In proton–
proton collisions we observe both pair production and interference effects. In contrast,
the Planck limit implies that in everyday, macroscopic situations, such as car–car collisions, we cannot observe embryo–antiembryo pair production and quantum interference effects.
Another way to convince oneself that points have no meaning is to observe that a
point is an entity with vanishing volume; however, the minimum volume possible in
3
nature is the Planck volume ?Pl = ?Pl
.
We conclude that the Planck units not only provide natural units; they also provide –
within a factor of order one – the limit values of space and time intervals.
In summary, from two simple properties common to all length-measuring devices,
namely that they are discrete and that they can be read, we arrive at the conclusion that
Motion Mountain – The Adventure of Physics
Furthermore, the measurement cannot be performed if signals cannot leave the objects;
thus, they cannot be black holes. Therefore their masses must be small enough for their
Schwarzschild radius ?S = 2??/?2 to be less than the distance ?? separating them. Again
omitting the factor of 2, we get
ℏ?
?? ⩾ √ 3 = ?Pl .
(80)
?
68
4 d oes m at ter d iffer from vacuum?
⊳ Lengths smaller than the Planck length cannot be measured.
Whatever method is used, be it a metre rule or time-of-flight measurement, we cannot
overcome this fundamental limit. It follows that the concept of a ‘point in space’ has no
experimental or theoretical basis. In other terms,
⊳ In nature there is a minimum length interval.
The limitations on length measurements imply that we cannot speak of continuous space,
except in an approximate sense. As a result of the lack of measurement precision at Planck
scales, the concepts of spatial order, of translation invariance, of isotropy of the vacuum
and of global coordinate systems have no experimental basis.
The generalized indeterminacy rel ation
The limit values for length and time measurements are often expressed by the so-called
generalized indeterminacy relation
Δ?Δ? ⩾ ℏ/2 + ?
?
(Δ?)2
?3
(81)
Δ?Δ? ⩾ ℏ/2 + ?
2
?Pl
(Δ?)2 ,
ℏ
(82)
or
Ref. 24
Ref. 64, Ref. 65
Ref. 66, Ref. 67
Ref. 68
Farewell to space-time continuit y
“
Ich betrachte es als durchaus möglich, dass die Physik nicht auf dem Feldbegriff
begründet werden kann, d.h. auf kontinuierlichen Gebilden. Dann bleibt von
meinem ganzen Luftschloss inklusive Gravitationstheorie nichts bestehen.*
Albert Einstein, 1954, in a letter to Michele Besso.
”
* ‘I consider it as quite possible that physics cannot be based on the field concept, i.e., on continuous structures. In that case, nothing remains of my castle in the air, gravitation theory included.’
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Challenge 37 e
where ? is a numerical factor of order unity. A similar expression holds for the time–
energy indeterminacy relation. The first term on the right-hand side is the usual
quantum-mechanical indeterminacy. The second term is negligible for everyday energies, and is significant only near Planck energies; it is due to the changes in space-time
induced by gravity at these high energies. You should be able to show that the generalized
principle (81) implies that Δ? can never be smaller than ?1/2 ?Pl .
The generalized indeterminacy relation is derived in exactly the same way in which
Heisenberg derived the original indeterminacy relation Δ?Δ? ⩾ ℏ/2, namely by studying
the scattering of light by an object under a microscope. A careful re-evaluation of the
process, this time including gravity, yields equation (81). For this reason, all descriptions
that unify quantum mechanics and gravity must yield this relation, and indeed all known
approaches do so.
Motion Mountain – The Adventure of Physics
Ref. 24
d oes m at ter d iffer from vacuum?
Ref. 69
69
The classical description of nature is based on continuity: it involves and allows differences of time and space that are as small as can be imagined. Between any two points
in time or space, the existence of infinitely many other points is assumed. Measurement
results of arbitrary small values are deemed possible. The same is valid for action values.
However, quantum mechanics begins with the realization that the classical concept of
action makes no sense below the value of ℏ/2; similarly, unified theories begin with the
realization that the classical concepts of time and length make no sense below Planck
scales. Therefore, the continuum description of space-time has to be abandoned in favour
of a more appropriate description.
The minimum length distance, the minimum time interval, and equivalently, the new,
generalized indeterminacy relation appearing at Planck scales show that space, time and
in particular, space-time, are not well described as a continuum. Inserting ?Δ? ⩾ Δ? ⩾
ℏ/Δ? into equation (81), we get
(83)
which of course has no counterpart in standard quantum mechanics. This shows that
also space-time events do not exist. The concept of an ‘event’, being a combination of a
‘point in space’ and an ‘instant of time’, loses its meaning for the description of nature
at Planck scales.
Interestingly, the view that continuity must be abandoned is almost one hundred years
old. Already in 1917, Albert Einstein wrote in a letter to Werner Dällenbach:
Ref. 70
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Wenn die molekulare Auffassung der Materie die richtige (zweckmässige) ist, d.h. wenn ein Teil Welt durch eine endliche Zahl bewegter Punkte
darzustellen ist, so enthält das Kontinuum der heutigen Theorie zu viel Mannigfaltigkeit der Möglichkeiten. Auch ich glaube, dass dieses zu viel daran
schuld ist, dass unsere heutige Mittel der Beschreibung an der Quantentheorie scheitern. Die Frage scheint mir, wie man über ein Diskontinuum
Aussagen formulieren kann, ohne ein Kontinuum (Raum-Zeit) zu Hilfe zu
nehmen; letzteres wäre als eine im Wesen des Problems nicht gerechtfertigte
zusätzliche Konstruktion, der nichts “Reales” entspricht, aus der Theorie zu
verbannen. Dazu fehlt uns aber leider noch die mathematische Form. Wie
viel habe ich mich in diesem Sinne schon geplagt!
Allerdings sehe ich auch hier prinzipielle Schwierigkeiten. Die Elektronen (als Punkte) wären in einem solchen System letzte Gegebenheiten (Bausteine). Gibt es überhaupt letzte Bausteine? Warum sind diese alle von gleicher Grösse? Ist es befriedigend zu sagen: Gott hat sie in seiner Weisheit alle
gleich gross gemacht, jedes wie jedes andere, weil er so wollte; er hätte sie
auch, wenn es ihm gepasst hätte, verschieden machen können. Da ist man
bei der Kontinuum-Auffassung besser dran, weil man nicht von Anfang an
die Elementar-Bausteine angeben muss. Ferner die alte Frage vom Vakuum!
Aber diese Bedenken müssen verblassen hinter der blendenden Tatsache:
Das Kontinuum ist ausführlicher als die zu beschreibenden Dinge...
Lieber Dällenbach! Was hilft alles Argumentieren, wenn man nicht bis zu
einer befriedigenden Auffassung durchdringt; das aber ist verteufelt schwer.
Motion Mountain – The Adventure of Physics
Δ?Δ? ⩾ ℏ?/?4 = ?Pl ?Pl ,
70
4 d oes m at ter d iffer from vacuum?
Es wird einen schweren Kampf kosten, bis man diesen Schritt, der uns da
vorschwebt, wirklich gemacht haben wird. Also strengen Sie Ihr Gehirn an,
vielleicht zwingen Sie es.*
The second half of this text will propose a way to rise to the challenge. At this point
however, we first complete the exploration of the limitations of continuum physics.
In 20th century physics, space-time points are idealizations of events – but this idealization is inadequate. The use of the concept of ‘point’ is similar to the use of the concept
of ‘aether’ a century ago: it is impossible to measure or detect.
⊳ Like the ‘aether’, also ‘points’ lead reason astray.
⊳ Between two points there is not always a third.
Page 86
* ‘If the molecular conception of matter is the right (appropriate) one, i.e., if a part of the world is to be
represented by a finite number of moving points, then the continuum of the present theory contains too
great a manifold of possibilities. I also believe that this ‘too great’ is responsible for our present means of
description failing for quantum theory. The questions seems to me how one can formulate statements about
a discontinuum without using a continuum (space-time) as an aid; the latter should be banned from the
theory as a supplementary construction not justified by the essence of the problem, which corresponds to
nothing “real”. But unfortunately we still lack the mathematical form. How much have I already plagued
myself in this direction!
Yet I also see difficulties of principle. In such a system the electrons (as points) would be the ultimate
entities (building blocks). Do ultimate building blocks really exist? Why are they all of equal size? Is it satisfactory to say: God in his wisdom made them all equally big, each like every other one, because he wanted
it that way; he could also have made them, if he had wanted, all different. With the continuum viewpoint
one is better off, because one doesn’t have to prescribe elementary building blocks from the outset. Furthermore, the old question of the vacuum! But these considerations must pale beside the dazzling fact: The
continuum is more ample than the things to be described...
Dear Dällenbach! All arguing does not help if one does not achieve a satisfying conception; but this is
devilishly difficult. It will cost a difficult fight until the step that we are thinking of will be realized. Thus,
squeeze your brain, maybe you can force it.’
Compare this letter to what Einstein wrote almost twenty and almost forty years later.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
This results again means that space and time are not continuous. Of course, at large scales
they are – approximately – continuous, in the same way that a piece of rubber or a liquid
seems continuous at everyday scales, even though it is not at a small scale. But in nature,
space, time and space-time are not continuous entities.
Motion Mountain – The Adventure of Physics
All paradoxes resulting from the infinite divisibility of space and time, such as Zeno’s
argument on the impossibility of distinguishing motion from rest, or the Banach–Tarski
paradox, are now avoided. We can dismiss them straight away because of their incorrect
premises concerning the nature of space and time.
The consequences of the Planck limits for measurements of time and space can be
expressed in other ways. It is often said that given any two points in space or any two
instants of time, there is always a third in between. Physicists sloppily call this property
continuity, while mathematicians call it denseness. However, at Planck scales this property cannot hold, since there are no intervals smaller than the Planck time. Thus points
and instants are not dense, and
d oes m at ter d iffer from vacuum?
71
But there is more to come. The very existence of a minimum length contradicts the
theory of special relativity, in which it is shown that lengths undergo Lorentz contraction
when the frame of reference is changed. There is only one conclusion: special relativity
(and general relativity) cannot be correct at very small distances. Thus,
⊳ Space-time is not Lorentz-invariant (nor diffeomorphism-invariant) at
Planck scales.
Even the number of spatial dimensions makes no sense at Planck scales. Let us remind
ourselves how to determine this number experimentally. One possible way is to determine how many points we can choose in space such that all the distances between them
are equal. If we can find at most ? such points, the space has ? − 1 dimensions. But if
reliable length measurement at Planck scales is not possible, there is no way to determine
reliably the number of dimensions of space with this method.
Another way to check for three spatial dimensions is to make a knot in a shoe string
and glue the ends together: since it stays knotted, we know that space has three dimensions, because there is a mathematical theorem that in spaces with greater or fewer than
three dimensions, knots do not exist. Again, at Planck scales, we cannot say whether a
string is knotted or not, because measurement limits at crossings make it impossible to
say which strand lies above the other.
There are many other methods for determining the dimensionality of space.* In all
cases, the definition of dimensionality is based on a precise definition of the concept of
* For example, we can determine the dimension using only the topological properties of space. If we draw a
so-called covering of a topological space with open sets, there are always points that are elements of several
sets of the covering. Let ? be the maximal number of sets of which a point can be an element in a given
covering. The minimum value of ? over all possible coverings, minus one, gives the dimension of the space.
In fact, if physical space is not a manifold, the various methods for determining the dimensionality may
give different answers. Indeed, for linear spaces without norm, the dimensionality cannot be defined in a
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Farewell to dimensionalit y
Motion Mountain – The Adventure of Physics
Ref. 24, Ref. 71
All the symmetries that are at the basis of special and general relativity are only approximately valid at Planck scales.
The imprecision of measurement implies that most familiar concepts used to describe
spatial relations become useless. For example, the concept of a metric loses its usefulness
at Planck scales, since distances cannot be measured with precision. So it is impossible
to say whether space is flat or curved. The impossibility of measuring lengths exactly is
equivalent to fluctuations of the curvature, and thus of gravity.
In short, space and space-time are not smooth at Planck scales. This conclusion has
important implications. For example, the conclusion implies that certain mathematical
solutions found in books on general relativity, such as the Eddington–Finkelstein coordinates and the Kruskal–Szekeres coordinates do not describe nature! Indeed, these
coordinate systems, which claim to show that space-time goes on behind the horizon of
a black hole, are based on the idea that space-time is smooth everywhere. However,
quantum physics shows that space-time is not smooth at the horizon, but fluctuates
wildly there. In short, quantum physics confirms what common sense already knew:
Behind a horizon, nothing can be observed, and thus there is nothing there.
72
4 d oes m at ter d iffer from vacuum?
neighbourhood. At Planck scales, however, length measurements do not allow us to say
whether a given point is inside or outside a given region. In short, whatever method we
use, the lack of precise length measurements means that
⊳ At Planck scales, the dimensionality of physical space is not defined.
Farewell to the space-time manifold
Ref. 72
Ref. 73
But there are more surprises. At Planck scales, since both temporal and spatial order
break down, there is no way to say if the distance between two nearby space-time regions
unique way. Different definitions (fractal dimension, Lyapunov dimension, etc.) are possible.
* Where does the incorrect idea of continuous space-time have its roots? In everyday life, as well as in physics, space-time is a book-keeping device introduced to describe observations. Its properties are extracted
from the properties of observables. Since observables can be added and multiplied, like numbers, we infer that they can take continuous values, and, in particular, arbitrarily small values. It is then possible to
define points and sets of points. A special field of mathematics, topology, shows how to start from a set of
points and construct, with the help of neighbourhood relations and separation properties, first a topological
space, then, with the help of a metric, a metric space. With the appropriate compactness and connectedness
relations, a manifold, characterized by its dimension, metric and topology, can be constructed.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
⊳ Physical space-time cannot be a set of mathematical points.
Motion Mountain – The Adventure of Physics
Vol. I, page 57
The reasons for the problems with space-time become most evident when we remember
Euclid’s well-known definition: ‘A point is that which has no part.’ As Euclid clearly understood, a physical point, as an idealization of position, cannot be defined without some
measurement method. Mathematical points, however, can be defined without reference
to a metric. They are just elements of a set, usually called a ‘space’. (A ‘measurable’ or
‘metric’ space is a set of points equipped with a measure or a metric.)
In the case of physical space-time, the concepts of measure and of metric are more
fundamental than that of a point. Confusion between physical and mathematical space
and points arises from the failure to distinguish a mathematical metric from a physical
length measurement.*
Perhaps the most beautiful way to make this point is the Banach–Tarski theorem,
which clearly shows the limits of the concept of volume. The theorem states that a sphere
made up of mathematical points can be cut into five pieces in such a way that the pieces
can be put together to form two spheres, each of the same volume as the original one.
However, the necessary ‘cuts’ are infinitely curved and detailed: the pieces are wildly disconnected. For physical matter such as gold, unfortunately – or fortunately – the existence of a minimum length, namely the atomic distance, makes it impossible to perform
such a cut. For vacuum, the puzzle reappears. For example, the energy of zero-point
fluctuations is given by the density times the volume; following the Banach–Tarski theorem, the zero-point energy content of a single sphere should be equal to the zero-point
energy of two similar spheres each of the same volume as the original one. The paradox
is resolved by the Planck length, which provides a fundamental length scale even for vacuum, thus making infinitely complex cuts impossible. Therefore, the concept of volume
is only well defined at Planck scales if a minimum length is introduced.
To sum up:
d oes m at ter d iffer from vacuum?
73
is space-like or time-like.
⊳ At Planck scales, time and space cannot be distinguished from each other.
In addition, we cannot state that the topology of space-time is fixed, as general relativity
implies. The topology changes, mentioned above, that are required for particle reactions
do become possible. In this way another of the contradictions between general relativity
and quantum theory is resolved.
In summary, space-time at Planck scales is not continuous, not ordered, not endowed
with a metric, not four-dimensional, and not made up of points. It satisfies none of the
defining properties of a manifold.* We conclude that the concept of a space-time manifold
has no justification at Planck scales. This is a strong result. Even though both general
relativity and quantum mechanics use continuous space-time, the combined theory does
not.
Vol. V, page 358
* A manifold is what looks locally like a Euclidean space. The exact definition can be found in the previous
volume.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
If space and time are not continuous, no quantities defined as derivatives with respect
to space or time are precisely defined. Velocity, acceleration, momentum, energy and
so on are only well defined under the assumption of continuity. That important tool,
the evolution equation, is based on derivatives and can thus no longer be used. Therefore the Schrödinger and Dirac equations lose their basis. Concepts such as ‘derivative’,
‘divergence-free’ and ‘source free’ lose their meaning at Planck scales.
All physical observables are defined using length and time measurements. Each physical unit is a product of powers of length and time (and mass) units. (In the SI system,
electrical quantities have a separate base quantity, the ampere, but the argument still
holds: the ampere is itself defined in terms of a force, which is measured using the three
base units of length, time and mass.) Since time and length are not continuous, at Planck
scales, observables cannot be described by real numbers.
In addition, if time and space are not continuous, the usual expression for an observable field, ?(?, ?), does not make sense: we have to find a more appropriate description.
Physical fields cannot exist at Planck scales. Quantum mechanics also relies on the possibility to add wave functions; this is sometimes called the superposition principle. Without
fields and superpositions, all of quantum mechanics comes crumbling down.
The lack of real numbers has severe consequences. It makes no sense to define multiplication of observables by real numbers, but only by a discrete set of numbers. Among
other implications, this means that observables do not form a linear algebra. Observables are not described by operators at Planck scales. In particular, the most important
observables are the gauge potentials. Since they do not form an algebra, gauge symmetry
is not valid at Planck scales. Even innocuous-looking expressions such as [??, ??] = 0 for
?? ≠ ??, which are at the root of quantum field theory, become meaningless at Planck
scales. Since at those scales superpositions cannot be backed up by experiment, even
the famous Wheeler–DeWitt equation, sometimes assumed to describe quantum gravity, cannot be valid.
Motion Mountain – The Adventure of Physics
Farewell to observables, symmetries and measurements
74
4 d oes m at ter d iffer from vacuum?
Similarly, permutation symmetry is based on the premise that we can distinguish two
points by their coordinates, and then exchange particles between those locations. As we
have just seen, this is not possible if the distance between the two particles is very small.
We conclude that permutation symmetry has no experimental basis at Planck scales.
Even discrete symmetries, like charge conjugation, space inversion and time reversal,
cannot be correct in this domain, because there is no way to verify them exactly by measurement. CPT symmetry is not valid at Planck scales.
Finally we note that all types of scaling relations break down at small scales, because
of the existence of a smallest length. As a result, the renormalization group breaks down
at Planck scales.
In summary, due to the impossibility of accurate measurements,
⊳ All symmetries break down at Planck scales.
⊳ The concept of measurement has no significance at Planck scales.
This results from the limitations on time and length measurements.
Can space or space-time be a l at tice?
Ref. 75
Ref. 76
Ref. 77
Ref. 78
⊳ Space is not discrete. Neither is space-time.
We will discover more evidence for this negative conclusion later on.
But in fact, many discrete models of space and time have a much bigger limitation.
Any such model has to answer a simple question: Where is a particle during the jump
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 74
Let us take a breath. Can a space or even a space-time lattice be an alternative to continuity?
Discrete models of space-time have been studied since the 1940s. Recently, the idea
that space or space-time could be described as a lattice – like a crystal – has been explored
most notably by David Finkelstein and by Gerard ’t Hooft. The idea of space as a lattice
is based on the idea that, if there is a minimum distance, then all distances are multiples
of this minimum.
In order to get an isotropic and homogeneous situation for large, everyday scales,
the structure of space cannot be periodic, but must be random. But not only must it be
random in space, it must also be fluctuating in time. In fact, any fixed structure for spacetime would violate the result that there are no lengths smaller than the Planck length: as
a result of the Lorentz contraction, any moving observer would find lattice distances
smaller than the Planck value. Worse still, the fixed lattice idea conflicts with general
relativity, in particular with the diffeomorphism-invariance of the vacuum.
Thus, neither space nor space-time can be a lattice. A minimum distance does exist in
nature; however, we cannot hope that all other distances are simple multiples of it.
Motion Mountain – The Adventure of Physics
For example, supersymmetry cannot be valid at Planck scale. All mentioned conclusions
are consistent: if there are no symmetries at Planck scales, there are also no observables,
since physical observables are representations of symmetry groups. And thus,
d oes m at ter d iffer from vacuum?
75
from one lattice point to the next? This simple question eliminates most naive space-time
models.
A glimpse of quantum geometry
Vol. V, page 261
In every example of motion, some object is involved. One of the important discoveries
of the natural sciences was that all objects are composed of small constituents, called
elementary particles. Quantum theory shows that all composite, non-elementary objects
have a finite, non-vanishing size. The naive statement is: a particle is elementary if it behaves like a point particle. At present, only the leptons (electron, muon, tau and the neutrinos), the quarks, the radiation quanta of the electromagnetic, weak and strong nuclear
interactions (the photon, the W and Z bosons, and the gluons) and the Higgs boson have
been found to be elementary. Protons, atoms, molecules, cheese, people, galaxies and so
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Farewell to p oint particles
Motion Mountain – The Adventure of Physics
Ref. 27
Given that space-time is not a set of points or events, it must be something else. We have
three hints at this stage. The first is that in order to improve our description of motion
we must abandon ‘points’, and with them, abandon the local description of nature. Both
quantum mechanics and general relativity assume that the phrase ‘observable at a point’
has a precise meaning. Because it is impossible to describe space as a manifold, this
expression is no longer useful. The unification of general relativity and quantum physics
forces the adoption of a non-local description of nature at Planck scales. This is the first
hint.
The existence of a minimum length implies that there is no way to physically distinguish between locations that are even closer together. We are tempted to conclude that
no pair of locations can be distinguished, even if they are one metre apart, since on any
path joining two points, no two locations that are close together can be distinguished.
The problem is similar to the question about the size of a cloud or of an atom. If we
measure water density or electron density, we find non-vanishing values at any distance
from the centre of the cloud or the atom; however, an effective size can still be defined,
because it is very unlikely that the effects of the presence of a cloud or of an atom can
be seen at distances much larger than this effective size. Similarly, we can guess that two
points in space-time at a macroscopic distance from each other can be distinguished because the probability that they will be confused drops rapidly with increasing distance.
In short, we are thus led to a probabilistic description of space-time. This is the second
hint. Space-time becomes a macroscopic observable, a statistical or thermodynamic limit
of some microscopic entities. This is our second hint.
We note that a fluctuating structure for space-time also avoids the problems of fixed
structures with Lorentz invariance. In summary, the experimental observations of special relativity – Lorentz invariance, isotropy and homogeneity – together with the notion
of a minimum distance, point towards a description of space-time as fluctuating. This is
the third hint.
Several research approaches in quantum gravity have independently confirmed that
a non-local and fluctuating description of space-time at Planck scales resolves the contradictions between general relativity and quantum theory. These are our first results on
quantum geometry. To clarify the issue, we turn to the concept of the particle.
76
Page 60
Vol. IV, page 107
Ref. 79
4 d oes m at ter d iffer from vacuum?
on are all composite, as shown in Table 2.
Although the naive definition of ‘elementary particle’ as point particle is all we need in
the following argument, the definition is not precise. It seems to leave open the possibility
that future experiments could show that electrons or quarks are not elementary. This is
not so! In fact, the precise definition is the following:
⊳ Any particle is elementary if it is smaller than its own Compton wavelength.
ℏ?
= ?Pl .
?3
(84)
In other words, there is no way to observe that an object is smaller than the Planck length.
Thus,
⊳ There is no way to deduce from observations that a particle is point-like.
The term ‘point particle’ makes no sense at all.
Of course, there is a relation between the existence of a minimum length for empty
space and the existence of a minimum length for objects. If the term ‘point of space’
is meaningless, then the term ‘point particle’ is also meaningless. And again, the lower
limit on particle size results from the combination of quantum theory and general relativity.*
The minimum size for particles can be tested. A property connected with the size is
the electric dipole moment. This describes the deviation of its charge distribution from
* We note that the existence of a minimum size for a particle has nothing to do with the impossibility, in
quantum theory, of localizing a particle to within less than its Compton wavelength.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
?>√
Motion Mountain – The Adventure of Physics
If such a small particle were composite, there would be a lighter particle inside it, which
would have a larger Compton wavelength than the composite particle. This is impossible,
since the size of a composite particle must be larger than the Compton wavelength of its
components. (The alternative possibility that all components are heavier than the composite does not lead to satisfying physical properties: for example, it leads to intrinsically
unstable components.)
The size of an object, such as those given in Table 2, is defined as the length at which
differences from point-like behaviour are observed. The size ? of an object is determined
by measuring how it scatters a beam of probe particles. For example, the radius of the
atomic nucleus was determined for the first time in Rutherford’s experiment using alpha
particle scattering. In daily life as well, when we look at objects, we make use of scattered
photons. In general, in order for scattering to be useful, the effective wavelength ? =
ℏ/?? of the probe must be smaller than the object size ? to be determined. We thus
need ? > ? = ℏ/?? ⩾ ℏ/??. In addition, in order for a scattering experiment to be
possible, the object must not be a black hole, since, if it were, it would simply swallow
the approaching particle. This means that its mass ? must be smaller than that of a
black hole of the same size; in other words, from equation (72) we must have ? < ??2 /?.
Combining this with the previous condition we get, for the size ? of an object, the relation
d oes m at ter d iffer from vacuum?
Ref. 80
spherical. Some predictions from the standard model of elementary particles give as an
upper limit for the electron dipole moment ?? a value of
|?? |
< 10−39 m ,
?
Ref. 81
(85)
where ? is the charge of the electron. This predicted value is ten thousand times smaller
than the Planck length ?Pl . Since the Planck length is the smallest possible length, we
seem to have a contradiction here. However, a more careful and recent prediction from
the standard model only states
|?? |
< 3 ⋅ 10−23 m ,
?
Ref. 82
77
(86)
which is not in contradiction with the minimal length. The experimental limit in 2013 is
(87)
In the coming years, the experimental limit value will approach the Planck length. In
summary, no point particle is known. In fact, not even a particle smaller than the Planck
length is known.
Farewell to particle properties
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Planck scales have other strange consequences. In quantum field theory, the difference
between a virtual particle and a real particle is that a real particle is ‘on shell’, obeying
?2 = ?2 ?4 + ?2 ?2 , whereas a virtual particle is ‘off shell’. Because of the fundamental
limits of measurement precision, at Planck scales we cannot determine whether a particle
is real or virtual.
That is not all. Antimatter can be described as matter moving backwards in time.
Since the difference between backwards and forwards cannot be determined at Planck
scales, matter and antimatter cannot be distinguished at Planck scales.
Every particle is characterized by its spin. Spin describes two properties of a particle:
its behaviour under rotations (and thus, if the particle is charged, its behaviour in magnetic fields) and its behaviour under particle exchange. The wave function of a particle
with spin 1 remains invariant under a rotation of 2π, whereas that of a particle with spin
1/2 changes sign. Similarly, the combined wave function of two particles with spin 1 does
not change sign under exchange of particles, whereas for two particles with spin 1/2 it
does.
We see directly that both transformations are impossible to study at Planck scales.
Given the limit on position measurements, the position of a rotation axis cannot be well
defined, and rotations become impossible to distinguish from translations. Similarly,
positional imprecision makes it impossible to determine precise separate positions for
exchange experiments; at Planck scales it is impossible to say whether particle exchange
has taken place or not, and whether the wave function has changed sign or not. In short,
Motion Mountain – The Adventure of Physics
|?? |
< 8.7 ⋅ 10−31 m .
?
78
4 d oes m at ter d iffer from vacuum?
F I G U R E 3 Andrei Sakharov (1921–1989).
⊳ At Planck scales, spin cannot be defined or measured, and neither fermion
nor boson behaviour can be defined or measured.
A mass limit for elementary particles
?<
Ref. 28
ℏ
ℏ?
=√
= ?Pl = 2.2 ⋅ 10−8 kg = 1.2 ⋅ 1019 GeV/c2 .
? ?Pl
?
(88)
The limit ?Pl , the so-called Planck mass, corresponds roughly to the mass of a human
embryo that is ten days old, or equivalently, to that of a small flea. In short, the mass
of any elementary particle must be smaller than the Planck mass. This fact was already
noted as ‘well known’ by Andrei Sakharov* in 1968; he explains that these hypothetical
particles are sometimes called ‘maximons’. And indeed, the known elementary particles
all have masses well below the Planck mass. (In fact, the question why their masses are
so very much smaller than the Planck mass is one of the most important questions of
high-energy physics. We will come back to it.)
* Andrei Dmitrievich Sakharov, Soviet nuclear physicist (b. 1921 Moscow, d. 1989 Moscow). One of the
keenest thinkers in physics, Sakharov, among others, invented the Tokamak, directed the construction of
nuclear bombs, and explained the matter–antimatter asymmetry of nature. Like many others, he later campaigned against nuclear weapons, a cause for which he was put into jail and exile, together with his wife,
Yelena Bonner. He received the Nobel Peace Prize in 1975.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
The size ? of any elementary particle must by definition be smaller than its own (reduced)
Compton wavelength ℏ/??. Moreover, the size of a particle is always larger than the
Planck length: ? > ?Pl . Combining these two requirements and eliminating the size ?,
we get a constraint on the mass ? of any elementary particle, namely
Motion Mountain – The Adventure of Physics
Challenge 38 e
In particular, this implies that supersymmetry cannot be valid at Planck scales.
And we can continue. Due to measurement limitations, also spatial parity cannot be
defined or measured at Planck scales.
We have thus shown that at Planck scales, particles do not interact locally, are not
point-like, cannot be distinguished from antiparticles, cannot be distinguished from virtual particles, have no definite spin and have no definite spatial parity. We deduce that
particles do not exist at Planck scales. Let us explore the remaining concept: particle mass.
d oes m at ter d iffer from vacuum?
Ref. 83
79
There are many other ways to arrive at the mass limit for particles. For example, in
order to measure mass by scattering – and that is the only way for very small objects –
the Compton wavelength of the scatterer must be larger than the Schwarzschild radius;
otherwise the probe will be swallowed. Inserting the definitions of the two quantities and
neglecting the factor 2, we again get the limit ? < ?Pl . In fact it is a general property
of descriptions of nature that a minimum space-time interval leads to an upper limit for
masses of elementary particles.
Farewell to massive particles – and to massless vacuum
The Planck mass divided by the Planck volume, i.e., the Planck density, is given by
?Pl =
(89)
and is a useful concept in the following. One way to measure the (gravitational) mass ?
enclosed in a sphere of size ?, and thus (roughly) of volume ?3 , is to put a test particle
in orbit around it at that same distance ?. Universal gravitation then gives for the mass
? the expression ? = ??2 /?, where ? is the speed of the orbiting test particle. From
? < ?, we deduce that ? < ?2 ?/?; since the minimum value for ? is the Planck distance,
we get (again neglecting factors of order unity) a limit for the mass density ?, namely
? < ?Pl .
(90)
Δ? ⩾
ℏ
.
??
(91)
Note that for everyday situations, this error is extremely small, and other errors, such as
the technical limits of the balance, are much larger.
To check this result, we can explore another situation. We even use relativistic expressions, in order to show that the result does not depend on the details of the situation or
the approximations. Imagine having a mass ? in a box of size ?, and weighing the box
with a scale. (It is assumed that either the box is massless or that its mass is subtracted by
the scale.) The mass error is given by Δ? = Δ?/?2 , where Δ? is due to the indeterminacy
in the kinetic energy of the mass inside the box. Using the expression ?2 = ?2 ?4 + ?2 ?2 ,
we get that Δ? ⩾ Δ?/?, which again reduces to equation (91). Now that we are sure of
the result, let us continue.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
In other words, the Planck density is the maximum possible value for mass density.
Interesting things happen when we try to determine the error Δ? of a mass measurement in a Planck volume. Let us return to the mass measurement by an orbiting probe.
From the relation ?? = ??2 we deduce by differentiation that ?Δ? = ?2 Δ? + 2??Δ? >
2??Δ? = 2??Δ?/?. For the error Δ? in the velocity measurement we have the indeterminacy relation Δ? ⩾ ℏ/?Δ? + ℏ/?? ⩾ ℏ/??. Inserting this in the previous inequality,
and again forgetting the factor of 2, we find that the mass measurement error Δ? of a
mass ? enclosed in a volume of size ? is subject to the condition
Motion Mountain – The Adventure of Physics
Challenge 39 e
?5
= 5.2 ⋅ 1096 kg/m3
?2 ℏ
80
4 d oes m at ter d iffer from vacuum?
M
R
R
F I G U R E 4 A thought experiment
showing that matter and vacuum
cannot be distinguished when the
size of the enclosing box is of the
order of a Planck length.
⩾ ? ⩾
ℏ
.
??
(empty box)
(92)
We see directly that for sizes ? of the order of the Planck scales, the two limits coincide;
in other words, we cannot distinguish a full box from an empty box in that case.
To be sure of this strange result, we check whether it also occurs if, instead of measuring the gravitational mass, as we have just done, we measure the inertial mass. The
inertial mass for a small object is determined by touching it: physically speaking, by
performing a scattering experiment. To determine the inertial mass inside a region of
size ?, a probe must have a wavelength smaller than ?, and a correspondingly high energy. A high energy means that the probe also attracts the particle through gravity. (We
thus find the intermediate result that at Planck scales, inertial and gravitational mass can-
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
?2 ?
?
(full box)
Motion Mountain – The Adventure of Physics
From equation (91) we deduce that for a box of Planck dimensions, the mass measurement error is given by the Planck mass. But from above we also know that the mass
that can be put inside such a box must not be larger than the Planck mass. Therefore,
for a box of Planck dimensions, the mass measurement error is larger than (or at best
equal to) the mass contained in it: Δ? ⩾ ?Pl . In other words, if we build a balance
with two boxes of Planck size, one empty and the other full, as shown in Figure 4, nature
cannot decide which way the balance should hang! Note that even a repeated or a continuous measurement will not resolve the situation: the balance will change inclination
at random, staying horizontal on average.
The argument can be rephrased as follows. The largest mass that we can put in a box
of size ? is a black hole with a Schwarzschild radius of the same value; the smallest mass
present in such a box – corresponding to what we call a vacuum – is due to the indeterminacy relation and is given by the mass with a Compton wavelength that matches
the size of the box. In other words, inside any box of size ? we have a mass ?, the limits
of which are given by:
d oes m at ter d iffer from vacuum?
81
not be distinguished. Even the balance experiment shown in Figure 4 illustrates this: at
Planck scales, the two types of mass are always inextricably linked.) Now, in any scattering experiment, for example in a Compton-type experiment, the mass measurement
is performed by measuring the wavelength change ?? of the probe before and after the
scattering. The mass indeterminacy is given by
Δ? Δ??
=
.
?
??
(93)
Mat ter and vacuum are indistinguishable
Vol. V, page 128
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 84
We can put these results in another way. On the one hand, if we measure the mass of
a piece of vacuum of size ?, the result is always at least ℏ/??: there is no possible way
to find a perfect vacuum in an experiment. On the other hand, if we measure the mass
of a particle, we find that the result is size-dependent: at Planck scales it approaches the
Planck mass for every type of particle, be it matter or radiation.
To use another image, when two particles approach each other to a separation of the
order of the Planck length, the indeterminacy in the length measurements makes it impossible to say whether there is something or nothing between the two objects. In short,
matter and vacuum are interchangeable at Planck scales. This is an important result: since
mass and empty space cannot be differentiated, we have confirmed that they are made
of the same ‘fabric’, of the same constituents. This idea, already suggested above, is now
common to all attempts to find a unified description of nature.
This approach is corroborated by attempts to apply quantum mechanics in highly
curved space-time, where a clear distinction between vacuum and particles is impossible,
as shown by the Fulling–Davies–Unruh effect. Any accelerated observer, and any observer in a gravitational field, detects particles hitting him, even if he is in a vacuum. The
effect shows that for curved space-time the idea of vacuum as particle-free space does
not work. Since at Planck scales it is impossible to say whether or not space is flat, it is
impossible to say whether it contains particles or not.
In short, all arguments lead to the same conclusion: vacuum, i.e., empty space-time,
cannot be distinguished from matter at Planck scales. Another common way to express
this state of affairs is to say that when a particle of Planck energy travels through space
it will be scattered by the fluctuations of space-time itself, as well as by matter, and the
Motion Mountain – The Adventure of Physics
In order to determine the mass in a Planck volume, the probe has to have a wavelength of
the Planck length. But we know from above that there is always a minimum wavelength
indeterminacy, given by the Planck length ?Pl . In other words, for a Planck volume the
wavelength error – and thus the mass error – is always as large as the Planck mass itself:
Δ? ⩾ ?Pl . Again, this limit is a direct consequence of the limit on length and space
measurements.
This result has an astonishing consequence. In these examples, the measurement error
is independent of the mass of the scatterer: it is the same whether or not we start with a
situation in which there is a particle in the original volume. We thus find that in a volume
of Planck size, it is impossible to say whether or not there is something there when we
probe it with a beam!
82
4 d oes m at ter d iffer from vacuum?
Nature's energy scale
Eeveryday
EPlanck
F I G U R E 5 Planck effects make the
energy axis an approximation.
Curiosities and fun challenges on Pl anck scales
“
There is nothing in the world but matter in
motion, and matter in motion cannot move
otherwise than in space and time.
Lenin, Materialism and empirio-criticism.
∗∗
Observers are made of matter. Observers are not made of radiation. Observers are not
made of vacuum. Observers are thus biased, because they take a specific standpoint.
But at Planck scales, vacuum, radiation and matter cannot be distinguished. Two consequences follow: first, only at Planck scales would a description be free of any bias in
favour of matter. Secondly, on the other hand, observers do not exist at all at Planck
energy. Physics is thus only possible below Planck energy.
∗∗
If measurements become impossible near Planck energy, we cannot even draw a diagram
with an energy axis reaching that value. A way out is shown Figure 5. The energy of
elementary particles cannot reach the Planck energy.
∗∗
Challenge 40 s
By the standards of particle physics, the Planck energy is rather large. Suppose we wanted
to impart this amount of energy to protons using a particle accelerator. How large would
a Planck accelerator have to be?
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
”
Lenin’s statement is wrong. And this is not so much because the world contains moving
matter, moving radiation, moving vacuum and moving horizons, which is not exactly
what Lenin claimed. Above all, his statement is wrong because at Planck scales, there is
no matter, no radiation, no horizon, no space and no time. These concepts only appear
at low energy. The rest of our adventure clarifies how.
Motion Mountain – The Adventure of Physics
two cases are indistinguishable. These surprising results rely on a simple fact: whatever
definition of mass we use, it is always measured via combined length and time measurements. (This is even the case for normal weighing scales: mass is measured by the
displacement of some part of the machine.) Mass measurement is impossible at Planck
scales. The error in such mass measurements makes it impossible to distinguish vacuum
from matter. In particular, the concept of particle is not applicable at Planck scale.
d oes m at ter d iffer from vacuum?
83
∗∗
Challenge 41 s
By the standards of everyday life, the Planck energy is rather small. Measured in litres of
gasoline, how much fuel does it correspond to?
∗∗
The usual concepts of matter and of radiation are not applicable at Planck scales. Usually,
it is assumed that matter and radiation are made up of interacting elementary particles.
The concept of an elementary particle implies an entity that is discrete, point-like, real
and not virtual, has a definite mass and a definite spin, is distinct from its antiparticle,
and, most of all, is distinct from vacuum, which is assumed to have zero mass. All these
properties are lost at Planck scales. At Planck scales, the concepts of ‘mass’, ‘vacuum’,
‘elementary particle’, ‘radiation’ and ‘matter’ do not make sense.
∗∗
∗∗
We now have a new answer to the old question: why is there something rather than
nothing? At Planck scales, there is no difference between something and nothing. We
can now honestly say about ourselves that we are made of nothing.
∗∗
Ref. 85
Page 279
∗∗
Ref. 86
Ref. 67
Challenge 44 s
Quantum mechanics alone gives, via the Heisenberg indeterminacy relation, a lower
limit to the spread of measurements, but, strangely enough, not on their precision, i.e.,
not on the number of significant digits. Wolfgang Jauch gives an example: atomic lattice
constants are known to a much higher precision than the positional indeterminacy of
single atoms inside the crystal.
It is sometimes claimed that measurement indeterminacies smaller than the Planck
values are possible for large enough numbers of particles. Can you show why this is
incorrect, at least for space and time?
∗∗
The idea that vacuum is not empty is not new. More than two thousand years ago,
Aristotle argued for a filled vacuum, although his arguments were incorrect as seen
from today’s perspective. Also in the fourteenth century there was much discussion
on whether empty space was composed of indivisible entities, but the debate died down
again.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Challenge 43 r
Special relativity implies that no length or energy can be invariant. Since we have come
to the conclusion that the Planck energy and the Planck length are invariant, it appears
that there must be deviations from Lorentz invariance at high energy. What effects would
follow? What kind of experiment could measure them? If you have a suggestion, publish it! Several attempts are being explored. We will settle the issue later on, with some
interesting insights.
Motion Mountain – The Adventure of Physics
Challenge 42 s
Do the large errors in mass measurements imply that mass can be negative at Planck
energy?
84
4 d oes m at ter d iffer from vacuum?
∗∗
Challenge 45 s
A Planck-energy particle falling in a gravitational field would gain energy. But the Planck
energy is the highest energy in nature. What does this apparent contradiction imply?
∗∗
Ref. 59
One way to generalize the results presented here is to assume that, at Planck energy,
nature is event-symmetric, i.e., symmetric under exchange of any two events. This idea,
developed by Phil Gibbs, provides an additional formulation of the strange behaviour of
nature at extreme scales.
∗∗
Vol. II, page 274
Because there is a minimum length in nature, so-called singularities do not exist. The
issue, hotly debated for decades in the twentieth century, thus becomes uninteresting.
∗∗
Ref. 87
∗∗
Vol. I, page 26
If vacuum and matter cannot be distinguished, we cannot distinguish between objects
and their environment. However, this was one of the starting points of our journey. Some
interesting adventures still await us!
∗∗
Vol. III, page 322
We have seen earlier that characterizing nature as made up of particles and vacuum creates problems when interactions are included. On the one hand interactions are the
difference between the parts and the whole, while on the other hand interactions are
exchanges of quantum particles. This apparent contradiction can be used to show that
something is counted twice in the usual characterization of nature. Noting that matter
and space-time are both made of the same constituents resolves the issue.
∗∗
Challenge 46 d
Is there a smallest possible momentum? And a smallest momentum error?
∗∗
Given that time becomes an approximation at Planck scales, can we still ask whether
nature is deterministic?
Let us go back to the basics. We can define time, because in nature change is not ran-
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Because mass and energy density are limited, any object of finite volume has only a finite number of degrees of freedom. The calculation of the entropy of black holes has
confirmed that entropy values are always finite. This implies that perfect baths do not
exist. Baths play an important role in thermodynamics (which must therefore be viewed
as only an approximation), and also in recording and measuring devices: when a device
measures, it switches from a neutral state to a state in which it shows the result of the
measurement. In order not to return to the neutral state, the device must be coupled to
a bath. Without a bath, a reliable measuring device cannot exist. In short, perfect clocks
and length-measuring devices do not exist, because nature puts a limit on their storage
ability.
Motion Mountain – The Adventure of Physics
Vol. V, page 145
d oes m at ter d iffer from vacuum?
Challenge 47 s
Page 410
85
dom, but gradual. What is the situation now that we know that time is only approximate?
Is non-gradual change possible? Is energy conserved? In other words, are surprises possible in nature?
It is correct to say that time is not defined at Planck scales, and that therefore that
determinism is an undefinable concept, but it is not a satisfying answer. What happens at
‘everyday’ scales? One answer is that at our everyday scales, the probability of surprises
is so small that the world indeed is effectively deterministic. In other words, nature is not
really deterministic, but the departure from determinism is not measurable, since every
measurement and observation, by definition, implies a deterministic world. The lack of
surprises would be due to the limitations of our human nature – more precisely, of our
senses and brain.
Can you imagine any other possibility? In truth, it is not possible to prove these answers at this point, even though the rest of our adventure will do so. We need to keep
any possible alternative in mind, so that we remain able to check the answers.
Page 115
If matter and vacuum cannot be distinguished, then each has the properties of the other.
For example, since space-time is an extended entity, matter and radiation are also extended entities. Furthermore, as space-time is an entity that reaches the borders of the
system under scrutiny, particles must also do so. This is our first hint at the extension of
matter; we will examine this argument in more detail shortly.
∗∗
∗∗
Challenge 48 s
When can matter and vacuum be distinguished? At what energy? This issue might be
compared to the following question: Can we distinguish between a liquid and a gas by
looking at a single atom? No, only by looking at many. Similarly, we cannot distinguish
between matter and vacuum by looking at one point, but only by looking at many. We
must always average. However, even averaging is not completely successful. Distinguishing matter from vacuum is like distinguishing clouds from the clear sky: like clouds,
matter has no precise boundary.
∗∗
Challenge 49 e
If the dimensionality of space is undefined at Planck scales, what does this mean for
superstrings?
∗∗
Vol. I, page 27
Since vacuum, particles and fields are indistinguishable at Planck scales, we also lose
the distinction between states and permanent, intrinsic properties of physical systems
at those scales. This is a strong statement: the distinction was the starting point of our
exploration of motion; the distinction allowed us to distinguish systems from their en-
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 287
The impossibility of distinguishing matter and vacuum implies a lack of information at
Planck scales. In turn, this implies an intrinsic basic entropy associated with any part of
the universe at Planck scales. We will come back to this topic shortly, when we discuss
the entropy of black holes.
Motion Mountain – The Adventure of Physics
∗∗
86
4 d oes m at ter d iffer from vacuum?
vironment. In other words, at Planck scales we cannot talk about motion! This is a strong
statement – but it is not unexpected. We are searching for the origin of motion, and we
are prepared to encounter such difficulties.
C ommon constituents
“
One can give good reasons why reality cannot at all be represented by a
continuous field. From the quantum phenomena it appears to follow with
certainty that a finite system of finite energy can be completely described by a
finite set of numbers (quantum numbers). This does not seem to be in
accordance with a continuum theory, and must lead to an attempt to find a
purely algebraic theory for the description of reality. But nobody knows how to
obtain the basis of such a theory.
Albert Einstein, 1955, the last sentences of The Meaning of Relativity – Including
the Relativistic Theory of the Non-Symmetric Field, fifth edition. These were also
his last published words.
”
”
Page 69
* ‘Yet it has been suggested that the introduction of a space-time continuum, in view of the molecular
structure of all events in the small, may possibly be considered as contrary to nature. Perhaps the success of
Heisenberg’s method may point to a purely algebraic method of description of nature, to the elimination of
continuous functions from physics. Then, however, one must also give up, in principle, the use of the spacetime continuum. It is not inconceivable that human ingenuity will some day find methods that will make it
possible to proceed along this path. Meanwhile, however, this project resembles the attempt to breathe in
an airless space.’
See also what Einstein thought twenty years before. The new point is that he believes that an algebraic
description is necessary. He repeats the point in the next quote.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 69
In this rapid journey, we have destroyed all the experimental pillars of quantum theory:
the superposition of wave functions, space-time symmetry, gauge symmetry, renormalization symmetry and permutation symmetry. We also have destroyed the foundations
of special and general relativity, namely the concepts of the space-time manifold, fields,
particles and mass. We have even seen that matter and vacuum cannot be distinguished.
It seems that we have lost every concept used for the description of motion, and thus
made its description impossible. It seems that we have completely destroyed our two
‘castles in the air’, general relativity and quantum theory. And it seems that we are trying
to breathe in airless space. Is this pessimistic view correct, or can we save the situation?
First of all, since matter and radiation are not distinguishable from vacuum, the quest
for unification in the description of elementary particles is correct and necessary. There
is no alternative to tearing down the castles and to continuing to breathe.
Motion Mountain – The Adventure of Physics
“
Es ist allerdings darauf hingewiesen worden, dass bereits die Einführung eines
raum-zeitlichen Kontinuums angesischts der molekularen Struktur allen
Geschehens im Kleinen möglicherweise als naturwidrig anzusehen sei.
Vielleicht weise der Erfolg von Heisenbergs Methode auf eine rein algebraische
Methode der Naturbeschreibung, auf die Ausschaltung kontinuierlicher
Funktionen aus der Physik hin. Dann aber muss auch auf die Verwendung des
Raum-Zeit-Kontinuums prinzipiell verzichtet werden. Es ist nicht undenkbar,
dass der menschliche Scharfsinn einst Methoden finden wird, welche die
Beschreitung dieses Weges möglich machen. Einstweilen aber erscheint dieses
Projekt ähnlich dem Versuch, in einem luftleeren Raum zu atmen.*
Albert Einstein, 1936, in Physik und Realität.
d oes m at ter d iffer from vacuum?
87
Secondly, after tearing down the castles, the invariant Planck limits ?, ℏ and ?4 /4? still
remain as a foundation.
Thirdly, after tearing down the castles, one important result appears. Since the concepts of ‘mass’, ‘time’ and ‘space’ cannot be distinguished from each other, a new, single
entity or concept is necessary to define both particles and space-time. In short, vacuum
and particles must be made of common constituents. In other words, we are not in airless space, and we uncovered the foundation that remains after we tore down the castles.
Before we go on exploring these common constituents, we check what we have deduced
so far against experiment.
Experimental predictions
Challenge 50 r
Vol. V, page 146
Ref. 89, Ref. 90
Ref. 90
Ref. 92, Ref. 93
Ref. 94
This energy value is between 1.4 ⋅ 1019 GeV and over 1022 GeV for the best measurement
to date. This is between just above the Planck energy and over one thousand times the
Planck energy. However, despite this high characteristic energy, no dispersion has been
found: even after a trip of ten thousand million years, all light arrives within one or two
seconds.
Another candidate experiment is the direct detection of distance fluctuations between
bodies. Gravitational wave detectors are sensitive to extremely small noise signals in
length measurements. There should be a noise signal due to the distance fluctuations
induced near Planck energy. The indeterminacy in measurement of a length ? is predicted to be
? 2/3
??
(95)
⩾ ( Pl ) .
?
?
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 92, Ref. 91
Motion Mountain – The Adventure of Physics
Ref. 88
A race is going on both in experimental and in theoretical physics: to be the first to
suggest and to be the first to perform an experiment that detects a quantum gravity effect
– apart possibly from (a part of) the Sokolov–Ternov effect. Here are some proposals.
At Planck scales, space fluctuates. We might think that the fluctuations of space could
blur the images of faraway galaxies, or destroy the phase relation between the photons.
However, no blurring is observed, and the first tests show that light from extremely distant galaxies still interferes. The precise prediction of the phase washing effect is still
being worked out; whatever the exact outcome, the effect is too small to be measured.
Another idea is to measure the speed of light at different frequencies from faraway
light flashes. There are natural flashes, called gamma-ray bursts, which have an extremely
broad spectrum, from 100 GeV down to visible light at about 1 eV. These flashes often
originate at cosmological distances ?. Using short gamma-ray bursts, it is thus possible
to test precisely whether the quantum nature of space-time influences the dispersion of
light signals when they travel across the universe. Planck-scale quantum gravity effects
might produce a dispersion. Detecting a dispersion would confirm that Lorentz symmetry breaks down at Planck scales.
The difference in arrival time Δ? between two photon energies ?1 and ?2 defines a
characteristic energy by
(? − ?2 ) ?
?char = 1
.
(94)
? Δ?
88
Page 66
Ref. 95
Ref. 92
Ref. 96
summary on particles and vacuum
Combining quantum theory and general relativity leads us to several important results
on the description of nature:
Ref. 98
— Vacuum and particles mix at Planck scales, because there is no conceivable way to distinguish whether a Planck-sized region is part of a particle or of empty space. Matter,
radiation and vacuum cannot be distinguished at Planck scales. Equivalently, empty
space and particles are made of fluctuating common constituents.
— We note that all arguments of this chapter equally imply that vacuum and particles
mix near Planck scales. For example, matter, radiation and vacuum cannot be distinguished near Planck scales.
— The constituents of vacuum and particles cannot be points. There is no conceivable
way to prove that points exist, because the smallest measurable distance in nature is
the Planck length.
— Particles, vacuum and continuous space do not exist at Planck scales. They disappear
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 97
This expression is deduced simply by combining the measurement limit of a ruler, from
quantum theory, with the requirement that the ruler not be a black hole. The sensitivity
of the detectors to noise might reach the required level in the twenty-first century. The
noise induced by quantum gravity effects has also been predicted to lead to detectable
quantum decoherence and vacuum fluctuations. So far, no such effect has been found.
A further candidate experiment for measuring quantum gravity effects is the detection of the loss of CPT symmetry at high energies. Especially in the case of the decay of
certain elementary particles, such as neutral kaons, the precision of experimental measurement is approaching the detection of Planck-scale effects. However, no such effect
has been found yet.
Another possibility is that quantum gravity effects may change the threshold energy at
which certain particle reactions become possible. It may be that extremely high-energy
photons or cosmic rays will make it possible to prove that Lorentz invariance is indeed
broken at Planck scales. However, no such effect has been found yet.
In the domain of atomic physics, it has also been predicted that quantum gravity
effects will induce a gravitational Stark effect and a gravitational Lamb shift in atomic
transitions. However, no such effect has been found yet.
Other proposals start from the recognition that the bound on the measurability of
observables also puts a bound on the measurement precision for each observable. This
bound is of no importance in everyday life, but it is important at Planck energy. One
proposal is to search for a minimal noise in length measurements, e.g., in gravitational
wave detectors. But no such noise has been found yet.
In summary, the experimental detection of quantum gravity effects might be possible,
despite their weakness, at some time during the twenty-first century. The successful prediction and detection of such an effect would be one of the highlights of physics, as it
would challenge the usual description of space and time even more than general relativity did. On the other hand, most unified models of physics predict the absence of any
measurable quantum gravity effect.
Motion Mountain – The Adventure of Physics
Ref. 95
4 d oes m at ter d iffer from vacuum?
sum m ary on pa rticles a nd vacuum
89
in a yet unclear Planck scale mixture.
— The three independent Planck limits ?, ℏ and ?4 /4? remain valid also in domains
where quantum theory and general relativity are combined.
Page 54
All these results must be part of the final theory that we are looking for. Generally speaking, we found the same conclusions that we found already in the chapter on limit statements. We thus continue along the same path that we took back then: we explore the
universe as a whole.
Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Chapter 5
WHAT IS T HE DIFFER ENCE BET WEEN
T HE UNIVERSE AND NOT HING?
C osmological scales
”
Antiquity
The description of motion requires the application of general relativity whenever the
scale ? of the situation is of the order of the Schwarzschild radius, i.e., whenever
? ≈ ?S = 2??/?2 .
Challenge 51 s
(96)
It is straightforward to confirm that, with the usually quoted mass ? and size ? of
everything visible in the universe, this condition is indeed fulfilled. We do need general
relativity, and thus curved space-time, when talking about the whole of nature.
Similarly, quantum theory is required for the description of the motion of an object
whenever we approach it within a distance ? of the order of the (reduced) Compton
wavelength ?C , i.e., whenever
ℏ
? ≈ ?C =
.
(97)
??
Obviously, for the total mass of the universe this condition is not fulfilled. However, we
are not interested in the motion of the universe itself; we are interested in the motion
of its components. In the description of these components, quantum theory is required
whenever pair production and annihilation play a role. This is the case in the early his** ‘The frontier is the really productive place of understanding.’ Paul Tillich (b. 1886 Starzeddel,
d. 1965 Chicago), theologian, socialist and philosopher.
*** ‘Here are lions.’ This was written across unknown and dangerous regions on ancient maps.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
“
Hic sunt leones.***
”
his strange question is the topic of the current leg of our mountain ascent. In
he last section we explored nature in the vicinity of Planck scales. In fact,
he other limit, namely the description of motion at large, cosmological scales, is
equally fascinating. As we proceed, many incredible results will appear, and at the end
we will discover a surprising answer to the question in the title.
Motion Mountain – The Adventure of Physics
T
“
Die Grenze ist der eigentlich fruchtbare Ort der
Erkenntnis.**
Paul Tillich, Auf der Grenze.
w hat is the d ifference b et w een the univ erse a nd nothing?
91
tory of the universe and near the horizon, i.e., for the most distant events that we can
observe in space and time. We are thus obliged to include quantum theory in any precise description of the universe.
Since at cosmological scales we need both quantum theory and general relativity, we
start our investigation with the study of time, space and mass, by asking at large scales
the same questions that we asked above at Planck scales.
Maximum time
Is it possible to measure time intervals of any imaginable size? General relativity shows
that in nature there is a maximum time interval, with a value of about
13 800 million years, or 435 Ps,
“
Vol. II, page 308
One should never trust a woman who tells one
her real age. A woman who would tell one that,
would tell one anything.
Oscar Wilde**
”
In light of all measurements, it may seem silly to question the age of the universe. The
age value is found in many books and tables and its precise determination is one of the
most important quests in modern astrophysics. But is this quest reasonable?
In order to measure the duration of a movement or the age of a system, we need a
clock that is independent of that movement or system, and thus outside the system. How* This implies that so-called ‘oscillating universe’ models, in which it is claimed that ‘before’ the big bang
there were other phenomena, cannot be justified on the basis of nature or observations. They are based on
beliefs.
** Oscar Wilde, (b. 1854 Dublin, d. 1900 Paris), poet and playwright, equally famous for his wit.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
D oes the universe have a definite age?
Motion Mountain – The Adventure of Physics
Challenge 52 s
providing an upper limit to the measurement of time. It is called the age of the universe,
and has been deduced from two sets of measurements: the expansion of space-time and
the age of matter.
We are all familiar with clocks that have been ticking for a long time: the hydrogen
atoms in our body. All hydrogen atoms were formed just after the big bang. We can
almost say that the electrons in these atoms have been orbiting their nuclei since the
dawn of time. In fact, the quarks inside the protons in these atoms have been moving a
few hundred thousand years longer than the electrons.
We thus have an upper time limit for any clock made of atoms. Even ‘clocks’ made of
radiation (can you describe one?) yield a similar maximum time. Now, the study of the
spatial expansion of the universe leads to the same maximum age. No clock or measurement device was ticking longer ago than this maximum time, and no clock could provide
a record of having done so.
In summary, it is not possible to measure time intervals greater than the maximum
time, either by using the history of space-time or by using the history of matter or radiation.* The maximum time is thus rightly called the age of the universe. Of course, this
is not a new idea; but looking at the age issue in more detail does reveal some surprises.
92
5 w hat is the d ifference b et ween the univ erse a nd nothing?
Challenge 53 s
?3
= 0.39 ⋅ 1070 m−2
?ℏ
(98)
as a limit for the surface curvature ? in nature. In other words, the universe has never
been as small as a point, never had zero age, never had infinite density, and never had
infinite curvature. It is not difficult to get a similar limit for temperature or any other
physical quantity near the big bang. In short, since events do not exist,
⊳ The big bang cannot have been an event.
There never was an initial singularity or a beginning of the universe.
In short, the situation is consistently muddled. Neither the age of the universe nor
its origin makes sense. What is going wrong? Or rather, how are things going wrong?
What happens if instead of jumping directly to the big bang, we approach it as closely as
possible? To clarify the issue, we ask about the measurement error in our statement that
the universe is fourteen thousand million years old. This turns out to be a fascinating
topic.
How precise can age measurements be?
“
No woman should ever be quite accurate about
her age. It looks so calculating.
Oscar Wilde
”
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
?<
Motion Mountain – The Adventure of Physics
ever, there are no clocks outside the universe, and no clock inside it can be independent.
In fact, we have just seen that no clock inside the universe can run throughout its full
history. In particular, no clock can run through its earliest history.
Time can be defined only if it is possible to distinguish between matter and space.
Given this distinction, we can talk either about the age of space, by assuming that matter
provides suitable and independent clocks – as is done in general relativity – or about the
age of matter, such as stars or galaxies, by assuming that the extension of space-time, or
possibly some other matter, provides a good clock. Both possibilities are being explored
experimentally in modern astrophysics – and both give the same result, of about fourteen
thousand million years, which was mentioned above. Despite this correspondence, for
the universe as a whole, an age cannot be defined, because there is no clock outside it!
The issue of the starting point of time makes this difficulty even more apparent. We
may imagine that going back in time leads to only two possibilities: either the starting
instant ? = 0 is part of time or it is not. (Mathematically, this means that the segment
representing time is either closed or open.) Both these possibilities imply that it is possible to measure arbitrarily small times; but we know from the combination of general
relativity and quantum theory that this is not the case. In other words, neither possibility
is correct: the beginning cannot be part of time, nor can it not be part of it. There is only
one solution to this contradiction: there was no beginning at all.
The lack of a beginning is consistent with a minimum length or a minimum action.
Indeed, both imply that there is a maximum curvature for space-time. Curvature can be
measured in several ways: for example, surface curvature is an inverse area. Within a
factor of order one, we find
w hat is the d ifference b et w een the univ erse a nd nothing?
Ref. 55, Ref. 56
The first way to measure the age of the universe* is to look at clocks in the usual sense of
the word, namely at clocks made of matter. As explained in the part on quantum theory,
Salecker and Wigner showed that a clock built to measure a total time ? with a precision
Δ? has a minimum mass ? given by
?>
Ref. 94
93
ℏ ?
.
?2 (Δ?)2
(99)
A simple way to incorporate general relativity into this result was suggested by Ng and
van Dam. Any clock of mass ? has a minimum resolution Δ? due to the curvature of
space that it introduces, given by
??
Δ? > 3 .
(100)
?
?<
(101)
where ?Pl = √ℏ?/?5 = 5.4 ⋅ 10−44 s is the Planck time. (As usual, we have omitted factors
of order one in this and in all the following results of this chapter.) In other words, the
higher the accuracy of a clock, the shorter the time during which it works dependably.
The precision of a clock is limited not only by the expense of building it, but also by
nature itself. Nevertheless, it is easy to check that for clocks used in daily life, this limit
is not even remotely approached. For example, you may wish to calculate how precisely
your own age can be specified.
As a consequence of the inequality (101), a clock trying to achieve an accuracy of one
Planck time can do so for at most one Planck time! A real clock cannot achieve Plancktime accuracy. If we try to go beyond the limit (101), fluctuations of space-time hinder the
working of the clock and prevent higher precision. With every Planck time that passes,
the clock accumulates a measurement error of at least one Planck time. Thus, the total
measurement error is at least as large as the measurement itself. This conclusion is also
valid for clocks based on radiation.
In short, measuring age with a clock always involves errors. Whenever we try to reduce these errors to the smallest possible level, the Planck level, the clock becomes so
imprecise over large times that age measurements become impossible.
D oes time exist?
“
Ref. 99
Time is waste of money.
Oscar Wilde
”
* The age ?0 is not the same as the Hubble time ? = 1/?0 . The Hubble time is only a computed quantity and
(almost) always larger than the age; the relation between the two depends on the values of the cosmological
constant, the density and other properties of the universe. For example, for the standard ‘hot big bang’
scenario, i.e., for the matter-dominated Einstein–de Sitter model, we have the simple relation ? = (3/2) ?0.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Challenge 54 e
(Δ?)3
,
?2Pl
Motion Mountain – The Adventure of Physics
If ? is eliminated, these two results imply that a clock with a precision Δ? can only measure times ? up to a certain maximum value, namely
94
Vol. I, page 40
Challenge 55 e
What is the error in the measurement of the age of the universe?
Challenge 57 e
It is now straightforward to apply our discussion about the measurement of time to the
age of the universe. The inequality (101) implies that the highest precision possible for a
clock is about 10−23 s, or about the time light takes to move across a proton. The finite
age of the universe also yields a maximum relative measurement precision. Inequality
(101) can be written as
Δ?
? 2/3
(102)
> ( Pl ) .
?
?
Inserting the age of the universe for ?, we find that no time interval can be measured
with a precision of more than about 40 decimals.
To clarify the issue, we can calculate the error in measurement as a function of the
observation energy ?meas , the energy of the measurement probe. There are two limit
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 68
Challenge 56 e
Ever since people began to study physics, the concept of ‘time’ has designated what is
measured by a clock. But the inequality (101) for a maximum clock time implies that
perfect clocks do not exist, and thus that time is only an approximate concept: perfect
time does not exist. Thus, in nature there is no ‘idea’ of time, in the Platonic sense.
In fact, the discussion so far can be seen as proof that combining quantum theory and
general relativity, because of the resulting measurement errors, prevents the existence of
perfect or ‘ideal’ examples of any classical observable or any everyday concept.
Time does not exist. Yet it is obviously a useful concept in everyday life. The key to
understanding this is measurement energy. Any clock – in fact, any system of nature –
is characterized by a simple number, namely the highest ratio of its kinetic energy to the
rest energy of its components. In daily life, this ratio is about 1 eV/10 GeV = 10−10 . Such
low-energy systems are well suited for building clocks. The more precisely the motion
of the main moving part – the pointer of the clock – can be kept constant and monitored, the higher the precision of the clock. To achieve very high precision, the pointer
must have very high mass. Indeed, in any clock, both the position and the speed of the
pointer must be measured, and the two measurement errors are related by the quantummechanical indeterminacy relation Δ? Δ? > ℏ/?. High mass implies low intrinsic fluctuation. Furthermore, in order to screen the pointer from outside influences, even more
mass is needed. This connection between mass and accuracy explains why more accurate
clocks are usually more expensive.
The standard indeterminacy relation ?Δ? Δ? > ℏ is valid only at everyday energies.
However, we cannot achieve ever higher precision simply by increasing the mass without
limit, because general relativity changes the indeterminacy relation to Δ? Δ? > ℏ/? +
?(Δ?)2 ?/?3 . The additional term on the right-hand side, negligible at everyday scales, is
proportional to energy. Increasing it by a large amount limits the achievable precision of
the clock. The smallest measurable time interval turns out to be the Planck time.
In summary, time exists, as a good approximation, only for low-energy systems. Any
increase in precision beyond a certain limit requires an increase in the energy of the
components; at Planck energy, this increase will prevent an increase in precision.
Motion Mountain – The Adventure of Physics
Page 58
5 w hat is the d ifference b et ween the univ erse a nd nothing?
w hat is the d ifference b et w een the univ erse a nd nothing?
95
Relative
measurement error
1
quantum
error
total
error
quantum
gravity
error
ΔEmin
E
Energy
Eopt
EPl
FIGURE 6
cases. For low energies, the error is due to quantum effects and is given by
Δ?
1
∼
?
?meas
(103)
Δ? ?meas
∼
?
?Pl
(104)
so that the total error varies as shown in Figure 6. In particular, very high energies do not
reduce measurement errors: any attempt to reduce the measurement error for the age of
the universe below 10−23 s would require energies so high that the limits of space-time
would be reached, making the measurement itself impossible. We reached this conclusion through an argument based on clocks made of particles. We will see below that
trying to determine the age of the universe from its expansion leads to the same limitation.
Imagine observing a tree which, as a result of some storm or strong wind, has fallen
towards second tree, touching it at the very top, as shown in Figure 7. It is possible to
determine the heights of both trees by measuring their separation and the angles at the
base. The error in the heights will depend on the errors in measurement of the separation
and angles.
Similarly, the age of the universe can be calculated from the present distance and
speed of objects – such as galaxies – observed in the night sky. The present distance
? corresponds to separation of the trees at ground level, and the speed ? to the angle
between the two trees. The Hubble time ? of the universe (which is usually assumed to
be larger than the age of the universe) then corresponds to the height at which the two
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
which decreases with increasing measurement energy. For high energies, however, the
error is due to gravitational effects and is given by
Motion Mountain – The Adventure of Physics
Measurement
errors as a
function of
measurement
energy.
96
5 w hat is the d ifference b et ween the univ erse a nd nothing?
v
T
d
trees meet. This age – in a naive sense, the time since the galaxies ‘separated’ – is given,
within a factor of order one, by
?
?= .
(105)
?
Let us explore this in more detail. For any measurement of ?, we have to choose the
object, i.e., a distance ?, as well as an observation time Δ?, or, equivalently, an observation
energy Δ? = 2πℏ/Δ?. We will now investigate the consequences of these choices for
equation (106), always taking into account both quantum theory and general relativity.
At everyday energies, the result of the determination of the age of the universe ?0 is
about (13.8 ± 0.1) ⋅ 109 Ga. This value is deduced by measuring red-shifts, i.e., velocities,
and distances, using stars and galaxies in distance ranges, from some hundred thousand
light years up to a red-shift of about 1. Measuring red-shifts does not produce large velocity errors. The main source of experimental error is the difficulty in determining the
distances of galaxies.
What is the smallest possible error in distance? Obviously, inequality (102) implies
Δ?
? 2/3
> ( Pl )
?
?
(107)
* At higher red-shifts, the speed of light, as well as the details of the expansion, come into play. To continue
with the analogy of the trees, we find that the trees are not straight all the way up to the top and that they
grow on a slope, as suggested by Figure 8.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
In simple terms, this is the method used to determine the age of the universe from the
expansion of space-time, for galaxies with red-shifts below unity.* The (positive) measurement error Δ? becomes
Δ? Δ? Δ?
=
+
.
(106)
?
?
?
Motion Mountain – The Adventure of Physics
F I G U R E 7 Trees and galaxies.
w hat is the d ifference b et w een the univ erse a nd nothing?
big bang
97
space
4ct0/9
light cone:
what we
can see
time
other galaxies
in the night sky
t0
our galaxy
Challenge 58 e
Challenge 59 e
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 99
thus giving the same indeterminacy in the age of the universe as the one we found above
in the case of material clocks.
We can try to reduce the age error in two ways: by choosing objects at either small or
large distances. Let us start with small distances. In order to get high precision at small
distances, we need high observation energies. It is fairly obvious that at observation
energies near the Planck value, Δ?/? approaches unity. In fact, both terms on the righthand side of equation (106) become of order one. At these energies, Δ? approaches ?
and the maximum value for ? approaches the Planck length, for the same reason that
at Planck energy the maximum measurable time is the Planck time. In short, at Planck
scales it is impossible to say whether the universe is old or young.
Let us consider the other extreme, namely objects extremely far away, say with a redshift of ? ≫ 1. Relativistic cosmology requires the diagram of Figure 7 to be replaced by
the more realistic diagram of Figure 8. The ‘light onion’ replaces the familiar light cone
of special relativity: light converges near the big bang. In this case the measurement
error for the age of the universe also depends on the distance and velocity errors. At
the largest possible distances, the signals an object sends out must be of high energy,
because the emitted wavelength must be smaller than the universe itself. Thus, inevitably,
we reach Planck energy. However, we have seen that in such high-energy situations,
both the emitted radiation and the object itself are indistinguishable from the space-time
background. In other words, the red-shifted signal we would observe today would have
a wavelength as large as the size of the universe, with a correspondingly small frequency.
There is another way to describe the situation. At Planck energy or near the horizon,
the original signal has an error of the same size as the signal itself. When measured at
the present time, the red-shifted signal still has an error of the same size as the signal. As
a result, the error in the horizon distance becomes as large as the value to be measured.
In short, even if space-time expansion and large scales are used, the instant of the socalled beginning of the universe cannot be determined with an error smaller than the
Motion Mountain – The Adventure of Physics
F I G U R E 8 The speed and distance of remote
galaxies.
98
Challenge 60 ny
5 w hat is the d ifference b et ween the univ erse a nd nothing?
age of the universe itself: a result we also found at Planck distances. If we aim for perfect
precision, we just find that the universe is 13.8 ± 13.8 thousand million years old! In
other words, in both extremal situations, it is impossible to say whether the universe has a
non-vanishing age.
We have to conclude that the anthropocentric concept of ‘age’ does not make any
sense for the universe as a whole. The usual textbook value is useful only for ranges of
time, space and energy in which matter and space-time are clearly distinguished, namely
at everyday, human-scale energies; the value has no more general meaning.
You may like to examine the issue of the fate of the universe using the same arguments.
But we will now continue on the path outlined at the start of this chapter; the next topic
on this path is the measurement of length.
Maximum length
Ref. 99
Is the universe really a big pl ace?
Ref. 100
Vol. II, page 308
Ref. 99
Astronomers and Hollywood films answer this question in the affirmative. Indeed, the
distance to the horizon of the universe is often included in tables. Cosmological models specify that the scale factor ?, which fixes the distance to the horizon, grows with
* In cosmology, we need to distinguish between the scale factor ?, the Hubble radius ?/? = ??/?,̇ the
horizon distance ℎ and the size ? of the universe. The Hubble radius is a computed quantity giving the
distance at which objects move away with the speed of light. The Hubble radius is always smaller than the
horizon distance, at which in the standard Einstein–de Sitter model, for example, objects move away with
twice the speed of light. However, the horizon itself moves away with three times the speed of light.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 99
Motion Mountain – The Adventure of Physics
Vol. II, page 243
General relativity shows that the horizon distance, i.e., the distance of objects with infinite
red-shift, is finite. In the usual cosmological model, for hyperbolic (open) and parabolic
(marginal) evolutions of the universe, the size of the universe is assumed infinite.* For
elliptical evolution, the total size is finite and depends on the curvature. However, in this
case also the present measurement limit yields a minimum size for the universe many
times larger than the horizon distance.
Quantum field theory, on the other hand, is based on flat and infinite space-time. Let
us see what happens when the two theories are combined. What can we say about measurements of length in this case? For example, would it be possible to construct and use
a metre rule to measure lengths larger than the distance to the horizon?
Admittedly, we would have no time to push the metre rule out up to the horizon,
because in the standard big bang model the horizon moves away from us faster than the
speed of light. (We should have started using the metre rule right at the big bang.) But
just for fun, let us assume that we have actually managed to do this. How far away can
we read off distances? In fact, since the universe was smaller in the past, and since every
observation of the sky is an observation of the past, Figure 8 shows that the maximum
spatial distance away from us at which an object can be seen is only 4??0/9. Obviously,
for space-time intervals, the maximum remains ??0 .
Thus, in all cases it turns out to be impossible to measure lengths larger than the horizon distance, even though general relativity sometimes predicts such larger distances.
This result is unsurprising, and in obvious agreement with the existence of a limit for
measurements of time intervals. The real surprises come next.
w hat is the d ifference b et w een the univ erse a nd nothing?
99
time ?; for the case of the standard mass-dominated Einstein–de Sitter model, i.e., for a
vanishing cosmological constant and flat space, we have
?(?) = ? ?2/3 ,
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Challenge 62 e
where the numerical constant ? relates the commonly accepted horizon distance to the
commonly accepted age. Indeed, observation shows that the universe is large, and is
getting larger. But let us investigate what happens if we add some quantum theory to this
result from general relativity. Is it really possible to measure the distance to the horizon?
We look first at the situation at high (probe) energies. We saw above that space-time
and matter are not distinguishable at Planck scales. Therefore, at Planck energy we cannot state whether or not objects are localized. At Planck scales, the distinction between
matter and vacuum – so basic to our thinking – disappears.
Another way to say this is that we cannot claim that space-time is extended at Planck
scales. Our concept of extension derives from the possibility of measuring distances and
time intervals, and from observations such as the ability to align several objects behind
one another. Such observations are not possible at Planck scales and energies, because
of the inability of probes to yield useful results. In fact, all of the everyday observations
from which we deduce that space is extended are impossible at Planck scales, where the
basic distinction between vacuum and matter, namely between extension and localization,
disappears. As a consequence, at Planck energy the size of the universe cannot be measured. It cannot even be called larger than a matchbox.
The problems encountered with probes of high probe energies have drastic consequences for the size measurement of the universe. All the arguments given above for
the errors in measurement of the age can be repeated for the distance to the horizon. To
reduce size measurement errors, a measurement probe needs to have high energy. But
at high energy, measurement errors approach the value of the measurement results. At
the largest distances and at Planck energy, the measurement errors are of the same magnitude as the measured values. If we try to determine the size of the universe with high
precision, we get no precision at all.
The inability to get precise values for the size of the universe should not come unexpected. For a reliable measurement, the standard must be different, independent, and
outside the system to be measured. For the universe this is impossible.
Studying the size of the big bang also produces strange results. The universe is said to
have been much smaller near the big bang because, on average, all matter is moving away
from all other matter. But if we try to follow the path of matter into the past with high
precision, using Planck energy probes, we get into trouble: since measurement errors are
as large as measurement data, we cannot claim that the universe was smaller near the big
bang than it is today: there is no way to reliably distinguish size values.
There are other confirmations too. If we had a metre rule spanning the whole universe,
even beyond the horizon, with zero at the place where we live, what measurement error
would it produce for the horizon? It does not take long to work out that the expansion
of space-time, from Planck scales to the present size, implies an expansion in the error
from Planck size to a length of the order of the present distance to the horizon. Again,
the error is as large as the measurement result. And again, the size of the universe turns
Motion Mountain – The Adventure of Physics
Challenge 61 e
(108)
100
5 w hat is the d ifference b et ween the univ erse a nd nothing?
out not to be a meaningful property.
Since this reasoning also applies if we try to measure the diameter of the universe
instead of its radius, it is impossible to say whether the antipodes in the sky really are
distant from each other!
We can summarize the situation by noting that anything said about the size of the
universe is as limited as anything said about its age. The height of the sky depends on
the observation energy. If we start measuring the sky at standard observation energies,
and try to increase the precision of measurement of the distance to the horizon, the
measurement error increases beyond all bounds. At Planck energy, the volume of the
universe is indistinguishable from the Planck volume – and vice versa.
The boundary of space – is the sky a surface?
Challenge 63 ny
Page 85
* The measurement errors also imply that we cannot say anything about translational symmetry at cosmological scales. Can you confirm this? In addition, at the horizon it is impossible to distinguish between
space-like and time-like distances. Even worse, concepts such as ‘mass’ or ‘momentum’ become muddled
at the horizon. This means that, as at Planck energy, we are unable to distinguish between object and background, and between state and intrinsic properties. We thus confirm the point made above.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 58
Motion Mountain – The Adventure of Physics
Challenge 64 s
The horizon of the universe – essentially, the black part of the night sky – is a fascinating entity. Everybody interested in cosmology wants to know what happens there. In
newspapers the horizon is sometimes called the boundary of space. Some surprising insights – which have not yet made it to the newspapers – appear when we combine general
relativity and quantum mechanics.
We have seen that the errors in measuring the distance of the horizon are substantial.
They imply that we cannot pretend that all points of the sky are equally far away from us.
Thus we cannot say that the sky is a surface; it could be a volume. In fact, there is no way
to determine the dimensionality of the horizon, nor the dimensionality of space-time
near the horizon.*
Thus measurements do not allow us to determine whether the boundary is a point, a
surface, or a line. It may be a very complex shape, even knotted. In fact, quantum theory
tells us that it must be all of these from time to time: that the sky fluctuates in height and
shape.
In short, measurement errors prevent the determination of the topology of the sky.
In fact, this is not a new result. As is well known, general relativity is unable to describe
particle–antiparticle pair creation particles with spin 1/2. The reason for this inability is
the change in space-time topology required by such processes. The universe is full of
these and many other quantum processes; they imply that it is impossible to determine
or define the microscopic topology for the universe and, in particular, for the horizon.
Can you find at least two other arguments to confirm this conclusion?
Worse still, quantum theory shows that space-time is not continuous at a horizon:
this can easily be deduced using the Planck-scale arguments from the previous section.
Time and space are not defined at horizons.
Finally, there is no way to decide by measurement whether the various points on the
horizon are different from each other. On the horizon, measurement errors are of the
same order as the size of the horizon. The distance between two points in the night sky
is thus undefined. Therefore it is unclear what the diameter of the horizon is.
w hat is the d ifference b et w een the univ erse a nd nothing?
101
In summary, the horizon has no specific distance or shape. The horizon, and thus the
universe, cannot be shown to be manifolds. This unexpected result leads us to a further
question.
D oes the universe have initial conditions?
Ref. 101
Page 19
D oes the universe contain particles and stars?
Vol. II, page 308
Vol. IV, page 115
The total number of stars in the universe, about 1023±1, is included in every book on
cosmology. A smaller number can be counted on clear nights. But how dependable is the
statement? We can ask the same question about particles instead of stars. The commonly
quoted numbers are 1080±1 baryons and 1089±1 photons. However, the issue is not simple.
Neither quantum theory nor general relativity alone make predictions about the number
of particles. What happens if we combine the two theories?
In order to define the number of particles in a region, quantum theory first of all requires a vacuum state to be defined. The number of particles is defined by comparing
the system with the vacuum. If we neglect or omit general relativity by assuming flat
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 306
Motion Mountain – The Adventure of Physics
Page 58
One often reads about the quest for the initial conditions of the universe. But before
joining this search, we should ask whether and when such initial conditions make any
sense.
Obviously, our everyday description of motion requires knowledge of initial conditions, which describe the state of a system, i.e., all those aspects that differentiate it from
a system with the same intrinsic properties. Initial conditions – like the state – are attributed to a system by an outside observer.
Quantum theory tells us that initial conditions, or the state of a system, can only be
defined by an outside observer with respect to an environment. It is already difficult
to be outside the universe – but even inside the universe, a state can only be defined if
matter can be distinguished from vacuum. This is impossible at Planck energy, near the
big bang, or at the horizon. Thus the universe has no state. This means also that it has
no wave function.
The limits imposed by the Planck values confirm this conclusion in other ways. First
of all, they show that the big bang was not a singularity with infinite curvature, density
or temperature, because infinitely large values do not exist in nature. Secondly, since
instants of time do not exist, it is impossible to define the state of any system at a given
time. Thirdly, as instants of time do not exist, neither do events, and so the big bang was
not an event, and neither an initial state nor an initial wave function can be ascribed to
the universe. (Note that this also means that the universe cannot have been created.)
In short, there are no initial conditions for the universe. Initial conditions make sense
only for subsystems, and only far from Planck scales. Thus, for initial conditions to exist,
the system must be far from the horizon and it must have evolved for some time ‘after’
the big bang. Only when these two requirements are fulfilled can objects move in space.
Of course, this is always the case in everyday life. The lack of initial conditions means
that we have solved the first issue from the millennium list.
At this point in our mountain ascent, where neither time nor length is clearly defined
at cosmological scales, it should come as no surprise that there are similar difficulties
concerning the concept of mass.
102
Challenge 65 e
Challenge 66 s
5 w hat is the d ifference b et ween the univ erse a nd nothing?
Vol. II, page 308
Mass distinguishes objects from the vacuum. The average mass density of the universe,
about 10−26 kg/m3 , is often cited in texts. Is it different from a vacuum? Quantum theory
shows that, as a result of the indeterminacy relation, even an empty volume of size ? has
a mass. For a zero-energy photon inside such a vacuum, we have ?/? = Δ? > ℏ/Δ?, so
that in a volume of size ?, we have a minimum mass of at least ?min(?) = ℎ/??. For a
spherical volume of radius ? there is thus a minimal mass density given approximately
by
? (?)
ℏ
?min ≈ min3
= 4 .
(109)
?
??
For the universe, if the standard horizon distance ?0 of 13 800 million light years is inserted, the value becomes about 10−142 kg/m3 . This describes the density of the vacuum.
In other words, the universe, with a textbook density of about 10−26 kg/m3 , seems to be
clearly different from vacuum. But are we sure?
We have just deduced that the radius of the horizon is undefined: depending on the
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D oes the universe have mass?
Motion Mountain – The Adventure of Physics
space-time, this procedure poses no problem. However, if we include general relativity,
and thus a curved space-time, especially one with a strangely behaved horizon, the answer is simple: there is no vacuum state with which we can compare the universe, for two
reasons. First, nobody can explain what an empty universe would look like. Secondly,
and more importantly, there is no way to define a state of the universe. The number of
particles in the universe thus becomes undefinable. Only at everyday energies and for
finite dimensions are we able to speak of an approximate number of particles.
In the case of the universe, a comparison with the vacuum is also impossible for purely
practical reasons. The particle counter would have to be outside the system. (Can you
confirm this?) In addition, it is impossible to remove particles from the universe.
The impossibility of defining a vacuum state, and thus the number of particles in the
universe, is not surprising. It is an interesting exercise to investigate the measurement
errors that appear when we try to determine the number of particles despite this fundamental impossibility.
Can we count the stars? In principle, the same conclusion applies as for particles.
However, at everyday energies the stars can be counted classically, i.e., without taking
them out of the volume in which they are enclosed. For example, this is possible if
the stars are differentiated by mass, colour or any other individual property. Only near
Planck energy or near the horizon are these methods inapplicable. In short, the number
of stars is only defined as long as the observation energy is low, i.e., as long as we stay
away from Planck energy and from the horizon.
So, despite appearances on human scales, there is no definite number of particles in
the universe. The universe cannot be distinguished from vacuum by counting particles.
Even though particles are necessary for our own existence and functioning, a complete
count of them cannot be made.
This conclusion is so strange that we should try to resist it. Let us try another method
of determining the content of matter in the universe: instead of counting particles in the
universe, let us weigh the universe.
w hat is the d ifference b et w een the univ erse a nd nothing?
Vol. I, page 100
Vol. II, page 65
Challenge 68 e
observation energy, it can be as small as the Planck length. This implies that the density
of the universe lies somewhere between the lowest possible value, given by the density of
vacuum ?min just mentioned, and the highest possible one, namely the Planck density.*
In short, the total mass of the universe depends on the energy of the observer.
Another way to measure the mass of the universe would be to apply the original definition of mass, as given in classical physics and as modified by special relativity. Thus, let
us try to collide a standard kilogram with the universe. It is not hard to see that whatever
we do, using either low or high energies for the standard kilogram, the mass of the universe cannot be constrained by this method. We would need to produce or to measure a
velocity change Δ? for the rest of the universe after the collision. To hit all the mass in
the universe at the same time, we need high energy; but then we are hindered by Planck
energy effects. In addition, a properly performed collision measurement would require
a mass outside the universe, which is rather difficult to achieve.
Yet another way to measure the mass would be to determine the gravitational mass
of the universe through straightforward weighing. But the lack of balances outside the
universe makes this an impractical solution, to say the least.
Another way out might be to use the most precise definition of mass provided by general relativity, the so-called ADM mass. However, the definition of this requires a specified
behaviour at infinity, i.e., a background, which the universe lacks.
We are then left with the other general-relativistic method: determining the mass of
the universe by measuring its average curvature. Let us take the defining expressions for
average curvature ? for a region of size ?, namely
Challenge 69 e
Ref. 102
1
2
?curvature
=
3 4π?2 − ? 15 4π?3 /3 − ?
=
.
4π
?4
4π
?5
(111)
We have to insert the horizon radius ?0 and either its surface area ?0 or its volume ?0 .
However, given the error margins on the radius and the volume, especially at Planck
energy, we again find no reliable result for the radius of curvature.
An equivalent method starts with the usual expression provided by Rosenfeld for the
indeterminacy Δ? in the scalar curvature for a region of size ?, namely
Δ? >
2
16π?Pl
.
?4
(112)
However, this expression also shows that the error in the radius of curvature behaves like
the error in the distance to the horizon.
We find that at Planck energy, the average radius of curvature of nature lies between
infinity and the Planck length. This implies that the mass density of the universe lies
Challenge 67 e
Vol. V, page 144
* In fact, at everyday energies the density of the universe lies midway between the two values, yielding the
strange relation
?20 /?20 ≈ ?2Pl /?2Pl = ?4 /?2 .
(110)
But this fascinating relation is not new. The approximate equality can be deduced from equation 16.4.3
(p. 620) of Steven Weinberg, Gravitation and Cosmology, Wiley, 1972, namely ????? = 1/?20 . The
relation is required by several cosmological models.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
?=
Motion Mountain – The Adventure of Physics
Vol. II, page 189
103
104
Challenge 70 s
5 w hat is the d ifference b et ween the univ erse a nd nothing?
between the minimum value and the Planck value. There is thus no method to determine
the mass of the universe at Planck energy. (Can you find one?)
In summary, mass measurements of the universe vary with the energy scale. Both at
the lowest and at the highest energies, a precise mass value cannot be determined. The
concept of mass cannot be applied to the universe as a whole: the universe has no mass.
D o symmetries exist in nature?
It is common to take ‘boundary’ and ‘horizon’ as synonyms in the case of the universe,
because they are the same for all practical purposes. Knowledge of mathematics does
not help us here: the properties of mathematical boundaries – for example, that they
themselves have no boundary – are not applicable to the universe, since space-time is
not continuous. We need other, physical arguments.
The boundary of the universe is supposed to represent the boundary between something and nothing. There are three possible interpretations of ‘nothing’:
— ‘Nothing’ could mean ‘no matter’. But we have just seen that this distinction cannot be made at Planck scales. So either the boundary will not exist at all or it will
encompass the horizon as well as the whole universe.
— ‘Nothing’ could mean ‘no space-time’. We then have to look for those domains
where space and time cease to exist. These occur at Planck scales and at the horizon.
Again, either the boundary will not exist or it will encompass the whole universe.
— ‘Nothing’ could mean ‘neither space-time nor matter’. The only possibility is a
boundary that encloses domains beyond the Planck scales and beyond the horizon;
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D oes the universe have a boundary?
Motion Mountain – The Adventure of Physics
Challenge 71 e
We have already seen that at the horizon, space-time translation symmetry breaks down.
Let us have a quick look at the other symmetries.
What happens to permutation symmetry? Permutation is an operation on objects in
space-time. It thus necessarily requires a distinction between matter, space and time. If
we cannot distinguish positions, we cannot talk about exchange of particles. Therefore, at
the horizon, general relativity and quantum theory together make it impossible to define
permutation symmetry.
Let us explore CPT symmetry. As a result of measurement errors or of limiting maximum or minimum values, it is impossible to distinguish between the original and the
transformed situations. Therefore we cannot claim that CPT is a symmetry of nature at
horizon scales. In other words, matter and antimatter cannot be distinguished at the
horizon.
Also gauge symmetry is not valid at horizon scale, as you may wish to check in detail
yourself. For its definition, the concept of gauge field requires a distinction between time,
space and mass; at the horizon this is impossible. We therefore also deduce that at the
horizon, concepts such as algebras of observables cannot be used to describe nature.
Renormalization breaks down too.
All symmetries of nature break down at the horizon. None of the vocabulary we use
to talk about observations – including terms such as such as ‘magnetic field’, ‘electric
field’, ‘potential’, ‘spin’, ‘charge’, or ‘speed’ – can be used at the horizon.
w hat is the d ifference b et w een the univ erse a nd nothing?
105
but again, such a boundary would also encompass all of nature.
Challenge 72 s
Ref. 103
Is the universe a set?
“
”
Cicero
We are used to thinking of the universe the sum of all matter and all space-time. In doing so, we imply that the universe is a set of mutually distinct components. This idea
has been assumed in three situations: in claiming that matter consists of particles; that
space-time consists of events (or points); and that different states consist of different initial conditions. However, our discussion shows that the universe is not a set of such
distinguishable elements. We have encountered several proofs: at the horizon, at the big
bang and at Planck scales, it becomes impossible to distinguish between events, between
particles, between observables, and between space-time and matter. In those domains,
distinctions of any kind become impossible. We have found that distinguishing between
two entities – for example, between a toothpick and a mountain – is only approximately possible. It is approximately possible because we live at energies well below the
Planck energy. The approximation is so good that we do not notice the error when we
distinguish cars from people and from toothpicks. Nevertheless, our discussion of the
situation at Planck energy shows that a perfect distinction is impossible in principle. It
is impossible to split the universe into separate parts.
Another way to reach this result is the following. Distinguishing between two entities requires different measurement results: for example, different positions, masses
or sizes. Whatever quantity we choose, at Planck energy the distinction becomes impossible. Only at everyday energies is it approximately possible.
In short, since the universe contains no distinguishable parts, there are no (mathematical) elements in nature. Simply put: the universe is not a set. We envisaged this possibility earlier on; now it is confirmed. The concepts of ‘element’ and ‘set’ are already too
* ‘The mistress and queen of all things is reason.’ Tusculanae Disputationes, 2.21.47. Marcus Tullius Cicero
(106–43 bce), was an influential lawyer, orator and politician at the end of the Roman republic.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Vol. III, page 327
Domina omnium et regina ratio.*
Motion Mountain – The Adventure of Physics
This is puzzling. When combining quantum theory and relativity, we do not seem to
be able to find a conceptual definition of the horizon that distinguishes it from what
it includes. A distinction is possible in general relativity alone, and in quantum theory
alone; but as soon as we combine the two, the boundary becomes indistinguishable from
its content. The interior of the universe cannot be distinguished from its horizon. There is
no boundary of the universe.
The difficulty in distinguishing the horizon from its contents suggests that nature may
be symmetric under transformations that exchange interiors and boundaries. This idea
is called holography, because it vaguely recalls the working of credit-card holograms. It
is a busy research field in high-energy physics.
We note that if the interior and the boundary of the universe cannot be distinguished,
the constituents of nature can neither be points nor tiny objects of any kind. The constituents of nature must be extended. But before we explore this topic, we continue with
our search for differences between the universe and nothing. The search leads us to our
next question.
106
Challenge 74 s
* Some people knew this long before physicists. For example, the belief that the universe is or contains
information was ridiculed most thoroughly in the popular science-fiction parody by Douglas Adams,
The Hitchhiker’s Guide to the Galaxy, 1979, and its sequels.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Challenge 73 e
specialized to describe the universe. The universe must be described by a mathematical
concept that does not contain any set. The new concept must be more general than that
of a set.
This is a powerful result: a precise description of the universe cannot use any concept
that presupposes the existence of sets. But all the concepts we have used so far to describe
nature, such as space-time, metric, phase space, Hilbert space and its generalizations, are
based on elements and sets. They must all be abandoned at Planck energies, and in any
precise description.
Elements and sets must be abandoned. Note that this radical conclusion is deduced
from only two statements: the necessity of using quantum theory whenever the dimensions are of the order of the Compton wavelength, and of using general relativity
whenever the dimensions are of the order of the Schwarzschild radius. Together, they
mean that no precise description of nature can contain elements and sets. The difficulties in complying with this result explain why the unification of the two theories has
not so far been successful. Not only does unification require that we stop using space,
time and mass for the description of nature; it also requires that all distinctions, of any
kind, should be only approximate. But all physicists have been educated on the basis of
exactly the opposite creed!
Many past speculations about the final unified description of nature depend on sets.
In particular, all studies of quantum fluctuations, mathematical categories, posets, involved mathematical spaces, computer programs, Turing machines, Gödel’s incompleteness theorem, creation of any sort, space-time lattices, quantum lattices and Bohm’s unbroken wholeness presuppose sets. In addition, all speculations by cosmologists about the
origin of the universe presuppose sets. But since these speculations presuppose sets, they
are wrong. You may also wish to check the religious explanations you know against this
criterion. In fact, no approach used by theoretical physicists up to the year 2000 satisfied
the requirement that elements and sets must be abandoned.
The task of abandoning sets is not easy. This is shown with a simple test: do you know
of a single concept not based on elements or sets?
In summary, the universe is not a set. In particular, the universe is not a physical
system. Specifically, it has no state, no intrinsic properties, no wave function, no initial
conditions, no energy, no mass, no entropy and no cosmological constant. The universe
is thus neither thermodynamically closed nor open; and it contains no information. All
thermodynamic quantities, such as entropy, temperature and free energy, are defined
using ensembles. Ensembles are limits of systems which are thermodynamically either
open or closed. As the universe is neither open nor closed, no thermodynamic quantity
can be defined for it.* All physical properties are defined only for parts of nature. Only
parts of nature are approximated or idealized as sets, and thus only parts of nature are
physical systems.
Motion Mountain – The Adventure of Physics
Ref. 104
5 w hat is the d ifference b et ween the univ erse a nd nothing?
w hat is the d ifference b et w een the univ erse a nd nothing?
107
Curiosities and fun challenges abou t the universe
“
“
Insofern sich die Sätze der Mathematik auf die
Wirklichkeit beziehen, sind sie nicht sicher, und
sofern sie sicher sind, beziehen sie sich nicht auf
die Wirklichkeit.*
Albert Einstein
”
”
Die ganzen Zahlen hat der liebe Gott gemacht,
alles andere ist Menschenwerk.**
Leopold Kronecker
∗∗
∗∗
Ref. 105
In 2002, Seth Lloyd estimated how much information the universe can contain, and how
many calculations it has performed since the big bang. This estimate is based on two
ideas: that the number of particles in the universe is a well-defined quantity, and that the
universe is a computer, i.e., a physical system. We now know that neither assumption
is correct. The universe contains no information. Conclusions such as this one show the
power of the criteria that we have deduced for any precise or complete description of
motion.
∗∗
Challenge 75 s
Astronomers regularly take pictures of the cosmic background radiation and its variations. Is it possible that these photographs will show that the spots in one direction of
the sky are exactly the same as those in the diametrically opposite direction?
∗∗
* ‘In so far as mathematical statements describe reality, they are not certain, and as far as they are certain,
they are not a description of reality.’
** ‘Gracious god made the integers, all else is the work of man.’ Leopold Kronecker (b. 1823 Liegnitz,
d. 1891 Berlin) was a well-known mathematician. Among others, the Kronecker delta and the Kronecker
product are named for him.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
If vacuum cannot be distinguished from matter or radiation, and if the universe cannot
be distinguished from nothing, then it is incorrect to claim that “the universe appeared
from nothing.” The naive idea of creation is a logical impossibility. “Creation” results
from a lack of imagination.
Motion Mountain – The Adventure of Physics
In mathematics, 2 + 2 = 4. This statement is an idealization of statements such as ‘two
apples plus two apples makes four apples.’ However, we now know that at Planck energy,
the statement about apples is not a correct statement about nature. At Planck energy,
objects cannot be counted or even defined, because separation of objects is not possible
at that scale. We can count objects only because we live at energies much lower than the
Planck energy.
The statement by Kronecker must thus be amended. Since all integers are low-energy
approximations, and since we always use low-energy approximations when talking or
thinking, we provokingly conclude: man also makes the integers.
108
Ref. 106
Challenge 76 s
5 w hat is the d ifference b et ween the univ erse a nd nothing?
In 1714, the famous scientist and thinker Leibniz (b. 1646 Leipzig, d. 1716 Hannover)
published his Monadologie. In it he explores what he calls a ‘simple substance’, which
he defined to be a substance that has no parts. He called it a monad and describes some
of its properties. However, mainly thanks to his incorrect deductions, the term has not
been generally adopted. What is the physical concept most closely related to that of a
monad?
∗∗
Challenge 77 s
We usually speak of the universe, implying that there is only one of them. Yet there is
a simple case to be made that ‘universe’ is an observer-dependent concept, since the
idea of ‘all’ is observer-dependent. Does this mean that there are many universes, or a
‘multiverse’?
∗∗
Is the ‘radius’ of the universe observer-invariant?
∗∗
Challenge 79 e
Is the cosmological constant Λ observer-invariant?
∗∗
Challenge 80 s
If all particles were removed (assuming one knew where to put them), there wouldn’t be
much of a universe left. True?
∗∗
Can you show that the distinction between matter and antimatter is not possible at the
cosmic horizon? And the distinction between real and virtual particles?
∗∗
Challenge 82 s
At Planck energy, interactions cannot be defined. Therefore, ‘existence’ cannot be
defined. In short, at Planck energy we cannot say whether particles exist. True?
Hilbert ’ s sixth problem set tled
Vol. III, page 281
Ref. 107
In the year 1900, David Hilbert* gave a famous lecture in which he listed 23 of the great
challenges facing mathematics in the twentieth century. Most of these provided challenges to many mathematicians for decades afterwards. A few are still unsolved, among
them the sixth, which challenged mathematicians and physicists to find an axiomatic
treatment of physics. The problem has remained in the minds of many physicists since
that time. Scholars have developed axiomatic treatments if classical mechanics, electrodynamics and special relativity. Then they did this for quantum theory, quantum field
theory and general relativity.
Whenever we combine quantum theory and general relativity, we must abandon the
concept of point particle, of space point and of event. Mathematically speaking, when
we combine quantum theory and general relativity, we find that nature does not contain
* David Hilbert (b. 1862 Königsberg, d. 1943 Göttingen) was the greatest mathematician of his time. His
textbooks are still in print.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Challenge 81 e
Motion Mountain – The Adventure of Physics
Challenge 78 e
w hat is the d ifference b et w een the univ erse a nd nothing?
109
sets, and that the universe is not a set. However, all mathematical systems – be they
algebraic systems, order systems, topological systems or a mixture of these – are based
on elements and sets. Mathematics does not have axiomatic systems without elements
or sets. The reason for this is simple: every (mathematical) concept contains at least one
element and one set. However, nature is different. And since nature does not contain
sets, an axiomatic description of nature is impossible.
All concepts used in physics before the year 2000 depend on elements and sets. For
humans, it is difficult even to think without first defining a set of possibilities. Yet nature
does not contain sets.
⊳ There is no axiomatic description of nature.
The perfect physics bo ok
Vol. I, page 435
A perfect physics book describes all of nature with full precision. In particular, a perfect
physics book describes itself, its own production, its own author, its own readers and its
own contents. Can such a book exist?
Since the universe is not a set, a perfect physics book can exist, as it does not contradict
any property of the universe. Since the universe is not a set and since it contains no
information, the paradox of the perfect physics book disappears. Indeed, any existing
physics book attempts to be perfect. But now a further question arises.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 218
Motion Mountain – The Adventure of Physics
Vol. I, page 436
And since an axiomatic formulation of physics is impossible, we conclude that the final,
unified theory cannot be based on axioms. This is surprising at first, because separate
axiomatic treatments of quantum theory and general relativity are possible. However,
axiomatic systems in physics are always approximate. The need to abandon axioms is one
of the reasons why reaching a unified description of nature is a challenge.
The impossibility of an axiomatic system for physics is also confirmed in another way.
Physics starts with a circular definition: space-time and vacuum are defined with the help
of objects and objects are defined with the help of space-time and vacuum. In fact, physics has never been axiomatic! Physicists have always had to live with circular definitions.
The situation is similar to a child’s description of the sky as ‘made of air and clouds’.
Looking closely, we discover that clouds are made up of water droplets. We find that
there is air inside clouds, and that there is also water vapour away from clouds. When
clouds and air are viewed through a microscope, there is no clear boundary between the
two. We cannot define either of the terms ‘cloud’ and ‘air’ without the other.
Like clouds and air, also objects and vacuum are indistinguishable. Virtual particles
are found in vacuum, and vacuum is found inside objects. At Planck scales there is no
clear boundary between the two; we cannot define either of the terms ‘particle’ and
‘vacuum’ without the other. But despite the lack of a clean definition, and despite the
logical problems that can ensue, in both cases the description works well at large, everyday scales.
In summary, an axiomatic description of nature is impossible. In particular, the final,
unified theory must contain circular definitions. We will find out how to realize the
requirement later on.
110
5 w hat is the d ifference b et ween the univ erse a nd nothing?
TA B L E 3 Statements about the universe when explored at highest precision, i.e., at Planck scales
The universe has no beginning.
The universe has no volume.
The universe’s particle number is undefined.
The universe has no energy.
The universe contains no matter.
The universe has no initial conditions.
The universe has no wave function.
The universe contains no information.
The universe is not open.
The universe does not interact.
The universe cannot be said to exist.
The universe cannot be distinguished from a
single event.
The universe is not composite.
The universe is not a concept.
There is no plural for ‘universe’.
The universe was not created.
D oes the universe make sense?
Vol. III, page 252
Challenge 83 r
Challenge 84 s
”
Is the universe really the sum of matter–energy and space-time? Or of particles and
vacuum? We have heard these statements so often that we may forget to check them.
We do not need magic, as Faust thought: we only need to list what we have found so far,
especially in this section, in the section on Planck scales, and in the chapter on brain and
language. Table 3 shows the result.
Not only are we unable to state that the universe is made of space-time and matter;
we are unable to say anything about the universe at all! It is not even possible to say that
it exists, since it is impossible to interact with it. The term ‘universe’ does not allow us to
make a single sensible statement. (Can you find one?) We are only able to list properties
it does not have. We are unable to find any property that the universe does have. Thus,
the universe has no properties! We cannot even say whether the universe is something
or nothing. The universe isn’t anything in particular. The term universe has no content.
By the way, there is another well-known, non-physical concept about which nothing
can be said. Many scholars have explored it in detail. What is it?
* ‘Thus I have devoted myself to magic, [ ... ] that I understand how the innermost world is held together.’
Goethe was a German poet.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
“
Drum hab ich mich der Magie ergeben,
[ ... ]
Daß ich erkenne, was die Welt
Im Innersten zusammenhält.*
Goethe, Faust.
Motion Mountain – The Adventure of Physics
The universe has no age.
The universe has no size.
The universe has no shape.
The universe has no mass.
The universe has no density.
The universe has no cosmological constant.
The universe has no state.
The universe is not a physical system.
The universe is not isolated.
The universe has no boundaries.
The universe cannot be measured.
The universe cannot be distinguished from
nothing.
The universe contains no moments.
The universe is not a set.
The universe cannot be described.
The universe cannot be distinguished from
vacuum.
w hat is the d ifference b et w een the univ erse a nd nothing?
Vol. III, page 283
111
In short, the term ‘universe’ is not at all useful for the description of motion. We
can obtain a confirmation of this strange conclusion from an earlier chapter. There we
found that any concept needs defined content, defined limits and a defined domain of
application. In this section, we have found that the term ‘universe’ has none of these;
there is thus no such concept. If somebody asks why the universe exists, the answer is:
not only does the use of the word ‘why’ wrongly suggest that something may exist outside
the universe, providing a reason for it and thus contradicting the definition of the term
‘universe’ itself; but more importantly, the universe does not exist, because there is no
such concept as a ‘universe’.
In summary, any sentence containing the word ‘universe’ is meaningless. The word only
seems to express something, but it doesn’t.* This conclusion may be interesting, even
strangely beautiful, but does it help us to understand motion more precisely? Yes, it
does.
Extremal scales and open questions in physics
Page 19
At the beginning of this volume, we listed all the fundamental properties of nature that
are unexplained either by general relativity or by quantum theory. We called it the millennium list. The results of this chapter provide us with surprising statements on many
of the items. In fact, many of the statements are not new at all, but are surprisingly familiar. Let us compare systematically the statements from this chapter, on the universe,
with those of the previous chapter, on Planck scales. The comparison is given in Table 4.
* Of course, the term ‘universe’ still makes sense if it is defined more restrictively: for example, as everything
interacting with a particular human or animal observer in everyday life. But such a definition, equating
‘universe’ and ‘environment’, is not useful for our quest, as it lacks the precision required for a description
of motion.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Challenge 85 e
Our discussion of the term ‘universe’ shows that the term cannot include any element or
set. And the same applies to the term ‘nature’. Nature cannot be made of atoms. Nature
cannot be made of space-time points. Nature cannot be made of separate, distinct and
discrete entities.
The difficulties in giving a sharp definition of ‘universe’ also show that the fashionable
term ‘multiverse’ makes no sense. There is no way to define such a term, since there is
no empirical way and also no logical way to distinguish ‘one’ universe from ‘another’:
the universe has no boundary. In short, since the term ‘universe’ has no content, the
term ‘multiverse’ has even less. The latter term has been created only to trick the media
and various funding agencies. In fact, the same might be said of the former term...
So far, by taking into account the limits on length, time, mass and all the other quantities we have encountered, we have reached a number of almost painful conclusions about
nature. However, we have also received something in exchange: all the contradictions
between general relativity and quantum theory that we mentioned at the beginning of
this chapter are now resolved. We changed the contradictions to circular definitions. Although we have had to leave many cherished habits behind us, in exchange we have the
promise of a description of nature without contradictions. But we get even more.
Motion Mountain – The Adventure of Physics
Aband oning sets and discreteness eliminates contradictions
112
5 w hat is the d ifference b et ween the univ erse a nd nothing?
TA B L E 4 Properties of nature at maximal, everyday and minimal scales
At
every d ay
scale
At P l anck
scales
requires quantum theory and relativity
intervals can be measured precisely
length and time intervals appear
space-time is not continuous
points and events cannot be distinguished
space-time is not a manifold
space is 3-dimensional
space and time are indistinguishable
initial conditions make sense
space-time fluctuates
Lorentz and Poincaré symmetry
CPT symmetry
renormalization
permutation symmetry
interactions and gauge symmetries
number of particles
algebras of observables
matter indistinguishable from vacuum
boundaries exist
nature is a set
true
false
limited
true
true
true
false
true
false
true
do not apply
does not apply
does not apply
does not apply
do not exist
undefined
undefined
true
false
false
false
true
unlimited
false
false
false
true
false
true
false
apply
applies
applies
applies
exist
defined
defined
false
true
true
true
false
limited
true
true
true
false
true
false
true
do not apply
does not apply
does not apply
does not apply
do not exist
undefined
undefined
true
false
false
First, Table 4 shows that each unexplained property listed there is unexplained at both
limits of nature, the small and the large limit. Worse, many of these unexplained general
properties do not even make sense at the two limits of nature!
Secondly, and more importantly, nature behaves in the same way at the cosmological
horizon scale and at the Planck scale. In fact, we have not found any difference between
the two cases. (Can you discover one?) We are thus led to the hypothesis that nature
does not distinguish between the large and the small. Nature seems to be characterized
by extremal identity.
Is extremal identit y a principle of nature?
The idea of extremal identity incorporates some rather general points:
—
—
—
—
—
All open questions about nature appear at both size extremes.
Any description of nature requires both general relativity and quantum theory.
Nature, or the universe, is not a set.
Initial conditions and evolution equations make no sense at nature’s limits.
There is a relation between local and global issues in nature.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
At hori zon scale
Motion Mountain – The Adventure of Physics
Challenge 86 r
P hysical p ropert y of nature
sum m ary on the univ erse
113
— The concept of ‘universe’ has no content.
Challenge 87 e
Ref. 108
Extremal identity thus looks like a useful hypothesis in the search for a unified description of nature. To be a bit more provocative, it seems that extremal identity may be the
only hypothesis incorporating the idea that the universe is not a set. Therefore, extremal
identity seems to be essential in the quest for unification.
Extremal identity is beautiful in its simplicity, in its unexpectedness and in the richness of its consequences. You might enjoy exploring it by yourself. In fact, the exploration of extremal identity is currently the subject of much activity in theoretical physics,
although often under different names.
The simplest approach to extremal identity – in fact, one that is too simple to be correct – is inversion. Indeed, extremal identity seems to imply a connection such as
?↔
Ref. 103
Page 85
?? ↔
2
??
?Pl
?? ??
(113)
relating distances ? or coordinates ?? with their inverse values using the Planck length
?Pl . Can this mapping be a symmetry of nature? At every point of space? For example,
if the horizon distance is inserted, the relation (113) implies that lengths smaller than
?Pl /1061 ≈ 10−96 m never appear in physics. Is this the case? What would inversion imply
for the big bang?
More involved approaches to extremal identity come under the name of space-time
duality and holography. They are subject of intense research. Numerous fascinating questions are contained in extremal identity; there is a lot of fun ahead of us.
Above all, we need to find the correct version of the inversion relation (113). Inversion
is neither sufficient nor correct. It is not sufficient because it does not explain any of the
millennium issues left open by general relativity and quantum theory. It only relates some
of them, but it does not solve any of them. (You may wish to check this for yourself.)
In other words, we need to find the precise description of quantum geometry and of
elementary particles.
However, inversion is also simply wrong. Inversion is not the correct description of
extremal identity because it does not realize a central result discovered above: it does
not connect states and intrinsic properties, but keeps them distinct. In particular, inversion does not take interactions into account. And most open issues at this point of our
mountain ascent concern the properties and the appearance of interactions.
summary on the universe
The exploration of the universe allows us to formulate several additional requirements
for the final theory that we are looking for:
— Whenever we combine general relativity and quantum theory, the universe teaches
us that it is not a set of parts. For this reason, any sentence or expression containing
the term ‘universe’ is meaningless whenever full precision is required.
— We learned that a description of nature without sets solves the contradictions between
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Challenge 89 e
or
Motion Mountain – The Adventure of Physics
Challenge 88 s
2
?Pl
?
114
5 w hat is the d ifference b et ween the univ erse a nd nothing?
general relativity and quantum theory.
— We found, again, that despite the contradictions between quantum theory and general relativity, the Planck limits ?, ℏ and ?4 /4? remain valid.
— We then found an intriguing relation between Planck scales and cosmological scales:
they seem to pose the same challenges to their description. There is a close relationship between large and small scales in nature.
We can now answer the question in the chapter title: there seems to be little difference
– if any at all – between the universe and nothing. We can express this result in the
following catchy statement:
⊳ The universe cannot be observed.
A physical aphorism
Ref. 55, Ref. 56
Inserting numbers, we find rather precisely that the time ? is the present age of the universe.
With the right dose of humour we can see this result as a sign that time is now ripe,
after so much waiting, for us to understand the universe down to the Planck scales. We
are thus getting nearer to the top of Motion Mountain. Be prepared for a lot of fun.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Challenge 90 s
Here is a humorous ‘proof’ that we really are near the top of Motion Mountain. Salecker
and Wigner, and then Zimmerman, formulated the fundamental limit for the measurement precision ? attainable by a clock of mass ?. It is given by ? = √ℏ?/?2 ? , where
? is the time to be measured. We can then ask what maximum time ? can be measured
with a precision of a Planck time ?Pl , given a clock of the mass of the whole universe. We
get a maximum time of
?2 ?2
? = Pl ? .
(114)
ℏ
Motion Mountain – The Adventure of Physics
In our journey, the confusion and tension are increasing. But in fact we are getting close
to our goal, and it is worth continuing.
Chapter 6
T HE SHAPE OF POINT S – EX T ENSION
IN NAT UR E
”
he usual expressions for the reduced Compton wavelength ? = ℏ/?? and for
he Schwarzschild radius ?s = 2??/?2 , taken together, imply the conclusion
hat at Planck energies, what we call ‘space points’ and ‘point particles’ must actually be described by extended constituents that are infinite and fluctuating in size. We
will show this result with the following arguments:
We conclude the chapter with some experimental and theoretical checks of extension
** ‘Nothing is so difficult that it could not be investigated.’ Terence is Publius Terentius Afer (b. c. 190
Carthago, d. 159 bce Greece), important Roman poet. He writes this in his play Heauton Timorumenos,
verse 675.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
1. Any experiment trying to measure the size or the shape of an elementary particle
with high precision inevitably leads to the result that at least one dimension of the
particle is of macroscopic size.
2. There is no evidence that empty space consists of points, as they cannot be measured
or detected. In addition, in order to build up a measurable entity, such as the vacuum,
that is extended in three dimensions, its constituents must also be extended.
3. The existence of minimum measurable distances and time intervals implies the existence of space-time duality: a symmetry between very large and very small distances.
Space-time duality in turn implies that the fundamental constituents that make up
vacuum and matter are extended.
4. The constituents of the universe, and thus of vacuum, matter and radiation, cannot
form a (mathematical) set. And any precise description of nature without sets must
use extended constituents.
5. The Bekenstein–Hawking expression for the entropy of black holes – in particular
its surface dependence – confirms that both vacuum and particles are composed of
extended constituents.
6. The attempt to extend statistical properties to Planck scales shows that both particles
and space points behave as extended constituents at high energies, namely as braids
or tangles.
7. The belt trick provides a model for fermions that matches observations and again
suggests extended constituents in matter.
Motion Mountain – The Adventure of Physics
T
“
Nil tam difficile est, quin quaerendo investigari
possiet.**
Terence
116
6 the sha pe of p oints
and an overview of present research efforts.
“
Also, die Aufgabe ist nicht zu sehen, was noch
nie jemand gesehen hat, sondern über dasjenige
was jeder schon gesehen hat zu denken was
noch nie jemand gedacht hat.*
Erwin Schrödinger
”
the size and shape of elementary particles
D o boxes exist?
Can the Greeks help? – The limitations of knives
The Greeks deduced the existence of atoms by noting that matter cannot be divided indefinitely. There must be uncuttable particles, which they called atoms. Twenty-five centuries later, experiments in the field of quantum physics confirmed the conclusion, but
* ‘Our task is not to see what nobody has ever seen, but to think what nobody has ever thought about that
which everybody has seen already.’ Erwin Schrödinger (b. 1887 Vienna, d. 1961 Vienna) discovered the
equation that brought him international fame and the Nobel Prize in Physics.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
The first and simplest way to determine the size of a compact particle such as a sphere,
or to find at least an upper limit, is to measure the size of a box it fits in. To be sure that
the particle is inside, we must first be sure that the box is tight: that is, whether anything
(such as matter or radiation) can leave the box.
But there is no way to ensure that a box has no holes! We know from quantum physics
that any wall is a finite potential hill, and that tunnelling is always possible. In short, there
is no way to make a completely tight box.
Let us cross-check this result. In everyday life, we call particles ‘small’ when they
can be enclosed. Enclosure is possible in daily life because walls are impenetrable. But
walls are only impenetrable for matter particles up to about 10 MeV and for photons up
to about 10 keV. In fact, boxes do not even exist at medium energies. So we certainly
cannot extend the idea of ‘box’ to Planck energy.
Since we cannot conclude that particles are of compact size by using boxes, we need
to try other methods.
Motion Mountain – The Adventure of Physics
Page 88
Size is the length of vacuum taken by an object. This definition comes naturally in everyday life, quantum theory and relativity. To measure the size of an object as small as an
elementary particle, we need high energy. The higher the energy, the higher the precision
with which we can measure the size.
However, near the Planck energy, vacuum and matter cannot be distinguished: it is
impossible to define the boundary between the two, and thus it is impossible to define
the size of an object. As a consequence, every object, and in particular every elementary
particle, becomes as extended as the vacuum! There is no measurement precision at all
at Planck scales. Can we save the situation? Let us take a step back. Do measurements at
least allow us to say whether particles can be contained inside small spheres?
the siz e a nd sha pe of elem entary pa rticles
117
modified it: nowadays, the elementary particles are the ‘atoms’ of matter and radiation.
Despite the huge success of the concept of elementary particle, at Planck energy, we
have a different situation. The use of a knife, like any other cutting process, is the insertion of a wall. Walls and knives are potential hills. All potential hills are of finite height,
and allow tunnelling. Therefore a wall is never perfect, and thus neither is a knife. In
short, any attempt to divide matter fails to work when we approach Planck scales. At
Planck energy, any subdivision is impossible.
The limitations of knives and walls imply that at Planck energy, an attempted cut does
not necessarily lead to two separate parts. At Planck energy, we can never state that the
two parts have been really, completely separated: the possibility of a thin connection
between the two parts to the right and left of the blade can never be excluded. In short,
at Planck scales we cannot prove compactness by cutting objects.
Are cross sections finite?
Vol. I, page 25
That is quite a statement. Are particles really not of finite, bounded size? Right at the
start of our mountain ascent, we distinguished objects from their environment. Objects
are by definition localized, bounded and compact. All objects have a boundary, i.e., a
surface which does not itself have a boundary. Objects are also bounded in abstract
ways: also the set of symmetries of an object, such as its geometric symmetry group or
its gauge group, is bounded. In contrast, the environment is not localized, but extended
and unbounded. But all these basic assumptions fail us at Planck scales. At Planck energy,
it is impossible to determine whether something is bounded or compact. Compactness
and locality are only approximate properties; they are not applicable at high energies.
In particular, the idea of a point particle is an approximate concept, valid only at low
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
⊳ Quantum particles are extended.
Motion Mountain – The Adventure of Physics
Ref. 109
To sum up: despite all attempts, we cannot show that elementary particles are point-like.
Are they, at least, of finite size?
To determine the size of a particle, we can try to determine its departure from pointlikeness. Detecting this departure requires scattering. For example, we can suspend the
particle in a trap and then shoot a probe at it. What happens in a scattering experiment at
highest energies? This question has been studied by Leonard Susskind and his colleagues.
When shooting at the particle with a high-energy probe, the scattering process is characterized by an interaction time. Extremely short interaction times imply sensitivity to
size and shape fluctuations, due to the quantum of action. An extremely short interaction time also provides a cut-off for high-energy shape and size fluctuations, and thus
determines the measured size. As a result, the size measured for any microscopic, but
extended, object increases when the probe energy is increased towards the Planck value.
In summary, even though at experimentally achievable energies the size of an elementary particle is always smaller than the measurement limit, when we approach the
Planck energy, the particle size increases beyond all bounds. So at high energies we cannot give an upper limit to the size of a particle – except the universe itself. In other
words, since particles are not point-like at everyday energies, at Planck energy they are
enormous:
118
6 the sha pe of p oints
TA B L E 5 Effects of various camera shutter times on photographs
O b s e rvat i o n p o s s i b i l i t i e s a n d e f f e c t s
1h
1s
20 ms
10 ms
high
high
lower
lower
0.25 ms
lower
1 μs
c. 10 ps
very low
lowest
10 fs
100 zs
shorter times
higher
high
very high
10−43 s
highest
Ability to see faint quasars at night if motion is compensated
Everyday motion completely blurred
Interruption by eyelids; small changes impossible to see
Effective eye/brain shutter time; tennis ball impossible to see while
hitting it
Shortest commercial photographic camera shutter time; ability to
photograph fast cars
Ability to photograph flying bullets; strong flashlight required
Study of molecular processes; ability to photograph flying light
pulses; laser light required to get sufficient illumination
Light photography impossible because of wave effects
X-ray photography impossible; only ?-ray imaging left over
Photographs get darker as illumination decreases; gravitational effects significant
Imaging impossible
energies.
We conclude that particles at Planck scales are as extended as the vacuum. Let us
perform another check.
“
Ref. 110
Ref. 47, Ref. 24
Καιρὸν γνῶθι.*
”
Pittacus
Humans – or any other types of observers – can only observe the world with finite resolution in time and in space. In this respect, humans resemble a film camera. Every
camera has a resolution limit: it can only distinguish two events if they are a certain
minimum distance apart and separated by a certain minimum time. What is the best
resolution possible? The value was (almost) discovered in 1899: the Planck time and the
Planck length. No human, no film camera and no apparatus can measure space or time
intervals smaller than the Planck values. But what would happen if we took photographs
with shutter times that approach the Planck time?
Imagine that you have the world’s best shutter and that you are taking photographs
at shorter and shorter times. Table 5 gives a rough overview of the possibilities. When
shutter times are shortened, photographs get darker and sharper. When the shutter time
reaches the oscillation time of light, strange things happen: light has no chance to pass
undisturbed; signal and noise become indistinguishable; and the moving shutter will
produce colour shifts. In contrast to our everyday experience, the photograph would get
* ‘Recognize the right moment.’ Also rendered as: ‘Recognize thy opportunity.’ Pittacus (Πιττακος) of
Mytilene (c. 650–570 BCE ), was a Lesbian tyrant and lawmaker; he was also one of the ‘Seven Sages’ of
ancient Greece.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Can we take a photo graph of a p oint?
Motion Mountain – The Adventure of Physics
D uration Blur
the siz e a nd sha pe of elem entary pa rticles
119
Since particles are not point-like, they have a shape. How can we determine it? We determine the shape of an everyday object by touching it from all sides. This works with
plants, people or machines. It even works with molecules, such as water molecules. We
can put them (almost) at rest, for example in ice, and then scatter small particles off them.
Scattering is just a higher-energy version of touching. However, scattering cannot determine shapes of objects smaller than the wavelength of the probes used. To determine
the shape of an object as small as an electron, we need the highest energies available. But
we already know what happens when approaching Planck scales: the shape of a particle
becomes the shape of all the space surrounding it. In short, the shape of an electron
cannot be determined in this way.
Another way to determine the shape is to build a tight box around the system under
investigation and fill it with molten wax. We then let the wax cool and observe the hollow
part. However, near Planck energy, boxes do not exist. We are unable to determine the
shape in this way.
A third way to measure the shape of an object is to cut it into pieces and then study
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
What is the shape of an electron?
Motion Mountain – The Adventure of Physics
more blurred and incorrect at extremely short shutter times. Photography is impossible
not only at long but also at short shutter times.
The difficulty of taking photographs is independent of the wavelength used. The limits
move, but do not disappear. With a shutter time of ?, photons of energy lower than ℏ/?
cannot pass the shutter undisturbed.
In short, the blur decreases when shutter times usual in everyday life are shortened,
but increases when shutter times are shortened further towards Planck times. As a result,
there is no way to detect or confirm the existence of point objects by taking pictures.
Points in space, as well as instants of time, are imagined concepts: they do not belong in
a precise description of nature.
At Planck shutter times, only signals with Planck energy can pass through the shutter.
Since at these energies matter cannot be distinguished from radiation or from empty
space, all objects, light and vacuum look the same. It is impossible to say what nature
looks like at very short times.
But the situation is worse than this: a Planck shutter cannot exist at all, as it would
need to be as small as a Planck length. A camera using it could not be built, as lenses do
not work at this energy. Not even a camera obscura – without any lens – would work,
as diffraction effects would make image production impossible. In other words, the idea
that at short shutter times a photograph of nature shows a frozen image of everyday life,
like a stopped film, is completely wrong. In fact, a shutter does not exist even at medium
energy: shutters, like walls, stop existing at around 10 MeV. At a single instant of time,
nature is not frozen at all. Zeno criticized this idea in his discussions of motion, though
not as clearly as we can do now. At short times, nature is blurred. In particular, point
particles do not exist.
In summary, whatever the intrinsic shape of what we call a ‘point’ might be, we know
that, being always blurred, it is first of all a cloud. Whatever method is used to photograph an elementary particle, the picture is always extended. Therefore we need to study
its shape in more detail.
120
Ref. 47
Is the shape of an electron fixed?
Page 85
Only an object composed of localized constituents, such as a house or a molecule, can
have a fixed shape. The smaller the system, the more quantum fluctuations play a role.
No small entity of finite size – in particular, no elementary particle – can have a fixed
shape. In every thought experiment involving a finite shape, the shape itself fluctuates.
But we can say more.
The distinction between particles and environment rests on the idea that particles
have intrinsic properties. In fact, all intrinsic properties, such as spin, mass, charge, and
parity, are localized. But we have seen that no intrinsic property is measurable or definable at Planck scales. Thus it is impossible to distinguish particles from the environment.
In addition, at Planck energy particles have all the properties that the environment has.
In particular, particles are extended.
In short, we cannot prove by experiments that at Planck energy elementary particles
are finite in size in all directions. In fact, all experiments we can think of are compatible
with extended particles, with ‘infinite’ size. More precisely, a particle always reaches the
borders of the region of space-time under exploration. In simple words, we can also say
that particles have tails.
Not only are particles extended, but their shape cannot be determined by the methods
just explored. The only remaining possibility is that suggested by quantum theory: the
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Challenge 91 e
the pieces. As is well known, the term ‘atom’ just means ‘uncuttable’ or ‘indivisible’.
However, neither atoms nor indivisible particles can exist. Indeed, cutting is just a lowenergy version of a scattering process. And the process does not work at high energies.
Therefore, there is no way to prove that an object is indivisible at Planck scales. Our
everyday intuition leads us completely astray at Planck energy.
We could try to distinguish transverse and longitudinal shape, with respect to the direction of motion. However, for transverse shape we get the same issues as for scattering;
transverse shape diverges for high energy. And to determine longitudinal shape, we need
at least two infinitely high potential walls. We already know that this is impossible.
A further, indirect way of measuring shapes is to measure the moment of inertia.
A finite moment of inertia means a compact, finite shape. But when the measurement
energy is increased towards Planck scales, rotation, linear motion and exchange become
mixed up. We do not get meaningful results.
Yet another way to determine shapes is to measure the entropy of a collection of
particles we want to study. This allows us to determine the dimensionality and the number of internal degrees of freedom. But at high energies, a collection of electrons would
become a black hole. We will study this issue separately below, but again we find no new
information.
Are these arguments watertight? We assumed three dimensions at all scales, and that
the shape of the particle itself is fixed. Maybe these assumptions are not valid at Planck
scales? Let us check the alternatives. We have already shown that because of the fundamental measurement limits, the dimensionality of space-time cannot be determined at
Planck scales. Even if we could build perfect three-dimensional boxes, holes could remain in other dimensions. It does not take long to see that all the arguments against
compactness work even if space-time has additional dimensions.
Motion Mountain – The Adventure of Physics
Ref. 47
6 the sha pe of p oints
the sha pe of p oints in vacuum
Vol. IV, page 108
Ref. 28
121
Summary of the first argument for extension
Ref. 47
the shape of points in vacuum
Ref. 112
“
Thus, since there is an impossibility that [finite]
quantities are built from contacts and points, it
is necessary that there be indivisible material
elements and [finite] quantities.
Aristotle,*** Of Generation and Corruption.
”
We are used to the idea that empty space is made of spatial points. However, at Planck
scales, no measurement can give zero length, zero mass, zero area or zero volume. There
Ref. 111
* Examples are the neutron, positronium, or the atoms. Note that the argument does not change when the
elementary particle itself is unstable, like the muon. The possibility that all components are heavier than
the composite, which would avoid this argument, does not seem to lead to satisfying physical properties:
for example, it leads to intrinsically unstable composites.
** Thus at Planck scales there is no quantum Zeno effect.
*** Aristotle (b. 384/3 Stageira, d. 322 bce Chalkis), Greek philosopher and scientist.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Challenge 92 e
Point particles do not exist at Planck scales. At Planck scales, all thought experiments
with partciles suggest that matter and radiation are made of extended and fluctuating
constituents of infinite size.
For extended constituents, the requirement of a non-local description is satisfied. The
argument forbidding composition of elementary particles is circumvented, as extended
constituents have no mass. Thus the concept of Compton wavelength cannot be defined
or applied to extended constituents, and elementary particles can have constituents if
these constituents are extended and massless. However, if the constituents are infinitely
extended, how can compact, point-like particles be formed from them? We will look at
a few options shortly.
Motion Mountain – The Adventure of Physics
shape of a particle fluctuates.
We reach the same conclusions for radiation particles. The box argument shows that
radiation particles are also extended and fluctuating.
Incidentally, we have also settled an important question about elementary particles.
We have already seen that any particle that is smaller than its own Compton wavelength
must be elementary. If it were composite, there would be a lighter component inside
it; this lighter particle would have a larger Compton wavelength than the composite
particle. This is impossible, since the size of a composite particle must be larger than
the Compton wavelength of its components.*
However, an elementary particle can have constituents, provided that they are not
compact. The difficulties of compact constituents were described by Andrei Sakharov
in the 1960s. If the constituents are extended, the previous argument does not apply, as
extended constituents have no localized mass. As a result, if a flying arrow – Zeno’s
famous example – is made of extended constituents, it cannot be said to be at a given
position at a given time. Shortening the observation time towards the Planck time makes
an arrow disappear in the cloud that makes up space-time.**
122
* Imagining the vacuum as a collection of compact constituents, such as spheres, with Planck size in all
directions would avoid the Banach–Tarski paradox, but would not allow us to deduce the number of dimensions of space and time. It would also contradict all the other results of this section. Therefore we do
not explore it further.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 113
Challenge 93 s
is no way to state that something in nature is a point without contradicting experimental
results.
Furthermore, the idea of a point is an extrapolation of what is found in small empty
boxes getting smaller and smaller. But we have just seen that at high energies small boxes
cannot be said to be empty. In fact, boxes do not exist at all, as they can never have
impenetrable walls at high energies.
Also, the idea of a point as a limiting subdivision of empty space is untenable. At
small distances, space cannot be subdivided, as division requires some sort of dividing
wall, which is impossible.
Even the idea of repeatedly putting a point between two others cannot be applied. At
high energy, it is impossible to say whether a point is exactly on the line connecting the
outer two points; and near Planck energy, there is no way to find a point between them
at all. In fact, the term ‘in between’ makes no sense at Planck scales.
We thus find that space points do not exist, just as point particles do not exist. But
there are other reasons why space cannot be made of points. In order to form space.
points need to be kept apart somehow. Indeed, mathematicians have a strong argument
for why physical space cannot be made of mathematical points: the properties of mathematical spaces described by the Banach–Tarski paradox are quite different from those
of the physical vacuum. The Banach–Tarski paradox states that a sphere made of mathematical points can be cut into five pieces which can be reassembled into two spheres
each of the same volume as the original sphere. Mathematically, there are sets of points
for which the concept of volume makes no sense. Physically speaking, we conclude that
the concept of volume does not exist for continuous space; it is only definable if an intrinsic length exists. And in nature, an intrinsic length exists for matter and for vacuum:
the Planck length. And any concept with an intrinsic length must be described by one
or several extended constituents.* In summary, in order to build up space, we need extended constituents.
Also the number of space dimensions is problematic. Mathematically, it is impossible
to define the dimension of a set of points on the basis of the set structure alone. Any
compact one-dimensional set has as many points as any compact three-dimensional set
– indeed, as any compact set of any dimensionality greater than zero. To build up the
physical three-dimensional vacuum, we need constituents that organize their neighbourhood. The fundamental constituents must possess some sort of ability to form bonds,
which will construct or fill precisely three dimensions. Bonds require extended constituents. A collection of tangled constituents extending to the maximum scale of the region
under consideration would work perfectly. Of course, the precise shape of the fundamental constituents is not yet known. In any case, we again find that any constituents of
physical three-dimensional space must be extended.
In summary, we need extension to define dimensionality and to define volume. This
result is not surprising. We deduced above that the constituents of particles are extended.
Since vacuum is not distinguishable from matter, we would expect the constituents of
Motion Mountain – The Adventure of Physics
Ref. 47
6 the sha pe of p oints
the sha pe of p oints in vacuum
123
vacuum to be extended as well. Stated simply, if elementary particles are not point-like,
then points in the vacuum cannot be either.
Measuring the void
What is the maximum number of particles that fit inside a piece
of vacuum?
Summary of the second argument for extension
Vol. I, page 338
Ref. 114
Planck scales imply that space is made of fluctuating extended constituents of huge size.
Like particles, also space and vacuum are not made of points, but of a web. Vacuum
requires a statistical description.
More than two thousand years ago, the Greeks argued that matter must be made of
particles because salt can be dissolved in water and because fish can swim through water.
Now that we know more about Planck scales, we have to reconsider this argument. Like
fish swimming through water, particles can move through vacuum; but since vacuum
has no bounds and cannot be distinguished from matter, vacuum cannot be made of
localised particles. However, another possibility allows for motion of particles through
a vacuum: both vacuum and particles might be made of a web of extended constituents.
Let us study this possibility in more detail.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Another approach to counting the number of points in a volume is to fill a piece of vacuum with point particles.
The maximum mass that fits into a piece of vacuum is a black hole. But in this case
too, the maximum mass depends only on the surface of the given region of vacuum. The
maximum mass increases less rapidly than the volume. In other words, the number of
physical points inside a region of space is only proportional to the surface area of the
region. We are forced to conclude that vacuum must be made of extended constituents
crossing the whole region, independently of its shape.
Motion Mountain – The Adventure of Physics
To check whether the constituents of the vacuum are extended, let us perform a few
additional thought experiments. First, let us measure the size of a point in space. The
clearest definition of size is in terms of the cross section. How can we determine the cross
section of a point? We can determine the cross section of a piece of vacuum and then
determine the number of points inside it. However, at Planck energy, we get a simple
result: the cross section of a volume of empty space is independent of depth. At Planck
energy, vacuum has a surface, but no depth. In other words, at Planck energy we can only
state that a Planck layer covers the surface of a region. We cannot say anything about its
interior. One way to picture this result is to say that what we call ‘space points’ are in
fact long tubes.
Another way to determine the size of a point is to count the points found in a given
volume of space-time. One approach is to count the possible positions of a point particle
in a volume. However, at Planck energy point particles are extended and indistinguishable from vacuum. At Planck energy, the number of points is given by the surface area
of the volume divided by the Planck area. Again, the surface dependence suggests that
particles and the constituents of space are long tubes.
124
6 the sha pe of p oints
the l arge, the small and their connection
“
”
Is small l arge?
Ref. 116
“
[Zeno of Elea maintained:] If the existing are
many, it is necessary that they are at the same
time small and large, so small to have no size,
and so large to be without limits.
Simplicius***
”
* William Shakespeare (1564 Stratford upon Avon–1616 Stratford upon Avon) wrote theatre plays that are
treasures of world literature.
** There is also an S-duality, which connects large and small coupling constants, and a U-duality, which is
the combination of S- and T-duality.
*** Simplicius of Cilicia (c. 499 – 560), neoplatonist philosopher.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
If two observables cannot be distinguished, there is a symmetry transformation connecting them. For example, by a change of observation frame, an electric field may (partially)
change into a magnetic one. A symmetry transformation means that we can change the
viewpoint (i.e., the frame of observation) in such a way that the same observation is described by one quantity from one viewpoint and by the corresponding quantity from the
other viewpoint.
When measuring a length at Planck scales it is impossible to say whether we are
measuring the length of a piece of vacuum, the Compton wavelength of a body, or the
Schwarzschild diameter of a body. For example, the maximum size for an elementary
object is its Compton wavelength. The minimum size for an elementary object is its
Schwarzschild radius. The actual size of an elementary object is somewhere in between.
If we want to measure the size precisely, we have to go to Planck energy; but then all these
quantities are the same. In other words, at Planck scales, there is a symmetry transformation between Compton wavelength and Schwarzschild radius. In short, at Planck scales
there is a symmetry between mass and inverse mass.
As a further consequence, at Planck scales there is a symmetry between size and inverse size. Matter–vacuum indistinguishability means that there is a symmetry between
length and inverse length at Planck energy. This symmetry is called space-time duality
or T-duality in the research literature of superstrings.** Space-time duality is a symmetry between situations at scale ? ?Pl and at scale ??Pl /?, or, in other words, between
? and (??Pl )2 /?, where the number ? is usually conjectured to have a value somewhere
between 1 and 1000.
Duality is a genuine non-perturbative effect. It does not exist at low energy, since
duality automatically also relates energies ? and ?2Pl /? = ℏ?3 /??, i.e., it relates energies
below and above Planck scale. Duality is not evident in everyday life. It is a quantum
symmetry, as it includes Planck’s constant in its definition. It is also a general-relativistic
effect, as it includes the gravitational constant and the speed of light. Let us study duality
in more detail.
Motion Mountain – The Adventure of Physics
Ref. 115
I could be bounded in a nutshell and count
myself a king of infinite space, were it not that I
have bad dreams.
William Shakespeare,* Hamlet.
the l a rge, the sm a ll a nd their connection
125
Vol. III, page 92
Ref. 117
Ref. 115
So far, we have shown that at Planck energy, time and length cannot be distinguished,
and that vacuum and matter cannot be distinguished. Duality shows that mass and inverse mass cannot be distinguished. As a consequence, we deduce that length, time, and
mass cannot be distinguished from each other at all energies and scales! And since every
observable is a combination of length, mass and time, space-time duality means that there
is a symmetry between all observables. We call it the total symmetry.*
Total symmetry implies that there are many specific types of duality, one for each
pair of quantities under investigation. Indeed, the number of duality types discovered is
increasing every year. It includes, among others, the famous electric–magnetic duality we
first encountered in electrodynamics, coupling constant duality, surface–volume duality,
space-time duality, and many more. All this confirms that there is an enormous amount
of symmetry at Planck scales. In fact, similar symmetries have been known right from
the beginning of research in quantum gravity.
Most importantly, total symmetry implies that gravity can be seen as equivalent to all
other forces. Space-time duality thus shows that unification is possible. Physicists have
* A symmetry between size and Schwarzschild radius, i.e., a symmetry between length and mass, leads
to general relativity. Additionally, at Planck energy there is a symmetry between size and Compton
wavelength. In other words, there is a symmetry between length and inverse mass. This implies a symmetry between coordinates and wave functions, i.e., a symmetry between states and observables. It leads to
quantum theory.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Unification and total symmetry
Motion Mountain – The Adventure of Physics
To explore the consequences of duality, we can compare it to rotational symmetry in
everyday life. Every object in daily life is symmetrical under a full rotation of 2π. For the
rotation of an observer, angles make sense only as long as they are smaller than 2π. If a
rotating observer were to insist on distinguishing angles of 0, 2π, 4π etc., he would get a
new copy of the universe at each full turn.
2
Similarly, in nature, scales ? and ?Pl
/? cannot be distinguished. Lengths make no
sense when they are smaller than ?Pl . If, however, we insist on using even smaller values
and on distinguishing them from large ones, we get a new copy of the universe at those
small scales. Such an insistence is part of the standard continuum description of motion, where it is assumed that space and time are described by the real numbers, which
are defined over arbitrarily small intervals. Whenever the (approximate) continuum de2
scription with infinite extension is used, the ? ↔ ?Pl
/? symmetry pops up.
Duality implies that diffeomorphism invariance is only valid at medium scales, not at
extremal ones. At extremal scales, quantum theory has to be taken into account in the
proper manner. We do not yet know how to do this.
Space-time duality means that introducing lengths smaller than the Planck length (as
when one defines space points, which have size zero) means at the same time introducing
things with very large (‘infinite’) value. Space-time duality means that for every small
enough sphere the inside equals the outside.
Duality means that if a system has a small dimension, it also has a large one, and vice
versa. There are thus no small objects in nature. So space-time duality is consistent with
the idea that the basic constituents are extended.
126
Summary of the third argument for extension
Challenge 95 e
Unification implies thinking in terms of duality and the concepts that follow from it.
The large and the small are connected. Duality points to one single type of extended
constituents that defines all physical observables.
We still need to understand exactly what happens to duality when we restrict ourselves
to low energies, as we do in everyday life. We explore this now.
* Renormalization energy does connect different energies, but not in the correct way; in particular, it does
not include duality.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
always dreamt about unification. Duality tells us that this dream can indeed be realized.
It may seem that total symmetry completely contradicts what was said in the previous
section, where we argued that all symmetries are lost at Planck scales. Which result is
correct? Obviously, both of them are.
At Planck scales, all low-energy symmetries are indeed lost. In fact, all symmetries
that imply a fixed energy are lost. However, duality and its generalizations combine both
small and large dimensions, or large and small energies. Most of the standard symmetries of physics, such as gauge, permutation and space-time symmetries, are valid at each
fixed energy separately. But nature is not made this way. The precise description of nature
requires us to take into consideration large and small energies at the same time. In everyday life, we do not do that. The physics of everyday life is an approximation to nature
valid at low and fixed energies. For most of the twentieth century, physicists tried to
reach higher and higher energies. We believed that precision increases with increasing
energy. But when we combine quantum theory and gravity we are forced to change this
approach. To achieve high precision, we must take high and low energy into account at
the same time.*
The great differences between the phenomena that occur at low and high energies are
the main reason why unification is so difficult. We are used to dividing nature along
a scale of energies: high-energy physics, atomic physics, chemistry, biology, and so on.
But we are not allowed to think in this way any more. We have to take all energies into
account at the same time. That is not easy, but we do not have to despair. Important
conceptual progress was made in the last decade of the twentieth century. In particular,
we now know that we need only one constituent for all things that can be measured.
Since there is only one constituent, total symmetry is automatically satisfied. And
since there is only one constituent, there are many ways to study it. We can start from any
(low-energy) concept in physics and explore how it looks and behaves when we approach
Planck scales. In the present section, we are looking at the concept of ‘point’. Obviously,
the conclusions must be the same whatever concept we start with, be it electric field, spin,
or any other. Such studies thus provide a check for the results in this section.
Motion Mountain – The Adventure of Physics
Challenge 94 d
6 the sha pe of p oints
d oes nature hav e pa rts?
127
does nature have parts?
“
Ref. 118
”
Pluralitas non est ponenda sine necessitate.*
William of Occam
Another argument, independent of those given so far, points towards a model of nature
based on extended constituents. We know that any concept for which we can distinguish parts is described by a set. We usually describe nature as a set of objects, positions,
instants and so on. The most famous set-theoretic description of nature is the oldest
known, given by Democritus:
The world is made of indivisible particles and void.
Vol. III, page 327
The conclusion does not come as a surprise. We have already encountered another reason
to doubt that nature is a set. Whatever definition we use for the term ‘particle’, Democritus cannot be correct for a purely logical reason. The description he provided is not
complete. Every description of nature that defines nature as a set of parts fails to explain
the number of these parts. In particular, the number of particles and the number of dimensions of space-time must be specified if we describe nature as made from particles
and vacuum. For example, we saw that it is rather dangerous to make fun of the famous
statement by Arthur Eddington
* ‘Multitude should not be introduced without necessity.’ This famous principle is commonly called Occam’s razor. William of Ockham (b. 1285/1295 Ockham, d. 1349/50 Munich), or Occam in the common
Latin spelling, was one of the great thinkers of his time. In his famous statement he expresses that only
those concepts which are strictly necessary should be introduced to explain observations. It can be seen as
the requirement to abandon beliefs when talking about nature. But at this stage of our mountain ascent it
has an even more direct interpretation: the existence of any multitude in nature is questionable.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
⊳ Nature cannot be a set.
Motion Mountain – The Adventure of Physics
This description was extremely successful in the past: there are no discrepancies with
observations. However, after 2500 years, the conceptual difficulties of this approach are
obvious.
We know that Democritus was wrong, first of all, because vacuum and matter cannot be distinguished at Planck scales. Thus the word ‘and’ in his sentence is already a
mistake. Secondly, because of the existence of minimal scales, the void cannot be made
of ‘points’, as we usually assume. Thirdly, the description fails because particles are not
compact objects. Finally, total symmetry implies that we cannot distinguish parts in
nature. Nothing can be distinguished from anything else with complete precision, and
thus the particles or points in space that make up the naive model of the world cannot
exist.
In summary, quantum theory and general relativity together show that in nature, all
partitions and all differences are only approximate. Nothing can really be distinguished
from anything else with complete precision. In other words, there is no way to define a
‘part’ of nature, whether for matter, space, time, or radiation.
128
Ref. 119
Ref. 120
In fact, practically all physicists share this belief, although they usually either pretend to
favour some other number, or worse, keep the number unspecified.
In modern physics, many specialized sets are used to describe nature. We have used
vector spaces, linear spaces, topological spaces and Hilbert spaces. But so far, we consistently refrained, like all physicists, from asking about the origin of their sizes (mathematically speaking, of their dimensionality or cardinality). In fact, it is just as unsatisfying
to say that the universe contains some specific number of atoms as it is to say that spacetime is made of point-like events arranged in 3+1 dimensions. Both are statements about
set sizes, in the widest sense. In a complete, unified description of nature the number of
smallest particles and the number of space-time points must not be fixed beforehand,
but must result from the description.
Any part of nature is by definition smaller than the whole of nature, and different from
other parts. As a result, no description of nature by a set can possibly yield the number
of particles or the dimensionality of space-time. As long as we insist on using spacetime or Hilbert spaces for the description of nature, we cannot understand the number
of dimensions or the number of particles.
That is not too bad, as we know already that nature is not made of parts. We know
that parts are only approximate concepts. In short, if nature were made of parts, it could
not be a unity, or a ‘one.’ On the other hand, if nature is a unity, it cannot have parts.*
Nature cannot be separable exactly. It cannot be made of particles.
To sum up, nature cannot be a set. Sets are lists of distinguishable elements. When
general relativity and quantum theory are unified, nature shows no elements: nature
stops being a set at Planck scales. This result clarifies a discussion we started earlier in
relation to classical physics. There we discovered that matter objects were defined using
space and time, and that space and time were defined using objects. Along with the
results of quantum theory, this implies that in modern physics particles are defined in
terms of the vacuum and the vacuum in terms of particles. Circularity is not a good
idea, but we can live with it – at low energy. But at Planck energy, vacuum and particles
are indistinguishable from each other. Particles and vacuum – thus everything – are
the same. We have to abandon the circular definition. This is a satisfactory outcome;
however, it also implies that nature is not a set.
Also space-time duality implies that space is not a set. Space-time duality implies
that events cannot be distinguished from each other, and thus do not form elements of
some space. Phil Gibbs has given the name event symmetry to this property of nature.
This thought-provoking term, although still containing the term ‘event’, emphasizes the
impossibility to use a set to describe space-time.
In short,
* As a curiosity, practically the same discussion can already be found in Plato’s Parmenides, written in the
fourth century bce. There, Plato musically ponders different arguments on whether nature is or can be a
unity or a multiplicity, i.e., a set. It seems that the text is based on the real visit to Athens by Parmenides
and Zeno. (Their home city, Elea, was near Naples.) Plato does not reach a conclusion. Modern physics,
however, does.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 121
I believe there are 15,747,724,136,275,002,577,605,653,961,181,555,468,044,
717,914,527,116,709,366,231,425,076,185,631,031,296 protons in the universe
and the same number of electrons.
Motion Mountain – The Adventure of Physics
Page 101
6 the sha pe of p oints
d oes nature hav e pa rts?
129
⊳ Nature cannot be made of vacuum and particles.
This is a bizarre result. Atomists, from Democritus to Galileo, have been persecuted
throughout history. Were their battles all in vain? Let us continue to clarify our thoughts.
D oes the universe contain anything?
An amoeba
“
”
We have found that parts are approximate concepts. The parts of nature are not strictly
smaller than nature itself. As a result, any ‘part’ must be extended. Let us try to extract
some more information about the constituents of nature.
In any unified theory, all the concepts that appear must be only approximately parts
of the whole. Thus we need an entity Ω, describing nature, which is not a set but which
can be approximated by one. This is strange. We are all convinced very early in our lives
that we are a part of nature. Our senses provide us with this information. We are not
used to thinking otherwise. But now we have to.
Let us straight away eliminate a few options for Ω. One concept without parts is the
empty set. Perhaps we need to construct a description of nature from the empty set? We
could be inspired by the usual construction of the natural numbers from the empty set.
However, the empty set makes only sense as the opposite of some full set. So the empty
set is not a candidate for Ω.
Another possible way to define approximate parts is to construct them from multiple
copies of Ω. But in this way we would introduce a new set through the back door. Furthermore, new concepts defined in this way would not be approximate.
We need to be more imaginative. How can we describe a whole which has no parts,
but which has parts approximately? Let us recapitulate. The world must be described by
a single entity, sharing all properties of the world, but which can be approximated as a set
of parts. For example, the approximation should yield a set of space points and a set of
particles. But also, whenever we look at any ‘part’ of nature, without any approximation,
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Vol. III, page 290
A theory of everything describing nothing is
not better than a theory of nothing describing
everything.
Anonymous
Motion Mountain – The Adventure of Physics
To state that the universe contains something implies that we are able to distinguish the
universe from its contents. However, we now know that precise distinctions are impossible. If nature is not made of parts, it is wrong to say that the universe contains
something.
Let us go further. We need a description of nature that allows us to state that at Planck
energy nothing can be distinguished from anything else. For example, it must be impossible to distinguish particles from each other or from the vacuum. There is only one
solution: everything – or at least, what we call ‘everything’ in everyday life – must be
made of the same single constituent. All particles are made of one ‘piece’. Every point
in space, every event, every particle and every instant of time must be made of the same
single constituent.
130
6 the sha pe of p oints
Summary of the fourth argument for extension
Challenge 96 r
* This is the simplest model; but is it the only way to describe nature?
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 146
The lack of particles and of sets in nature leads to describing nature by a single constituent. Nature is thus modelled by an entity which is one single ‘object’ (to eliminate distinguishability), which is extended (to eliminate localizability) and which is fluctuating (to
ensure approximate continuity). Nature is a far-reaching, fluctuating fold. Nature is similar to an amoeba. The tangled branches of the amoeba allow a definition of length via
counting of the folds. In this way, discreteness of space, time, and particles could also be
realized; the quantization of space-time, matter and radiation thus follows. Any flexible
and deformable entity is also a perfect candidate for the realization of diffeomorphism
invariance, as required by general relativity.
A simple candidate for the extended fold is the image of a fluctuating, flexible tube
of Planck diameter. Counting tubes implies determining distances or areas. The minimum possible count (one) gives the minimum distance, from which quantum theory
is derived. In fact, at this point we can use as a model any flexible object with a small
dimension, such as a tube, a thin sheet, a ball chain or a woven collection of rings. We
will explore these options below.
Motion Mountain – The Adventure of Physics
we should not be able to distinguish it from the whole world. Composite objects are
not always larger than their constituents. On the other hand, composed objects must
usually appear to be larger than their constituents. For example, space ‘points’ or ‘point’
particles are tiny, even though they are only approximations. Which concept without
boundaries can be at their origin? Using usual concepts, the world is everywhere at
the same time; if nature is to be described by a single constituent, this entity must be
extended.
The entity has to be a single one, but it must seem to be multiple: it has to be multiple
approximately, as nature shows multiple aspects. The entity must be something folded.
It must be possible to count the folds, but only approximately. (An analogy is the question of how many grooves there are on an LP or a CD: depending on the point of view,
local or global, one gets different answers.) Counting folds would correspond to a length
measurement.
The simplest model would be a single entity which is extended and fluctuating,
reaches spatial infinity, allows approximate localization, and thus allows approximate
definition of parts and points.* In more vivid imagery, nature could be described by
some deformable, folded and tangled entity: a giant, knotted amoeba. An amoeba slides
between the fingers whenever we try to grab a part of it. A perfect amoeba flows around
any knife trying to cut it. The only way to hold it would be to grab it in its entirety. However, for someone himself made of amoeba strands, this is impossible. He can only grab
it approximately, by catching part of it and approximately blocking it, for example using
a small hole, so that the escape takes a long time.
the entropy of b l ack holes
131
the entropy of bl ack holes
We are still collecting arguments to determining the shape of fundamental constituents.
Another approach is to study situations where particles appear in large numbers. Systems
composed of many particles behave differently depending on whether the particles are
point-like or extended. In particular, their entropy is different. Studying large-number
entropy thus allows us to determine the shape of components. The most revealing situations are those in which large numbers of particles are crammed in a small volume.
Therefore we are led to study the entropy of black holes. Indeed, black holes tell us a lot
about the fundamental constituents of nature.
A black hole is a body whose gravity is so strong that even light cannot escape. It is
easily deduced from general relativity that any body whose mass ? fits inside the socalled Schwarzschild radius
?S = 2??/?2
(115)
?=
Ref. 35
Ref. 123
Ref. 125
Ref. 124
?=?
4π??2
ℏ?
(116)
where ? is the Boltzmann constant and ? = 4π?S2 is the surface of the black hole horizon.
This important result has been derived in many different ways. The various derivations
confirm that space-time and matter are equivalent: they show that the entropy value can
be interpreted as an entropy either of matter or of space-time. In the present context, the
two main points of interest are that the entropy is finite, and that it is proportional to the
area of the black hole horizon.
In view of the existence of minimum lengths and times, the finiteness of the entropy
is not surprising: it confirms the idea that matter is made of a finite number of discrete
constituents per given volume (or area). It also shows that these constituents behave statistically: they fluctuate. In fact, quantum gravity implies a finite entropy for any object,
not only for black holes. Jacob Bekenstein has shown that the entropy of an object is
always smaller than the entropy of a (certain type of) black hole of the same mass.
The entropy of a black hole is proportional to its horizon area. Why? This question
has been the topic of a stream of publications.* A simple way to understand the entropy–
surface proportionality is to look for other systems in nature whose entropy is proportional to system surface instead of system volume. In general, the entropy of a collection
of flexible one-dimensional objects, such as polymer chains, shares this property. Indeed,
the entropy of a polymer chain made of ? monomers, each of length ?, whose ends are
* The result can be derived from quantum statistics alone. However, this derivation does not yield the
proportionality coefficient.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 122
?
??3
?
=
? or
2
4?Pl
4ℏ?
Motion Mountain – The Adventure of Physics
Ref. 57, Ref. 58
is a black hole. A black hole can be formed when a whole star collapses under its own
weight. Such a black hole is a macroscopic body, with a large number of constituents.
Therefore it has an entropy. The entropy ? of a macroscopic black hole was determined
by Bekenstein and Hawking, and is given by
132
Ref. 126
6 the sha pe of p oints
kept a distance ? apart, is given by
?(?) = ?
for ?? ≫ √?? ≫ ? .
(117)
Summary of the fifth argument for extension
Page 287
Black hole entropy is best understood as resulting from extended constituents that tangle
and fluctuate. And black hole entropy confirms that vacuum and particles are made of
common constituents.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
This formula can be derived in a few lines from the properties of a random walk on a lattice, using only two assumptions: the chains are extended; and they have a characteristic
internal length ? given by the smallest straight segment. Expression (117) is only valid
if the polymers are effectively infinite: in other words, if the length ?? of the chain and
the elongation ?√? , are much larger than the radius ? of the region of interest. If the
chain length is comparable to or smaller than the region of interest, we get the usual extensive entropy, satisfying ? ∼ ?3 . Thus only flexible extended constituents yield an ? ∼ ?2
dependence.
However, there is a difficulty. From the expression for the entropy of a black hole we
deduce that the elongation ?√? is given by ?√? ≈ ?Pl ; thus it is much smaller than
the radius of a general macroscopic black hole, which can have a diameter of several
kilometres. On the other hand, the formula for long constituents is only valid when the
chains are longer than the distance ? between the end points.
This difficulty disappears when we remember that space near a black hole is strongly
curved. All lengths have to be measured in the same coordinate system. It is well known
that for an outside observer, any object of finite size falling into a black hole seems to
cover the complete horizon for long times (whereas for an observer attached to the object
it falls into the hole in its original size). In short, an extended constituent can have a
proper length of Planck size but still, when seen by an outside observer, be as long as the
horizon of the black hole.
We thus find that black holes are made of extended constituents. Another viewpoint
can confirm this result. Entropy is (proportional to) the number of yes-or-no questions
needed to know the exact state of the system. But if a system is defined by its surface, as
a black hole is, its components must be extended.
Finally, imagining black holes as made of extended constituents is also consistent with
the so-called no-hair theorem: black holes’ properties do not depend on what falls into
them – as long as all matter and radiation particles are made of the same extended components. The final state of a black hole only depends on the number of extended constituents.
Motion Mountain – The Adventure of Physics
Ref. 109
3?2
2??2
excha nging space p oints or pa rticles at pl a nck sca les
133
exchanging space points or particles at pl anck
scales
Let us now focus on the exchange behaviour of fundamental constituents in nature. We
saw above that ‘points’ in space have to be abandoned in favour of continuous, fluctuating constituents common to space, time and matter. Is such a constituent a boson or a
fermion? If we exchange two points of empty space, in everyday life, nothing happens.
Indeed, at the basis of quantum field theory is the relation
[?, ?] = ?? − ?? = 0
(118)
Ref. 47
Ref. 127
At Planck energy this cannot be correct. Quantum gravity effects modify the right-hand
side: they add an energy-dependent term that is negligible at experimentally accessible
energies but becomes important at Planck energy. We know from our experience with
* The same reasoning applies to the so-called fermionic or Grassmann coordinates used in supersymmetry.
They cannot exist at Planck energy.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
This means that ‘points’ are neither bosons nor fermions.* ‘Points’ have more complex
exchange properties. In fact, the term on the right-hand side will be energy-dependent,
to an increasing extent as we approach Planck scales. In particular, as we have seen,
gravity implies that a double exchange does not lead back to the original situation at
Planck scales.
Constituents obeying this or similar relations have been studied in mathematics for
many decades: they are called braids. Thus space is not made of points at Planck scales,
but of braids or their generalizations, namely tangles. We find again that quantum theory
and general relativity taken together imply that the vacuum must be made of extended
constituents.
We now turn to particles. All particles in nature behave in a similar way: we know that
at low, everyday energies, particles of the same type are identical. Experiments sensitive
to quantum effects show that there is no way to distinguish them: any system of several identical particles has permutation symmetry. On the other hand, we know that at
Planck energy all low-energy symmetries disappear. We also know that at Planck energy
permutation cannot be carried out, as it implies exchanging positions of two particles.
At Planck energy, nothing can be distinguished from vacuum; thus no two entities can
be shown to have identical properties. Indeed, no two particles can be shown to be indistinguishable, as they cannot even be shown to be separate.
What happens when we slowly approach Planck energy? At everyday energies, permutation symmetry is defined by commutation or anticommutation relations between
particle creation operators
?† ?† ± ?† ?† = 0 .
(120)
Motion Mountain – The Adventure of Physics
between any two points with coordinates ? and ?, making them bosons. But at Planck
scales, because of the existence of minimal distances and areas, this relation must at least
be changed to
2
[?, ?] = ?Pl
+ ... .
(119)
134
Ref. 47
Ref. 127
6 the sha pe of p oints
Planck scales that, in contrast to everyday life, exchanging particles twice cannot lead
back to the original situation. A double exchange at Planck energy cannot have no effect, because at Planck energy such statements are impossible. The simplest extension of
the commutation relation (120) for which the right-hand side does not vanish is braid
symmetry. This again suggests that particles are made of extended constituents.
Summary of the sixth argument for extension
Extrapolating both point and particle indistinguishability to Planck scales suggests extended, braided or tangled constituents.
the meaning of spin
* With a flat (or other) background, it is possible to define a local energy–momentum tensor. Thus particles
can be defined. Without a background, this is not possible, and only global quantities can be defined.
Without a background, even particles cannot be defined. Therefore, in this section we assume that we have
a slowly varying space-time background.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 172
Motion Mountain – The Adventure of Physics
Vol. IV, page 112
As last argument we will now show that the extension of particles makes sense even
at everyday energy. Any particle is a part of the universe. A part is something that is
different from anything else. Being ‘different’ means that exchange has some effect. Distinction means detection of exchange. In other words, any part of the universe is also
described by its exchange behaviour.
In nature, particle exchange is composed of rotations. In other words, parts of nature
are described by their rotation behaviour. This is why, for microscopic particles, exchange
behaviour is specified by spin. Spin distinguishes particles from vacuum.*
We note that volume does not distinguish vacuum from particles; neither does rest
mass or charge: nature provides particles without measurable volume, rest mass or
charge, such as photons. The only observables that distinguish particles from vacuum
are spin and momentum. In fact, linear momentum is only a limiting case of angular
momentum. We thus find again that rotation behaviour is the basic aspect distinguishing particles from vacuum.
If spin is the central property that distinguishes particles from vacuum, finding a
model for spin is of central importance. But we do not have to search for long. A model
for spin 1/2 is part of physics folklore since almost a century. Any belt provides an example, as we discussed in detail when exploring permutation symmetry. Any localized
structure with any number of tails attached to it – tails that reach the border of the region
of space under consideration – has the same properties as a spin 1/2 particle. The only
condition is that the tails themselves are unobservable. It is a famous exercise to show that
such a model, like one of those shown in Figure 9, is indeed invariant under 4π rotations
but not under 2π rotations, and that two such particles get entangled when exchanged,
but get untangled when exchanged twice. Such a tail model has all the properties of spin
1/2 particles, independently of the precise structure of the central region, which is not
important at this point. The tail model even has the same problems with highly curved
space as real spin 1/2 particles have. We will explore the issues in more detail shortly.
curiosities a nd fun cha llenges a b ou t extension
position
of spin 1/2
particle
135
flexible bands
in unspecified
number
reaching the
border
of space
particle.
Ref. 128
Summary of the seventh argument for extension
Exploring the properties of particle spin suggests the existence of extended constituents
in elementary fermions. We note that gravitation is not used explicitly in the argument.
It is used implicitly, however, in the definition of the locally flat space-time and of the
asymptotic region to where the tails are reaching.
curiosities and fun challenges abou t extension
“
No problem is so big or complicated that it
can’t be run away from.
Charles Schulz
”
In case that this section has not provided enough food for thought, here is some more.
∗∗
Challenge 98 s
Quantum theory implies that even if tight walls exist, the lid of a box made of them could
never be tightly shut. Can you provide the argument?
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Challenge 97 e
The tail model thus confirms that rotation is partial exchange. More interestingly, it
shows that rotation implies connection with the border of space. Extended particles can
be rotating. Particles can have spin 1/2 provided that they have tails going to the border
of space. If the tails do not reach the border, the model does not work. Spin 1/2 thus
even seems to require extension.
It is not hard to extend this idea to include spin 1 particles. In short, both bosons and
fermions can be modelled with extended constituents.
Motion Mountain – The Adventure of Physics
F I G U R E 9 Possible models for a spin 1/2
136
6 the sha pe of p oints
∗∗
Challenge 99 e
Can you provide an argument against the idea of extended constituents in nature? If so,
publish it!
∗∗
Challenge 100 s
Does duality imply that the cosmic background fluctuations (at the origin of galaxies and
clusters) are the same as vacuum fluctuations?
∗∗
Challenge 101 s
Does duality imply that a system with two small masses colliding is equivalent to a system
with two large masses gravitating?
∗∗
It seems that in all arguments so far we have assumed that time is continuous, even
though we know it is not. Does this change the conclusions?
∗∗
Duality also implies that in some sense large and small masses are equivalent. A mass
2
? in a radius ? is equivalent to a mass ?2Pl /? in a radius ?Pl
/?. In other words, duality
2
transforms mass density from ? to ?Pl /?. Vacuum and maximum density are equivalent!
Vacuum is thus dual to black holes.
∗∗
Total symmetry and space-time duality together imply that there is a symmetry between
all values an observable can take. Do you agree?
∗∗
Challenge 104 s
Any description is a mapping from nature to mathematics, i.e., from observed differences
(and relations) to thought differences (and relations). How can we do this accurately, if
differences are only approximate? Is this the end of physics?
∗∗
Challenge 105 d
Duality implies that the notion of initial conditions for the big bang makes no sense,
as we saw earlier by considering the minimal distance. As duality implies a symmetry
between large and small energies, the big bang itself becomes a vague concept. What else
do extended constituents imply for the big bang?
∗∗
Challenge 106 d
Can you show that going to high energies or selecting a Planck-size region of space-time
is equivalent to visiting the big bang?
∗∗
Ref. 129
Challenge 107 s
In 2002, Andrea Gregori made a startling prediction for any model using extended constituents that reach the border of the universe: if particles are extended in this way, their
mass should depend on the size of the universe. Thus particle masses should change with
time, especially around the big bang. Is this conclusion unavoidable?
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Challenge 103 s
Motion Mountain – The Adventure of Physics
Challenge 102 d
checks of extension
137
∗∗
Challenge 108 s
What is wrong with the following argument? We need lines to determine areas, and we
need areas to determine lines. This implies that at Planck scales, we cannot distinguish
areas from lengths at Planck scales.
∗∗
Ref. 129, Ref. 130
Challenge 109 s
We need a description for the expansion of the universe in terms of extended constituents. Various approaches are being explored. Can you speculate about the solution?
Gender preferences in physics
Vol. I, page 338
The idea that nature is described by extended constituents is taken for granted in all
current research approaches to unification. How can we be sure that extension is correct?
The arguments presented so far provide several possible checks. We start with some
options for theoretical falsification.
Challenge 110 e
Page 111
— Any explanation of black hole entropy without extended constituents would invalidate the need for extended constituents.
— A single thought experiment invalidating extended constituents would prove extension wrong.
— Extended constituents must appear if we start from any physical (low-energy)
concept – not only from length measurements – and study how the concept behaves
at Planck scales.
— Invalidating the requirement of extremal identity, or duality, would invalidate the
need for extended constituents. As Edward Witten likes to say, any unified model of
nature must include duality.
— If the measurement of length could be shown to be unrelated to the counting of folds
of extended constituents, extension would become unnecessary.
— Finding any property of nature that contradicts extended constituents would spell the
end of extension.
Any of these options would signal the end for almost all current unification attempts.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
checks of extension
Motion Mountain – The Adventure of Physics
Why has extension appeared so late in the history of physics? Here is a not too serious
answer. When we discussed the description of nature as made of tiny balls moving in a
void, we called this as a typically male idea. This implies that the female part is missing.
Which part would that be?
From a general point of view, the female part of physics might be the quantum description of the vacuum, the container of all things. We can speculate that if women
had developed physics, the order of its discoveries might have been different. Instead
of studying matter first, as men did, women might have studied the vacuum first. And
women might not have needed 2500 years to understand that nature is not made of a
void and little balls, but that everything in nature is made of extended constituents. It is
curious that (male) physics took so long for this discovery.
138
6 the sha pe of p oints
Fortunately, theoretical falsification has not yet occurred. But physics is an experimental
science. What kind of data could falsify the idea of extended constituents?
Ref. 131
Ref. 132
Page 19
— Observing a single particle in cosmic rays with energy above the corrected Planck
energy would invalidate the invariant limits and thus also extension. However, the
present particle energy record, about 0.35 ZeV, is a million times lower than the
Planck energy.
— Paul Mende has proposed a number of checks on the motion of extended objects in
space-time. He argues that an extended object and a mass point move differently; the
differences could be noticeable in scattering or dispersion of light near masses.
Current research based on extended constituents
Ref. 114
”
The Greeks deduced the existence of atoms from the observation that fish can swim
through water. They argued that only if water is made of atoms could a fish make its
way through it, by pushing the atoms aside. We can ask a similar question of a particle
flying through a vacuum: why is it able to do so? A vacuum cannot be a fluid or a solid
composed of small constituents, as its dimensionality would not then be fixed. Only one
possibility remains: both vacuum and particles are made of extended constituents.
The idea of describing matter as composed of extended constituents dates from the
1960s. That of describing nature as composed of ‘infinitely’ extended constituents dates
from the 1980s. In addition to the arguments presented so far, current research provides
several other approaches that arrive at the same conclusion.
∗∗
Ref. 133
Bosonization, the construction of fermions using an infinite number of bosons, is a central aspect of modern unification attempts. It also implies coupling duality, and thus the
extension of fundamental constituents.
∗∗
Ref. 134, Ref. 135
Research into quantum gravity – in particular the study of spin networks, spin foams
and loop quantum gravity – has shown that the vacuum can be thought of as a collection
of extended constituents.
* Isaiah Berlin (b. 1909 Riga, d. 1997 Oxford) was an influential political philosopher and historian of ideas.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
“
To understand is to perceive patterns.
Isaiah Berlin*
Motion Mountain – The Adventure of Physics
Experimental falsification of extension has not yet occurred. In fact, experimental falsification is rather difficult. It seems easier and more productive to confirm extension.
Confirmation is a well-defined project: it implies to deduce all those aspects of nature
that are given in the millennium list of unexplained properties. Among others, confirmation requires to find a concrete model, based on extended constituents, for the electron,
the muon, the tau, the neutrinos, the quarks and all bosons. Confirmation also requires
using extended constituents to realize an old dream of particle physics: to deduce the
values of the coupling constants and particle masses. Before we attempt this deduction,
we have a look at some other attempts.
checks of extension
139
∗∗
Ref. 136
In the 1990s, Dirk Kreimer showed that high-order QED Feynman diagrams are related
to knot theory. He thus proved that extension arrives by the back door even when electromagnetism is described in terms of point particles.
∗∗
Ref. 137
A popular topic in particle physics, ‘holography’, relates the surface and the volume of
physical systems at high energy. It implies extended constituents of nature.
∗∗
Vol. IV, page 157
It is long known that wave function collapse can be seen as the result of extended constituents. We will explore the details below.
∗∗
Ref. 140, Ref. 141
Ref. 142, Ref. 143
Page 347
At the start of the twenty-first century, a number of new approaches to describe elementary particles appeared, such as models based on string nets, models based on bands,
models based on ribbons, and models based on knots. All these attempts make use of
extended constituents. Several of them are discussed in more detail below.
Despite the use of extension, none of these attempts solved a single problem from the
millennium list. One approach – especially popular between the years 1984 and 2010 –
merits a closer look.
Superstrings – extension plus a web of dualities
Ref. 144
Ref. 145
”
Superstrings and supermembranes – often simply called strings and membranes – are extended constituents in the most investigated physics conjecture ever. The approach contains a maximum speed, a minimum action and a maximum force (or tension). The approach thus incorporates special relativity, quantum theory and general relativity. This
attempt to achieve the final description of nature uses four ideas that go beyond standard
general relativity and quantum theory:
1. Particles are conjectured to be extended. Originally, particles were conjectured to be
one-dimensional oscillating superstrings. In a subsequent generalization, particles are
conjectured to be fluctuating higher-dimensional supermembranes.
2. The conjecture uses higher dimensions to unify interactions. A number of space-time
dimensions much higher than 3+1, typically 10 or 11, is necessary for a mathematically
consistent description of superstrings and membranes.
3. The conjecture is based on supersymmetry. Supersymmetry is a symmetry that relates
matter to radiation, or equivalently, fermions to bosons. Supersymmetry is the most
general local interaction symmetry that can be constructed mathematically. Supersymmetry is the reason for the terms ‘superstring’ and ‘supermembrane’.
4. The conjecture makes heavy use of dualities. In the context of high-energy physics,
dualities are symmetries between large and small values of physical observables. Important examples are space-time duality and coupling constant duality. Dualities are
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
“
Throw physic to the dogs; I’ll none of it.
William Shakespeare, Macbeth.
Motion Mountain – The Adventure of Physics
Ref. 138, Ref. 139
140
6 the sha pe of p oints
global interaction and space-time symmetries. They are essential for the inclusion of
gauge interaction and gravitation in the quantum description of nature. Dualities also
express a fundamental equivalence between space-time and matter–radiation. Dualities also imply and contain holography, the idea that physical systems are completely
fixed by the states on their bounding surface.
Ref. 146
By incorporating these four ideas, the superstring conjecture – named so by Brian Greene,
one of its most important researchers – acquires a number of appealing characteristics.
Why superstrings and supermembranes are so appealing
* This argument is questionable, because general relativity already cures that divergence.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 122
Motion Mountain – The Adventure of Physics
Ref. 147
First of all, the superstring conjecture is unique: the Lagrangian is claimed to be unique
and to have no adjustable parameters. Furthermore, as we would expect from a description involving extended constituents, the conjecture includes gravity. In addition, the
conjecture describes interactions: it describes gauge fields. The conjecture thus expands
quantum field theory, while retaining all its essential points. In this way, the conjecture
fulfils most of the requirements for a unified description of motion that we have deduced
so far. For example, particles are not point-like, there are minimal length and time intervals, and all other limit quantities appear. (However, sets are still used.)
The superstring conjecture has many large symmetries, which arise from the many
dualities it contains. These symmetries connect many situations that seem intuitively to
be radically different: this makes the conjecture extremely fascinating, but also difficult
to picture.
The conjecture shows special cancellations of anomalies and of other inconsistencies. Historically, the first example was the Green–Schwarz anomaly cancellation; superstrings also solve other anomalies and certain inconsistencies of quantum field theory.
Edward Witten, the central figure of the field, liked to say that quantum theory cures
the infinities that appear in ?2 /? when the distance ? goes to zero; in the same way, superstrings cure the infinities that appear in ?2 /? when the distance ? goes to zero.*
Also following Witten, in the superstring conjecture, the interactions follow from the
particle definitions: interactions do not have to be added. That is why the superstring
conjecture predicts gravity, gauge theory, supersymmetry and supergravity.
About gravity, one of the pretty results of the superstring conjecture is that superstrings and black holes are complementary to each other. This was argued by Polchinsky,
Horowitz and Susskind. As expected, superstrings explain the entropy of black holes.
Strominger and Vafa showed this in 1996.
The superstring conjecture naturally includes holography, the idea that the degrees of
freedom of a physical system are determined by its boundary. In particular, holography
provides for a deep duality between gauge theory and gravity. More precisely, there is a
correspondence between quantum field theory in flat space and the superstring conjecture in certain higher-dimensional spaces that contain anti-de Sitter space.
In short, the superstring conjecture implies fascinating mathematics. Conformal invariance enters the Lagrangian. Concepts such as the Virasoro algebra, conformal field
theory, topological field theory and many related ideas provide vast and fascinating generalizations of quantum field theory.
checks of extension
141
Why the mathematics of superstrings is difficult
Ref. 148
One of the main results of quantum chromodynamics or QCD, the theory of strong interactions, is the explanation of mass relations such as
?proton ∼ e−?/?Pl ?Pl
Page 378
Ref. 149
and
? = 11/2π , ?Pl ≈ 1/25 .
(121)
Here, the value of the strong coupling constant ?Pl is taken at the Planck energy. In
other words, a general understanding of masses of bound states of the strong interaction,
such as the proton, requires little more than a knowledge of the Planck energy and the
coupling constant at that energy. The approximate value ?Pl ≈ 1/25 is an empirical value
based on experimental data.
Any unified theory must allow us to calculate the three gauge coupling constants as a
function of energy, thus also ?Pl . At present, most researchers regard the search for the
vacuum state – the precise embedding of four dimensions in the total ten – as the main
difficulty facing the superstring conjecture. Without knowledge of the vacuum state, no
calculations of coupling constants or masses are possible.
The vacuum state of the superstring conjecture is expected to be one of an rather
involved set of topologically distinct manifolds. It was first estimated that there are only
10500 possible vacuum states; recent estimates raised the number to 10272 000 candidate
vacuum states. Since the universe contains 1080 atoms, it seems easier to find a particular
atom somewhere in the universe than to find the correct vacuum state. The advantages
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Testing superstrings: couplings and masses
Motion Mountain – The Adventure of Physics
The superstring conjecture, like all modern descriptions of physics, is claimed to be described by a Lagrangian. The Lagrangian is constructed starting from the Lagrangian for
the motion of a classical superstring of matter. Then the Lagrangian for the corresponding quantum superstring fields is constructed, and then higher dimensions, supersymmetry, dualities and membranes are incorporated. This formulation of the superstring
conjecture takes for granted the existence of a space-time background.
The Lagrangian of the superstring conjecture is extremely complex, much too complex to write it down here. It is not as simple as the Lagrangian of the standard model
of particle physics or the Lagrangian of general relativity. But the complexity of the Lagrangian is not the only reason why the studying the superstring conjecture is difficult.
It turns out that exploring how the known 4 dimensions of space-time are embedded
in the 10 or 11 dimensions of the superstring conjecture is extremely involved. The topology and the size of the additional dimensions is unclear. There are only few people who
are able to study these options.
Indeed, a few years ago a physicist and a mathematician listened to a talk on superstrings, describing nature in eleven dimensions. The mathematician listened intensely
and obviously enjoyed the talk. The physicist did not understand anything and got more
and more annoyed. At the end, the physicist had a terrible headache, whereas the mathematician was full of praise. ‘But how can you even understand this stuff?’, asked the
physicist. ‘I simply picture it in my head!’ ‘But how do you imagine things in eleven
dimensions?’ ‘Easy! I first imagine them in ? dimensions and then let ? go to 11.’
142
6 the sha pe of p oints
due to a unique Lagrangian are thus lost.
We can also describe the problems with the calculation of particle masses in the following way. The superstring conjecture predicts states with Planck mass and with zero
mass. The zero-mass particles are then thought to get their actual mass, which is tiny
compared with the Planck mass, from the Higgs mechanism. However, the Higgs mechanism and its measured properties – or any other parameter of the standard model –
have not yet been deduced from superstrings.
The status of the superstring conjecture
“
Ref. 150
* ‘Nothing great has been achieved without passion, nor can it be achieved without it.’ Hegel, an influential
philospher, writes this towards the end of the third and last part of his Enzyklopädie der philosophischen
Wissenschaften im Grundrisse, §474, 296.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 151
”
It is fair to say that nowadays, superstring researchers are stuck. Despite the huge collective effort, not a single calculation of an experimentally measurable value has been
performed. For example, the superstring conjecture has not predicted the masses of any
elementary particle, nor the value of any coupling constant, nor the number of gauge
interactions, nor the number of particle generations. In fact, none of the open issues in
physics that are listed the millennium list has been solved by the superstring conjecture.
This disappointing situation is the reason that many scholars, including several Nobel
Prize winners, dismiss the superstring conjecture altogether.
What are the reasons that the superstring conjecture, like several other approaches
based on extended constituents, was unsuccessful? First of all, superstrings and supermembranes are complex: superstrings and supermembranes move in many dimensions,
carry mass, have tension and carry (supersymmetric) fields. In fact, the precise mathematical definition of a superstring or a supermembrane and their features is so complex
that already understanding the definition is beyond the capabilities of most physicists.
But a high complexity always nourishes the doubt that some of the underlying assumptions do not apply to nature.
Superstrings are complex entities. And no researcher tried to make them simple. Put
in different terms, the superstring conjecture was not successful because its basic principles have never been clarified. It is estimated that, from 1984 to 2010, over 10 000 manyears have been invested in the exploration of the superstring conjecture. Compare this
with about a dozen man-years for the foundations and principles of electrodynamics,
a dozen man-years for the foundations and principles of general relativity, and a dozen
man-years for the foundation and principles of quantum theory. The lack of clear foundations of the superstring conjecture is regularly underlined even by its supporters, such
as Murray Gell-Mann. And despite this gap, no research papers on the basic principles
exist – to this day.
Apart from the complexity of the conjecture, a further aspect about superstrings and
supermembranes has been getting growing attention: the original claim that there is a
unique well-defined Lagrangian has been retracted; it not even made by the most outgo-
Motion Mountain – The Adventure of Physics
Page 19
Es ist nichts Großes ohne Leidenschaft vollbracht worden, noch kann es ohne
solche vollbracht werden.*
Friedrich Hegel, Enzyklopädie.
sum m ary on extension in nature
143
ing proponents any more. In other words, it is not clear which specific supermembrane
conjecture should be tested against experiment in the first place.
These developments effectively dried out the research field. At the latest in 2014, during the Strings conference, it became clear that the research community has quietly given
up its quest to achieve a unified theory with the help of superstrings or supermembranes.
Researchers are now looking for other microscopic models of nature.
summary on extension in nature
“
”
We have explored nature at her limits: we have studied the Planck limits, explored threedimensionality, curvature, particle shape, renormalization, spin and bosonization; we
have investigated the cosmological constant problem and searched for a ‘backgroundfree’ description of nature. As a result, we have found that at Planck scales, all these
explorations lead to the same conclusions:
— Points and sets do not describe nature correctly.
— Matter and vacuum are two sides of the same medal.
— What we usually call space-time points and point particles are in fact made up of
common and, above all, extended constituents.
* ‘We must know, we will know.’ This was Hilbert’s famous personal credo.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
We can reach the conclusions in an even simpler way. What do quantum theory and
black holes have in common? They both suggest that nature is made of extended entities.
We will confirm below that both the Dirac equation and black hole entropy imply that
particles, space and horizons are built from extended constituents.
Despite using extension as fundamental aspect, and despite many interesting results,
all the attempts from the twentieth century – including the superstring conjecture and all
quantum gravity models, but also supersymmetry and supergravity – have not been successful in understanding or in describing nature at the Planck scale. The reasons for this
lack of success were the unclear relation to the Planck scale, the lack of clear principles,
the use of incorrect assumptions, the use of sets and, above all, the unclear connection
to experiment.
To be successful, we need a different approach to calculations with extended constituents. We need an approach that is built on Planck units, is based on clear principles, has
few but correct assumptions, and provides predictions that stand up against experimental
tests.
In our quest for a final theory of physics, one way to advance is by raising the following issue. The basis for the superstring conjecture is formed by four assumptions:
extension, duality, higher dimensions and supersymmetry. Can we dispense with any of
them? Now, duality is closely related to extension, for which enough theoretical and experimental evidence exists, as we have argued above. On the other hand, the expressions
for the Schwarzschild radius and for the Compton wavelength imply, as we found out
Motion Mountain – The Adventure of Physics
Ref. 152
Wir müssen wissen, wir werden wissen.*
David Hilbert
144
Page 71, page 77
Page 74, page 77
6 the sha pe of p oints
earlier on, that the dimensionality of space and the statistics of particles are undefined at
Planck scales. In other words, nature does not have higher dimensions nor supersymmetry at Planck scales. Indeed, all known experiments confirm this conclusion. In our
quest for a final theory of motion, we therefore drop the two incorrect assumptions and
continue our adventure.
In summary, extension is the central property of the fundamental entities of nature
that make up space, horizons, particles and interactions at Planck scales. We can thus
phrase our remaining quest in the following specific way:
⊳ How do extended entities relate the Planck constants ?, ℏ, ? and ? to the
electromagnetic, the weak and the strong interactions?
Challenge 111 e
Motion Mountain – The Adventure of Physics
This question is rarely asked so specifically. Attempts to answer it are even rarer. (Can
you find one?) Up to this point, we discovered: Finding the Planck origin of the gauge
interactions using extension means finding the final theory. To be successful in this quest,
we need three resources: simplicity, playfulness and intrepidity.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Chapter 7
T HE BA SIS OF T HE ST R AND MODEL
”
he two extremely precise descriptions of motion that were discovered in
he twentieth century – quantum field theory and general relativity – are
he low-energy approximations of how nature behaves at Planck scales. In order
to understand nature at Planck scales, and thus to find the unified and final description
of motion, we follow the method that has been the most effective during the history of
physics: we search for the simplest possible description. Simplicity was used successfully,
for example, in the discovery of special relativity, in the discovery of quantum theory,
and in the discovery of general relativity. We therefore use the guidance provided by
simplicity to deduce a promising speculation for the unified and final theory of motion.
Page 161
Page 163
The central requirement for any unified description is that it leads from Planck scales,
and thus from Planck units, to quantum field theory, to the standard model of elementary particles and to general relativity. In simple terms, as detailed below, the unified
description must be valid for all observations and provide complete precision.
From the preceding chapters, we know already quite a bit about the unified description. In particular, any unified description of general relativity and quantum theory must
use extended constituents. We discovered a number of reasons that are central for this
conclusion. All these reasons appear only when quantum theory and general relativity
are combined. First of all, only constituents that are extended allow us to deduce black
hole entropy. Secondly, only extended constituents allow us to model that elementary
particles are not point-like or that physical space is not made of points. Thirdly, only
extended constituents allow us to model a smallest measurable space and time interval.
Fourthly, only extended constituents allow us to model spin 1/2 in locally flat space-time.
But we are not only looking for a unified theory; we are also looking for the final
theory. This implies a second requirement: the final theory must be unmodifiable. As
we will show below, if a candidate for a final theory can be modified, or generalized, or
reduced to special cases, or varied in any other way, it is not final.
In the preceding chapters we have deduced many additional, requirements that a final
** Ernest Rutherford (b. 1871 Brightwater, d. 1937 Cambridge) was an important physicist and researcher;
he won the Nobel Prize in Chemistry for his work on atoms and radioactivity.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
R equirements for a final theory
Motion Mountain – The Adventure of Physics
T
“
We haven’t got the money, so we have to think.
Ernest Rutherford**
the basis of the stra nd m odel
147
The fundamental principle of the strand model
Strand model :
?1
Observation :
Some
deformation,
but no
passing
through
? = ℏ/2
Δ? = ?Pl
Δ? = ?Pl
? = ?/2
?2
F I G U R E 10 The fundamental principle of the strand model: the simplest observation in nature, a
‘‘point-like’’ fundamental event, is defined by a crossing switch in three spatial dimensions. The crossing
switch defines the action ℏ/2, the Planck length, the Planck time and half the Boltzmann constant ?/2.
TA B L E 6 General requirements for a final and unified description of nature and of motion.
R equirements for the final and unified
description
Precision
must be complete; the unified description must precisely describe
all motion – everyday, quantum and relativistic – and explain all
open issues from the millennium list, given (again) in Table 8 on
page 162, including the fine structure constant.
must be impossible, as explained on page 163.
must be clear. (Otherwise the unified description is not
falsifiable.)
must not differ at Planck scales because of limits of measurement
precision; vacuum and particles therefore must be described by
common fundamental constituents.
must determine all observables.
must be as simple as possible, to satisfy Occam’s razor.
must be extended and fluctuating, to explain black hole entropy,
spin, minimum measurement intervals, space-time homogeneity
and isotropy of space.
must be the only unobservable entities. (If they were observable,
the theory would not be final, because the properties of the
entities would need explanation; if additional unobservable
entities would exist, the theory would be fiction, not science.)
Modification
Fundamental principles
Vacuum and particles
Fundamental constituents
Fundamental constituents
Fundamental constituents
Fundamental constituents
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Aspect
Motion Mountain – The Adventure of Physics
Challenge 112 e
theory must realize. The full list of requirements is given in Table 6. Certain requirements
follow from the property that the description must be final, others from the property
that it must be unified, and still others from the property that it must describe nature
with quantum theory and general relativity. More specifically, every requirement appears
when the expressions for the Compton wavelength and for the Schwarzschild radius are
combined. So far, the table is not found elsewhere in the research literature.
148
7 the basis of the stra nd m od el
TA B L E 6 (Continued) General requirements for a final and unified description of nature and of motion.
R equirements for the final and unified
description
Non-locality
must be part of the description; non-locality must be negligible at
everyday scales, but important at Planck scales.
must not exist, due to limits of measurement precision; points and
sets only exist approximately, at everyday scales.
must not exist at Planck scales, due to the lack of points and sets.
must not exist at Planck scales, due to limits of measurement
precision; systems only exist approximately at everyday scales.
must not be a system, due to limits of measurement precision.
must not be an event, and thus not be a beginning, as this would
contradict the non-existence of points and sets in nature.
must not exist, due to the limits of measurements.
must be limit values for each observable (within a factor of order
one); infinitely large or small measurement values must not exist.
must imply quantum field theory, the standard model of particle
physics, general relativity and cosmology.
must follow from the final unified theory by eliminating ?.
Physical points and sets
Evolution equations
Physical systems
Universe
Big bang
Singularities
Planck’s natural units
Planck scale description
Circularity of definitions
Axiomatic description
Dimensionality of space
Symmetries
Large and small scales
Challenge 113 e
must follow from the final unified theory by eliminating ℏ.
must define all observables, including coupling constants.
must be as simple as possible, to satisfy Occam’s razor.
is required, as background independence is logically impossible.
must be equal to physical space-time at everyday scale, but must
differ globally and at Planck scales.
of physical concepts must be part of the final, unified description,
as a consequence of being ‘precise talk about nature’.
must be impossible, as nature is not described by sets; Hilbert’s
sixth problem must have no solution.
must be undefined at Planck scales, as space is undefined there.
must be undefined at Planck scales, due to the limits to
measurement precision.
must be similar, due to the limits to measurement precision.
The requirement list given in Table 6 can be considerably shortened. Shortening the
list of requirements is possible because the various requirements are consistent with each
other. In fact, shortening is possible because a detailed check confirms a suspicion that
arose during the last chapters: extension alone is sufficient to explain all those requirements that seem particularly surprising or unusual, such as the lack of points or the lack
of axioms. Such a shortened list also satisfies our drive for simplicity. After shortening,
two requirements for a final theory remain:
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Quantum field theory,
including QED, QAD,
QCD
General relativity
Planck’s natural units
Relation to experiment
Background dependence
Background space-time
Motion Mountain – The Adventure of Physics
Aspect
the basis of the stra nd m odel
149
⊳ The final theory must describe nature at and below the Planck scale* as made
of extended constituents fluctuating in a background. Extended constituents
must explain particles, space, interactions and horizons.
⊳ In the final theory, the fluctuations of the extended constituents must explain all motion. The Planck-scale fluctuations must describe all observed
examples of everyday, quantum and relativistic motion with complete precision, imply all interactions, all concepts of physics and explain all fundamental constants.
Challenge 114 s
“
Nur die ergangenen Gedanken haben Wert.***
Friedrich Nietzsche
”
* The final theory must not describe nature beyond the Planck scales. A more relaxed requirement is that the
predictions of the theory must be independent of any fantasies of what might occur beyond Planck scales.
** In Dutch: draden, in French: fils, in German: Fäden, in Italian: fili.
*** ‘Only thoughts conceived while walking have value.’ Friedrich Nietzsche (b. 1844 Röcken,
d. 1900 Weimar) was philologist, philosopher and sick.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 165
Motion Mountain – The Adventure of Physics
Ref. 153
This requirement summary is the result of our journey up to this point. The summary
forms the starting point for the final leg of our adventure. If you do not agree with these
two requirements, take a rest and explore your disagreement in all its details.
Looking at the table of requirements for the final theory – both the full one and the
shortened one – we note something astonishing. Even though all requirements appear
when quantum physics and general relativity are combined, each of these requirements
contradicts both quantum physics and general relativity. The final theory thus differs from
both pillars of modern physics. A final theory cannot be found if we remain prisoners
of either quantum theory or general relativity. To put it bluntly, each requirement for
the final theory contradicts every result of twentieth century physics! This unexpected
conclusion is the main reason that past attempts failed to discover the final theory. In fact,
most attempts do not fulfil the requirements because various scholars explicitly disagree
with one or several of them.
The requirement of the extension of the fundamental constituents is the central result. A final theory must make a statement about these constituents. The fundamental
constituents, also called fundamental degrees of freedom, must explain everything we observe and know about nature. In particular, the constituents must explain the curvature
of space, the entropy of black holes, the origin of gauge interactions and the spectrum,
mass and other properties of all elementary particles. The fundamental constituents must
be extended. Extension is the reason that the final theory contradicts both general relativity and quantum theory; but extension must allows these theories as excellent approximations. In short, extension is the key to finding the final theory.
The requirement for fluctuating extended constituents resulted from our drive for extreme simplicity. With this requirement, the search for a candidate final theory does not
take long. Of the few candidates that satisfy the requirement, it appears that the simplest
is the one based on fluctuating featureless strands. In this approach, strands,** not points,
are assumed to be the fundamental constituents of vacuum, horizons, matter and radiation.
150
7 the basis of the stra nd m od el
Introducing strands
The strand model starts with a simple idea:
⊳ Nature is made of unobservable, fluctuating, featureless strands.
We will discover that everything observed in nature – vacuum, fermions, bosons and
horizons – is made of strands. Strands are the common and extended constituents of
everything. Even though strands are unobservable and featureless, all observations are
due to strands.
⊳ All observations, all change and all events are composed of the fundamental
event, the crossing switch.
⊳ Planck units are defined through crossing switches of strands.
Page 147
The definition of the Planck units with the crossing switch is illustrated in Figure 10. All
measurements are consequence of this definition. All observations and everything that
happens are composed of fundamental events. The fundamental principle thus specifies
why and how Planck units are the natural units of nature. In particular, the four basic
Planck units are associated in the following way:
⊳ The (corrected) Planck length ?Pl = √4?ℏ/?3 appears as the effective diameter of strands. Since the Planck length is a limit that cannot be achieved
by measurements, strands with such a diameter remain unobservable.*
⊳ The Planck entropy, i.e., the Boltzmann constant ?, is the natural unit associated to the counting and statistics of crossings.*
⊳ The (corrected) Planck time ?Pl = √4?ℏ/?5 appears as the shortest possible
duration of a crossing switch.*
Crossing switches that are faster than the Planck time do not play a role, as they are
unobservable and unmeasurable. Let us see why.
How can we imagine a minimum time interval in nature? A crossing switch could be
arbitrarily fast, couldn’t it? So how does the Planck time arise? To answer, we must recall
the role of the observer. The observer is a physical system, also made of strands. The
* In other words, the strand model sets ℏ = ?Pl = ?Pl = ? = 1. The strange numerical values that these
constants have in the SI, the international system of units, follow from the strange definitions of the metre,
second, kilogram and kelvin.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
⊳ Planck’s quantum of action ℏ/2 appears as the action value associated to a
crossing switch. The action ℏ corresponds to a double crossing switch, or
full turn of one strand segment around another.*
Motion Mountain – The Adventure of Physics
To describe all observations with precision, the strand model uses only one fundamental
principle:
the basis of the stra nd m odel
151
A twist :
t2
t1
A twirl :
t2
F I G U R E 11 An example of strand deformation leading to a crossing switch (above) and one that does
not lead to a crossing switch (below).
Page 36
Challenge 115 r
** The issue of time remains subtle also in the strand model. The requirement of consistency with macroscopic experience, realized with shivering space or space-time, allows us to side-step the issue. An alternative approach might be to picture a crossing switch and its fluctuations in 4 space-time dimensions, thus
visualizing how the minimum time interval is related to minimum distance. This might be worth exploring.
But also in this approach, the fuzziness due to shivering is at the basis of minimum time.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Challenge 116 s
Challenge 117 e
observer cannot define a really continuous background space-time; careful consideration
tells us that the space-time defined by the observer is somewhat fuzzy: it is effectively
shivering. The average shivering amplitude is, in the best possible case, of the order of a
Planck time and length. Therefore, crossing switches faster than the Planck time are not
observable by an observer made of strands.**
Strands are impenetrable; the switch of a crossing thus always requires the motion of
strand segments around each other. The simplest example of a deformation leading to a
crossing switch is shown in Figure 11.
Can you deduce the strand processes for the Planck momentum, the Planck force and
the Planck energy?
Exploring strand processes we find: the fundamental principle implies that every
Planck unit is an observer-invariant limit value. Therefore, the fundamental principle
naturally contains special and general relativity, quantum theory and thermodynamics
(though not elementary particle physics!). In theory, this argument is sufficient to show
that the fundamental principle contains all these parts of twentieth century physics. In
practice, however, physicists do not change their thinking habits that quickly; thus we
need to show this result in more detail.
Motion Mountain – The Adventure of Physics
t1
152
7 the basis of the stra nd m od el
A strand crossing
phase
shortest distance
position
orientation
F I G U R E 12 The definition of a crossing, its position, its orientation and its phase. The shortest distance
defines a local density.
Events, pro cesses, interactions and colours
⊳ Any event, any observation, any measurement and any interaction is composed of switches of crossings between two strand segments.
⊳ Particle masses, the elementary electric charge ? and the fine structure constant ? = 137.036(1) are due to crossing switches.
The value of the fine structure constant and the standard model are not evident consequences of the fundamental principle; nevertheless, they are natural consequences –
as we will find out.
From strands to modern physics
Every observation and every process is a sequence of crossing switches of unobservable
strands. In turn, crossing switches are automatic consequences of the shape fluctuations
of strands. We will show below that all the continuous quantities we are used to – physical space, physical time, gauge fields and wave functions – result from averaging crossing
switches over the background space. The main conceptual tools necessary in the following are:
⊳ A crossing of strands is a local minimum of strand distance. The position,
orientation and phase of a crossing are defined by the space vector corres-
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
The crossing switch is the fundamental process in nature. We will show that describing
events and interactions with the help of crossing switches leads, without alternative, to
the complete standard model of particle physics, with all its known gauge interactions
and all its known particle spectrum.
In particular, we will show:
Motion Mountain – The Adventure of Physics
In the strand model, every physical process is described as a sequence of crossing
switches. But every physical process is also a sequence of events. We thus deduce that
events are processes:
the basis of the stra nd m odel
153
ponding to the local minimum of distance, as shown in Figure 12.
Page 172
The position, orientation and phase of crossings will lead, as shown later on, to the position, orientation and phase of wave functions. The sign of the orientation is defined by
arbitrarily selecting one strand as the starting strand. The even larger arbitrariness in the
definition of the phase will be of great importance later on: it implies the existence of the
three known gauge groups.
⊳ A crossing switch is the rotation of the crossing orientation by an angle π at
a specific position. More precisely, a crossing switch is the inversion of the
orientation at a specific position.
⊳ Events are (one or several) observable crossing switches of unobservable
strands.
Page 165
Since all observations are events, all experimental observations should follow from the
strand definition of an event. We will confirm this in the rest of this text. The strands
are featureless: they have no mass, no tension, no stiffness, no branches, no fixed length,
no ends, and they cannot be pulled, cut or pushed through each other. Strands have
no measurable property at all: strands are unobservable. Only crossing switches are observable. Featureless strands are thus among the simplest possible extended constituents.
How simple are they? We will discuss this issue shortly.
⊳ Strands are one-dimensional curves in three-dimensional space that reach
the border of space.
In practice, the border of space has one of two possible meanings. Whenever space is
assumed to be flat, the border of space is spatial infinity. Whenever we take into account
the properties of the universe as a whole, the border of space is the cosmic horizon.
Imagining the strands as having Planck diameter does not make them observable,
as this measurement result cannot be realized. (We recall that the Planck length is the
lower bound on any length measurement.) In low energy situations, a vanishing strand
diameter is an excellent approximation.
⊳ In a purist definition, featureless strands have no diameter – neither the
Planck length nor zero. They are better thought as long thin clouds.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 353
Motion Mountain – The Adventure of Physics
We note that the definitions make use of all three dimensions of space; therefore the number of crossings and of crossing switches is independent of the direction of observation.
This contrasts with the definition of crossing used in two-dimensional knot diagrams; in
such two-dimensional projections, the number of crossings does depend on the direction of the projection.
We note that strand fluctuations do not conserve the number of crossings; due to
fluctuations, crossings disappear and appear and disappear over time. This appearance
and disappearance will turn out to be related to virtual particles.
The fundamental principle declares that events are not points on manifolds; instead,
154
7 the basis of the stra nd m od el
Vacuum :
Elementary
spin 1/2
fermion :
spin
Elementary
spin 1 boson :
spin
Horizon :
detail below.
Page 166
⊳ Physical space, or vacuum, is a physical system made of tangles that has size,
curvature and other measurable properties.
⊳ Continuous background space is introduced by the observer only to be able
to describe observations. Every observer introduces his own background.
It does not need to coincide with physical space, and it does not do so at
the location of matter or black holes. But every observer’s background is
continuous and has three spatial and one temporal dimension.
At this point of the discussion, we simply assume background space. Later on we will
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 169
Strands are unobservable and featureless, and thus have no diameter. Due to shape fluctuations, or equivalently, due to the shivering of space-time, the strands can be thought
as having an effective diameter, akin to the diameter of a long thin cloud; this effective
diameter is just a guide to our thinking. Since it is due to the shivering of the background space-time, the strand diameter is invariant under boosts. Funnels, mentioned
below, might be a better visualization of the purist definition of strand. To keep this introduction as intuitive as possible, however, we stick with the idea of strands having an
effective, invariant Planck diameter.
The strand model distinguishes physical space from background space. We will show
shortly why both concepts are required. With this distinction, the strand model asserts
that matter and radiation, vacuum and horizons, are all built from fluctuating strands in
a continuous background. We first clarify the two basic space concepts.
Motion Mountain – The Adventure of Physics
F I G U R E 13 A first illustration of the basic physical systems found in nature; they will be explored in
the basis of the stra nd m odel
155
TA B L E 7 Correspondences between all known physical systems and mathematical tangles.
Physical system
Strand content
Ta n g l e t y p e
Vacuum and dark energy
many unknotted and untangled
infinite strands
two infinite twisted strands
many infinite twisted strands
many woven infinite strands
one infinite curved strand
unlinked, trivial tangle
Graviton
Gravity wave
Horizon
Elementary vector boson
(radiation)
Classical electromagnetic
wave (radiation)
Elementary quark (matter)
Elementary lepton (matter)
two infinite linked strands
three infinite linked strands
see why background space appears and why it needs to be three-dimensional. The size
of the background space is assumed to be large; larger than any physical scale under
discussion. In most situations of everyday life, when space is flat, background space and
physical space coincide. However, they differ in situations with curvature and at Planck
energy.
All strand fluctuations are possible, as long as strands do not interpenetrate. For example, there is no speed limit for strands. Whenever strand fluctuations lead to a crossing
switch, they lead to an observable effect – be it a vacuum fluctuation, a particle reaction
or a horizon fluctuation.
⊳ Fluctuations are a consequence of the embedding of strands in a continuous
background.
In the strand model, even isolated physical systems are surrounded by a bath of fluctuating vacuum strands. The properties of fluctuations, such as their spectrum, their
density etc., are fixed once and for all by the embedding. Fluctuations are necessary for
the self-consistency of the strand model.
Due to the impenetrability of strands – which itself is a consequence of the embedding in a continuous background – any disturbance of the vacuum strands at one location propagates. We will see below what disturbances exist and how they differ from
fluctuations.
Fluctuating strands that lead to crossing switches explain everything that does happen,
and explain everything that does not happen. Our main aim in the following is to classify
all possible strand fluctuations and all possible strand configurations, in particular, all
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
⊳ Fluctuations change the position, shape and length of strands; fluctuations
thus change position, orientation and phase of strand crossings. However,
fluctuations never allow one strand to pass through another.
Motion Mountain – The Adventure of Physics
Page 206
many infinite curved strands
rational tangle
many rational tangles
woven, web-like tangle
any one from a family of
tangled curves
many helically
deformed/tangled curves
rational tangle
braided tangle
156
7 the basis of the stra nd m od el
Vacuum
The strand model :
Observation :
time average
of crossing
switches
Nothing
(only for long
observation
times)
F I G U R E 14 An illustration of the strand model for the vacuum.
Page 154
Page 313
Page 150
Some examples of important tangles are given in Figure 13. They will be discussed in
detail in the following. Among others, we will discover that knots and knotted tangles
do not play a role in the strand model; only linked, but unknotted tangles do.
We observe that vacuum, matter and radiation are all made of the same fundamental
constituents, as required for a final theory. We will discover below that classifying localized tangles naturally leads to the elementary particles that make up the standard model
of particle physics – and to no other elementary particle.
We will also discover that strand fluctuations and the induced crossing switches in
every physical system lead to the evolution equations and the Lagrangians of quantum
field theory and of general relativity. In this way, strands describe every physical process
observed in nature, including all known interactions and every type of motion.
The fundamental principle relates crossing switches and observations. The fundamental principle was discovered because it appears to be the only simple definition
of Planck units that on the one hand yields space-time, with its continuity, local isotropy and curvature, and on the other hand realizes the known connection between the
quantum of action, spin and rotation.
Vacuum
We now construct, step by step, all important physical systems, concepts and processes
from tangles. We start with the most important.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
⊳ A tangle is a configuration of one or more strands that are linked or knotted.
Tangles are characterized by their topology, i.e., by the precise way that they
are linked or knotted.
Motion Mountain – The Adventure of Physics
states that differ from flat vacuum states. By doing so, we will be able to classify every
process and every system that we observe in nature.
We will discover that all physical systems can be constructed from strands. Table 7
gives a first overview of how vacuum, particles and horizons result from strand tangles.
the basis of the stra nd m odel
157
⊳ Vacuum, or physical space, is formed by the time average of many unknotted
fluctuating strands.
Page 236
With the definition of the vacuum as a time average, the strand model yields a minimum
length and a continuous vacuum at the same time. In this way, many issues about the
alleged contradiction between continuity and minimum length are put to rest. In particular, physical space is not fundamentally discrete: a minimum length appears, though
only in domains where physical space is undefined. On the other hand, the continuity of physical space results from an averaging process. Therefore, physical space is not
fundamentally continuous: the strand model describes physical space as a homogeneous
distribution of crossing switches. This is the strand version of Wheeler’s idea space-time
foam.
Observable values and limits
The fundamental principle implies the following definitions of the basic observables:
* We recall that since over a century, the concept of aether is superfluous, because it is indistinguishable
from the concept of vacuum.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
⊳ We do not make any statement on the numerical density of strands in vacuum, or, equivalently, on their average spacing. Since strands are not observable, such a statement is not sensible. In particular, strands in vacuum
are not tightly packed.
Motion Mountain – The Adventure of Physics
Page 279
Figure 14 visualizes the definition. In the following, vacuum and physical space are always taken to be synonyms; the exploration will show that this is the most sensible use
of the two concepts.* However, as mentioned, the strand model distinguishes physical
space from background space. In particular, since matter and vacuum are made of the
same constituents, it is impossible to speak of physical space at the location of matter. At
the location of matter, it is only possible to use the concept of background space.
When the strand fluctuations in flat vacuum are averaged over time, there are no
crossing switches. Equivalently, if we use concepts to be introduced shortly, flat vacuum
shows, averaged over time, no knots and no tangles, so that it is observed to be empty of
matter and radiation. Temporary tangles that appear for a short time through vacuum
fluctuations will be shown later to represent virtual particles.
We note that the (flat) physical vacuum state, which appears after averaging the strand
crossings, is continuous, Lorentz invariant and unique. These are important points for the
consistency of the model. Later we will also discover that curvature and horizons have
a natural description in terms of strands; exploring them will yield the field equations
of general relativity. The strand model thus replaces what used to be called ‘space-time
foam’ or ‘quantum foam’.
We also note that Figure 14 implies, despite appearances, that vacuum is isotropic. To
see this, we need to recall that the observables are the crossing switches, not the strands,
and that the observed vacuum isotropy results from the isotropy of the time-averaged
strand fluctuations.
158
7 the basis of the stra nd m od el
A fermion
Strand model :
time average
of crossing
switches
tails
Observed
probability
density :
crossing
orientations
core
Page 172
positions
phases
spin
orientation
position
phase
F I G U R E 15 The tangle model of a spin 1/2 particle. More details will be given below.
⊳ The time interval between two events is the maximum number of crossing
switches that can be measured between them. Time measurement is thus
defined as counting Planck times.
⊳ The entropy of any physical system is related to the logarithm of the number of possible measurable crossing switches. Entropy measurement is thus
defined through the counting of crossing switches. The strand model thus
states that any large physical system – be it made of matter, radiation, empty
space or horizons – has entropy.
Page 147
It is well-known that all other physical observables are defined using these four basic
ones. In other words, all physical observables are defined with crossing switches. We
also note that even though counting always yields an integer, the result of a physical
measurement is often an average of many counting processes. As a result of averaging
and fluctuations, measured values can be non-integer multiples of Planck units. Therefore, space, time, action, entropy and all other observables are effectively real numbers,
and thus continuous. Continuity is thus reconciled with the existence of a minimum
measurable length and time interval.
Finally, we note that defining observables with the help of crossing switches automatically makes the Planck units ?, ℏ, ?4 /4?, ? and all their combinations both observerinvariant and limit values. All these conclusions agree with the corresponding requirements for a final theory of nature.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
⊳ The physical action of a physical system evolving from an initial to a final
state is the number of crossing switches that can be measured. Action measurement is thus defined as counting crossing switches. Physical action is
thus a measure for the change that a system undergoes.
Motion Mountain – The Adventure of Physics
⊳ The distance between two particles is the maximum number of crossing
switches that can be measured between them. Length measurement is thus
defined as counting Planck lengths.
the basis of the stra nd m odel
159
Particles and fields
Strands also define particles, as illustrated in Figure 15:
⊳ A quantum particle is a tangle of fluctuating strands. The tangle core, the
region where the strands are linked, defines position, speed, phase and spin
of the particle. The tangle tails reach up to the border of space.
Page 174
Page 222
Page 212
Page 226
Why do crossing switches have such a central role in the strand model? An intuitive explanation follows from their role in the definition of observables. All measurements – be
they measurements of position, speed, mass or any other observable – are electromagnetic. In other words, all measurements in nature are, in the end, detection of photons.
And the strand model shows that photon absorption and detection are intimately related
to the crossing switch, as we will find out below.
∗∗
Is there a limit to the fluctuations of strands? Yes and no. On the one hand, the ‘speed’
of fluctuations is unlimited. On the other hand, fluctuations with a ‘curvature radius’
smaller than a Planck length do not lead to observable effects. Note that the terms ‘speed’
and ‘radius’ are between quotation marks because they are unobservable. Care is needed
when talking about strands and their fluctuations.
∗∗
What are strands made of? This question tests whether we are really able to maintain
the fundamental circularity of the unified description. Strands are featureless. They have
no measurable properties: they have no branches, carry no fields and, in particular, they
cannot be divided into parts. The ‘substance’ that strands are made of has no properties.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Curiosities and fun challenges abou t strands
Motion Mountain – The Adventure of Physics
Page 314
As shown in more detail soon, this definition of quantum particles yields, depending on
the tangle details, either fermion or boson behaviour, and reproduces the spin–statistics
theorem.
Boson tangles will allow us to model field intensities. In particular, boson tangles
allow us to deduce the electromagnetic and the two nuclear fields, as well as the corresponding gauge symmetries of the standard model of particle physics.
Modelling fermions as tangles will allow us to deduce Dirac’s equation for relativistic quantum particles (and the Schrödinger equation for non-relativistic particles). Still
later, by classifying all possible tangles, we will discover that only a finite number of possible elementary particles exist, and that the topological type of tangle determines the
mass, mixings, quantum numbers, charges and couplings of each elementary particle.
We could also speak of a tangle model.
In the 1960s, John Wheeler stated that a unified description of nature must explain
‘mass without mass, charge without charge, field without field’. The strand model realizes
this aim, as we will find out.
Before we deduce modern physics, we first take a break and explore some general
issues of the strand model.
160
Challenge 118 e
Page 304
7 the basis of the stra nd m od el
Thus strands are not made of anything. This may seem surprising at first. Strands are
extended, and we naturally imagine them as sequence of points. But this is a fallacy.
Given the way that observations and events are defined, there is no way to observe, to
label or to distinguish points on strands. Crossing switches do not allow doing so, as is
easily checked: the mathematical points we imagine on a strand are not physical points.
‘Points’ on strands are unobservable: they simply do not exist.
But strands must be made of something, we might insist. Later we will find out that
in the strand model, the universe is made of a single strand folded in a complicated way.
Nature is one strand. Therefore, strands are not made of something, they are made of
everything. The substance of strands is nature itself.
∗∗
Since there is only one strand in nature, strands are not a reductionist approach. At
Planck scale, nature is one and indivisible.
∗∗
Challenge 119 e
Can macroscopic determinism arise at all from randomly fluctuating strands?
∗∗
Challenge 120 s
Do parallel strands form a crossing? Do two distant strands form a crossing?
∗∗
Challenge 121 s
Is a crossing switch defined in more than three dimensions?
∗∗
Challenge 122 s
Can you find a way to generalize or to modify the strand model?
∗∗
In hindsight, the fundamental principle resembles John Wheeler’s vision ‘it from bit’.
He formulated it, among others, in 1989 in his often-cited essay Information, physics,
quantum: the search for links.
∗∗
Looking back, we might equally note a relation between the strand model and the ex-
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 77
What are particles? In the strand model, elementary particles are (families of) tangles of
strands. In other words, elementary particles are not the basic building blocks of matter –
strands are. If particles could really be elementary, it would be impossible to understand
their properties.
In the strand model, particles are not really elementary, but neither are they, in the
usual sense, composed. Particles are tangles of unobservable strands. In this way, the
strand model retains the useful aspects of the idea of elementary particle but gets rid of
its limitations. In a sense, the strand model can be seen as eliminating the concepts of
elementariness and of particle. This confirms and realizes another requirement that we
had deduced earlier on.
Motion Mountain – The Adventure of Physics
∗∗
the basis of the stra nd m odel
Challenge 123 e
161
pression ‘it from qubit’ that is propagated by David Deutsch. A qubit is a quantummechanical two-level system. What is the difference between the fundamental principle
and a qubit?
∗∗
Ref. 154
Page 299
D o strands unify? – The millennium list of open issues
Page 64
Page 73
Page 127
Challenge 124 e
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 90
Does the strand model reproduce all the paradoxical results we found in the first
chapters? Yes, it does. The strand model implies that vacuum cannot be distinguished
from matter at Planck scales: both are made of strands. The strand model implies that
observables are not real numbers at Planck scales. The strand model implies that the
universe and the vacuum are the same, when explored at high precision: both are made
of one strand. The strand model also implies that the number of particles in the universe
is not clearly defined and that nature is not a set. You can check by yourself that all other
paradoxes appear automatically. Furthermore, almost all requirements for a final theory
listed in Table 6 are fulfilled. Only two requirements of the table must be discussed in
more detail: the requirements of complete precision and of unmodifiability. We start
with complete precision.
If strands really describe all of nature, they must explain the inverse square dependence with distance of the electrostatic and of the gravitational interaction. But that is
not sufficient. If the strand model is a final, unified description, it must provide complete
precision. This requires, first of all, that the model describes all experiments. As will be
shown below, this is indeed the case, because the strand model contains both general relativity and the standard model of particle physics. But secondly and most importantly,
the model must also settle all those questions that were left unanswered by twentiethcentury fundamental physics. Because the questions, the millennium list of open issues,
are so important, they are given, again, in Table 8.
Motion Mountain – The Adventure of Physics
Page 395
Is the strand model confirmed by other, independent research? Yes, a few years after the
strand model appeared, this started to happen. For example, in a long article exploring
the small scale structure of space-time from various different research perspectives in
general relativity, Steven Carlip comes to the conclusion that all these perspectives suggest the common result that ‘space at a fixed time is thus threaded by rapidly fluctuating
lines’. This is exactly what the strand model states.
Other theoretical approaches that confirm the strand model are mentioned in various
places later in the text. Despite such developments, the essential point remains to check
how the strand model compares with experiment. Given that the strand model turns out
to be unmodifiable, there are no ways to amend predictions that turn out to be wrong. If
a single prediction of the strand model turns out to be incorrect, the model is doomed.
But so far, no experimental prediction of the strand model contradicts experiments.
162
7 the basis of the stra nd m od el
TA B L E 8 The millennium list: everything the standard model and general relativity cannot explain; thus,
also the list of the only experimental data available to test the final, unified description of motion.
O b s e rva b l e P r o p e rt y u n e x p l a i n e d s i n c e t h e y e a r 2 0 0 0
Local quantities unexplained by the standard model: particle properties
Concepts unexplained by the standard model
SU(2)
SU(3)
Renorm. group
?? = 0
? = ∫? SM d?
the origin of the invariant Planck units of quantum field theory
the number of dimensions of physical space and time
the origin of Poincaré symmetry, i.e., of spin, position, energy, momentum
the origin and nature of wave functions
the origin of particle identity, i.e., of permutation symmetry
the origin of the gauge groups, in particular:
the origin of the electromagnetic gauge group, i.e., of the quantization of
electric charge, of the vanishing of magnetic charge, and of minimal coupling
the origin of weak interaction gauge group, its breaking and P violation
the origin of strong interaction gauge group and its CP conservation
the origin of renormalization properties
the origin of the least action principle in quantum theory
the origin of the Lagrangian of the standard model of particle physics
Global quantities unexplained by general relativity and cosmology
0
1.2(1) ⋅ 1026 m
?de = Λ?4 /(8π?)
≈ 0.5 nJ/m3
(5 ± 4) ⋅ 1079
?dm
the observed flatness, i.e., vanishing curvature, of the universe
the distance of the horizon, i.e., the ‘size’ of the universe (if it makes sense)
the value and nature of the observed vacuum energy density, dark energy or
cosmological constant
the number of baryons in the universe (if it makes sense), i.e., the average
visible matter density in the universe
the density and nature of dark matter
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
?, ℏ, ?
3+1
SO(3,1)
Ψ
?(?)
Gauge symmetry
U(1)
Motion Mountain – The Adventure of Physics
? = 1/137.036(1) the low energy value of the electromagnetic coupling or fine structure constant
?w or ?w
the low energy value of the weak coupling constant or the value of the weak
mixing angle
?s
the value of the strong coupling constant at one specific energy value
?q
the values of the 6 quark masses
?l
the values of 6 lepton masses
?W
the value of the mass of the ? vector boson
?H
the value of the mass of the scalar Higgs boson
?12 , ?13 , ?23
the value of the three quark mixing angles
?
the value of the CP violating phase for quarks
?
?
?
?12
, ?13
, ?23
the value of the three neutrino mixing angles
?
? , ?1 , ?2
the value of the three CP violating phases for neutrinos
3⋅4
the number of fermion generations and of particles in each generation
J, P, C, etc.
the origin of all quantum numbers of each fermion and each boson
the basis of the stra nd m odel
163
TA B L E 8 (Continued) The millennium list: everything the standard model and general relativity cannot
explain; also the only experimental data available to test the final, unified description of motion.
O b s e rva b l e P r o p e rt y u n e x p l a i n e d s i n c e t h e y e a r 2 0 0 0
?0 (1, ..., c. 1090 )
the initial conditions for c. 1090 particle fields in the universe (if or as long as
they make sense), including the homogeneity and isotropy of matter distribution, and the density fluctuations at the origin of galaxies
Concepts unexplained by general relativity and cosmology
?, ?
R × S3
???
?? = 0
? = ∫? GR d?
the origin of the invariant Planck units of general relativity
the observed topology of the universe
the origin and nature of curvature, the metric and horizons
the origin of the least action principle in general relativity
the origin of the Lagrangian of general relativity
— Reproduce quantum theory, the standard model, general relativity and cosmology.
— Explain masses, mixing angles and coupling constants.
Are strands final? – On generalizations and modifications
“
The chief attraction of the theory lies in its
logical completeness. If a single one of the
conclusions drawn from it proves wrong, it
must be given up; to modify it without
destroying the whole structure seems
impossible.
Albert Einstein, The Times, 28. 11. 1919.
”
If a description of motion claims to be final, it must explain all aspects of motion. To
be a full explanation, such a description must not only be logically and experimentally
complete, it must also be unmodifiable. Even though Einstein made the point for general relativity, this important aspect is rarely discussed with clarity. In particular, any
unmodifiable explanation has two main properties: first, it cannot be generalized, and
second, it is not itself a generalization.
Generalizing models is a sport among theoretical and mathematical physicists. If you
have a description of a part of nature, they will try to find more general cases. For any
candidate unified description, they will try to explore the model in more than three dimensions, with more than three generations of quarks, with more complicated gauge
symmetries, with different types of supersymmetry, with more Higgs bosons, or with
additional heavy neutrinos. In the case of the strand model, researchers will also ex-
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Of course, only the second point is the definite test for a final, unified description. But
we need the first point as well. The following chapters deal with both points.
Motion Mountain – The Adventure of Physics
The open issues in the millennium list must be resolved by any final, unified model
of nature, and thus also by the strand model. All the open issues can be summarized in
two general points:
164
7 the basis of the stra nd m od el
plore models with more complicated entities than strands, such as bands or bifurcating
entities, and any other generalization they can imagine.
⊳ Can a final description of nature have generalizations? No.
⊳ Can the unified theory be a generalization of existing theories? No.
Ref. 155
* Independently, David Deutsch made a similar point with his criterion that an explanation is only correct
if it is hard to vary. Used in the case of a final theory, we can say that the final theory must be an explanation
of general relativity and of the standard model. This implies that the final theory must be hard to vary. This
matches the above conclusion that the final theory must be unmodifiable.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Challenge 125 e
Because neither general relativity nor the standard model of particle physics are able to
explain the millennium issues, any generalization of them would also be unable to do so.
Generalizations have no explanatory power. If the unified theory were a generalization
of the two existing theories, it could not explain any of the millennium issues of Table 8!
Therefore, general relativity and the standard model of particle physics must be approximations, but not special cases, of the final theory. In particular, if the strand model is a
final description, approximations of the strand model must exist, but special cases must
not. This is indeed the case, as we will find out.
To summarize, a final theory must be an explanation of all observations. An explanation of an observation is the recognition that it follows unambiguously, without alternative, from a general property of nature. We conclude that the final, unified description
of motion must neither allow generalization nor must it be a generalization of either
the standard model or general relativity. The unified theory cannot be generalized and
cannot be ‘specialized’; the unified theory must be unmodifiable.* This requirement is
extremely strong; you may check that it eliminates most past attempts at unification. For
example, this requirement eliminates grand unification, supersymmetry and higher dimensions as aspects of the final theory: indeed, these ideas generalize the standard model
Motion Mountain – The Adventure of Physics
Indeed, if it were possible to generalize the final description, it would lose the ability to
explain any of the millennium issues! If a candidate unified theory could be generalized,
it would not be final. In short, if the strand model is a final description, the efforts of
theoretical and mathematical physicists just described must all be impossible. So far,
investigations confirm this prediction: no generalization of the strand model has been
found yet.
Where does this fondness for generalization come from? In the history of physics,
generalizations often led to advances and discoveries. In the past, generalizations often
led to descriptions that had a wider range of validity. As a result, generalizing became
the way to search for new discoveries. Indeed, in the history of physics, the old theory
often was a special case of the new theory. This relation was so common that usually,
approximation and special case were taken to be synonyms. This connection leads to a
second point.
General relativity and the standard model of particle physics must indeed be approximations of the final theory. But can either general relativity or the standard model be
special cases of the final, unified theory? Or, equivalently:
the basis of the stra nd m odel
Challenge 126 e
165
of elementary particles and they are modifiable. Therefore, all these ideas lack explanatory power.
A final and unified theory must be an unmodifiable explanation of general relativity
and the standard model. Because neither supersymmetry, nor the superstring conjecture,
nor loop quantum gravity explain the standard model of particle physics, they are not
unified theories. Because these models are modifiable, they are not final theories. In
fact, at least one of these two aspects is lacking in every candidate final theory proposed
in the twentieth century.
We will discover below that the strand model is unmodifiable. Its fundamental principle cannot be varied in any way without destroying the whole description. Indeed, no
modification of the strand model or of the fundamental principle has been found so
far. We will also discover that the strand model explains the standard model of particle
physics and explains general relativity. The strand model is thus a candidate for the final
theory.
“
Page 115
Ref. 140
Ref. 156
Ref. 157
Ref. 158
”
Antiquity
Let us assume that we do not know yet whether the strand model can be modified
or not. Two other reasons still induce us to explore featureless strands as basis for a
unified description. First, featureless strands are the simplest known model that unifies quantum field theory and general relativity. Second, featureless strands are the only
known model that realizes an important requirement: a unified description must not be
based on points, sets or any axiomatic system. Let us explore the issue of simplicity first.
In order to reproduce three-dimensional space, Planck units, spin, and black-hole entropy, the fundamental constituents must be extended and fluctuating. We have deduced
this result in detail in the previous chapter. The extension must be one-dimensional, because this is the simplest option, and it is also the only option compatible with threedimensional space. In fact, one-dimensional strands explain the three-dimensionality of
space, because tangles of one-dimensional strands exist only in three spatial dimensions.
In four or more dimensions, any tangle or knot can be undone; this is impossible in three
spatial dimensions.
No simpler model than featureless strands is possible. All other extended constituents
that have been explored – ribbons, bands, strings, membranes, posets, branched lines,
networks, crystals and quantum knots – increase the complexity of the model. In fact
these constituents increase the complexity in two ways: they increase the number of features of the fundamental constituents and they complicate the mapping from the model
to observation.
First, no other model based on extension uses featureless constituents. In all other
models, the fundamental constituents have properties such as tension, field values, coordinates, quantum numbers, shape, twists, orientation, non-trivial topological information, etc. In some models, space-time is non-commutative or fermionic. All these features are assumed; they are added to the model by fiat. As such, they allow alternatives
* ‘Simplicity is the seal of truth.’
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 105
Simplex sigillum veri.*
Motion Mountain – The Adventure of Physics
Why strands? – Simplicit y
166
Ref. 150
7 the basis of the stra nd m od el
“
Without the concepts place, void and time,
change cannot be. [...] It is therefore clear [...]
that their investigation has to be carried out, by
studying each of them separately.
Aristotle Physics, Book III, part 1.
”
The strand model describes strands as fluctuating in a background space-time of three
plus one space-time dimensions. The background space-time is introduced by the observer. The background is thus different for every observer; however, all such backgrounds have three dimensions of space and one of time. The observer – be it a machine,
an animal or a human – is itself made of strands, so that in fact, the background space is
itself the product of strands.
We therefore have a fundamental circular definition: we describe strands with a back-
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Why strands? – The fundamental circul arit y of physics
Motion Mountain – The Adventure of Physics
and are difficult if not impossible to justify. In addition, these features increase the complexity of the possible processes. In contrast, the strand model has no justification issue
and no complexity issue.
Secondly, the link between more complicated models and experiment is often intricate and sometimes not unique. As an example, the difficulties to relate superstrings to
experiments are well-known. In contrast, the strand model argues that the experimentally accessible Dirac equation of quantum field theory and the experimentally accessible
field equations of general relativity arise directly, from an averaging procedure of crossing switches. Indeed, the strand model proposes to unify these two halves of physics with
only one fundamental principle: strand crossing switches define Planck units. In fact, we
will find out that the strand model describes not only vacuum and matter, but also gauge
interactions and particle properties as natural consequences of the structure of nature at
Planck scales. The comparable ideas in other models are much more elaborate.
We remark that building three-dimensional physical space from strands is even simpler than building it from points! In order to build three-dimensional space from points,
we need concepts such as sets, neighbourhoods, topological structures and metric structures. And despite all these intricate concepts, the concept of space defined in this way
still has no defined physical length scale; in short, it is not the same as physical space.
In contrast, in order to build three-dimensional physical space from strands, we need
no fundamental points, sets, or metric structures; we only need long-time averages of
strands and their crossings. And the length scale is built in.
All this suggests that the strand model, based on featureless, one-dimensional and
fluctuating constituents, might be the model for unification with the smallest number
of concepts, thus satisfying Occam’s razor. In fact, we will discover that strands indeed
are the simplest way to model particles, interactions and the vacuum, while fulfilling the
requirements of a final theory.
The simplicity of a model helps in two ways. First, the simpler a model is, the freer it
is of ideology, preconceptions and beliefs. Secondly, the simpler a model is, the easier it
can be checked against observation. In particular, a simple model allows simple checking
of its solution of paradoxes. Above all, we can resolve the most important paradox of
physics.
the basis of the stra nd m odel
167
TA B L E 9 The differences between nature and any description.
Description
Nature is not a set.
Descriptions need sets to allow talking and
thinking.
Descriptions need events, points and
continuous 3 + 1-dimensional space-time to
allow formulating them.
Descriptions need point particles to allow
talking and thinking.
Descriptions need locality to allow talking and
thinking.
Descriptions need a background to allow
talking and thinking.
Descriptions need to break duality to allow
talking and thinking.
Axiomatic descriptions are needed for precise
talking and thinking.
Nature has no events, no points and no
continuity.
Nature has no point particles.
Nature is not local.
Nature has no background.
Nature shows something akin to ? ↔ 1/?
duality.
Nature is not axiomatic but contains circular
definitions.
Page 108
⊳ We use both continuous background space-time and discrete strands to describe nature.
In a few words: A unified model of physics allows talking about motion with highest
precision; this requirement forces us to use, at the same time, both continuous spacetime and discrete strands. This double use is not a contradiction but, as just explained,
the result of a circular definition. Since we, the talkers, are part of nature, a unified model
means that we, the talkers, talk about ourselves.
We stress that despite the circularity of physics, Gödel’s incompleteness theorem does
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
ground, and the background with strands. Strands thus do not provide an axiomatic
system in the mathematical sense. This fulfils one of the requirements for the unified
description.
Why does the fundamental circular definition arise? Physics is talking (and thinking) about nature and motion. A unified model of physics is talking about motion with
highest precision. This implies that on the one hand, as talkers, we must use concepts that
allow us to talk. Talking and thinking requires that we use continuous space and time:
on short, we must use a background. The background must be continuous, without minimum length. On the other hand, to talk with precision, we must have a minimum length,
and use strands. There is no way to get rid of this double and apparently contradictory
requirement. More such contradictory requirements are listed in Table 9. We know that
nature is not a set, has no points, no point particles and no locality, but that it is dual.
But in order to talk about nature, we need a background that lacks all these properties.
Because there is no way to get rid of these apparently contradictory requirements, we
don’t:
Motion Mountain – The Adventure of Physics
Nature
168
7 the basis of the stra nd m od el
Universe’s
horizon or
‘border
of space’
(pink)
Universe’s
tangle
(blue
lines)
Background
space
(grey)
Background
space
(grey)
Physical
space or
vacuum
(white)
Particle
tangle
(tangled
blue
lines)
Physical
space or
vacuum
(white)
Vol. III, page 308
Page 108
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 159
not apply to the situation. In fact, the theorem does not apply to any unified theory
of physics for two reasons. First, the incompleteness theorem applies to self-referential
statements, not to circular definitions. Self-referential statements do not appear in physics, not in sensible mathematics and not in the strand model. Secondly, Gödel’s theorem
applies to mathematical structures based on sets, and the final theory is not based on
sets.
We do not state that background space and time exist a priori, as Immanuel Kant
states, but only that background space and time are necessary for thinking and talking, as
Aristotle states. In fact, physical space and time result from strands, and thus do not exist
a priori; however, background space and time are required concepts for any description
of observations, and thus necessary for thinking and talking. Figure 16 illustrates the
solution proposed by the strand model.
We have always to be careful to keep the fundamental circular definition of strands
and backgrounds in our mind. Any temptation to resolve it leads astray. For example, if
we attempt to define sets or elements (or points) with the help of measurements, we are
hiding or forgetting the fundamental circularity. Indeed, many physicists constructed
and still construct axiomatic systems for their field. The fundamental circularity implies
that axiomatic systems are possible for parts of physics, but not for physics as a whole.
Indeed, there are axiomatic descriptions of classical mechanics, of electrodynamics, of
quantum theory, of quantum field theory, and even of general relativity. But there is no
axiomatic system for all of physics – i.e., for the description of all motion – and there
cannot be one.
A further issue must be discussed in this context. As mentioned, strands fluctuate
in a background space, and only crossing switches can be observed. In particular, this
implies that the mathematical points of the background space cannot be observed. In
other words, despite using mathematical points to describe the background space (and
strands themselves), none of them have physical significance. Physical points do not exist
Motion Mountain – The Adventure of Physics
F I G U R E 16 In the strand model, physical space – or vacuum – and background space are distinct, both
near the horizon and near particles.
the basis of the stra nd m odel
A strand :
169
A funnel :
F I G U R E 17 Two equivalent depictions of the fundamental constituents of nature: strands and funnels.
Another type of constituent also fulfils all the conditions for a unified description. As
shown in Figure 17, as an alternative to fluctuating strands, we can use fluctuating funnels
as fundamental constituents. In the description with funnels, nature resembles a complicated tangle of a three-dimensional space that is projected back into three dimensions.
Funnels show that the strand model only requires that the effective minimal effective
diameter of a strand is the Planck length; it could have other diameters as well. Funnels
also show that due to varying diameters, strands can, through their fluctuations, literally
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Funnels – an equivalent alternative to strands
Motion Mountain – The Adventure of Physics
Ref. 160
in the strand model. Physical locations of events are due to crossing switches, and can
at best be localized to within a Planck length. The same limitation applies to physical
events and to physical locations in time. A natural Planck-scale non-locality is built into
the model. This realizes a further requirement that any unified description has to fulfil.
The situation for physicists working on unification is thus harder – and more fascinating – than that for biologists, for example. Biology is talking about living systems.
Biologists are themselves living systems. But in the case of biologists, this does not lead
to a circular definition. Biology does not use concepts that contain circular definitions: a
living being has no problems describing other living beings. Even neurobiologists, who
aim to explore the functioning of the brain, face no fundamental limit doing so, even
though they explore the human brain using their own brain: a brain has no problem
describing other brains. In contrast, physicists working on unification need to live with
circularity: a fundamental, precise description of motion requires to be conscious of our
own limitations as describing beings. And our main limitation is that we cannot think
without continuous space and time, even though these concepts do not apply to nature.
We conclude: A unified description cannot be axiomatic, cannot be based on observable physical points, must distinguish physical space from background space, and cannot be
background-independent. Many models based on extended constituents also use backgrounds. However, most models also allow the definition of sets and axiomatic descriptions. Such models thus cannot be candidates for a unified description of nature. In
contrast, the strand model keeps the fundamental circularity of physics intact; it does
not allow an axiomatic formulation of fundamental physics, and only allows points or
sets as approximate concepts.
170
Page 129
Challenge 127 e
7 the basis of the stra nd m od el
Knots and the ends of strands
summary on the fundamental principle – and on
continuit y
Page 147
We have introduced featureless, fluctuating strands as common constituents of space,
matter, radiation and horizons. We defined fundamental events as crossing switches of
strands. All physical processes are composed of fundamental events. Events and the
values of all physical observables are defined with the help of Planck units, which in turn
are due to crossing switches of strands. The definition of all physical observables through
Planck units with the help of crossing switches of strands is the fundamental principle.
Using the fundamental principle, continuity of any kind – of space, fields, wave functions or time – results from the time averaging of crossing switches. This issue will be
explored in detail below.
The strand model fulfils the general requirements for the final and unified description
listed in Table 6, provided that it describes all motion with full precision and that it is
* Two issues that put this equivalence into question are ending funnels and diameter behaviour under
boosts. The first issue is subject of research, but it is expected that it poses no problem. The second issue is mitigated by the shivering of the background space.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
In the original strand model, developed in the year 2008, strands that contain knots were
part of the allowed configurations. This has the disadvantage that the formation of a knot
requires at least one loose end that is pulled through a strand configuration. Such loose
ends, however, produce a number of issues that are difficult to explain, especially during
the emission and absorption of knotted tangles.
Later, in 2015, it became clear that strands without knots are sufficient to recover the
standard model of particle physics. This approach is shown in the following. Knots and
the ends of strands play no role in the model any more; the aim of highest possible simplicity is now realized.
Motion Mountain – The Adventure of Physics
be everywhere in space and thus effectively fill space, even if their actual density is low.
Funnels resemble many other research topics. Funnels are similar to wormholes; however, both their ends lead, at the border of space, ‘into’ usual three-dimensional space.
Funnels are also similar to D-branes, except that they are embedded in three spatial dimensions, not ten. Funnels also resemble a part of an exotic manifold projected into
three-dimensional space. Fluctuating funnels also remind us of the amoeba mentioned
above. However, the similarities with wormholes, D-branes or exotic manifolds are of
little help: so far, none of these approaches has led to viable models of unification.
A first check shows that the funnel alternative seems completely equivalent to strands.*
You might enjoy testing that all the conclusions deduced in the following pages appear
unchanged if strands are replaced by funnels. In particular, also funnels allow us to deduce quantum field theory, the standard model and general relativity. Due to the strict
equivalence between strands and funnels, the choice between the two alternatives is a
matter of taste or of visualization, but not a matter of physics. We use strands in the
following, as they are simpler to draw.
sum m ary on the fundamental principle – a nd on continuity
Page 162
171
unmodifiable.
At this point, therefore, we must start the comparison with experiment. We need
to check whether strands describe all motion with complete precision. Fortunately, the
task is limited: we only need to check whether strands solve each of the millennium
issues listed in Table 8. If the strand model can solve those issues, then it reproduces all
observations about motion and provides a final and unified description of nature. If the
issues are not solved, the strand model is worthless.
Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Chapter 8
QUANT UM T HEORY OF MAT T ER
DEDUCED FR OM ST R ANDS
W
In nature, particles move in the vacuum. The vacuum is free of matter and energy. In the
strand model,
⊳ Vacuum is a collection of fluctuating, unknotted and untangled strands.
Page 156
Page 236
The vacuum is illustrated in Figure 14. The time average of unknotted and untangled
strands has no energy and no matter content, because there are – averaged over time –
no crossing switches and no tangles. The temporary crossing switches that can appear
through fluctuations of the vacuum will turn out to be virtual particles; we will explore
them below. We note that the physical vacuum, being a time average, is continuous. The
flat physical vacuum is also unique: it is the same for all observers. The strand model
thus contains both a minimum length and a continuous vacuum. The two aspects do
not contradict each other.
In nature, quantum particles move: quantum particles change position and phase over
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Strands, vacuum and particles
Motion Mountain – The Adventure of Physics
e show in this chapter that featureless strands that fluctuate, together
ith the fundamental principle – defining ℏ/2 as a crossing switch – imply
ithout alternative that matter is described by quantum theory. More precisely,
we deduce that tangles of fluctuating strands reproduce the spin 1/2 behaviour of matter
particles, allow us to define wave functions, and imply the Dirac equation for the motion of matter. In particular, we first show that the components and phases of the wave
function at a point in space are due to the orientation and phase of strand crossings at
that point. Then we show that the Dirac equation follows from the belt trick (or string
trick).
Furthermore, we show that strands imply the least action principle and therefore, that
tangles of fluctuating strands are described by the Lagrangian of relativistic quantum
particles. So far, it seems that the strand model is the only microscopic model of relativistic quantum theory that is available in the research literature.
In the present chapter, we derive the quantum theory of matter: we show that strands
reproduce all observations about fermions and their motion. We leave for later the derivation of the quantum theory of light and the nuclear interactions, the standard model of
elementary particles, and the quantum description of gravitation. As usual in quantum
theory, we work in flat space-time.
q ua ntum theory d ed uced from stra nds
A fermion
Strand model :
time average
of crossing
switches
tails
belt trick
orientation
core
positions
phases
crossing
orientations
173
Observed
probability
density :
spin
orientation
position
phase
F I G U R E 18 A fermion is described by a tangle of two or three strands. The crossings in the tangle core
and their properties lead, after averaging, to the wave function and the probability density.
⊳ An elementary matter particle, or fermion, is a tangle of two or more strands
that realizes the belt trick.
⊳ The position of a particle is given by the centre of the averaged tangle core.
The particle position is thus the average of all its crossing positions.
⊳ The phase of a matter particle is given by half the angle that describes the
orientation of the tangle core around the spin axis. The particle phase is thus
the average of all its crossing phases.
⊳ The spin orientation of a matter particle is given by the rotation axis of the
core. The spin orientation is thus the average of all its crossing orientations.
⊳ The wave function of a matter particle is a blurred rendering of the crossing
of its fluctuating strands.
Page 184
These definitions are illustrated in Figure 18 and will be explored in detail below. We
note that all these definitions imply a short-time average over tangle fluctuations. With
the definitions, we get:
⊳ Motion of any quantum particle is the change of the position and orientation
of its tangle core.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
The details of this definition will become clear shortly, including the importance of the
related tangle family. In every tangle, the important structure is the tangle core: the core
is the that region of the tangle that contains all the links and crossings. The core is connected to the border of space by the tails of the tangle.
Motion Mountain – The Adventure of Physics
time. We therefore must define these concepts. At this stage, as just explained, we concentrate on quantum matter particles and leave radiation particles for later on. As illustrated in Figure 18 and Figure 19, we define:
174
8 q ua ntum theory d ed uced from stra nds
In nature, quantum particle motion is described by quantum theory. The main property
of quantum theory is the appearance of the invariant quantum of action ℏ. In the strand
model, ℏ/2 is described by a single crossing switch; the value of the quantum of action
is thus invariant by definition.
We now explore in detail how the quantum of action ℏ determines the motion of
quantum particles. In particular, we will show that tangle fluctuations reproduce usual
textbook quantum theory. As an advance summary, we clarify that
⊳ Free quantum particle motion is due to fluctuations of tangle tails. The deformations of the tangle core are not important for free motion, and we can
neglect them in this case.
Rotation, spin 1/2 and the belt trick
⊳ Spin is core rotation.
Indeed, in the strand model, all quantum particles, including those with spin 1/2, differ
from everyday objects such as stones, and the essential difference is due to extension:
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
In nature, quantum particles are described by their behaviour under rotation and by their
behaviour under exchange. The behaviour of a particle under rotation is described by
its spin value, its spin axis and its phase. The behaviour of quantum particles under
exchange can be of two types: a quantum particle can be a fermion or a boson. In nature,
particles with integer spin are bosons, and particles with half-integer spin are fermions. This
is the spin–statistics theorem.
We now show that all properties of particle rotation and exchange follow from the
strand model. We start with the case of spin 1/2 particles, and first clarify the nature of
particle rotation. (We follow the usual convention to use ‘spin 1/2’ as a shorthand for
‘z-component of spin with value ℏ/2’.)
It is sometimes claimed that spin is not due to rotation. This misleading statement is
due to two arguments that are repeated so often that they are rarely questioned. First,
it is said, spin 1/2 particles cannot be modelled as small rotating stones. Secondly, it is
allegedly impossible to imagine rotating electric charge distributions with a speed of rotation below that of light and an electrostatic energy below the observed particle masses.
These statements are correct. Despite being correct, there is a way to get around them,
namely by modelling particles with strands; at the present stage, we focus on the first
argument: we will show that spin can be modelled as rotation.
In the strand model, for all quantum particles we have:
Motion Mountain – The Adventure of Physics
Page 222
In other words, when exploring quantum theory, we approximate tangle cores as being rigid. We will study core deformations in the next chapter, where we show that they
are related to interactions. Core deformations will lead to quantum field theory. In this
chapter we explore just the deformations of tangle tails; they produce the motion of free
(and stable) quantum particles. In short, tail deformations lead to quantum mechanics.
To deduce quantum mechanics from strands, we first study the rotation and then the
translation of free matter particles.
q ua ntum theory d ed uced from stra nds
175
The belt trick or string trick or plate trick or scissor trick explains the possibility of continuous
core rotation for any number of tails. A rotation by 4π is equivalent to none at all :
moving
tails
aside
moving
upper
tails
moving
lower
tails
moving
all
tails
The same equivalence can be shown with the original version of the belt trick :
or simply rearranging the belts,
independently of their number,
yields the other situation
F I G U R E 19 The belt trick – or string trick or plate trick or scissor trick – shows that a rotation by 4 π of a
central object with three or more tails (or with one or more ribbons) attached to spatial infinity is
equivalent to no rotation at all. This equivalence allows a suspended object, such as a belt buckle or a
tangle core, to rotate for ever. The belt trick thus shows that tangle cores made from two or more
strands behave as spin 1/2 particles.
⊳ Quantum particles are particles whose tails cannot be neglected.
For stones and other everyday objects, tails do not play an important role, because everyday objects are mixed states, and not eigenstates of angular momentum. In short, in
everyday objects, tails can be neglected. Therefore, everyday objects are neither fermions nor bosons. But for quantum particles, the tails are essential. Step by step we will see
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
rotating the buckle either by 4π,
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core
(or
belt
buckle)
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8 q ua ntum theory d ed uced from stra nds
F I G U R E 20 The belt trick: a double
rotation of the belt buckle is equivalent to
no rotation; te animation shows one way
in which the belt trick can be performed.
Not shown: the belt trick is also possible
with any number of belts attached to the
buckle. (QuickTime film © Greg Egan)
that
⊳ The tails of quantum particles explain their spin behaviour, their exchange
behaviour and their wave behaviour.
Ref. 161
In particular, we will see that in the strand model, wave functions are blurred tangles; we
can thus explore the general behaviour of wave functions by exploring the behaviour of
tangles.
The spin behaviour of quantum particles is a consequence of strand tails. Indeed, it
has been known for about a century that the so-called belt trick – illustrated in Figure 19,
Figure 20, Figure 21 and Figure 22 – can be used, together with its variations, to model
the behaviour of spin 1/2 particles under rotations. The belt trick is the observation that
a belt buckle rotated by two full turns – in contrast to a buckle rotated by only one full
turn – can be brought back into its original state without moving the buckle; only the
motion of the belt is necessary. The belt trick is also called the scissor trick, the plate trick,
the string trick, the Philippine wine dance or the Balinese candle dance. It is sometimes
incorrectly attributed to Dirac.
The belt trick is of central importance in the strand model of spin 1/2 particles. In
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
animation shows another way – another
direction – in which the trick can be
performed, for the same belt orientation
as in the previous figure. (QuickTime film
© Greg Egan)
Motion Mountain – The Adventure of Physics
F I G U R E 21 The belt trick again: this
q ua ntum theory d ed uced from stra nds
177
Page 198
Challenge 128 e
the strand model, all spin 1/2 particles are made of two (or more) tangled strands, and
thus have four (or more) tails to the ‘border’, as shown in Figure 19. For such tangles, a
rotation by 4π of the tangle core – thus a rotation by two full turns – can bring back the
tangle to the original state, provided that the tails can fluctuate. Any system that returns
to its original state after rotation by 4π is described by spin 1/2. In fact, the tails must be
unobservable for this equivalence to hold; in the strand model, tails are simple strands
and thus indeed unobservable. We will show below that the intermediate twisting of the
tails that appears after rotation by only 2π corresponds to a multiplication of the wave
function by −1, again as expected from a spin 1/2 particle.
If we replace each belt by its two coloured edges, Figure 22 shows how tails behave
when a spin 1/2 tangle is rotated. By the way, systems with tails – be they strands or
bands – are the only possible systems that realize the spin 1/2 property. Only systems with
tails to spatial infinity have the property that a rotation by 4π is equivalent to no rotation
at all. (Can you show this?) The fundamental connection between spin 1/2 and extension
is one of the properties that led to the strand model.
The animations show that the belt trick works with one and with two belts attached
to the buckle. In fact, belt trick works with any number of belts attached to the buckle.
The belt trick even works with infinitely many belts, and also with a full two-dimensional
sheet. The wonderful video www.youtube.com/watch?v=UtdljdoFAwg by Gareth Taylor
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 189
a particle. The animation then shows that such a particle (the square object) can return to the starting
position after rotation by 4π (but not after 2π). Such a ‘belted’ or ‘tethered’ particle thus fulfils the
defining property of a spin 1/2 particle: rotating it by 4π is equivalent to no rotation at all. The belted
square thus represents the spinor wave function; for example, a 2π rotation leads to a twist; this means
a change of the sign of the wave function. A 4π rotation has no influence on the wave function. The
equivalence is shown here with two attached belts, but the trick works with any positive number of
belts! You can easily repeat the trick at home, with a paper strip or one or several real belts. (QuickTime
film © Antonio Martos)
Motion Mountain – The Adventure of Physics
F I G U R E 22 Assume that the belt cannot be observed, but the square object can, and that it represents
178
8 q ua ntum theory d ed uced from stra nds
a rotating ball attached to a sheet
(QuickTime film © www.ariwatch.
com/VS/Algorithms/DiracStringTrick.
htm).
⊳ An object or a tangle core that is attached by (three or more) tails to the
border of space can rotate continuously.
Here we made the step from belts to strands. In other terms, the possibility of continuous
rotation allows us to describe spin 1/2 particles by rotating tangles. In other terms,
⊳ Rotating tangles model spin.
The tail fluctuations required to rearrange the tails after two full turns of the core can
be seen to model the average precession of the spin axis. We thus confirm that spin and
rotation are the same for spin 1/2 particles.
The belt trick is not unique
Ref. 161
Challenge 129 e
One aspect of the belt trick seems unmentioned in the research literature: after a rotation
of the belt buckle or tangle core by 4π, there are various options to untangle the tails. Two
different options are shown in Figure 20 and Figure 21. You can test this yourself, using a
real belt. In fact, there are two extreme ways to perform the belt trick, and a continuum
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
and the slightly different animation of Figure 23 both illustrate the situation. A sphere
glued to a flexible sheet can be rotated as often as you want: if you do this correctly, there
is no tangling and you can go on for ever.
The animations of the belt trick lead us to a statement on strands and tangles that is
central for the strand model:
Motion Mountain – The Adventure of Physics
F I G U R E 23 The belt trick realized as
q ua ntum theory d ed uced from stra nds
179
fresh blood in
welded seals
platelet-poor
blood out
plastic bag
rotates at over 100
revolutions per second
F I G U R E 24 In an apheresis machine, the central bag spins at high speed despite being connected with
tubes to a patient; this happens with a mechanism that continuously realizes the belt trick (photo
© Wikimedia).
Motion Mountain – The Adventure of Physics
of options in between. These options will be of central importance later on: the options
require a description of fermions with four complex functions. We will discover that
the various options of the belt trick are related to the difference between matter and
antimatter and to the parity violation of the weak interaction.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
F I G U R E 25 The basis of the
apheresis machine – and yet
another visualisation of the belt
trick, here with 6 belts
(QuickTime film © Jason Hise).
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8 q ua ntum theory d ed uced from stra nds
Without the belt trick, the apheresis machines found in many hospitals would not work.
When a person donates blood platelets, the blood is continuously extracted from one
arm and fed into a bag in a centrifuge, where the platelets are retained. The platelet-free
blood then flows back into the other arm of the donor. This happens continuously, for
about an hour or two. In order to be sterile, tubes and bag are used only once and are
effectively one piece, as shown in Figure 24. Apheresis machines need tethered rotation
to work. Topologically, this set-up is identical to a fermion tangle: each tube corresponds
to one belt, or two strand tails, and the rotating bag corresponds to the rotating core.
In such apheresis machines, the centrifugation of the central bag takes place at over
100 revolutions per second, in the way illustrated in Figure 25. To avoid tangling up the
blood tubes, a bracket moves the tubes during each rotation, alternatively up and down.
This so-called anti-twister mechanism produces precisely the motion along which the
belt moves when it is untangled after the buckle is rotated by 4π. An apheresis machine
thus performs the belt trick 50 times per second, with each rotation of the centrifugation.
Due to the centrifugation, the lighter platelets are retained in the bag, and the heavier
components of the blood are pumped back to the donor. The retained platelets are then
used to treat patients with leukaemia or severe blood loss due to injury. A single platelet
donation can sustain several lives.
In short, without the belt trick, platelet donations would not be sterile and would thus
be impossible. Only the belt trick, or tethered rotation, allows sterile platelet donations
that save other people’s lives.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
An aside: the belt trick saves lives
Motion Mountain – The Adventure of Physics
F I G U R E 26 A version of the
antitwister mechanism, or belt
trick, with 96 belts attached to a
black and white ball that rotates
continuously (QuickTime film
© Jason Hise).
q ua ntum theory d ed uced from stra nds
181
Any two sufficiently complex tangles behave as fermions
under (single or double) exchange of their cores (try it) :
Motion Mountain – The Adventure of Physics
is possible to rearrange their tails to yield the original situation. This is not possible when the tangles
are only exchanged once. Spin 1/2 tangles are thus fermions. The figure presents common systems that
show this behaviour.
Fermions and spin
In nature, fermions are defined as those particles whose wave function changes sign when
they are exchanged. Does the strand model reproduce this observation?
We will see below that in the strand model, wave functions are blurred tangles. We
thus can explore exchange properties of quantum particles and of their wave functions
by exploring the exchange properties of their tangles. Now, if we exchange two tangle
cores twice, while keeping all tails connections fixed, tail fluctuations alone can return the
situation back to the original state! The exchange properties of spin 1/2 tangles are easily
checked by playing around with some pieces of rope or bands, as shown in Figure 27, or
by watching the animation of Figure 28.
The simplest possible version of the experiment is the following: take two coffee cups,
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
F I G U R E 27 When two spin 1/2 tangles each made of several strands or bands, are exchanged twice, it
182
8 q ua ntum theory d ed uced from stra nds
one in each hand, and cross the two arms over each other (once). Keeping the orientation
of the cups fixed in space, uncross the arms by walking around the cups. This is possible,
but as a result, both arms are twisted. If you are intrepid, you can repeat this with two (or
more) people holding the cups. And you can check the difference with what is possible
after a double crossing of arms: in this case, all arms return to the starting situation.
All these experiments show:
⊳ A simple exchange of two spin 1/2 particles (tangles, cups on hands, belt
buckles) is equivalent to a multiplication by −1, i.e., to twisted tangles, arms
or belts.
⊳ In contrast, a double exchange of two spin 1/2 particles can always be untwisted and is equivalent to no exchange at all.
Spin 1/2 particles are thus fermions. In other words, the strand model reproduces the
spin–statistics theorem for spin 1/2: all elementary matter particles are fermions. In summary, a tangle core made of two or more tangled strands behaves – both under rotations
and under exchange – like a spin 1/2 particle.
We note that it is sometimes claimed that the appearance of spin 1/2 can only be mod-
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
represent particles. We know from above that belted buckles behave as spin 1/2 particles. The
animation shows that two such particles return to the original situation if they are switched in position
twice (but not once). Such particles thus fulfil the defining property of fermions. (For the opposite case,
that of bosons, a simple exchange would lead to the identical situation.) You can repeat the trick at
home using paper strips. The equivalence is shown here with two belts per particle, but the trick works
with any positive number of belts attached to each buckle. This animation is the essential part of the
proof that spin 1/2 particles are fermions. This is called the spin–statistics theorem. (QuickTime film
© Antonio Martos)
Motion Mountain – The Adventure of Physics
F I G U R E 28 Assume that the belts cannot be observed, but the square objects can, and that they
q ua ntum theory d ed uced from stra nds
A boson
Strand model :
tail
spin
time average
of crossing
switches
183
Observed
probability
density :
spin
core
F I G U R E 29 A massive spin 1 particle in the strand model (left) and the observed probability density
Boson exchange
Starting situation, before exchange
Final situation, after exchange
crossings, and thus without changing the sign of the phase.
Challenge 130 e
elled with the help of a topology change of space or space-time. The various belt trick
animations given above prove that this is not correct: spin 1/2 can be modelled in three
dimensions in all its aspects. No topology change is needed. You might want to model
the creation of a spin 1/2 particle–antiparticle pair as a final argument.
B osons and spin
Vol. IV, page 117
For tangles made of one strand – thus with two tails to the border – a rotation of the
tangle core by 2π restores the original state. Such a tangle, shown in Figure 29, thus
behaves like a spin 1 particle. The figure also shows the wave function that results from
time averaging the crossings.
Bosons are particles whose combined state does not change phase when two particles
are exchanged. We note directly that this is impossible with the tangle shown in Figure 29; the feat is only possible if the boson tangle is made of unknotted strands. Indeed,
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
F I G U R E 30 In the strand model, unknotted boson tangles can switch states without generating
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when averaging its crossings over long time scales (right).
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8 q ua ntum theory d ed uced from stra nds
for unknotted strands, the exchange process can easily switch the two deformations, as
illustrated in Figure 30.
⊳ Massive elementary particles thus can only be bosons if they also have an
unknotted tangle in the tangle family that represents them.
Page 314, page 331
Page 279
The simplest strand model for each elementary boson – the photon, the W boson, the Z
boson, the gluon and the Higgs boson – must thus be made of unknotted strands. We
will deduce the precise tangles below, in the chapter on the particle spectrum. The tangle
for the hypothetical graviton – also a boson, but in this case with spin 2 and invariant
under core rotations by π – will be introduced in the chapter on general relativity.
In summary, unknotted tangles realize the spin–statistics theorem for particles with
integer spin: radiation particles, which have integer spin, are automatically bosons.
Challenge 131 e
Tangle functions: blurred tangles
In the strand model, particle motion is due to the motion of tangle cores. But according to the fundamental principle, strands and tangles are not observable; only crossing
switches are. To explore the relation between crossing switches and motion, we first recall what a crossing is.
⊳ A crossing of strands is a local minimum of strand distance. The position,
orientation and the phase of a crossing are defined by the space vector corresponding to the local distance minimum, as shown in Figure 31. The sign
of the orientation is defined by arbitrarily selecting one strand as the starting strand. The even larger arbitrariness in the definition of the phase will
be of great importance later on, and lead to gauge invariance.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 162
We just saw that fluctuating strands reproduce the spin–statistics theorem for fermions
and for bosons, and thus for all elementary particles, if appropriate tangles are used.
Apart from this fundamental result, the strand model also implies that no spins lower
than ℏ/2 are possible, and that spin values are always an integer multiple of ℏ/2. All this
matches observations.
In the strand model, temporal evolution and particle reactions conserve spin, because
all interactions conserve the number of strands and tails. The details of the conservation
will become clear later on. Again, the result agrees with observations.
The strand model thus explains the origin of permutation symmetry in nature: permutation symmetry of particles is due the possibility to exchange tangle cores of identical
particles; and identical particles have tangle cores of identical topology. We have thus
already ticked off one item from the millennium list of unexplained properties of nature.
In summary, the strand model reproduces the rotation, the spin and the exchange
behaviour of elementary quantum particles – both fermions and bosons – in all its observed details. We now proceed to the next step: quantum mechanics of translational
motion.
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Spin and statistics
q ua ntum theory d ed uced from stra nds
185
A strand crossing
phase
shortest distance
position
orientation
F I G U R E 31 The definition of a crossing, its position, its orientation and its phase.
⊳ The tangle function of a system described by a tangle is the short-time average of the positions and the orientations of its crossings (and thus not of
crossing switches and not of the strands themselves).
⊳ For the definition of the tangle function, the short-time average of crossings
is taken over the typical time resolution of the observer. This is a time that
is much longer than the Planck time, but also much shorter than the typical
evolution time of the system. The time resolution is thus what the observer
calls an ‘instant’ of time. Typically – and in all known experiments – this
will be 10−25 s or more; the typical averaging will thus be over a time interval
with a value between 10−43 s, the Planck time, and around 10−25 s.
There are two ways to imagine tangle fluctuations and to deduce the short-time average
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
The tangle function can be called the ‘oriented crossing density’ or simply the ‘blurred
tangle’. As such, the tangle function is a continuous function of space, similar to a cloud;
we will see below what its precise mathematical description looks like. The tangle function captures the short-time average of all possible tangle fluctuations. For a tangle made
of two strands, Figure 32 illustrates the idea. However, the right-hand side of the figure
does not show the tangle function itself, but its probability density. We will see shortly
that the probability density is the (square of the) crossing position density, whereas the
tangle function is a density that describes both position and orientation of crossings.
The tangle function at any given time is not observable, as its definition is not based
on crossing switches, but only on crossings. However, since crossing switches only occur
at places with crossings, the tangle function is a useful tool to calculate observables. In
fact, we will show that the tangle function is just another name for what is usually called
the wave function. In short, the tangle function, i.e., the oriented crossing density, will
turn out to describe the quantum state of a system.
In summary, the tangle function is a blurred image of the tangle – with the important
detail that the crossings are blurred, not the strands.
Motion Mountain – The Adventure of Physics
To describe the motion of tangles, we need concepts that allow us to take the step from
general strand fluctuations to the motion of tangle cores. As a mathematical tool to describe crossing fluctuations, we define:
186
8 q ua ntum theory d ed uced from stra nds
Strand model :
A slowly moving strand :
time average
of crossing
switches
Observed
probability
density, i.e.,
crossing
switch
density :
precession
of spin axis
A rapidly moving strand :
Challenge 132 e
from a given tangle. The first, straightforward way is to average over all possible strand
fluctuations during the short time. Each piece of strand can change in shape, and as a
result, we get a cloud. This is the common Schrödinger picture of the wave function and of
quantum mechanics. The second, alternative way to average is to imagine that the tangle
core as a whole changes position and orientation randomly. This is easiest if the core with
all its crossings is imagined to be tightened to a small, almost ‘point-like’ region. Then
all observables are also localized in that region. It is often simpler to imagine an average
over all position and orientation fluctuations of such a tightened core, that to imagine an
average over all possible strand fluctuations. This alternate view leads to what physicists
call the path integral formulation of quantum mechanics. (Can you show the equivalence
of the two averaging methods?) Of course, in both cases the final result is that the tangle
function is a cloud, i.e., a probability amplitude.
Details on fluctuations and averages
In the strand model, the strand fluctuations of particle strands are a consequence of the
embedding of all particles in a background which itself is made of fluctuating vacuum
strands. Fluctuations randomly add detours to particle strands and randomly shift the
core position. Fluctuations do not keep the strand length constant. Fluctuations do not
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
probability density that results when averaging crossing switches over time. (The black dots are not
completely drawn correctly.)
Motion Mountain – The Adventure of Physics
F I G U R E 32 Some strand configurations, some of their short time fluctuations, and the corresponding
q ua ntum theory d ed uced from stra nds
187
conserve strand shape nor any other property of strands, as there is no mechanism that
enforces such rules. Strand fluctuations are thus quite wild. What then can be said about
the details of the averaging procedure for strand fluctuations?
The fluctuations of the vacuum are those strand fluctuations that lead to the definition
of the background space. This definition is possible in a consistent manner only if the
fluctuations are homogeneous and isotropic. The vacuum state can thus be defined as that
state for which the fluctuations are (locally) homogeneous and isotropic. In particular,
the fluctuations imply
⊳ Flat vacuum has a tangle function that vanishes everywhere.
Challenge 133 e
Page 108, page 166
Tangle functions are wave functions
In the following, we show that the tangle function, the blurred image of tangle crossings,
is the same as what is usually called the wave function. We recall what we know from
textbook quantum theory:
⊳ A single-particle wave function is, generally speaking, a rotating and diffusing cloud.
The rotation describes the evolution of the phase, and the diffusion describes the evolution of the density. We now show that tangle functions have these and all other known
properties of wave functions. We proceed by deducing all the properties from the definition of tangle functions. We recall that, being a short-time average, a tangle function is
a continuous function of space and time.
⊳ Using the tangle function, we define the strand crossing position density,
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Page 202
Motion Mountain – The Adventure of Physics
Page 294
The proof is an interesting exercise. The existence of a homogeneous and isotropic background space then implies conservation of energy, linear and angular momentum of
particles travelling through it.
The fluctuations of a tangle lead, after averaging, to the tangle function, i.e., as we will
see, to the wave function. The conservation of energy and momentum implies that the
time average of the tangle fluctuations also conserves these quantities.
Therefore we can continue our discussion without yet knowing the precise details of
the tangle fluctuations themselves. (We will provide these details below, in the section
on general relativity.) Here we only require that the average of the fluctuations behaves
in such a way as to be consistent with the definition of the background used by the observer. We thus make explicit use of the conviction that a background-free description
of nature is impossible, and that a fundamental description of nature must contain a circular definition that makes an axiomatic description of nature impossible. Despite this
limitation, such a circular description of nature must be self-consistent.
We will also show below that the definition of the tangle function does not introduce
hidden variables, even though first impression might suggest the opposite. In fact, it is
possible to define something akin to a strand evolution equation. However, it does not
deepen our understanding of the evolution equation of the wave function.
188
8 q ua ntum theory d ed uced from stra nds
or crossing density, for each point in space, by discarding the orientation
information, counting the crossings in a volume, and taking the square root.
The crossing density – more precisely, its square root – is a positive number,
more precisely, a positive real function ?(?, ?) of space and time.
We will see shortly that the crossing position density is the square root of what is usually
called the probability density.
⊳ A tangle function also defines an average crossing orientation and a average
phase at each point in space. The average crossing orientation and the average phase are related to the spin orientation and phase of the wave function.
The mathematical descriptions of these quantities depend on the approximation used.
⊳ The quantum phase of fermions is one half the core rotation angle ?.
Challenge 134 s
Page 202
If a system changes with time, the tangle function changes; this leads to crossing
switches; therefore, temporal evolution is expected to be observable through these crossing switches. As we will see shortly, this leads to an evolution equation for tangle functions.
Here is a fun challenge: how is the shortest distance between the strands, for a crossing
located at position ? and ?, related to the magnitude, i.e., the absolute value ?(?, ?), of
the wave function?
We note that if many particles need to be described, the many-particle tangle function
defines a separate crossing density for each particle tangle.
Tangle functions form a vector space. To show this, we need to define the linear combination or superposition ? = ?1 ?1 + ?2 ?2 of two tangle functions. This requires the
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Without the neglect of spin, and especially when the spin axis can change over space, the
description of orientation and phase averages require more details; we will study these
cases separately below. They will lead to the non-relativistic Pauli equation and to the
relativistic Dirac equation.
In short, in the simple approximation when spin effects can be neglected, the local
tangle function value can be described by one real number ? and by one quantum
phase ?. The tangle function can thus be described by a complex number ? at each point
in space and time:
?(?, ?) = ?(?, ?) e??(?,?)/2 .
(122)
Motion Mountain – The Adventure of Physics
The simplest approximation for a tangle function is to assume, in the physical situation
under study, that the spin direction is independent of spatial position and thus not taken
into consideration; this approximation will lead to the Schrödinger equation. In this
simplest approximation, at each point in space, the local average orientation of the fluctuations of the tangle core will just be described by a single angle. This quantum phase is
a function of time and space and describes how much the local average phase is rotated
around the fixed spin orientation.
q ua ntum theory d ed uced from stra nds
189
definition of two operations: scalar multiplication and addition. We can do this in two
ways. The first way is to define the operations for tangle functions directly, as is done in
quantum mechanics:
⊳ First, boring definition: The scalar multiplication ?? and the addition
?1 + ?2 of quantum states are taken by applying the relative operations on
complex numbers at each point in space, i.e., on the local values of the tangle
function.
The second way to deduce the vector space is more fun, because it will help us to visualize
quantum mechanics: we can define addition and multiplication for tangles, and take the
time average after the tangle operation is performed.
⊳ Second, fun definition: The addition of two tangles ?1 ?1 and ?2 ?2 , where
?1 and ?2 have the same topology and where ?12 + ?22 = 1, is defined by
connecting those tails that reach the border of space, and discarding all parts
of the tangles that were pushed to the border of space. The connection of
tangles must be performed in such a way as to maintain the topology of
the original tangles; in particular, the connection must not introduce any
crossings or linking. Time averaging then leads to the tangle function of the
superposition ? = ?1 ?1 + ?2 ?2 .
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
The scalar multiplication for strands is illustrated in Figure 33. The above definition of
scalar multiplication is only defined for factors ? ⩽ 1. Indeed, no other factors ever
appear in physical problems (provided all wave functions are normalized), so that scalar
multiplication is not required for other scalars.
The strand version of scalar multiplication is unique; indeed, even though there is
a choice about which fraction ? of a tangle is kept and which fraction 1 − ? is sent to
the border of space, the resulting tangle function, which is defined as an average over
fluctuations, is independent from this choice.
The scalar multiplication of strands behaves as expected for 1 and 0. By construction,
the strand version of scalar multiplication is associative: we have ?(??) = (??)?. The
strand multiplication by −1 is defined as the rotation of the full tangle core by 2π.
We also need to define the addition operation that appears in the linear combination
of two tangle functions. This is a straightforward complex addition at each point in space.
Again, for fun, we also define the operation on tangles themselves, and take the time
average that leads to the tangle function afterwards.
Motion Mountain – The Adventure of Physics
⊳ Second, fun definition: The scalar multiplication ?? of a state ? by a complex number ? = ?e?? is formed by taking a tangle underlying the tangle
function ?, then rotating the tangle core by the angle 2?, and finally pushing
a fraction 1 − ? of the tangle to the border of space, thus keeping the fraction
? of the original tangle at finite distances. Time averaging then leads to the
tangle function ??.
190
8 q ua ntum theory d ed uced from stra nds
Strand multiplication :
Strand model :
?
time average
of crossing
switches
Observed
probability
density :
√0.2 ?
functions.
Page 191
To visualize the result of addition and superposition, it is easiest to imagine that the
strands reaching the border of space have fluctuated back to finite distances. This is
possible because by definition, these connections are all unlinked. An example of superposition, for the case of two quantum states at different positions in space, is shown
in Figure 34. We note that despite the wording of the definition, no strand is actually cut
or re-glued in the operation of addition.
The definition of linear combination requires that the final strand ? has the same topology and the same norm as each of the two strands ?1 and ?2 to be combined. Physically,
this means that only states for the same particle can be added and that particle number
is preserved; this automatically implements the so-called superselection rules of quantum
theory. This result is pretty because in usual quantum mechanics the superselection rules
need to be added by hand. This is not necessary in the strand model.
The sum of two tangle functions is unique, for the same reasons given in the case of
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
F I G U R E 33 Scalar multiplication of localized tangles, visualizing the scalar multiplication of wave
Motion Mountain – The Adventure of Physics
√0.8 ?
q ua ntum theory d ed uced from stra nds
191
Linear combination of strands :
Strand model :
time average
of crossing
switches
Observed
probability
densities :
The two quantum states localized at different positions :
?1
?2
x2
x2
x1
x2
A linear combination ? = √0.8 ?1 + √0.2 ?2 :
untangled
“addition
region”
x1
x2
F I G U R E 34 A linear combination of strands, in this case for two states representing a particle at two
different position in space, visualizing the linear combination of wave functions.
Challenge 135 e
scalar multiplication. The definition of addition can also be extended to more than two
terms. Addition is commutative and associative, and there is a zero state, or identity
element, given by no strands at all. The definition of addition also implies distributivity
with respect to addition of states and with respect to addition of scalars. It is also possible
to extend the definitions of scalar multiplication and of addition to all complex numbers
and to unnormed states, but this leads us too far from our story.
In short, tangle functions form a vector space. We now define the scalar product and
the probability density in the same way as for wave functions.
⊳ The scalar product between two states ? and ? is ⟨?|?⟩ = ∫?(x)?(x) dx.
⊳ The norm of a state is ‖?‖ = √⟨?|?⟩ .
⊳ The probability density ? is ?(?, ?) = ?(?, ?)?(?, ?) = ?2 (?, ?). It thus ignores the orientation of the crossings and is the crossing position density.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
x1
Motion Mountain – The Adventure of Physics
x1
192
Challenge 136 e
Challenge 137 e
8 q ua ntum theory d ed uced from stra nds
The scalar product and the probability density are observables, because their definitions
can be interpreted in terms of crossing switches. Indeed, the scalar product ⟨?|?⟩ can
be seen as the (suitably normed) number of crossing switches required to transform the
tangle ? into the tangle ?, where the tangle ? is formed from the tangle ? by exchanging
the orientation of each crossing. A similar interpretation is possible for the probability
density, which therefore is at the same time the crossing density squared and the crossing
switch density. We leave this confirmation as fun for the reader.
It is also possible to define the scalar product, the norm and the probability density
using tangles, instead of using tangle functions. This is left as a puzzle to the reader.
In summary, we have shown that tangle functions form a Hilbert space. The next
steps are now obvious: We must first show that tangle functions obey the Schrödinger
equation. Then we must extend the definition of quantum states by including spin and
special relativity, and show that they obey the Dirac equation.
⊳ Particles with high energy have rapidly rotating tangles.
⊳ Particles with low energy have slowly rotating tangles.
The energy of a rotating tangle is the number of crossing switches per time. Rotating a
tangle core leads to crossing switches in its tails. In the strand model, the kinetic energy
? of a particle is thus due to the crossing switches formed in its tails. In other words, the
kinetic energy ? is related to the (effective) angular frequency ? of the core rotation by
? = ℏ? .
(123)
The local phase of the tangle function ? changes with the rotation. This implies that
? = ?∂? ? .
(124)
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
The Schrödinger equation, like all evolution equations in the quantum domain, results
when the definition of the wave function is combined with the energy–momentum relation. As already mentioned, the Schrödinger equation for a quantum particle also assumes that the orientation of particle spin is constant for all positions and all times. In
this case, the spin can be neglected, and the tangle function is a single complex number
at each point in space and in time, usually written ?(?, ?). How does the tangle function
evolve in time? To answer this question, we will only need the fundamental principle
that crossing switches define the quantum of action ℏ.
We start with a free particle. We assume a fixed, but unspecified rotation direction of
its tangle. Now, in the strand model, a localized particle with constant speed is described
by a localized tangle that rotates and advances. In other words, the strand fluctuations
produce a peak of probability density that changes position with constant speed.
Every tangle rotation leads to crossing switches. A rapid tangle rotation leads to many
crossing switches per time, and slow rotation to few crossing switches per time. Now, the
fundamental principle tells us that crossing switches per time are naturally measured in
action per time, or energy. In other words, tangle rotation is related to tangle energy.
Motion Mountain – The Adventure of Physics
Deducing the S chrödinger equation from tangles
q ua ntum theory d ed uced from stra nds
193
Strand model :
time average
of crossing
switches
Localized particle at rest :
Observed
probability
density :
t1
t1
t2
t2
Slow motion :
t2
t1
Motion Mountain – The Adventure of Physics
t1
t2
rotation,
precession and
displacement
t2
t1
t1
t2
rotation,
precession and
displacement
F I G U R E 35 Examples of moving tangles of free particles.
We will need the relation shortly.
The linear motion of a tangle implies that it makes also sense to pay attention to the
number of crossing switches per distance.
⊳ Rapidly moving tangles show many crossing switches per distance.
⊳ Slowly moving tangles show few crossing switches per distance.
The fundamental principle tells us that the natural observable to measure crossing
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Rapid motion :
194
8 q ua ntum theory d ed uced from stra nds
switches per distance is action per distance, or momentum. Linear motion of tangles
is thus related to momentum: The momentum of a moving tangle is the number of crossing switches per distance. The momentum ? is thus related to the (effective) wave number
? = 2π/? of the core motion by
? = ℏ? .
(125)
The local phase of the tangle function ? changes with the motion. This implies
? = −?∂? ? .
(126)
?=
?2
2?
and ? =
ℏ 2
? .
2?
(127)
?ℏ∂? ? = −
ℏ2
∂ ?.
2? ??
(128)
This is the famous Schrödinger equation for a free particle (written for just one space
dimension for simplicity). We thus have deduced the equation from the strand model
under the condition that spin can be neglected and that velocities are small compared to
the speed of light. In this way, we have also deduced, indirectly, Heisenberg’s indeterminacy relations.
We have thus completed the proof that tangle functions, in the case of negligible spin
effects and small velocities, are indeed wave functions. In fact, tangle functions are wave
functions also in the more general case, but then their mathematical description is more
involved, as we will see shortly. We can sum up the situation in a few simple terms: wave
functions are blurred tangles.
Mass from tangles
In quantum theory, particles spin while moving: the quantum phase rotates while a
particle advances. The coupling between rotation and translation has a name: it is called
the mass of a particle. We saw that the rotation is described by an average angular fre-
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
This is dispersion relation for masses moving at velocities much smaller than the speed
of light. The relation agrees with all experiments. The constant ? is a proportionality
factor that depends on the tangle core. We can now use the same argument that was
used already by Schrödinger. Substituting the tangle relations in the dispersion relation,
we get the evolution equation for the tangle function ? given by
Motion Mountain – The Adventure of Physics
This completes the description of matter wave functions without spin.
The belt trick for the fluctuating tails now has a fascinating consequence. To allow
the belt trick also for high linear momentum, the more the momentum increases, the
more the spin rotation axis has to align with the direction of motion. This is shown in
Figure 35. This leads to a quadratic increase of crossing switches with momentum ?:
one factor ? is due to the increase of the speed of rotation, the other factor is due to the
increase of the alignment. We thus get
q ua ntum theory d ed uced from stra nds
195
quency ?, and the translational motion is described by a wave number ?. The proportionality factor ? = ℏ?2 /2? = ?2 /2? is thus a quantity that relates rotation frequency
and wave number. In quantum theory,
⊳ The (inertial) mass ? describes the coupling between translation and rotation.
⊳ Linked and localized tangles have mass.
⊳ Unknotted or unlinked, unlocalized tangles, such as those of photons, are
predicted to be massless.
⊳ Particle masses are calculable – if the tangle topology is known.
Challenge 138 e
Page 358
This is an exciting prospect! To sum up, the strand model predicts that experiments in
viscous fluids can lead to a deeper understanding of the masses of elementary particles.
The tangle model also implies that the mass of elementary particles – thus of particles
made of few strands – will be much smaller than the Planck mass. This is the first hint
that the strand model solves the so-called mass hierarchy problem of particle physics.
At this point, however, we are still in the dark about the precise origin of particle
mass values. We do not know how to calculate them. Nevertheless, the missing steps
are clear: first, we need to determine the tangle topology for each elementary particle;
then we need to deduce their mass values, i.e., the relation between their rotation and
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 356
We also deduce that the more complicated a tangle is, the higher the mass value is.
In addition to the geometry effect due to the core, which is valid for massive bosons
and fermions, the rotation of fermions is also influenced by the tails. The effective volume
required by the belt trick will influence the coupling between translation and rotation.
This effective volume will depend on the topology of the tangle core, and on the number
of its tails. We again deduce that, for a given number of tails, a complicated core topology
implies a high mass value.
In other words, the strand model links the mass ? of a particle to its tangle topology:
large tangle cores have large mass. The strand model thus predicts
Motion Mountain – The Adventure of Physics
Ref. 162
We note that a large mass value implies, for a given momentum value, both a slow translation and a slow rotation.
In the strand model, particle translation and rotation are modelled by the translation
and rotation of the tangle core. Now, the strand model makes a point that goes beyond
usual quantum theory. The strand model explains why core translation and rotation are
coupled: When the core moves through the vacuum, the vacuum strands and the core
effectively push against each other, due to their impenetrability. The result is a motion
that resembles the motion of an asymmetrical body in a viscous fluid.
When an asymmetrical body is moved through a viscous fluid, it starts rotating. For
example, this happens when a stone falls through water or honey. The rotation results
from the asymmetrical shape of the body. All the tangle cores of elementary particles are
asymmetrical. The strand model thus predicts that tangle cores will rotate when they
move through vacuum. In other terms, the strand model predicts
196
Challenge 139 e
8 q ua ntum theory d ed uced from stra nds
translation. This is a central aim in the following.
An example of the issues that arise: How does the mass value depend on the number
of strands in a tangle? How does mass depend on the type of tangle?
Potentials
(?ℏ∂? − ??)? =
(129)
This equation is the simplest formulation of quantum theory. We saw in the fourth
volume that it describes and explains the size of atoms and molecules, and thus of all
objects around us; and we saw that it also explains the (relative) colours of all things.
The equation also explains interference, tunnelling and decay.
In summary, a non-relativistic fluctuating tangle reproduces the full Schrödinger
equation. An obvious question is: how does the strand model explain the influence of
interactions on the rotation speed and on the wavelength of tangles? In other words:
why do strands imply minimal coupling? We will answer this question in the following
chapter, on gauge interactions.
Q uantum interference from tangles
Page 189
Page 175
The observation of interference of quantum particles is due to the linear combination of
states with different phases at the same position in space. Tangle functions, being wave
functions, reproduce the effect. But again, it is both more fun and more instructive to
explain and visualize interference with the help of tangles.
As mentioned above, a pure change of phase of a state ? is defined by multiplication
by a complex number of unit norm, such as e?? . This corresponds to a rotation of the
tangle core by an angle 2?, where the factor 2 is due to the belt trick of Figure 19.
To deduce interference, we simply use the above definition of linear combinations
of tangles. This leads to the result shown in Figure 36. We find, for example, that a
symmetric sum of a tangle and the same tangle with the phase rotated by π/2 (thus a
core rotated by π) results in a tangle whose phase is rotated by the intermediate angle,
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 222
1
(−?ℏ∇ − ??)2 ? .
2?
Motion Mountain – The Adventure of Physics
Page 232
In quantum mechanics, interactions are described by potentials. An electric potential
?(?, ?) changes the total energy of a particle with charge ? at position ?, since in quantum
mechanics, electric potentials influence the rotation velocity of the wave function. As a
result, with an electric potential, the left-hand side of the Schrödinger equation (128), the
energy term, is changed from ℏ??(?, ?) to (ℏ? − ??)?(?, ?).
Another possibility is a potential that does not change the rotation velocity, but that
changes the wavelength of a charged particle. Such a magnetic vector potential ?(?, ?)
thus changes the momentum term ℏ? on the right-hand side of Schrödinger’s equation
to (ℏ?−??)?(?, ?). This double substitution, the so-called minimal coupling, is equivalent
to the statement that quantum electrodynamics has a U(1) gauge symmetry. We will
deduce it in detail in the next chapter.
In the strand model of quantum mechanics, potentials are introduced in precisely
the same way as in usual quantum mechanics, so that the full Schrödinger equation for
charged particles in external fields is recovered:
q ua ntum theory d ed uced from stra nds
197
Observed
probability
density :
Strand model :
time average
of crossing
changes
The two quantum states with different phase at the same position :
?2 = ?1 e?π/2
?1
?
?
?
A linear combination :
? = (?1 + ?2 )/√2
?
Extinction (requires situations with space-dependent phase) :
no possible
tangle
topology
?
zero
density
?
F I G U R E 36 Interference: the linear combination of strands with different phase, but located at the
same position.
thus π/4.
The most interesting case of interference is that of extinction. Scalar multiplication
of a tangle function ? by −1 gives the negative of the tangle function, the additive inverse −?. The sum of a tangle function with its negative is zero. This gives extinction
in usual quantum theory. Let us check the result in the strand model, using the tangle
definition of linear combinations. We have seen above that the negative of a tangle is a
tangle whose core is rotated by 2π. Using the tangle definition of linear combination,
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
?
Motion Mountain – The Adventure of Physics
?
198
8 q ua ntum theory d ed uced from stra nds
we find that it is topologically impossible to draw or construct a localized tangle for the
sum of a quantum state with its negative. The resulting particle tangle therefore must
have vanishing crossing density in spatial regions where this operation is attempted. In
short, particle tangles do explain extinction. And as expected from quantum particles,
the explanation of extinction directly involves the tangle structure.
Deducing the Pauli equation from tangles
cos(?/2)e??/2
Ψ(?, ?) = √? e??/2 (
) ,
? sin(?/2)e−??/2
(130)
0 1
0 −?
1 0
? = ((
),(
),(
)) .
1 0
? 0
0 −1
(131)
The three matrices are the well-known Pauli matrices.
We now take the description of the axis orientation and the description of the spinning and insert both, as we did for the Schrödinger equation, into the non-relativistic
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
which is the natural description of a tangle that includes the orientation of the axis. As
before, the crossing density is the square root of the probability density ?(?, ?). The angle
?(?, ?), as before, describes the phase, i.e., (one half of) the rotation around the axis. The
local orientation of the axis is described by a two-component matrix and uses the two
angles ?(?, ?) and ?(?, ?). Due to the belt trick, the expression for the tangle function only
contains half angles. And indeed, due to the half angles, the two-component matrix is
not a vector, but a spinor. (The term ‘spinor’ was coined by well-known physicist Paul
Ehrenfest in analogy to ‘vector’ and ‘tensor’; the English pronunciation is ‘spinnor’.)
For ? = ? = 0, the previous wave function ? is recovered.
The other ingredient we need is a description of the spinning motion of the tangle.
In contrast to the Schrödinger case, the spinning motion itself must be added in the
description. A spinning tangle implies that the propagation of the wave is described by
the wave vector ? multiplied with the spin operator ?. The spin operator ?, for the wave
function just given, is defined as the vector of three matrices
Motion Mountain – The Adventure of Physics
Ref. 163
As we have seen, the Schrödinger equation describes the motion of quantum particles
when their spin is neglected, by assuming that spin is constant over space and time. The
next step is thus to include the variations of spin over space and time. This turns out to
be quite straightforward.
In the strand model, spin is modelled by the continuous rotation of a tangle. We also
saw that we get wave functions from tangles if we average over short time scales. At a
given position in space, a tangle function will have a local average density of crossings, a
local average phase, and new, a local average orientation of the rotation axis of the tangle.
To describe the axis and orientation of the tangle core, we use the Euler angles ?, ?
and ?. This yields a description of the tangle function as
q ua ntum theory d ed uced from stra nds
199
dispersion relation ℏ? = ? = ?2 /2? = ℏ2 ?2 /2?. We then get the wave equation
?ℏ∂? Ψ = −
Challenge 140 s
(132)
This is Pauli’s equation for the evolution of a free quantum particle with spin 1/2.
As final step, we include the electric and the magnetic potentials, as we did in the
case of the Schrödinger equation. We again use minimal coupling, substituting ?ℏ∂? by
?ℏ∂? − ?? and −?ℏ∇ by −?ℏ∇ − ??, thus introducing electric charge ? and the potentials
? and ?. A bit of algebra involving the spin operator then leads to the famous complete
form of the Pauli equation
(?ℏ∂? − ??)Ψ =
?ℏ
1
(−?ℏ∇ − ??)2 Ψ −
??Ψ ,
2?
2?
(133)
Rotating arrows, path integrals and interference
Ref. 164
Another simple way to visualize the equivalence between the strand model and the Pauli
equation uses the formulation of quantum theory with path integrals. We recall that
tangle tails are not observable, and that the tangle core defines the position and phase of
the quantum particle. The motion of the core thus describes the ‘path’ of the particle.
Different paths are due to different core motions.
The continuous rotation of the tangle core corresponds to Feynman’s rotating little
arrow in his famous popular book on QED. The different paths then correspond to different motions of the tangle core. The tangle model also reproduces the path integral
formulation of quantum mechanics.
Also interference can be visualized with strands. Because of its tails, a fermion tangle
obeys spinor statistics and spinor rotation behaviour. This leads to the correct interference behaviour for spin 1/2 particles. Indeed, interference for fermions is visualized in
Figure 37. The corresponding visualization for photon interference is given in Figure 38.
Measurements and wave function coll apse
In nature, a measurement of a quantum system in a superposition is observed to yield one
of the possible eigenvalues and to prepare the system in the corresponding eigenstate. In
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
where now the magnetic field ? = ∇ × ? appears explicitly. The equation is famous for
describing, among others, the motion of silver atoms, which have spin 1/2, in the Stern–
Gerlach experiment. This is due to the new, last term on the right-hand side, which does
not appear in the Schrödinger equation. The new term is a pure spin effect and predicts
a ?-factor of 2. Depending on the spin orientation, the sign of the last term is either
positive or negative; the term thus acts as a spin-dependent potential. The two options
for the spin orientation then produce the upper and the lower beams of silver atoms that
are observed in the Stern–Gerlach experiment.
In summary, a non-relativistic tangle that rotates continuously reproduces the Pauli
equation. In particular, such a tangle predicts that the ?-factor of an elementary charged
fermion is 2.
Motion Mountain – The Adventure of Physics
Vol. IV, page 83
ℏ2
(?∇)2 Ψ .
2?
200
8 q ua ntum theory d ed uced from stra nds
destructive
interference
constructive
interference
interference (right).
constructive
interference
destructive
interference
interference (right).
Vol. IV, page 143
nature, the probability of each measurement outcome depends on the coefficient of that
eigenstate in the superposition.
To put the issue into context, here is a short reminder from quantum mechanics. Every
measurement apparatus shows measurement results. Thus, every measurement apparatus is a device with memory. (In short, it is classical.) All devices with memory contain
one or several baths. Thus, every measurement apparatus couples at least one bath to the
system it measures. The coupling depends on and defines the observable to be measured
by the apparatus. Every coupling of a bath to a quantum systems leads to decoherence.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
F I G U R E 38 The double-slit experiment with photons: constructive interference (left) and destructive
Motion Mountain – The Adventure of Physics
F I G U R E 37 A fermion tangle passing a double slit: constructive interference (left) and destructive
q ua ntum theory d ed uced from stra nds
Spin
measurement
direction :
201
Strand model :
Observed spin :
Basis states :
always
`up’
always
`down’
untangled
“addition
region”
either
`up’
or `down’
inwards.
Decoherence leads to probabilities and wave function collapse. In short, collapse and
measurement probabilities are necessary and automatic in quantum theory.
The strand model describes the measurement process in precisely the same way as
standard quantum theory; in addition, it visualizes the process.
⊳ A measurement is modelled as a strand deformation induced by the measurement apparatus that ‘pulls’ a tangle towards the resulting eigenstate.
⊳ This pulling of strands models and visualizes the collapse of the wave function.
An example of measurement is illustrated in Figure 39. When a measurement is performed on a superposition, the untangled ‘addition region’ can be imagined to shrink into
disappearance. For this to happen, one of the underlying eigenstates has to ‘eat up’ the
other: that is the collapse of the wave function. In the example of the figure, the addition
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
F I G U R E 39 Measurement of a spin superposition: the addition region disappears either outwards or
Motion Mountain – The Adventure of Physics
Superposition (one of
two equivalent states) :
202
8 q ua ntum theory d ed uced from stra nds
region can disappear either towards the outside or towards the inside. The choice is due
to the bath that is coupled to the system during measurement; the bath thus determines
the outcome of the measurement. We also deduce that the probability of measuring a
particular eigenstate will depend on the (weighed) volume that the eigenstate took up in
the superposition.
This visualization of the wave function collapse also makes clear that the collapse is
not limited by any speed limit, as no energy and no information is transported. Indeed,
the collapse happens by displacing strands and at most crossings, but does not produce
any crossing changes.
In summary, the strand model describes measurements in precisely the same way as
usual quantum theory. In addition, strands visualize the collapse of the wave function as
a shape deformation from a superposed tangle to an eigenstate tangle.
Hidden variables and the Ko chen–Specker theorem
Therefore, the strand model does not contradict the Kochen–Specker theorem.
In simple language, in quantum theory, hidden variables are not a problem if they are
properties of the environment, and not of the quantum system itself. This is precisely
the case for the strand model. For a quantum system, the strand model provides no
hidden variables. In fact, for a quantum system, the strand model provides no variables
beyond the usual ones from quantum theory. And as expected and required from any
model that reproduces decoherence, the strand model leads to a contextual, probabilistic
description of nature.
In summary, despite using fluctuating tangles as underlying structure, the strand
model is equivalent to usual quantum theory. The strand model contains nothing more
and nothing less than usual quantum theory.
Many-particle states and entanglement
In nature, the quantum states of two or more particles can be entangled. Entangled states
are many-particle states that are not separable. Entangled states are one of the most fascinating quantum phenomena; especially in the case of macroscopic entanglement, they
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
⊳ The evolution of strand shapes and crossing switches is contextual.
Motion Mountain – The Adventure of Physics
Ref. 165
At first sight, the strand model seems to fall into the trap of introducing hidden variables
into quantum theory. One could indeed argue that the shapes (and fluctuations) of the
strands play the role of hidden variables. On the other hand, it is well known that noncontextual hidden variables are impossible in quantum theory, as shown by the Kochen–
Specker theorem (for sufficiently high Hilbert-space dimensions). Is the strand model
flawed? No.
We recall that strands are not observable. In particular, strand shapes are not physical
observables and thus not physical variables either. Even if we tried promoting strand
shapes to physical variables, the evolution of the strand shapes would only be observable
through the ensuing crossing switches. And crossing switches evolve due to the influence of the environment, which consists of all other strands in nature, including those
of space-time itself. Thus
q ua ntum theory d ed uced from stra nds
Strand model :
203
Observation :
First separable basis state :
| ↑↓⟩
?1
?2
?1
?2
| ↓↑⟩
model.
are still being explored in many experiments. We will discover that the strand model
visualizes them simply and clearly.
To describe entanglement, we first need to clarify the notion of many-particle state.
In the strand model,
⊳ A many-particle state is composed of several tangles.
In this way, an ?-particle wave function defines ? values at every point in space, one
value for each particle. This is possible, because in the strand model, the strands of each
particle tangle are separate from these of other particles.
Usually, a ?-particle wave function is described by a single-valued function in 3?
dimensions. It is less known that a single-valued ?-particle wave function in 3? dimensions is mathematically equivalent to an ?-valued wave function in three dimensions. Usually, ?-valued functions are not discussed; we feel uneasy with the concept.
But the strand model naturally defines ? wave function values at each point in space:
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
F I G U R E 40 Two examples of two distant particles with spin in separable states: observation and strand
Motion Mountain – The Adventure of Physics
Second separable basis state :
204
8 q ua ntum theory d ed uced from stra nds
Entangled state
√90 % | ↑↓⟩ + √10 % | ↓↑⟩
Strand model :
Observation yields
either this eigenstate (90%) :
untangled
“addition
region”
or this eigenstate (10%) :
?2
?1
?2
F I G U R E 41 An entangled spin state of two distant particles.
⊳ An entangled state is a non-separable superposition of separable manyparticle states. State are separable when their tangles can be pulled away
without their tails being entangled.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 164
each particle has its own tangle, and each tangle yields, via short-term averaging, one
complex value, with magnitude and phase, at each point in space. In this way, the strand
model is able to describe ? particles in just 3 dimensions.
In other words, the strand model does not describe ? particles with 1 function in
3? dimensions; it describes many-particle states with ? functions in 3 dimensions. In
this way, the strand model remains as close to everyday life as possible. Many incorrect
statements on this issue are found in the research literature; many authors incorrectly
claim the impossibility of many-particle quantum theory in 3 dimensions. Some authors
even claim, in contrast to experiment, that it is impossible to visualize many-particle
states in 3 dimensions. These arguments all fail to consider the possibility to define
completely separate wave functions for each particle in three dimensions. (It must be
said that this unusual possibility is hard to imagine if wave functions are described as
continuous functions.) However, clear thinkers like Richard Feynman always pictured
many-particle wave functions in 3 dimensions. Also in this domain, the strand model
provides an underlying picture to Feynman’s approach. This is another situation where
the strand model eliminates incorrect thinking habits and supports the naive view of
quantum theory.
Now that we have defined many-particle states, we can also define entangled states.
Motion Mountain – The Adventure of Physics
?1
q ua ntum theory d ed uced from stra nds
205
The Aspect experiment
First separable basis state :
Entangled state (50% + 50%) :
source
source
Second separable basis state :
the untangled
addition region
expands with time
in this situation
F I G U R E 42 The basis states and an entangled state of two distant photons travelling in opposite
directions, with total spin 0.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
We will now show that the above definitions of superpositions and of measurements
using strands are sufficient to describe entanglement.
As first example, we explore entangled states of the spin of two distant massive fermions. This is the famous thought experiment proposed by David Bohm. In the strand
model, two distant particles with spin 1/2 in a separable state are modelled as two distant, separate tangles of identical topology. Figure 40 shows two separable basis states,
namely the two states with total spin 0 given by | ↑↓⟩ and by | ↓↑⟩. Such states can also be
produced in experiments. We note that to ensure total spin 0, the tails must be imagined
to cross somewhere, as shown in the figure.
We can now draw a superposition √90 % | ↑↓⟩ + √10 % | ↓↑⟩ of the two spin-0 basis
states. We simply use the definition of addition and find the state shown in Figure 41. We
can now use the definition of measurement to check that the state is indeed entangled.
If we measure the spin orientation of one of the particles, the untangled addition region
disappears. The result of the measurement will be either the state on the inside of the addition region or the state on the outside. And since the tails of the two particles are linked,
after the measurement, independently of the outcome, the spin of the two particles will
always point in opposite directions. This happens for every particle distance. Despite
this extremely rapid and apparently superluminal collapse, no energy travels faster than
light. The strand model thus reproduces exactly the observed behaviour of entangled
spin 1/2 states.
A second example is the entanglement of two photons, the well-known Aspect experiment. Also in this case, entangled spin 0 states, i.e., entangled states of photons of
opposite helicity (spin), are most interesting. Again, the strand model helps to visualize
Motion Mountain – The Adventure of Physics
source
206
Page 227
Ref. 166
8 q ua ntum theory d ed uced from stra nds
the situation. Here we use the strand model for the photon that we will deduce only later
on. Figure 42 shows the strand model of the two separable basis states and the strand
model of the entangled state. Again, the measurement of the helicity of one photon in
the entangled state will lead to one of the two basis states. And as soon as the helicity of one photon is measured, the helicity of its companion collapses to the opposite
value, whatever the distance! Experimentally, the effect has been observed for distances
of many kilometres. Again, despite the extremely rapid collapse, no energy travels faster
than light. And again, the strand model completely reproduces the observations.
Mixed states
Mixed states are statistical ensembles of pure states. In the strand model,
⊳ A mixed state is a (weighted) temporal alternation of pure states.
The dimensionalit y of space-time
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Page 166
‘Nature consists of particles moving in empty space.’ Democritus stated this 2500 years
ago. Today, we know that is a simplified description of one half of physics: it is a simplified description of quantum theory. In fact, Democritus’ statement, together with
strands, allows us to argue that physical space must have three dimensions, as we will
see now.
Deducing the dimensionality of physical space from first principles is an old and difficult problem. The difficulty is also due to the lack of alternative descriptions of nature.
Our exploration of the foundations of the strand model has shown that humans, animals
and machines always use three spatial dimensions to describe their environment. They
cannot do otherwise. Humans, animals and machines cannot talk and think without
three dimensions as background space.
But how can we show that physical space – not the background space we need for thinking – is three-dimensional and must be so? We need to show that (1) all experiments
reproduce the result and that (2) no other number of dimensions yields a consistent description of nature.
In nature, and also in the strand model, as long as particles can be defined, they
can be rotated around each other and they can be exchanged. No experiment has ever
been performed or has ever been proposed that changes this observation. The observed
properties of rotations, of spin 1/2, of particle exchange and all other observations confirm that space has three dimensions. Fermions only exist in three dimensions. In
the strand model, the position and the orientation of a particle is intrinsically a threedimensional quantity; physical space is thus three-dimensional, in all situations where
it can be defined. (The only situations where this definition is impossible are horizons
and the Planck scales.) In short, both nature and the strand model are found to be threedimensional at all experimentally accessible energy scales. Conversely, detecting an additional spatial dimension would directly invalidate the strand model.
Motion Mountain – The Adventure of Physics
Mixed states are important in discussions of thermodynamic quantities. We mention
them to complete the equivalence of the states that appear in quantum theory with those
provided by the strand model. We do not pursue this topic any further.
q ua ntum theory d ed uced from stra nds
Page 162
Nature has three dimensions. The only way to predict this result is to show that no
other number is possible. The number of dimensions of nature can only result from a
self-consistency argument. And interestingly, the strand model produces such an argument.
In the strand model, knots and tangles are impossible to construct in physical spaces
with dimensions other than three. Indeed, mathematicians can show that in four spatial
dimensions, every knot and every tangle can be undone. (In this argument, time is not
and does not count as a fourth spatial dimension, and strands are assumed to remain onedimensional entities.) Worse, in the strand model, spin does not exist in spaces that have
more or fewer than three dimensions. Also the vacuum and its quantum fluctuations do
not exist in more than three dimensions. Moreover, in other dimensions it is impossible
to formulate the fundamental principle. In short, the strand model of matter and of
observers, be they animals, people or machines, is possible in three spatial dimensions
only. No description of nature with a background or physical space of more or less than
three dimensions is possible with strands. Conversely, constructing such a description
would invalidate the strand model.
The same type of arguments can be collected for the one-dimensionality of physical
time. It can be fun exploring them – for a short while. In summary, the strand model only
works in 3+1 space-time dimensions; it does not allow any other number of dimensions.
We have thus ticked off another of the millennium issues. We can thus continue with our
adventure.
Operators and the Heisenberg picture
?? − ?? = ℏ?
(134)
is related to a crossing switch. The present section confirms that speculation.
In quantum mechanics, the commutation relation follows from the definition of the
momentum operator as ? = ℏ?, ? = −?∂? being the wave vector operator. The factor
ℏ defines the unit of momentum. The wave vector counts the number of wave crests of
a wave. Now, in the strand model, a rotation of a state by an angle π is described by a
multiplication by ?. Counting wave crests of a propagating state is only possible by using
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 167
In quantum theory, Hermitean operators play an important role. In the strand model,
Hermitean or self-adjoint operators are operators that leave the tangle topology invariant.
Also unitary operators play an important role in quantum theory. In the strand model,
unitary operators are operators that deform tangles in a way that the corresponding wave
function retains its norm, i.e., such that tangles retain their topology and their core shape.
Physicists know two ways to describe quantum theory. One is to describe evolution
with time-dependent quantum states – the Schrödinger picture we are using here – and
the other is to describe evolution with time-dependent operators. In this so-called Heisenberg picture, the temporal evolution is described by the operators.
The two pictures of quantum theory are equivalent. In the Heisenberg picture, the
fundamental principle, the equivalence of a crossing switch with ℏ, becomes a statement
on the behaviour of operators. Already in 1987, Louis Kauffman had argued that the
commutation relation for the momentum and position operators
Motion Mountain – The Adventure of Physics
Challenge 141 e
207
208
8 q ua ntum theory d ed uced from stra nds
the factor ?, as this factor is the only property that distinguishes a crest from a trough.
In short, the commutation relation follows from the fundamental principle of the strand
model.
L agrangians and the principle of least action
Before we derive the Dirac equation, we show that the strand model naturally leads to
describe motion with Lagrangians.
In nature, physical action is an observable measured in multiples of the natural unit,
the quantum of action ℏ. Action is the fundamental observable about nature, because
action measures the total change occurring in a process.
In the strand model,
⊳ The physical action ? of a physical process is the observed number of crossing switches of strands. Action values are multiples of ℏ.
⊳ Energy is the number of crossing switches per time in a system.
⊳ The kinetic energy ? of a particle is the number of crossing switches per time
induced by shape fluctuations of the continuously rotating tangle core.
We call T the corresponding volume density: T = ?/?. In nature, the Lagrangian is a
practical quantity to describe motion. For a free particle, the Lagrangian density L = T
is simply the kinetic energy density, and the action ? = ∫L d?d? = ?? is the product
of kinetic energy and time. In the strand model, a free particle is a constantly rotating
and advancing tangle. We see directly that this constant evolution minimizes the action
? for a particle, given the states at the start and at the end.
This aspect is more interesting for particles that interact. Interactions can be described
by a potential energy ?, which is, more properly speaking, the energy of the field that
produces the interaction. In the strand model,
⊳ Potential energy ? is the number of crossing switches per time induced by
an interaction field.
We call U the corresponding volume density: U = ?/?. In short, in the strand model,
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
In nature, when free quantum particles move, their phase changes linearly with time. In
other words, the ‘little arrow’ representing the free particle phase rotates with constant
angular frequency. We saw that in the strand model, the ‘little arrow’ is taken as (half)
the orientation angle of the tangle core, and the arrow rotation is (half) the rotation of
the tangle core.
Motion Mountain – The Adventure of Physics
We note that these multiples, if averaged, do not need to be integer multiples. We further
note that through this definition, action is observer-invariant. This important property is
thus automatic in the strand model.
In nature, energy is action per time. Thus, in the strand model we have:
q ua ntum theory d ed uced from stra nds
209
an interaction changes the rotation rate and the linear motion of a particle tangle.
In the strand model, the difference between kinetic and potential energy is thus a
quantity that describes how much a system consisting of a tangle and a field changes
at a given time. The total change is the integral over time of all instantaneous changes.
In other words, in the strand model we have:
⊳ The Lagrangian density L = T − U is the number of crossing switches per
volume and time, averaged over many Planck scales.
⊳ The physical action ? = ∫? d? = ∫∫L d?d? of a physical process is the observed number of crossing switches of strands. The action value ?if between
an initial state ?i and a final state ?f is given by
?if = ⟨?i |∫L d? | ?f ⟩ = ⟨?i |∫(T − U) d? | ?f ⟩ .
(135)
⊳ Evolution of tangles minimizes the action ?.
L=
iℏ
ℏ2
(? ∂? ? − ∂? ? ?) −
∇?∇? .
2
2?
(136)
In this way, the principle of least action can be used to describe the evolution of the
Schrödinger equation. The same is possible for situations with potentials, for the Pauli
equation, and for all other evolution equations of quantum particles.
We thus retain that the strand model explains the least action principle. It explains it
in the following way: quantum evolution minimizes the number of crossing switches.
Special rel ativit y: the vacuum
In nature, there is an invariant limit energy speed ?, namely the speed of light and of all
other massless radiation. Special relativity is the description of the consequences from
this observation, in the case of a flat space-time.
We remark that special relativity also implies and requires that the flat vacuum looks
exactly the same for all inertial observers. In the strand model, the idea of flat vacuum
as a set of fluctuating featureless strands that are unknotted and unlinked automatically
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Challenge 142 e
In the strand model, the least action principle appears naturally. In the strand model, an
evolution has least action when it occurs with the smallest number of crossing changes.
With this connection, one can also show that the strand model implies Schwinger’s
quantum action principle.
To calculate quantum motion with the principle of least action, we need to define the
kinetic and the potential energy in terms of strands. There are various possibilities for
Lagrangian densities for a given evolution equation; however, all are equivalent. In case
of the free Schrödinger equation, one possibility is:
Motion Mountain – The Adventure of Physics
Since energy is related to crossing switches, it is natural that strand fluctuations that do
not induce crossing switches are favoured. In short, the strand model states
210
8 q ua ntum theory d ed uced from stra nds
implies that for any inertial observer the flat vacuum has no matter content, has no energy content, is isotropic and is homogeneous. The strand model thus realizes this basic
requirement of special relativity. In the strand model, vacuum is Lorentz-invariant.
Many models of the vacuum, even fluctuating ones, have difficulties reproducing
Lorentz invariance. The strand model differs, because the strands are not the observable entities; only their crossing switches are. This topological definition, together with
the averaging of the fluctuations, makes the vacuum Lorentz-invariant.
We note that in the strand model, the vacuum is unique, and the vacuum energy of
flat infinite vacuum is exactly zero. In the strand model, there is no divergence of the
vacuum energy, and there is thus no contribution to the cosmological constant from
quantum field theory. In particular, there is no need for supersymmetry to explain the
small energy density of the vacuum.
Special rel ativit y: the invariant limit speed
⊳ The Planck speed ? is the observed average speed of crossing switches due
to photons.
Page 351
Page 351
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 168
Because the definition uses crossing switches and a massless particle, the speed of light
? is an energy speed. The speed of light ? is also an average for long times. Indeed, as is
well-known in quantum field theory, due to the indeterminacy relation, single photons
can travel faster or slower than light, but the probability for large deviations is extremely
low.
The linear motion of a helically deformed photon strand through the vacuum strands
is similar to the motion of a bottle opener through cork. It differs from the linear motion
of a matter tangle through vacuum, which makes use of the belt trick. The belt trick slows
fermions down, though the details are not simple, as we will discover below. In short, we
find that matter tangles always move more slowly than light. The speed ? is a limit speed.
In fact, we see that ultrarelativistic tangles move, as shown in Figure 43, almost like
light. We thus find that matter can almost reach the speed of light. The speed ? is indeed
a limit speed.
However, one problem remains open: how exactly do tangles move through the web
that describes the vacuum? We will clarify this issue later on. In a few words, the motion
of a photon requires that the strands of the surrounding space make room for it. This
requires favourable fluctuations, thus a finite time. The motion process of photons thus
makes it clear that the speed of light is finite.
The speed of light ? is defined as an average, because, as well-known in quantum
field theory, there are small probabilities that light moves faster or slower that ?. But
the average result ? will be the same for every observer. The value of the speed ? is thus
invariant.
In 1905, Einstein showed that the mentioned properties of the speed of light – energy
Motion Mountain – The Adventure of Physics
Page 227
In the strand model, massless particles are unknotted and untangled. Even though we
will deduce the strand model for photons only later on, we use it here already, to speed
up the discussion. In the strand model, the photon is described by a single, helically
deformed unknotted strand, as shown in Figure 50. Therefore, we can define:
q ua ntum theory d ed uced from stra nds
211
Strand model :
time average
of crossing
switches
Localized particle at rest :
Observed
probability
density :
t1
t1
t2
t2
Slow motion :
t2
t1
Motion Mountain – The Adventure of Physics
t1
t2
rotation,
precession and
displacement
t1
t2
t1
t2
rotation,
precession and
displacement
F I G U R E 43 Tangles at rest, at low speed and at relativistic speed.
speed, limit speed, finite speed and invariant speed – imply the Lorentz transformations.
In particular, the three properties of the speed of light ? imply that the energy ? of a
particle of mass ? is related to its momentum ? as
?2 = ?2 ?4 + ?2 ?2
Page 356
Page 150
or
ℏ2 ?2 = ?2 ?4 + ?2 ℏ2 ?2 .
(137)
This dispersion relation is thus also valid for massive particles made of tangled strands –
even though we cannot yet calculate tangle masses. (We will do this later on.)
Should we be surprised at this result? No. In the fundamental principle, the definition
of the crossing switch, we inserted the speed of light as the ratio between the Planck
length and the Planck time. Therefore, by defining the crossing switch in the way we did,
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Relativistic motion :
212
Page 206
Page 162
Page 194
8 q ua ntum theory d ed uced from stra nds
we have implicitly stated the invariance of the speed of light.
Fluctuating strands imply that flat vacuum has no matter or energy content, for every
inertial observer. Due to the strand fluctuations, flat vacuum is also homogeneous and
isotropic for every inertial observer. Therefore, together with the 3 + 1-dimensionality of
space-time deduced above, we have now definitely shown that flat vacuum has Poincaré
symmetry. This settles another issue from the millennium list.
The relativistic dispersion relation differs from the non-relativistic case in two ways.
First, the energy scale is shifted, and now includes the rest energy ?0 = ?2 ?. Secondly,
the spin precession is not independent of the particle speed any more; for relativistic
particles, the spin lies close to the direction of motion. Both effects follow from the
existence of a limit speed.
If we neglect spin, we can use the relativistic dispersion relation to deduce directly the
well-known Klein–Gordon equation for the evolution of a wave function:
(138)
In other words, the strand model implies that relativistic tangles follow the Klein–
Gordon equation. We now build on this result to deduce Dirac’s equation for relativistic
quantum motion.
Dirac ’ s equation deduced from tangles
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
The relativistic Klein–Gordon equation assumes that spin effects are negligible. This approximation fails to describe most experiments. A precise description of relativistic elementary particles must include spin.
So far, we deduced the Schrödinger equation using the relation between phase and
the quantum of action, using the non-relativistic energy–momentum relation, and neglecting spin. In the next step we deduced the Pauli equation by including the properties
of spin 1/2. The following step was to deduce the Klein–Gordon equation using again
the relation between phase and the quantum of action, this time the relativistic energy–
momentum relation, but assuming zero spin. The final and correct description of elementary fermions, the Dirac equation, results from combining all three ingredients: (1)
the relation between the quantum of action and the phase of the wave function, (2) the
relativistic mass–energy relation, and (3) the effects of spin 1/2. Now we can reproduce
this derivation because all three ingredients are reproduced by the strand model.
We first recall the derivation of the Dirac equation found in textbooks. The main
observation about spin in the relativistic context is the existence of states of right-handed
and of left-handed chirality: spin can precess in two opposite senses around the direction
of momentum. In addition, for massive particles, the two chiral states mix. The existence
of two chiralities requires a description of spinning particles with a wave function that
has four complex components, thus twice the number of components that appear in the
Pauli equation. Indeed, the Pauli equation implicitly assumes only one, given sign for the
chirality, even though it does not specify it. This simple description is possible because
in non-relativistic situations, states of different chirality do not mix.
Consistency requires that each of the four components of the wave function of a relativistic spinning particle must follow the relativistic energy–momentum relation, and
Motion Mountain – The Adventure of Physics
− ℏ2 ∂?? ? = ?2 ?4 − ?2 ℏ2 ∇2 ? .
q ua ntum theory d ed uced from stra nds
213
Motion Mountain – The Adventure of Physics
deduce the Dirac equation (© Springer Verlag, from Ref. 171).
Ref. 169
Ref. 170
thus the Klein–Gordon equation. This requirement is known to be sufficient to deduce
the Dirac equation. One of the simplest derivations is due to Lerner; we summarize it
here.
When a spinning object moves relativistically, we must take both chiralities into account. We call ? the negative chiral state and ? the positive chiral state. Each state is
described by two complex numbers that depend on space and time. The 4-vector for
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
F I G U R E 44 The belt trick for a rotating body with many tails, as used by Battey-Pratt and Racey to
214
8 q ua ntum theory d ed uced from stra nds
probability and current becomes
?? = ?† ?? ? + ?† ?? ? .
(139)
We now introduce the four-component spinor ? and the 4 × 4 spin matrices ??
?
?=( )
?
0
?
and ?? = ( ?
) ,
0 ??
(140)
where ?? = (?, ?) and ?? = (?, −?) and ? is the 2 × 2 identity matrix. The 4-current can
then be written as
?? = ?† ?? ? .
(141)
The three requirements of current conservation, Lorentz invariance and linearity then
yield the evolution equation
iℏ∂? (?? ?) + ???5 ? = 0 .
This is the Dirac equation in the (less usual) spinorial representation.* The last term
shows that mass mixes right and left chiralities. The equation can be expanded to include
potentials using minimal coupling, in the same way as done above for the Schrödinger
and Pauli equations.
The above textbook derivation of the Dirac equation from usual quantum theory can
be repeated and visualized also with the help of strands. There is no difference in arguments or results. The derivation with the help of strands was performed for the first
time by Battey-Pratt and Racey, in 1980. They explored a central object connected by
unobservable strands (or ‘tails’) to the border of space, as shown in Figure 44. In their
approach, the central object plus the tails correspond to a quantum particle. The central
object is assumed to be continuously rotating, thus reproducing spin 1/2. They also assumed that only the central object is observable. (In the strand model, the central object
becomes the tangle core.) Battey-Pratt and Racey then explored a relativistically moving object of either chirality. They showed that a description of such an object requires
four complex fields. Studying the evolution of the phases and axes for the chiral objects
yields the Dirac equation. The derivation by Battey-Pratt and Racey is mathematically
equivalent to the textbook derivation just given.
We can thus say that the Dirac equation follows from the belt trick. We will visualize this connection in more detail in the next section. When the present author found
this connection in 2008, Lou Kauffman pointed out the much earlier paper by BatteyPratt and Racey. In fact, Paul Dirac was still alive when they found this connection, but
unfortunately he did not answer their letter asking for comment.
* The matrix ?5 is defined here as
0
?5 = (
?
where ? is the 2 × 2 identity matrix.
?
) ,
0
(143)
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 171
(142)
Motion Mountain – The Adventure of Physics
Ref. 170
q ua ntum theory d ed uced from stra nds
215
In summary, tangles completely reproduce both the rotation and the linear motion
of elementary fermions. Therefore, the strand model provides a simple view on the
evolution equations of quantum theory. In the terms of the strand model, when spin
is neglected, the Schrödinger equation describes the evolution of crossing density. For
relativistic fermions, when the belt trick is included, the Dirac equation describes the
evolution of crossing density. In fact, strands visualize these evolution equations in the
most concrete way known so far.
Visualizing spinors and Dirac ’ s equation using tangles
Ref. 172
In total, these are eight real parameters; they correspond to one positive real number and
seven phases. They lead to the description of a spinor wave function as
? = √? e?? ?(?) ?(?/2, ?/2, ?/2) ,
Ref. 172
(144)
where the product ?? is an abbreviation for the boosted and rotated unit spinor and
all parameters depend on space and time. This expression is equivalent to the description with four complex parameters used in most textbooks. In fact, this description of a
spinor wave function and the related physical visualization of its density and its first six
phases dates already from the 1960s. The visualisation can be deduced from the study of
relativistic spinning tops or of relativistic fluids. Rotating tangles are more realistic, however. In contrast to all previous visualizations, the rotating tangle model explains also the
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
— There is an average density ?(?, ?); physically, this is the probability density. In the
strand model, this is the local crossing density.
— There is a set of three Euler angles ?, ? and ?; physically, they describe the average
local orientation and phase of the spin axis. In the strand model, this is the average
local orientation and phase of the tangle core.
— There is a second set of three parameters ? = (?? , ??, ?? ); physically, they describe, at
one’s preference, either the average local Lorentz boost or a second set of three Euler
angles. In the strand model, these parameters describe the average local deformation
of the core that is due to the Lorentz boost. It can also be seen as the axis around
which the belt trick is performed.
— There is a phase ?; physically, this represents the relative importance of particle and
antiparticle density. In the strand model, this phase describes with what probability
the average local belt trick is performed right-handedly or left-handedly.
Motion Mountain – The Adventure of Physics
Despite its apparent complexity, the Dirac equation makes only a few statements: spin
1/2 particles are fermions, obey the relativistic energy–momentum relation, keep the
quantum of action invariant, and thus behave like a wave. Each statement is visualized by the tangle model of fermions: tangles behave as spinors, the relativistic energy–
momentum relation is built-in, the fundamental principle holds, and rotating tangle
cores reproduce the evolution of the phase. Let us look at the details.
Given a particle tangle, the short-time fluctuations lead, after averaging of the crossings, to the wave function. The tangle model of fermions also provides a visualization
of the spinor wave function. Indeed, at each point in space, the wave function has the
following parameters:
216
Ref. 164
Ref. 173
Ref. 175
1. Tangles support the view that elementary particles are little rotating entities, also in
the relativistic case. This fact has been pointed out by many scholars over the years.
The strand model provides a consistent visualization for these discussions.
2. The belt trick can be seen as the mechanism underlying the famous Zitterbewegung
that is part of the Dirac equation. The limitations in the observing the belt trick translate directly into the difficulties of observing the Zitterbewegung.
3. The belt trick also visualizes why the velocity operator for a relativistic particle has
eigenvalues ±?.
4. The Compton length is often seen as the typical length at which quantum field effects
take place. In the tangle model, it would correspond to the average size needed for
the belt trick. The strand model thus suggests that the mass of a particle is related to
the average size needed for the belt trick.
5. Tangles support the – at first sight bizarre – picture of elementary particles as little
charges rotating around a centre of mass. Indeed, in the tangle model, particle rotation requires a regular application of the belt trick of Figure 19, and the belt trick can
be interpreted as inducing the rotation of a charge, defined by the tangle core, around
a centre of mass, defined by the average of the core position. It can thus be helpful to
use the strand model to visualize this description.
6. The tangle model can be seen as a vindication of the stochastic quantization research
programme; quantum motion is the result of underlying fluctuations. For example,
the similarity of the Schrödinger equation and the diffusion equation is modelled and
explained by the strand model: since crossings can be rotated, diffusion of crossings
leads to the imaginary unit that appears in the Schrödinger equation.
In short, rotating tangles are a correct underlying model for the propagation of fermions.
And so far, tangles are also the only known correct model. Tangles model propagators.
This modelling is possible because the Dirac equation results from only three ingredients:
— the relation between the quantum of action and the phase of the wave function (the
wave behaviour),
— the relation between the quantum of action and spinor behaviour (the exchange behaviour),
— and the mass–energy relation of special relativity (the particle behaviour), itself due
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 174
Page 175
last, seventh phase. This is the phase that describes matter and anti-matter, that explains
the appearance of the quantum of action ℏ, and that explains the fermion behaviour.
In short, only rotating tangles together with the fundamental principle provide a
simple, complete and precise visualisation of spinor wave functions and their evolution.
The tangle model for spinning relativistic quantum particles remains a simple extension
of Feynman’s idea to describe a quantum particle as a rotating little arrow. The arrow
can be imagined as being attached to the rotating tangle core. The tails are needed to
reproduce fermion behaviour. The specific type of tangle core determines the type of
particle. The blurring of the crossings defines the wave function. Rotating arrows describe non-relativistic quantum physics; rotating tangles describe relativistic quantum
physics.
Visualizing spinor wave functions with tangles of strands helps the understanding of
the Dirac equation in several ways.
Motion Mountain – The Adventure of Physics
Ref. 172
8 q ua ntum theory d ed uced from stra nds
q ua ntum theory d ed uced from stra nds
A hydrogen atom
Simplified
strand model :
time average
of crossing
switches
217
Observed
electron
probability
density :
proton
proton
electron
tangle
electron cloud
to the fundamental principle.
And all three ingredients are reproduced by the strand model. We see that the apparent
complexity of the Dirac equation hides its fundamental simplicity. The strand model
reproduces the ingredients of the Dirac equation, reproduces the equation itself, and
makes the simplicity manifest. In fact, we can say:
Page 178
The belt trick is fundamental for understanding the Dirac equation. In the strand model,
core rotations vary along two dimensions – the rotation is described by two angles – and
so does the belt trick. The resulting four combinations form the four components of the
Dirac spinor and of the Dirac equation.
In summary, tangles can be used as a precise visualization and explanation of
quantum physics. Wave functions, also those of fermions, are blurred tangles – with the
detail that not the strands, but their crossings are blurred.
Q uantum mechanics vs. quantum field theory
Quantum mechanics is the approximation to quantum physics in which fields are continuous and particles are immutable. In the strand model, quantum mechanics is thus
the approximation in which a particle is described by a tangle with a shape that is fixed
in time. This approximation allows us to derive the Dirac equation, the Klein–Gordon
equation, the Proca equation, the Pauli equation and the Schrödinger equation. In this
approximation, the strand model for the electron in a hydrogen atom is illustrated in
Figure 45. This approximation already will allow us to deduce the existence of the three
gauge interactions, as we will see in the next chapter.
In contrast, quantum field theory is the description in which fields are themselves described by bosons, and particles types can transform into each other. The strand model
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
⊳ The Dirac equation describes the relativistic infinitesimal belt trick or string
trick.
Motion Mountain – The Adventure of Physics
F I G U R E 45 A simple, quantum-mechanical view of a hydrogen atom.
218
8 q ua ntum theory d ed uced from stra nds
allows us to deduce the existence of all known gauge bosons, as shown in the next chapter.
In the strand description of quantum field theory, particles are not tangles with a fixed
shape of their core, but for each particle, the shape varies. This variation leads to gauge
boson emission and absorption.
A fl ashback: set tling three parad oxes of Galilean physics
Page 108
“
Challenge 143 s
Urlaub ist die Fortsetzung des Familienlebens
unter erschwerten Bedingungen.*
Dieter Hildebrandt
”
Are the definitions for the addition and multiplication of Schrödinger wave functions
that were given above also valid for spinor tangle functions?
∗∗
* ‘Vacation is the continuation of family life under aggravated conditions.’ Dieter Hildebrandt (b. 1927
Bunzlau, d. 2013 Munich) was a cabaret artist, actor and author.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Fun challenges abou t quantum theory
Motion Mountain – The Adventure of Physics
In all descriptions of physics, space and time are measured, explained and defined using
matter. This occurs, for example, with the help of metre bars and clocks. On the other
hand, matter is measured, explained and defined using space and time. This occurs, for
example, by following a localized body over space and time. The circularity of the two
definitions is at the basis of modern physics.
As already mentioned above, the circularity is a natural consequence of the strand
model. Both matter and space-time turn out to be approximations of the same basic
building blocks; this common origin explains the apparent circular reasoning of Galilean
physics. Most of all, the strand model changes it from a paradox to a logical necessity.
The strand model defines vacuum, and thus physical space, as a result of averaging
strand crossings. Space is thus a relative concept. Newton’s bucket experiment is sometimes seen as a counter-argument to this conclusion and as an argument for absolute
space. However, the strand model shows that any turning object is connected to the rest
of the universe through its tails. This connection makes every rotation an example of relative motion. Rotation is thus always performed relatively to the horizon of the universe.
On the other hand, the detection of tangles among the tails allows a local determination
of the rotation state, as is observed. Strands thus confirm that rotation and space are relative concepts. Strands thus also explain why we can turn ourselves on ice by rotating an
arm over our head, without outside help. Strands lie to rest all issues around the rotating
bucket.
A long time ago, Zeno of Elea based one of his paradoxes – the flying arrow that
cannot reach the target – on an assumption that is usually taken as granted: he stated
the impossibility to distinguish a short-time image (or state) of a moving body from the
image (or state) of a resting body. The flattening of the tangles involved shows that the
assumption is incorrect; motion and rest are distinguishable, even in (imagined) photographs taken with extremely short shutter times. The argument of Zeno is thus not
possible, and the paradox disappears.
q ua ntum theory d ed uced from stra nds
Challenge 144 e
219
The definition of tangle functions, or wave functions, did not take into account the crossings of the vacuum strands, but only those of the particle tangle. Why is this allowed?
∗∗
Challenge 145 e
Modelling the measurement of action at the quantum level as the counting of full turns
of a wheel is a well-known idea that is used by good teachers to take the mystery out
of quantum physics. The strand model visualizes this idea by assigning the quantum of
action ℏ to a full turn of one strand segment around another.
∗∗
Challenge 146 s
Is any axiomatic system of quantum theory in contrast with the strand model?
∗∗
∗∗
Ref. 176
Challenge 148 e
If you do not like the deduction of quantum mechanics given here, there is an alternative:
you can deduce quantum mechanics in the way Schwinger did in his course, using the
quantum action principle.
∗∗
Challenge 149 r
Modern teaching of the Dirac equation replaces the spinor picture with the vector
picture. Hrvoje Nikolić showed that the vector picture significantly simplifies the understanding of Lorentz covariance of the Dirac equation. How does the vector picture clarify
the relation between the belt trick and the Dirac equation?
∗∗
Challenge 150 s
In the strand description of quantum mechanics, strands are impenetrable: they cannot
pass through each other (at finite distances). Can quantum mechanics also be derived if
the model is changed and this process is allowed? Is entanglement still found?
∗∗
Challenge 151 e
A puzzle: Is the belt trick possible in a continuous and deformable medium – such as a
sheet or a mattress – in which a coloured sphere is suspended? Is the belt trick possible
with an uncountably infinite number of tails?
∗∗
Page 180
Challenge 152 s
At first sight, the apheresis machine diagram of Figure 24 suggests that, using the belt
trick, animals could grow and use wheels instead of legs, because rotating wheels could
be supplied with blood and connected to nerves. Why did wheels not evolve nevertheless?
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 177
Motion Mountain – The Adventure of Physics
Challenge 147 s
In the strand model, tangle energy is related to tangle core rotation. What is the difference between the angular frequency for tangles in the non-relativistic and in the relativistic case?
220
8 q ua ntum theory d ed uced from stra nds
summary on quantum theory of matter:
experimenta l predictions
In this chapter, we used the fundamental principle – crossing switches define the
quantum of action ℏ and the other Planck units – to deduce that particles are tangles
of strands and that wave functions are time-averaged rotating tangles. In simple words,
⊳ Both non-relativistic and relativistic wave functions are blurred rotating
tangles.
⊳ The Dirac equation is essentially the infinitesimal version of the belt trick
(or string trick).
L = ? (?ℏ?/
D − ?2 ?) ? ,
where
Page 39
/ = ?? D? = ?? (∂? − ??? ? ) .
D
(145)
(146)
We thus conclude that strands reproduce the quantum theory of matter.
The strand model predicts deviations from the relativistic matter Lagrangian, and thus
from the Dirac equation, only in three cases: first, when quantum aspects of electrodynamic field play a role, second, when nuclear interactions play a role, and third, when
space curvature, i.e., strong gravity, plays a role. All this agrees with observation.
We will deduce the description of quantum electrodynamics and of the nuclear interactions in the next chapter. In the case of gravity, the strand model predicts that
deviations from quantum theory occur exclusively when the energy–momentum of an
elementary particle approaches the Planck value, i.e., for really strong gravity. Such deviations are not accessible to experiment at present. We will explore this situation in the
subsequent chapter.
In addition, the strand model predicts that in nature, the Planck values for momentum
and energy are limit values that cannot be exceeded by a quantum particle. All experi-
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In other words, we have shown that strands reproduce the relativistic Lagrangian density
L of charged, elementary, relativistic fermions in an external electromagnetic field ?
Motion Mountain – The Adventure of Physics
More precisely, a wave function appears from the blurred crossings of a tangle. The components and phases of the wave function at a point in space are due to the orientation
and phase of crossings at that point. We also deduced that blurred tangles obey the least
action principle and the Dirac equation.
In other words, visualizing the quantum of action as a crossing switch implies
quantum theory. The strand model confirms Bohr’s statement: quantum theory is indeed a consequence of the quantum of action. Specifically, the strand model thus shows
that all quantum effects are consequences of extension and consequences of the three dimensions of space. More precisely, all quantum effects are due to tails, the tails of the
tangles that represent a quantum system. In particular, the strand model confirms that
sum m ary on q ua ntum theory of m at ter: experimental pred ictions
Page 162
221
ments agree with this prediction.
The deduction of quantum theory from strands given here is, at present, the only
known microscopic explanation for quantum physics. So far, no other microscopic
model, no different explanation nor any other Planck-scale deduction of quantum theory has been found. In particular, the extension of fundamental entities – together with
observability limited to crossing switches – is the key to understanding quantum physics.
Let us evaluate the situation. In our quest to explain the open issues of the millennium
list, we have explained the origin of Planck units, the origin of wave functions, the origin
of the least action principle, the origin of space-time dimensions, the Lorentz and Poincaré symmetries, the origin of particle identity, and the simplest part of the Lagrangian
of quantum field theory, namely, the Lagrangian of free fermions, such as the electron,
and that of fermions in continuous external fields. Therefore, for the next leg, we turn to
the most important parts of the standard model Lagrangian that are missing: those due
to gauge interactions.
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Chapter 9
GAUGE INT ER ACT IONS DEDUCED
FROM ST R ANDS
Ref. 178
Ref. 179
W
Page 174
Ref. 164
Experiments in the quantum domain show that interactions change the phase of wave
functions. But how precisely does this happen? The strand model will give us a simple
answer: the emission and the absorption of gauge bosons is only possible together with
a phase change. To explain this connection, we need to study the phase of tangle cores in
more detail.
When we explored spin and its connection to the belt trick, we pictured the rotation
of the tangle core in the same way as the rotation of a belt buckle. We assumed that the
core of the tangle rotates like a rigid object; the rotation is completed through the shape
fluctuations of the tails only. Why did we assume this?
In Feynman’s description of quantum theory, free particles are advancing rotating arrows. In the strand model, free particle motion is modelled as the change of position of
the tangle core and spin as the rotation of the core. We boldly assumed that the core
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Interactions and phase change
Motion Mountain – The Adventure of Physics
Page 18
hat are interactions? At the start of this volume, when we summarized
hat relates the Planck units to relativity and to quantum theory,
e pointed out that the nature of interactions at Planck scales was still in the
dark. In the year 2000, it was known for several decades that the essential properties
of the electromagnetic, the weak and the strong nuclear interaction are their respective
gauge symmetries: all three interactions are gauge interactions. But the underlying
reason for this property was still unknown.
In this chapter we discover that fluctuating strands in three spatial dimensions explain the existence of precisely three gauge interactions, each with precisely the gauge
symmetry group that is observed. This is the first time ever that such an explanation
is possible. In other terms, we will deduce quantum field theory from strands. Indeed,
strands provide a natural mechanism for interactions that explains and implies Feynman
diagrams. The term ‘mechanism’ has to be taken with a grain of salt, because there is
nothing mechanical involved; nevertheless, the term is not wrong, because we shall discover a surprisingly simple result: Gauge interactions and gauge symmetries are due to
specific strand deformations.
In this chapter, we work in flat space-time, as is always done in quantum field theory. We leave the quantum aspects of curved space-time and of gravitation for the next
chapter. We thus start by exploring the non-gravitational interactions in the quantum
domain.
g aug e interactions d ed uced from stra nds
Strand model :
tail
time average
of crossing
changes
spin
223
Observed
probability
density :
spin
position
phase
position
phase
core
F I G U R E 46 In the chapter on quantum theory, the phase was defined assuming a rigidly rotating core;
position
phase
such core deformations lead to gauge interactions.
remained rigid, attached the phase arrow to it, and described spin as the rotation of the
core with its attached arrow, as shown again in Figure 46. This bold simplification led us
to the Dirac equation. In short, the assumption of a rigid core works.
However, we swept a problem under the rug: what happens if the core is not rigid? It
turns out that the answer to this question automatically leads to the existence of gauge
interactions. Now, we know from usual quantum theory that
⊳ An interaction is a process that changes the phase of a wave function, but
differs from a rotation.
In the strand model, shape deformations of tangle cores also lead to phase changes. In
fact, we will discover that core deformations automatically lead to precisely those three
gauge interactions that we observe in nature.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
F I G U R E 47 A magnified tangle core shows that the phase can also change due to core deformations;
Motion Mountain – The Adventure of Physics
this approximation was also used in the description of particle translation.
224
9 g auge interactions d ed uced from stra nds
Tail deformations versus core deformations
We can summarize the previous chapter, on the free motion of matter tangles, as the
chapter that focused on shape fluctuations of tails. Indeed, the belt trick completed the
proof that
⊳ Space-time symmetries are due to tail deformations.
All space-time symmetries – translation, rotation, boost, spin and particle exchange – are
due to tail deformations; in such tail deformations, the tangle core is assumed to remain
unchanged and rigid (in its own rest frame).
In contrast, the present chapter focuses on shape fluctuations in tangle cores.* We will
discover that
⊳ Gauge symmetries are due to core deformations.
⊳ When the phase of a core changes through core shape deformation, we speak
of interaction.
Ref. 179
Ref. 181
Ref. 180
We thus need to understand two things: First, what kinds of core deformation exist?
Secondly, how precisely is the phase – i.e., each arrow definition – influenced by core
deformations? In particular, we have to check the answers and deductions with experiment.
The first question, on the classification of the core deformations, is less hard than
it might appear. The fundamental principle – events are crossing switches of strands –
implies that deformations are observable only if they induce crossing switches. Other
deformations do not have any physical effect. (Of course, certain deformations will have
crossing switches for one observer and none for another. We will take this fact into consideration.) Already in 1926, the mathematician Kurt Reidemeister classified all those
tangle deformations that lead to crossing switches. The classification yields exactly three
classes of deformations, today called the three Reidemeister moves. They are shown in
Figure 48.
* The contrast between tail deformations and core deformations has a remote similarity to gravity/gauge
duality, or AdS/CFT correspondence, and to space-time duality. For example, in the strand model, the three
Reidemeister moves on tangle cores represent the three gauge interactions, whereas the three Reidemeister
moves on the vacuum represent (also) gravitational effects.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
⊳ When the phase of a core changes through rigid orientation change, we speak
of core rotation.
Motion Mountain – The Adventure of Physics
Let us explore the tangle core in more detail. Figure 47 shows a magnified view of the
core and its phase arrow. The phase of the core results from the phases of all its crossings. Thus, the figure makes it clear that the phase arrow will be sensitive to the shape
fluctuations and deformations of the strand segments that make up the core.
In nature, any phase change of the wave function that is not due to a space-time symmetry is due to an interaction. For the strand model, this connection implies:
g aug e interactions d ed uced from stra nds
Reidemeister move I
or twist
Reidemeister move II
or poke
225
Reidemeister move III
or slide
F I G U R E 48 The Reidemeister moves: the three types of deformations that induce crossing switches – if
the moves are properly defined in three dimensions.
⊳ The first Reidemeister move, or type I move, or twist, is the addition or removal of a twist in a strand.
⊳ The third Reidemeister move, or type III move, or slide, is the displacement
of one strand segment under (or over) the crossing of two other strands.
⊳ The first Reidemeister move corresponds to electromagnetism.
⊳ The second Reidemeister move corresponds to the weak nuclear interaction.
⊳ The third Reidemeister move corresponds to the strong nuclear interaction.
We will prove this correspondence in the following.
For each Reidemeister move we will explore two types of core deformation processes:
One deformation type are core fluctuations, which correspond, as we will see, to the emission and absorption of virtual interaction bosons. The other deformations are externally
induced core disturbances, which correspond to the emission and absorption of real interaction bosons. As the first step, we show that both for fluctuations and for disturbances,
the first Reidemeister move, the twist, is related to the electromagnetic interaction.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
The type number of each Reidemeister move is also the number of involved strands.
We will discover that despite appearances, each Reidemeister move induces a crossing
switch. To find this connection, we have to generalize the original Reidemeister moves,
which were defined in a two-dimensional projection plane, to the three-dimensional
situation of tangle cores.
The three Reidemeister moves turn out to be related to the three gauge interactions:
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⊳ The second Reidemeister move, or type II move, or poke, is the addition or
removal of a bend of one strand under (or over) a second strand.
226
9 g auge interactions d ed uced from stra nds
photon
vacuum
twist
transfer
fermion
fermion
with
different
phase
F I G U R E 49 A single strand changes the rotation of a tangle: twist transfer is the basis of
electrodynamics and the first reidemeist er move
Strands and the t wist, the first R eidemeister move
In the strand model of electromagnetism, massless spin 1 bosons such as the photon are
made of a single strand. How can a single strand change the phase of a tangle? The
answer is given in Figure 49: a twisted loop in a single strand will influence the rotation
of a tangle because it changes the possible fluctuations of the tangle core. Due to the
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 5
Experiments show that all four fundamental interactions are described by potentials.
Experiments also show that potentials change the phase, the rotation frequency and the
wave number of wave functions. Experiments show that interactions result from the
absorption and the emission of gauge bosons. In particular, for electromagnetism, the
potentials are due to the flow of real and virtual, massless, uncharged spin-1 photons.
Photons are emitted from or absorbed by charged elementary particles; neutral elementary particles do not emit or absorb photons. There are two types of electric charge, positive and negative. The attraction and repulsion of static charges diminishes with the
inverse square of the distance. Charge is conserved. All charged particles are massive
and move slower than light. The Lagrangian of matter coupled to the electromagnetic
field has a U(1) gauge symmetry. Electromagnetism has a single fundamental Feynman
diagram. The electromagnetic coupling constant at low energy, the so-called fine structure constant, is measured to be ? = 1/137.035 999 139(31); its energy dependence is
described by renormalization.
The previous paragraph contains everything known about the electromagnetic interaction. For example, Maxwell’s field equations follow from Coulomb’s inverse square
relation, its relativistic generalization, and the conservation of charge. More precisely, all
experimental observations about electricity and magnetism follow from the Lagrangian
of quantum electrodynamics, or QED. In short, we now need to show that the Lagrangian
of QED follows from the strand model.
Motion Mountain – The Adventure of Physics
electromagnetism in the strand model. No strand is cut or reglued; the transfer occurs, statistically,
through the excluded volume due to the impenetrability of strands.
electrodynamics a nd the first reidemeister m ov e
227
The photon
Strand model:
Observation :
time average
of crossing
switches
electric
field
strength
phase
helicity
motion
wavelength
F I G U R E 50 The photon in the strand model.
Page 232
⊳ A photon is a twisted strand. An illustration is given in Figure 50.
⊳ The electromagnetic interaction is the transfer of twists, i.e., the transfer of
first Reidemeister moves, between two particles, as shown in Figure 49.
The transfer of a twist from a single strand to a tangle core thus models the absorption
of a photon. We stress again that this transfer results from the way that strands hinder
each other’s motion, because of their impenetrability. No strand is ever cut or reglued.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 174
impenetrability of strands, an approaching twisted loop will sometimes transfer its twist
to the tangle and thereby change its phase. The observed effect of an electromagnetic
field on the phase of a charged fermion is the time average of all such twist transfers.
Single strands represent bosons, as we saw above. Twisted loops are single strands and
can have two twist senses, or two polarizations. Single, twisted and unknotted strands
have no mass; in other words, twisted loops effectively move with the speed of light.
And twisted loops, being curved, carry energy.
Approaching twisted loops will change the phase, i.e., the orientation of a matter
tangle. Twisted loops correspond to a local rotation of a strand segment by π. But twists
can be generalized to arbitrary angles. These generalized twists can be concatenated. Because they are described by a single angle, and because a double twist is equivalent to no
twist at all, twists form a U(1) group. We show this in detail shortly.
In summary, twists behave like photons in all their properties. Therefore, the strand
model suggests:
Motion Mountain – The Adventure of Physics
or, equivalently:
228
9 g auge interactions d ed uced from stra nds
Can photons decay, disappear or break up?
Challenge 153 e
Surrounded by a bath of photon strands, not all fermion tangles will change their phase.
A tangle subject to randomly approaching virtual photons will feel a net effect over time
only if it lacks some symmetry. In other words, only tangles that lack a certain symmetry
will be electrically charged. Which symmetry will this be?
In a bath of photon strands, thus in a bath that induces random Reidemeister I moves,
only chiral fermion tangles are expected to be influenced. In other terms:
⊳ Electric charge is due to lack of mirror symmetry, i.e., to tangle chirality.
Conversely, we have:
⊳ Electrically charged particles randomly emit twisted strands. Due to the
tangle chirality, a random emission will lead to a slight asymmetry, so that
right-handed twists will be in the majority for particles of one charge, and
left-handed twists will be in the majority for particles of the opposite charge.
Equating electric charge with tangle chirality allows modelling several important observations. First, because chirality can be right-handed or left-handed, there are positive and
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Electric charge
Motion Mountain – The Adventure of Physics
Page 351
The strand model of the photon, as shown in Figure 50, might be seen to suggest that
photons can disappear. For example, if a photon strand is straightened out by pulling the
ends of the helical deformation, the helix might disappear. A helix might also disappear
by a shape fluctuation or transform into several helices. However, this is a fallacy.
A lone twist cannot disappear by pulling; ‘‘pulling’’ requires an apparatus that performs it. That is impossible. A lone twist cannot disappear by fluctuations either, because
a photon also includes the vacuum strands around it. In the strand model, the energy
of the photon is localized in the configuration formed by the photon strand and the surrounding vacuum strands. In the strand model, energy is localized in regions of strand
curvature. If the helical strands disappears, the surrounding vacuum strands are curved
instead, or more strongly, and the twist energy is taken up by these surrounding strands.
The net result is that the helix is transferred, permanently or for a short time, to another
strand. In other terms, in the strand model, photons can also move by hopping from one
strand to the next.
Also, a single photon strand cannot break up into several photon strands of smaller
helical diameters or of different rotation frequencies. Such a process is prevented by the
fundamental principle, when the vacuum is taken into account.
The only way in which a photon can disappear completely is by transferring its crossing, i.e., its energy to a tangle. Such a process is called the absorption of a photon by a
charged particle.
In short, due to energy and to topological restrictions, the strand model prevents the
decay, disappearance or splitting of photons, as long as no electric charge is involved.
Linear and angular momentum conservation also lead to the same conclusion. Photons
are stable particles in the strand model.
electrodynamics a nd the first reidemeister m ov e
The twist move (or first
Reidemeister move)
in textbook form :
The twist move (or first
Reidemeister move)
applied to an interacting
tangle and loop :
fermion
photon
229
The unique
generator
of the twist
move is a
rotation by π.
vacuum
The basic twist can be described
as a local rotation by π.
A full rotation, from -π to π,
produces a crossing switch.
affected
not affected
Emission of numerous random twists by
chiral tangles leads to Coulomb’s law :
strand
model
observed
time average
negative charges. Second, because strands are never cut or reglued in the strand model,
chirality, and thus electric charge, is a conserved quantity. Third, chirality is only possible for tangles that are localized, and thus massive. Therefore, chiral tangles – charged
particles – always move slower than light. Fourth, a chiral tangle at rest induces a twisted strand density around it that changes as 1/?2 , as is illustrated in Figure 51. Finally,
photons are uncharged; thus they are not influenced by other photons (to first order).
In short, all properties of electric charge found in nature are reproduced by the tangle
model. We now check this in more detail.
Challenge: What top ological invariant is electric charge?
Chirality explains the sign of electric charge, but not its magnitude in units of the elementary charge ?. A full definition of electric charge must include this aspect.
Mathematicians defined various topological invariants for knot and tangles. Topological invariants are properties that are independent of the shape of the knot or tangle, but
allow to distinguish knots or tangles that differ in the ways they are knotted or tangled up.
Several invariants are candidates as building blocks for electric charge: chirality ?, which
can be +1 or −1, minimal crossing number ?, or topological writhe ?, i.e., the signed min-
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
F I G U R E 51 Electromagnetism in the strand model: the electromagnetic interaction, electric charge and
Coulomb’s inverse square relation.
Motion Mountain – The Adventure of Physics
Large numbers of random twists
affect only chiral tangles :
230
Page 419
Page 385
9 g auge interactions d ed uced from stra nds
imal crossing number.
A definition of electric charge ?, proposed by Claus Ernst, is ? = ? (? mod 2). Another
option for the definition of charge is ? = ?/3. Equivalent definitions use the linking
number. At this point of our exploration, the issue is open. We will come back to the
detailed connection between charge, chirality and tangle topology later on.
Electric and magnetic fields and p otentials
The definition of photons with twisted strands leads to the following definition.
⊳ The electric field is the volume density of (oriented) crossings of twisted
loops.
⊳ The magnetic field is the flow density of (oriented) crossings of twisted
loops.
⊳ The magnetic potential is the flow density of twisted loops.
Challenge 154 e
The strand model thus reproduces electromagnetic energy.
We note that in the strand model, the definition of the fields implies that there is no
magnetic charge in nature. This agrees with observation.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 184
The simplest way to check these definitions is to note that the random emission of twisted
loops by electric charges yields Coulomb’s inverse square relation: the force between two
static spherical charges changes with inverse square of the distance. The strand model
implies that in this case, the crossing density is proportional to the square of the loop density; in other words, the potential falls of as the inverse distance, and the electric field as
the square distance.
The definition of the magnetic field simply follows from that of the electric field by
changing to moving frame of reference. The two field definitions are illustrated in Figure 52.
We note that the electric field is defined almost in the same way as the wave function:
both are oriented crossing densities. However, the electric field is defined with the crossing density of twisted loops, whereas the wave function is defined with the crossing density of tangles. The definitions differ only by the topology of the underlying strand structures.
In the strand model, energy, or action per time, is the number of crossing switches
per time. The electromagnetic field energy per volume is thus given by the density of
crossing switches per time that are due to twisted loops. Now, the strand model implies
that the crossing switch density per time is given by half the square of the crossing density
plus half the square of the crossing density flow. For twisted loops, we thus get that the
energy density is half the square of the electric plus half the square of the magnetic field.
Inserting the proportionality factors that lead from Planck units to SI units we get the
well-known expression
? ?0 2
1 2
? .
(147)
= ? +
?
2
2?0
Motion Mountain – The Adventure of Physics
⊳ The electric potential is the density of twisted loops.
electrodynamics a nd the first reidemeister m ov e
twisted
loop
average
loop
motion
t1
t2
small
volume
element
with
crossing
magnetic
field
B
charge
F I G U R E 52 Moving twists allow us to define electric fields – as the density of twisted loop crossings –
and magnetic fields – as the corresponding flow.
The strand model predicts limit values to all observables. They always appear when
strands are as closely packed as possible. This implies a maximum electric field value
?max = ?4 /4?? ≈ 1.9 ⋅ 1062 V/m and a maximum magnetic field value ?max = ?3 /4?? ≈
6.3 ⋅ 1053 T. All physical systems – including all astrophysical objects, such as gamma-ray
bursters or quasars – are predicted to conform to this limit. This strand model prediction
agrees with observations.
The L agrangian of the electromagnetic field
In classical electrodynamics, the energy density of the electromagnetic field is used to
deduce its Lagrangian density. The Lagrangian density describes the intrinsic, observerindependent change that occurs in a system. In addition, the Lagrangian density must
be quadratic in the fields and be a Lorentz-scalar.
A precise version of these arguments leads to the Lagrangian density of the electromagnetic field ?
?
1 2
1
LEM = 0 ?2 −
? =−
? ???
(148)
2
2?0
4?0 ??
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
velocity
Motion Mountain – The Adventure of Physics
electric
field
E
231
232
9 g auge interactions d ed uced from stra nds
Freedom of definition of the crossing phase / crossing orientation:
phase 1
phase 2
phase 3
F I G U R E 53 The definition of the phase or orientation of a single crossing is not unique: there is a
freedom of choice.
where the electromagnetic field ? is defined with the electromagnetic potential ? as
(149)
Since the strand model reproduces the electromagnetic energy, it also reproduces the
Lagrangian of classical electrodynamics. In particular, Maxwell’s equations for the electromagnetic field follow from this Lagrangian density. Maxwell’s field equations are thus
a consequence of the strand model. Obviously, this is no news, because any model that
reproduces Coulomb’s inverse square distance relation and leaves the speed of light invariant automatically contains Maxwell’s field equations.
In nature, the electromagnetic potential ? ? is not uniquely defined: one says that there
is a freedom in the choice of gauge. The change from one gauge to another is a gauge
transformation. Gauge transformations are thus transformations of the electromagnetic
potential that have no effect on observations. In particular, gauge transformations leave
unchanged all field intensities and field energies on the one hand and particle probabilities and particle energies on the other hand.
All these observations can be reproduced with strands. In the strand model, the following definitions are natural:
⊳ A gauge choice for radiation and for matter is the choice of definition of the
respective phase arrow.
⊳ A gauge transformation is a change of definition of the phase arrow.
In the case of electrodynamics, the gauge freedom is a result of allowing phase choices
that lie in a plane around the crossing orientation. (The other interactions follow from
the other possible phase choices.) The phase choice can be different at every point in
space. Changing the (local) phase definition is a (local) gauge transformation. Changing
the phase definition for a single crossing implies changing the phase of wave functions
and of the electromagnetic potentials. A schematic illustration of the choice of gauge is
given in Figure 53 and Figure 54.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
U(1) gauge invariance induced by t wists
Motion Mountain – The Adventure of Physics
??? = ∂? ? ? − ∂? ? ? .
electrodynamics a nd the first reidemeister m ov e
233
Freedom of definition for the crossing phase of tangles
Strand model :
Observation :
time average
of crossing
changes
Matter :
spin
spin
U(1) phase
definition
freedom
probability
amplitude
core
helicity
helicity
U(1) phase
definition
freedom
F I G U R E 54 The freedom in definition of the phase of crossings leads to the gauge invariance of
electrodynamics. Three exemplary choices of phase are shown.
Twists on tangle cores
form a U(1) group
axis
axis
π
The basic twist, or Reidemeister I
move, is a local rotation, by
an angle π around the axis,
of the core region enclosed
by a dashed circle.
Generalized to arbitary angles,
the basic twist generates
a U(1) group.
F I G U R E 55 How the set of generalized twists – the set of all local rotations of a single strand segment
around an axis – forms a U(1) gauge group.
We note that gauge transformations have no effect on the density or flow of crossings or crossing switches. In other words, gauge transformations leave electromagnetic
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
electromagnetic
potential
Motion Mountain – The Adventure of Physics
Photons :
234
9 g auge interactions d ed uced from stra nds
field intensities and electromagnetic field energy invariant, as observed. Similarly, gauge
transformations have no effect on the number of crossing switches of rotating tangles.
A rotation by 4π does not change the phase, independently of which definition of arrow is chosen. Therefore, gauge transformations leave probability densities – and even
observable phase differences – unchanged. This agrees with experiment.
A gauge transformation on a wave functions also implies a gauge transformation on
the electrodynamic potential. The strand model thus implies that the two transformations are connected, as is observed. This connection is called minimal coupling. In short,
minimal coupling is a consequence of the strand model.
U(1) gauge interactions induced by t wists
There is only a small step from a gauge choice to a gauge interaction. We recall:
The L agrangian of QED
Given the U(1) gauge invariance of observables, the Lagrangian of quantum electrodynamics, or QED, follows directly, because U(1) gauge invariance is equivalent to minimal
coupling. We start from the Lagrangian density L of a neutral, free, and relativistic fermion in an electromagnetic field. It is given by
L = Ψ(?ℏ?/∂ − ?2 ?)Ψ −
1
? ??? .
4?0 ??
(150)
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Challenge 155 e
In particular, electromagnetism results from the transfer of twists; twists are one of the
three types of core deformations that lead to a crossing switch.
The basic twist, or first Reidemeister move, corresponds to a local rotation of some
strand segment in the core by an angle π, as illustrated by Figure 55. Twists can be generalized to arbitrary angles: we simply define a generalized twist as a local rotation of a
strand segment by an arbitrary angle. The rotation axis is chosen as in shown by Figure 55. Generalized twists can be concatenated, and the identity twist – no local rotation
at all – also exists. Generalized twists thus form a group. Furthermore, a generalized twist
by 2π is equivalent to no twist at all, as is easily checked with a piece of rope: keeping
the centre region is it disappears by pulling the ends, in contrast to a twist by π. These
properties uniquely define the group U(1). In short, Figure 55 shows that generalized
twists define the group U(1), which has the topology of a circle.
In summary, the addition of a twist to a fermion tangle or to a photon strand changes
their phase, and thus represents a gauge interaction. We have shown that core fluctuations induced by twists produce a U(1) gauge symmetry. Electromagnetic field energy
and particle energy are U(1) invariant. In short, the strand model implies that the gauge
group of quantum electrodynamics is U(1). With this result, we are now able to deduce
the full Lagrangian of QED.
Motion Mountain – The Adventure of Physics
⊳ A gauge interaction is a change of phase resulting from a strand deformation
of the particle core.
electrodynamics a nd the first reidemeister m ov e
Strand model :
charged
fermion
Observation :
time average
of crossing
switches
photon
235
charged
fermion
t2
photon
t2
t1
charged
fermion
charged
fermion
t1
F I G U R E 56 The fundamental Feynman diagram of QED and its tangle version.
Page 220
LQED = Ψ(?ℏ?/
D − ?2 ?)Ψ −
1
? ??? .
4?0 ??
(151)
/ = ?? D? is the gauge covariant derivative that is defined through minimal coupHere, D
ling to the charge ?:
D? = ∂? − ??? ? .
(152)
Page 378
Minimal coupling implies that the Lagrangian density of QED is invariant under U(1)
gauge transformations. We will discuss the details of the charge ? later on.
We have thus recovered the Lagrangian density of quantum electrodynamics from
strands. Strands thus reproduce the most precisely tested theory of physics.
Feynman diagrams and renormalization
Feynman diagrams are abbreviations of formulas to calculate effects of quantum electrodynamics in perturbation expansion. Feynman diagrams follow from the Lagrangian of
QED. All Feynman diagrams of QED can be constructed from one fundamental diagram,
shown on the right-hand side of Figure 56. Important Feynman diagrams are shown on
the left-hand sides of Figure 57 and of Figure 58.
In the strand model, the fundamental Feynman diagram can be visualized directly
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
We deduced the fermion term in the chapter of quantum theory, and we deduced the
electromagnetic term just now, from the properties of twisted loops.
As we have seen, the strand model implies minimal coupling. This changes the Lagrangian density for a charged, i.e., interacting, relativistic fermion in the electromagnetic
field, into the Lagrangian density of QED:
Motion Mountain – The Adventure of Physics
(Only crossing switches
are observable, strands
are not.)
time
236
real
electron
9 g auge interactions d ed uced from stra nds
virtual
photon
real
electron
t2
t2
real
photon
virtual
electron
t2
t2
real
photon
t2
t2
vacuum
t1
t1
t1
real
electron
virtual
electron
time
virtual
electron
vacuum
real
photon
t1
vacuum
vacuum
t1
time
t1
virtual
electron
virtual
electron
t2
time
real
positron t
2
t2
t2
t1
t1
t1
real
electron
time
virtual
positron
t2
t2
t1
real
photon
real
electron
real
positron
virtual
electron
t2
t2
vacuum
time
real
positron t
2
t2
vacuum
t1
t1
real
photon
time
t1
t1
virtual
photon
time
t1
t1
virtual
photon
vacuum
time
F I G U R E 57 The different variations of the fundamental Feynman diagram of QED and their tangle
versions.
Page 314
in terms of strands, as shown on the left-hand side of Figure 56. This is the same diagram that we have explored right at the start of the section on electrodynamics, when we
defined electrodynamics as twist exchange. (The precise tangles for the charged fermions will be deduced later on.) Since all possible Feynman diagrams are constructed from
the fundamental diagram, the strand model allows us to interpret all possible Feynman
diagrams as strand diagrams. For example, the strand model implies that the vacuum is
full of virtual particle-antiparticle pairs, as shown in Figure 58.
In quantum field theory, Lagrangians must not only be Lorentz and gauge invariant, but must also be renormalizable. The strand model makes several statements on
this issue. At this point, we focus on QED only; the other gauge interactions will be
treated below. The strand model reproduces the QED Lagrangian, which is renormalizable. Renormalizability is a natural consequence of the strand model in the limit that
strand diameters are negligible. The reason for renormalizability that the strand model
reproduces the single, fundamental Feynman diagram of QED, without allowing other
types of diagrams.
The twist deformations underlying the strand model for QED also suggest new ways
to calculate higher order Feynman diagrams. Such ways are useful in calculations of ?factors of charged particles, as shown in the next section. In particular, the strand model
for QED, as shown in Figure 56, implies that higher order QED diagrams are simple strand
deformations of lower order diagrams. Taking statistical averages of strand deformations
up to a given number of crossings thus allows us to calculate QED effects up to a given
order in the coupling. The strand model thus suggests that non-perturbative calculations
are possible in QED. However, we do not pursue this topic in the present text.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 226
Motion Mountain – The Adventure of Physics
virtual
electron
vacuum
electrodynamics a nd the first reidemeister m ov e
237
Virtual pair creation:
t3
t3
t2
t2
electron
positron
electron
positron
t1
time
t1
Motion Mountain – The Adventure of Physics
Electron-positron annihilation:
t3
t3
t2
t2
t1
positron
t1
time
electron
positron
F I G U R E 58 Some Feynman diagrams of QED with their tangle versions.
For precise non-perturbative calculations, the effective diameter of the strands must
be taken into account. The diameter eliminates the Landau pole and all ultraviolet divergences of QED. In the strand model, the vacuum energy of the electromagnetic field
is automatically zero. In other words, the strand model eliminates all problems of QED;
in fact, QED appears as an approximation of the strand model for negligible strand diameter. In passing, we thus predict that perturbation theory for QED is valid and converges
if the strand model, and in particular the finite strand diameter, is taken into account.
(The diameter is the only gravitational influence predicted to affect QED.)
The strand model also suggests that the difference between renormalized and unrenormalized mass and charge is related to the difference between minimal and nonminimal crossing switch number, or equivalently, between tangle deformations with few
and with many crossings, where strands are deformed on smaller distance scales. In other
terms, unrenormalized quantities – the so-called bare quantities at Planck energy – can
be imagined as those deduced when the tangles are pulled tight, i.e., pulled to Planck
distances, whereas renormalized mass and charge values are those deduced for particles
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
electron
238
Page 382
9 g auge interactions d ed uced from stra nds
surrounded by many large-size fluctuations.
The strand model also suggests a visualization for the cut-off used in QED. The cutoff is a characteristic energy or length used in intermediate calculations. In the strand
model, the cut-off corresponds to the size of the image.
In summary, the strand model provides a new underlying picture or mechanism for
Feynman diagrams. The strand model does not change any physical result at any experimentally accessible energy scale. In particular, the measured change or ‘running’ with
energy of the fine structure constant and of the masses of charged particles are reproduced by the strand model, because Feynman diagrams of all orders are reproduced up
to energies just below the Planck scale. Deviations between QED and the strand model
are only expected near the Planck energy, when tangles of Planck diameter are pulled
tight.
The anomalous magnetic moment
Page 224
Here, ? is the magnetic moment and ? is the intrinsic angular momentum, or spin.
The mechanical or geometric rotation frequency is related to the ratio of the intrinsic
angular momentum ? and the mass ?. Using the definitions from classical physics, we
have ?/? = ? × ?. The magnetic rotation frequency is related to the ratio of the magnetic
moment ? and the electric charge ?. Classically, this ratio is ?/? = ? × ?. Therefore, in
classical physics – and also in the first order of the Pauli–Dirac description of the electron
– the two rotation frequencies coincide, and the factor ?/2 is thus equal to 1. However, as
mentioned, both experiment and QED show a slight deviation of ?/2 from unity, called
the anomalous magnetic moment.
In the strand model, the geometric or mechanical rotation of a charged elementary
particle is due to the rotation of the tangle core as a rigid whole, whereas the magnetic
rotation also includes phase changes due to the deformations of the tangle core. In particular, the magnetic rotation of a charged elementary particle includes phase changes
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
where ?/2 is half the so-called g-factor, with a measured value of 1.00116(1), and ? is
the fine structure constant, with a measured value of 1/137.036(1). Julian Schwinger
discovered this expression in 1948; the involved calculations that led Schwinger to this
and similar results in quantum field theory earned him the 1965 Nobel Prize in Physics.
The result is also inscribed on the memorial marker near his grave in Mount Auburn
Cemetery. The strand model proposes an intuitive explanation for this result.
Generally speaking, the factor ?/2 describes the ratio between the ‘mechanical’ or
‘geometric’ rotation frequency – the rotation of the particle mass that leads to spin –
and ‘magnetic’ rotation frequency – the rotation of the particle charge that leads to the
magnetic moment. More precisely, the definition of the g-factor of a particle with charge
? and mass ? is
?
?/?
=
.
(154)
2 ?/?
Motion Mountain – The Adventure of Physics
The anomalous magnetic moment ? of the electron and of the muon is given by the
well-known expression
?
?
(153)
=1+
− ?(?2 ) ,
2
2π
electrodynamics a nd the first reidemeister m ov e
239
The propagating electron
Conjecture 1
electrodynamics and the strand model for a propagating free electron. The lower strand model
configurations are shown for a single instant – marked in magenta – of the electron propagator drawn
above them. (For simplicity, the external field is not drawn.) In the first conjecture, the loops of the belt
trick are conjectured to correspond to the virtual photons in the propagator and to be responsible for
the anomalous magnetic moment. In the second conjecture, the deformations of the core correspond
to the virtual photons.
due to emission and reabsorption of virtual photons, i.e., of twisted loops.
In nature, the probability of the emission and reabsorption of a photon is determined by the fine structure constant ?. The emission and reabsorption process leads to an
additional angle that makes the ‘magnetic’ rotation angle differ from the ‘mechanical’
rotation angle. Since the fine structure constant describes the rotation of the phase due
to virtual photon exchange, the emission and reabsorption of a virtual photon leads to
an angle difference, and this angle difference is given by the fine structure constant itself.
The ratio between the purely mechanical or geometric and the full magnetic rotation
frequency is therefore not one, but increased by the ratio between the additional angle ?
and 2π. This is Schwinger’s formula.
In short, the strand model reproduces Schwinger’s celebrated formula for the an-
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
F I G U R E 59 Two conjectured correspondences between the Feynman diagrams of quantum
Motion Mountain – The Adventure of Physics
Conjecture 2
240
Challenge 156 e
omalous magnetic moment almost from thin air. The strand model also implies that
Schwinger’s formula is valid for all charged elementary particles, independently of their
mass; this is indeed observed. Higher order corrections also appear naturally in the
strand model. Finally, the strand model implies that the complete expression, with all
orders included, converges, because the full result is due to the shape and dynamics of
the tangle core. The discussions about the existence of the perturbation limit in QED are
thus laid to rest.
If we look into the details, it might be that the belt trick itself is at the origin of the
anomalous magnetic moment. A conjecture for this connection is proposed and illustrated in Figure 59: if the two loops formed by the belt trick are seen as virtual photons,
the factor 2?/4π arises naturally. So do the higher-order terms. This explanation would
relate the belt trick directly to the additional magnetic rotation angle. However, it might
also be that this correspondence of the strand images in the figure to the upper diagrams
is not fully correct. The topic is subject of research.
A second conjecture is also given in Figure 59. The virtual photons could correspond
to deformations of the tangle core. This conjecture is more in line with the distinction
between gravity and gauge interactions given above, where it was stated that gravity is
due to tail deformations and gauge interactions are due to core deformations. This conjecture is more in line with the distinction between a geometric and a magnetic rotation:
the geometric rotation would be due to the rigid rotation of the tangle core, and the
magnetic rotation would be due to an additional effect due to core deformation.
Both conjectures on the origin of the g-factor imply that 1 < ?/2 < 2; in fact, we can
even argue, using ? < 1, that the strand model implies
Page 389
1
.
2π
(155)
This is not a new result; it is already implied by ordinary quantum field theory. However,
the strand description of particle rotation suggests a way to calculate the g-factor and the
fine structure constant. We will explore this below.
Max well ’ s equations
The strand model also allows us to check Maxwell’s field equations of classical electrodynamics directly. The equations are:
?
,
?0
∇B =0 ,
∂B
∇×E=−
,
∂?
1 ∂E
∇×B = 2
+ ?0 J .
? ∂?
∇E=
(156)
The first of these equations is satisfied whatever the precise mechanism at the basis
of twisted loop emission by electric charges may be. Indeed, any mechanism in which a
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
1 < ?/2 < 1 +
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9 g auge interactions d ed uced from stra nds
electrodynamics a nd the first reidemeister m ov e
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 231
charge randomly sends out or swallows a twisted handle yields a 1/?2 dependence for the
electrostatic field and the required connection between charge and the divergence of the
electric field. This is not a deep result: any spherically-symmetric system that randomly
emits or swallows some entity produces the equation, including the underlying inversesquare dependence. The result can also be confirmed in another, well-known way. In any
exchange interaction between two charges, the exchange time is proportional to their
distance apart ?; in addition, quantum theory states that the exchanged momentum is
inversely proportional to the distance ?. Therefore, the force, or momentum per unit
time, varies as 1/?2 . This relation is valid independently of the underlying motion of the
twisted loops, because space has three dimensions: all localized sources automatically
fulfil the inverse square dependence.
The constant on the right-hand side of the first equation results from the definition
of the units; in the language of the strand model, the constant fixes the twisted loop
emission rate for an elementary charge.
The second of the field equations (156) expresses the lack of magnetic charges. This
equation is automatically fulfilled by the strand model, as the definition of the magnetic
field with strands does not admit any magnetic sources. In fact, strands suggest that no
localized entity can have a magnetic charge. Also this equation is valid independently of
the details of the motion of the strands. Again, this is a topological effect.
The third field equation relates the temporal change of the magnetic field to the curl
of the electric field. In the strand model, this is satisfied naturally, because a curl in
the electric field implies, by construction, a change of the magnetic field, as shown by
Figure 52. Again, this relation is valid independently of the details of the motion of the
strands, as long as the averaging scale is taken to be large enough to allow the definition
of electric and the magnetic fields.
The most interesting equation is the last of the four Maxwell equations (156): in particular, the second term on the right-hand side, the dependence on the charge current.
In the description of electrodynamics, the charge current J appears with a positive sign
and with no numerical factor. (This is in contrast to linearized gravity, where the current
has a numerical factor and a negative sign.) The positive sign means that a larger current
produces a larger magnetic field. The strand model reproduces this factor: strands lead
to an effect that is proportional both to charge (because more elementary charges produce more crossing flows) and to speed of movement of charge (large charge speed lead
to larger flows). Because of this result, the classical photon spin, which is defined as ?/?,
and which determines the numerical factor, namely 1, that appears before the charge current J, is recovered. Also this connection is obviously independent of the precise motion
of the underlying strands.
The first term on the right-hand side of the fourth equation, representing the connection between a changing electric field and the curl of the magnetic field, is automatically
in agreement with the model. This can again be checked from Figure 52 – and again,
this is a topological effect, valid for any underlying strand fluctuation. As an example,
when a capacitor is charged, a compass needle between the plates is deflected. In the
strand model, the accumulating charges on the plates lead to a magnetic field. The last
of Maxwell’s equations is thus also confirmed by the strand model.
In summary, the strand model reproduces Maxwell’s equations. However, this is not
a great feat. Maxwell-like equations appear in many places in field theory, for example
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241
242
9 g auge interactions d ed uced from stra nds
in solid-state physics and hydrodynamics. Mathematical physicists are so used to the
appearance of Maxwell-like equations in other domains that they seldom pay it much
attention. The real tests for any model of electrodynamics, quantum or classical, are the
deviations that the model predicts from electrodynamics, especially at high energies.
Curiosities and fun challenges abou t QED
Challenge 157 e
Can you show that the calculation of the vacuum energy density of an infinity flat vacuum, when using strands, yields exactly zero, as expected?
∗∗
Challenge 158 e
Can you confirm that the strand model of quantum electrodynamics does not violate
charge conjugation C nor parity P at any energy?
∗∗
Can you confirm that the strand model of quantum electrodynamics conserves colour
and weak charge at all energies, using the results of the next sections?
∗∗
Challenge 160 e
Can you determine whether the U(1) gauge group deduced here is that of electrodynamics or that of weak hypercharge?
∗∗
Challenge 161 d
Can you find a measurable deviation of the strand model from QED?
Page 162
Ref. 182
In the strand model, photons are single, helically twisted strands, randomly exchanged
between charges; charges are chiral tangles, and therefore they effectively emit and absorb real and virtual photons. This is the complete description of QED using strands.
In particular, we have shown that Reidemeister I moves – or twists – of tangle cores
lead to U(1) gauge invariance, Coulomb’s inverse square relation, Maxwell’s equations of
electrodynamics and to Feynman diagrams. In short, we have deduced all experimental
properties of quantum electrodynamics, except one: the strength of the coupling. Despite this open point, we have settled one line of the millennium list of open issues: we
know the origin of the electromagnetic interaction and of its properties.
Is there a difference between the strand model and quantum electrodynamics? The
precise answer is: there are no measurable differences between the strand model and QED.
For example, the ?-factor of the electron or the muon predicted by QED is not changed
by the strand model. The U(1) gauge symmetry and the whole of QED remain valid at all
energies. There are no magnetic charges. There are no other gauge groups. QED remains
exact in all cases – as long as gravity plays no role.
The strand model prediction of a lack of larger gauge symmetries is disconcerting.
There is thus no grand unification in nature; there is no general gauge group in nature,
be it SU(5), SO(10), E6, E7, E8, SO(32) or any other. This result indirectly also rules out
supersymmetry and supergravity. This unpopular result contrasts with many cherished
habits of thought.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Summary on QED and experimental predictions
Motion Mountain – The Adventure of Physics
Challenge 159 e
electrodynamics a nd the first reidemeister m ov e
243
Motion Mountain – The Adventure of Physics
In the strand model, the equivalence of Feynman diagrams and strand diagrams implies that deviations of the strand model from QED are expected only when gravity starts
to play a role. The strand model predicts that this will only happen just near the Planck
energy √ℏ?5/4? . At lower energies, QED is predicted to remain valid.
The strand model also confirms that the combination of gravity and quantum theory
turns all Planck units into limit values, because there is a maximum density of strand
crossings in nature, due to the fundamental principle. In particular, the strand model
confirms the maximum electric field value ?max = ?4 /4?? ≈ 1.9 ⋅ 1062 V/m and a maximum magnetic field value ?max = ?3 /4?? ≈ 6.3 ⋅ 1053 T. So far, these predictions are not
in contrast with observations.
Thus the strand model predicts that approaching the electric or magnetic field limit
values – given by quantum gravity – is the only option to observe deviations from QED.
But measurements are not possible in those domains. Therefore we can state that there
are no measurable differences between the strand model and QED.
Our exploration of QED has left open only two points: the calculation of the electromagnetic coupling constant and the determination of the spectrum of possible tangles for
the elementary particles. Before we clarify these points, we look at the next Reidemeister
move.
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244
9 g auge interactions d ed uced from stra nds
Reidemeister move II, or poke, in textbook form :
The basic poke
can be described
as a local rotation by π.
A full rotation, from -π to π,
produces crossing switches.
A poke transfer :
weak boson of
unbroken SU(2)
poke
transfer
fermion
with
different
phase
F I G U R E 60 Poke transfer is the basis of the weak interaction in the strand model. No strand is cut or
reglued; the transfer occurs only through the excluded volume due to the impenetrability of strands.
Page 253
In nature, the weak interaction is the result of the absorption and the emission of massive
spin-1 bosons that form a broken weak triplet. The W and the Z bosons are emitted or
absorbed by particles with weak charge; these are the left-handed fermions and righthanded antifermions. In other words, the weak interaction breaks parity P maximally. The W boson has unit electric charge, the Z boson has vanishing electric charge.
The emission or absorption of W bosons changes the particle type of the involved fermion. The weak bosons also interact among themselves. All weakly charged particles
are massive and move slower than light. The Lagrangian of matter coupled to the weak
field has a broken SU(2) gauge symmetry. There are fundamental Feynman diagrams
with triple and with quartic vertices. The weak coupling constant is determined by the
electromagnetic coupling constant and the weak boson masses; its energy dependence is
fixed by renormalization. The Higgs boson ensures full consistency of the quantum field
theory of the weak interaction.
The previous paragraph summarizes the main observations about the weak interaction. More precisely, all observations related to the weak interaction are described by
its Lagrangian. Therefore, we need to check whether the weak interaction Lagrangian
follows from the strand model.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
the weak nuclear interaction and the second
reidemeist er move
Motion Mountain – The Adventure of Physics
fermion
vacuum
the w ea k nuclea r interaction a nd the second reidemeister m ov e
Pokes on tangle cores
form an SU(2) group
245
The poke, or Reidemeister II move,
is a local rotation, by an angle π,
of the core region enclosed by a
dashed circle.
τ
The three basic pokes on tangle cores are
local rotations by an angle π around the three
coordinate axes of the core region (enclosed by
a dashed circle). The infinitesimal versions of
the three basic pokes generate an
SU(2) group. The SU(2) group appears
most clearly when the analogy to the
belt trick is highlighted.
axis
τx
τx
axis
τy
τy
π
τz
τz
π
F I G U R E 61 How the set of all pokes – the set of all deformations induced on tangle cores by the weak
interaction – forms an SU(2) gauge group: the three pokes lead to the belt trick, illustrated here with a
pointed buckle and two belts. For clarity, deformations of two strands are shown, instead of the
deformation of a single strand.
Strands, p okes and SU(2)
Page 222
As explained above, any gauge interaction involving a fermion is a deformation of the
tangle core that changes the phase and rotation of the fermion tangle. We start directly
with the main definition.
⊳ The weak interaction is the transfer of a poke, i.e., the transfer of a Reidemeister II move, between two particles. An illustration is given in Figure 60. Strands are not cut in this process; they simply transfer the deformation as a result of their impenetrability.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
axis
Motion Mountain – The Adventure of Physics
π
246
Challenge 162 e
Challenge 163 e
Page 189
9 g auge interactions d ed uced from stra nds
⋅
??
??
??
?? −1
??? −???
?? −??? −1
???
?? ??? −??? −1
(157)
Weak charge and parit y viol ation
A particle has weak charge if, when subject to many random pokes, a non-zero average
phase change occurs. Surrounded by a bath of strands that continuously induce random
pokes, not all tangles will change their phase on a long-time average: only tangles that
lack symmetry will. One symmetry that must be lacking is spherical symmetry. Therefore, only tangles whose cores lack spherical symmetry have the chance to be influenced
by random pokes. Since all tangles, independently of their core details, lack spherical
symmetry, all such tangles, i.e., all massive particles, are candidates to be influenced, and
thus are candidates for weakly charged particles. We therefore explore them in detail
now.
If a tangle is made of two or more linked strands, it represents a massive spin-1/2
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
In other terms, the three basic pokes – and in particular also their infinitesimal versions –
behave like the generators of an SU(2) group. Because pokes can be seen as local rotations
of a buckle region, they can be generalized to arbitrary angles. Such arbitrary pokes can
be concatenated. We thus find that arbitrary pokes form a full SU(2) group. This is the
reason for their equivalence with the belt trick.
The different gauge choices for a particle are not illustrated in Figure 61. The gauge
choices arise from the different ways in which the basic pokes ?? , ?? and ?? can be assigned to the set of deformations that describe the belt trick.
In summary, we can state that in any definition of the phase of a tangled fermion core,
there is an SU(2) gauge freedom; in addition, there exists an interaction with SU(2) gauge
symmetry. In other words, the strand model implies, through the second Reidemeister
move, the existence of the unbroken weak interaction with a gauge group SU(2).
Motion Mountain – The Adventure of Physics
Strands describe the weak interaction as exchange of pokes. In tangle cores, the basic
pokes induce local rotations by an angle π, as shown in Figure 61: each basic poke rotates
the region enclosed by the dotted circle. A full poke produces two crossings. There are
three, linearly independent, basic pokes, in three mutually orthogonal directions. The
three basic pokes ?? , ?? and ?? act on the local region in the same way as the three possible
mutually orthogonal rotations act on a belt buckle. For completeness, we note that the
following arguments do not depend on whether the two strands involved in a poke are
parallel, orthogonal, or at a general angle. The following arguments also do not depend
on whether the pokes are represented by deforming two strands or only one strand. Both
cases lead to crossing switches, for each possible poke type.
Figure 61 illustrates that the product of two different basic pokes gives the third basic
poke, together with a sign – which depends on whether the sequence is cyclic or not –
and a factor of ?. Using the definition of −1 as a local rotation of the buckle region by
2π, we also find that the square of each basic poke is −1. In detail, we can read off the
following multiplication table for the three basic pokes:
the w ea k nuclea r interaction a nd the second reidemeister m ov e
247
Pokes on cores generate an SU(2) group, like the belt trick does :
x
z
y
The three basic pokes
– shown here applied to
a tangle core – define, when
reduced to infinitesimal
angles, the 3 generators
of SU(2).
Random pokes affect only tangles of identical spin and handedness :
poke affects a
tangle and a
tangle differently.
F I G U R E 62 The three basic pokes and weak charge in the strand model.
Page 296
⊳ Non-vanishing weak charge for fermions appears only for tangle cores
whose handedness leads to average poke effects.
In other words, the strand model predicts that random pokes will only affect a core if
the core handedness and the randomly applied belt trick are of the same handedness. In
physical terms, random pokes will only affect left-handed particles or right-handed antiparticles. Thus, the strand model predicts that the weak interaction violates parity maximally, This is exactly as observed. In other terms, weak charge and the parity violation
of the weak interaction are consequences of the belt trick. This relation is summarized
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 217
particle (except for a simple twist, which represents the graviton). All such fermion cores
lack spherical and cylindrical symmetry. When a fermion spins, two things happen:
the core rotates and the belt trick occurs, which untangles the tails. Compared to the
direction of motion, the rotation and the untangling can be either left-handed or righthanded.
Every poke is a shape transformation of the core with a preferred handedness. The
chirality is of importance in the following.
A particle has weak charge if random pokes lead to a long-time phase change. In order
to feel any average effect when large numbers of random pokes are applied, a core must
undergo different effects for a poke and its reverse. As already mentioned, this requires
a lack of core symmetry. Whenever the core has no symmetry, non-compensating phase
effects will occur: if the core rotation with its tail untangling and the poke are of the
same handedness, the phase will increase, whereas for opposite handedness, the phase
will decrease a bit less.
Motion Mountain – The Adventure of Physics
A
248
9 g auge interactions d ed uced from stra nds
Weak bosons
⊳ Weak intermediate bosons are described by double strands. An illustration
is given in Figure 63.
Single strands that induce phase changes in fermions interacting weakly are shown on
the left side of Figure 63. They correspond to the three basic pokes ?? , ?? and ?? .
We note two additional points. First of all, the (unbroken) spin-1 bosons could also
be described by the motion of a single strand in a strand group. This makes them spin 1
particles.
Furthermore, unknotted tangles are massless. In the strand model, tangles that induce
pokes differ from the massive weak intermediate bosons, shown on the right of Figure 63.
This difference is due to the breaking of the SU(2) gauge symmetry, as we will find out
soon.
* Non-Abelian gauge theory was introduced by Wolfgang Pauli. In the 1950s, he explained the theory in
series of talks. Two physicists, Yang Chen Ning and Robert Mills, then wrote down his ideas. Yang later
received the Nobel Prize in Physics with Lee Tsung Dao for a different topic, namely for the violation of
parity of the weak interaction.
** This reworked strand model of the W and Z bosons arose in 2015.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Gauge bosons are those particles that are exchanged between interacting fermions: gauge
bosons induce phase changes of fermions. This implies that the (unbroken) weak bosons
are the particles** that induce the three poke moves:
Motion Mountain – The Adventure of Physics
in Figure 62.
If an elementary particle is described by a two tangled strands, we expect it to be influenced by average pokes. Such tangle cores are spin-1 bosons; their cores lack spherical
and cylindrical symmetry. The core rotation will induce a left-right asymmetry that will
lead to a higher effect of a poke than of its reverse. Two-stranded particles are thus predicted to carry weak charge. We therefore expect that quarks – to be explored below –
and the weak bosons themselves interact weakly.
Because the weak bosons interact weakly, the strand model implies that the weak interaction is a non-Abelian gauge theory, as is observed.*
If a tangle is made of a single unknotted strand, it is not affected by random pokes.
The strand model thus predicts that the photon has no weak charge, as is observed. The
same also holds for gluons.
The strand definition of weak charge leads to two conclusions that can be checked
by experiment. First, all electrically charged particles – having cores that are chiral and
thus lack cylindrical symmetry – are predicted to be weakly charged. Secondly, in the
strand model, only massive particles interact weakly; in fact, all massive particles interact weakly, because their cores lack cylindrical symmetry. In other words, all weakly
charged particles move more slowly than light and vice versa. Both conclusions agree
with observation.
In summary, all properties of weak charge found in nature are reproduced by the
tangle model.
the w ea k nuclea r interaction a nd the second reidemeister m ov e
249
Boson mass generation and SU(2) breaking :
weak bosons of
unbroken SU(2)
photon
a
+
b
Z boson
candidate
(massive)
c
Z0
photon
e
W0
B
A
+
Wx
W boson
candidate
(massive)
h
W–
F I G U R E 63 Poke-inducing strand motions (left) become massive weak vector tangles (right) through
symmetry breaking and tail braiding. Tail braiding is related to the Higgs boson, whose tangle model
will be clarified later on.
The L agrangian of the unbroken SU(2) gauge interaction
The energy of the weak field is given by the density of weak gauge boson strands. As
long as the SU(2) symmetry is not broken, the energy of the weak field and the energy of
fermions are both SU(2) invariant. As a consequence, we are now able to deduce a large
part of the Lagrangian of the weak interaction, namely the Lagrangian for the case that
the SU(2) symmetry is unbroken.
As long as SU(2) is unbroken, the vector bosons are described as unknotted tangles
that induce pokes, as shown on the left of Figure 63. There are three such bosons. Since
they can be described by a single strand that moves, they have spin 1; since they are
unknotted, they have zero mass and electric charge.
Energy is the number of crossing switches per time. As long as SU(2) is unbroken
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
g
B
bosons of U(1) and SU(2)
after symmetry breaking
Motion Mountain – The Adventure of Physics
–
f
+
W0
B
d
Higgs
250
Page 231
9 g auge interactions d ed uced from stra nds
and the weak bosons are massless, the energy of the weak boson field and thus their
Lagrangian density is given by the same expression as the energy of the photon field. In
particular, the strand model implies that energy density is quadratic in the field intensities. We only have to add the energies of all three bosons together to get:
1 3
L = − ∑ ?? ?? ?? ?? ,
4 ?=1
(158)
Lunbroken weak = ∑ Ψ? (?ℏ?/
D − ?? ?2 )Ψ? −
?
1 3
∑ ?? ?? ?? ?? ,
4 ?=1
(159)
SU(2) breaking
Page 255
In nature, the weak interaction does not have an SU(2) gauge symmetry. The symmetry
is only approximate; is said to be broken. The main effect of SU(2) symmetry breaking are
the non-vanishing – and different – masses for the W and Z bosons, and thus the weakness and the short range of the weak interaction. In addition, the symmetry breaking
implies a mixing of the weak and the electromagnetic interaction: it yields the so-called
electroweak interaction. This mixing is often called electroweak ‘unification’.
The strand model suggests the following description:
⊳ Mass generation for bosons and the related SU(2) symmetry breaking are
due to tail braiding at the border of space. Figure 63 illustrates the idea.
In this description, tail braiding* is assumed to occur at a distance outside the domain
of observation; in that region – which can be also the border of physical space – tail
braiding is not forbidden and can occur. The probability of tail braiding is low, because
* In the original strand model of the weak bosons, from the year 2008, the role of tail braiding was taken
by strand overcrossing.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
/ is now the SU(2) gauge covariant derivative and the first sum is taken over
where D
all fermions. In this Lagrangian, only the left-handed fermions and the right-handed
antifermions carry weak charge. This Lagrangian, however, does not describe nature:
the observed SU(2) breaking is missing.
Motion Mountain – The Adventure of Physics
This expression is SU(2) gauge invariant. Indeed, SU(2) gauge transformations have no
effect on the number of crossing switches due to weak bosons or to the motion of pokes.
Thus, gauge transformations leave weak field intensities and thus also the energy of the
weak fields invariant, as observed.
We can now write down the Lagrangian for weakly charged fermions interacting with
the weak vector bosons. Starting from the idea that tangle core deformations lead to
phase redefinitions, we have found that pokes imply that the unbroken weak Lagrangian
density for matter and radiation fields is SU(2) gauge invariant. In parallel to electrodynamics we thus get the Lagrangian
the w ea k nuclea r interaction a nd the second reidemeister m ov e
Page 356
Page 255
Page 254
Page 356
Page 358
Open issue: are the W and Z tangles correct?
In 2014, Sergei Fadeev raised an issue: A tangle version of the W and Z that does not
contain any knot and does not require an actual strand overcrossing process at spatial
infinity, the strand model would gain in simplicity and elegance. Thinking about the
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 319
the crossings have first to fluctuate to that distance and then fluctuate back. Nevertheless,
the process of tail braiding can take place.
Tail braiding appears only in the weak interaction. It does not appear in the other
two gauge interactions, as the other Reidemeister moves are not affected by processes at
the border of space. In the strand model, this is the reason that only SU(2) is broken
in nature. In short, SU(2) breaking is a natural consequence of the second Reidemeister
move.
Tail braiding transforms the unbraided, and thus massless, poke strands into the
braided, and thus massive W and Z strands. Tail braiding leads to particle cores: therefore is a mass-generating process. The precise mass values that it generates will be determined below. The strand model thus confirms that mass generation is related to the
breaking of the weak interaction.
Tail braiding mixes the W0 with the ‘original’ photon. This is shown in Figure 63. The
mixing is due to the topological similarities of the strand models of the two particles. The
resulting Z boson is achiral, and thus electrically neutral, as observed. We note that the
existence of a neutral, massive Z boson implies that elastic neutrino scattering in matter
occurs in nature, as was observed for the first time in 1974. Since any electrically charged
particle also has weak charge, the existence of a Z boson implies that any two electrically
charged particles can interact both by exchange of photons and by exchange of Z bosons.
In other words, SU(2) breaking implies electroweak mixing, or, as is it usually called,
electroweak ‘unification’.
Tail braiding takes place in several weak interaction processes, as shown in Figure 66.
Tail braiding thus can change particle topology, and thus particle type. The strand model
thus predicts that the weak interaction changes particle flavours (types), as is observed.
In fact, the strand model also predicts that only the weak interaction has this property.
This is also observed.
On the other hand, strands are never cut or glued back together in the strand model,
not even in the weak interaction. As a result, the strand model predicts that the weak interaction conserves electric charge, spin and, as we will see below, colour charge, baryon
number and lepton number. All this is observed.
Tail braiding also implies that the tangles for the Z boson and for the W boson shown
above are only the simplest tangles associated with each boson; more complicated tangles
are higher order propagating states of the same basic open knots. This will be of great
importance later on, for the proof that all gauge bosons of nature are already known
today.
In summary, the second Reidemeister move leads to tail braiding; tail braiding leads
to the observed properties of SU(2) symmetry breaking. (Equivalently, the strand model
implies that the simplest tangles of the weak interaction bosons show SU(2) symmetry,
whereas the more complicated, massive tangles break this symmetry.) The value of the
mixing angle and the particle masses have still to be determined. This will be done below.
Motion Mountain – The Adventure of Physics
Page 248
251
252
9 g auge interactions d ed uced from stra nds
Weak bosons of
unbroken SU(2)
Weak bosons of
broken SU(2)
Wx
Wy
W0
W+
W–
Z0
2008 (on the right side), now seen to be incorrect.
Challenge 164 ny
The electroweak L agrangian
Page 329
We can now use the results on SU(2) symmetry breaking to deduce the electroweak Lagrangian density. We have seen that symmetry breaking leaves the photon massless but
introduces masses to the weak vector bosons, as shown in Figure 63. The non-vanishing
boson masses ?? and ?? add kinetic terms for the corresponding fields in the Lagrangian.
Due to the symmetry breaking induced by tail braiding, the Z boson results from
the mixing with the (unbroken) photon. The strand model predicts that the mixing can
be described by an angle, the so-called weak mixing angle ?w . In particular, the strand
model implies that cos ?w = ?? /?? .
As soon as symmetry breaking is described by a mixing angle due to tail braiding,
we get the known electroweak Lagrangian, though at first without the terms due to the
Higgs boson. (We will come back to the Higgs boson later on.) We do not write down
the Lagrangian of the weak interaction predicted by the strand model, but the terms are
the same as those found in the standard model of elementary particles. There is one
important difference: the Lagrangian so derived does not yet contain quark and lepton
mixing. Indeed, experiments show that the weak fermion eigenstates are not the same as
the strong or electromagnetic eigenstates: quarks mix, and so do neutrinos. The reason
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 249
issue, it became clear that such a tangle could occur when vacuum strands were included,
as shown above.
In contrast, in 2008, in the first version of the strand model, the W boson after symmetry breaking was thought to be an open overhand knot, and the Z boson an open
figure-eight knot.
It might well be that the new, 2015/2016 strand models for the two intermediate vector
bosons, shown in Figure 63 are still not correct. The possibility remains intriguing and
a definitive issue still needs to be found.
Motion Mountain – The Adventure of Physics
F I G U R E 64 The supposed models for the massive weak gauge bosons after symmetry breaking, from
the w ea k nuclea r interaction a nd the second reidemeister m ov e
Neutral current processes
253
Charged current processes
quark 2 or
lepton 2
quark 1 or
lepton 1
Z
W
quark 1 or
lepton 1
quark 1 or
lepton 1
Triple boson coupling processes
W
Quartic coupling processes
W
Z, γ
W
W
W
Z, γ
W
W
Z, γ
F I G U R E 65 The fundamental Feynman diagrams of the weak interaction that do not involve the Higgs
boson.
for this observation, and the effect that mixing has on the weak Lagrangian, will become
clear once we have determined the tangles for each fermion.
In summary, the strand model implies the largest part of the Lagrangian of the weak
interaction. The issue of the Higgs boson is still open, and the electroweak Lagrangian
contains a number of constants that are not yet clarified. These unexplained constants are
the number of the involved elementary particles, their masses, couplings, mixing angles
and CP violation phases, as well as the value of the weak mixing angle.
The weak Feynman diagrams
Challenge 165 e
Page 382
In nature, the weak interaction is described by a small number of fundamental Feynman
diagrams. Those not containing the Higgs boson are shown in Figure 65. These Feynman
diagrams encode the corresponding Lagrangian of the weak interaction.
In the strand model, pokes lead naturally to strand versions of the fundamental Feynman diagrams. This happens as shown in Figure 66. We see again that the strand model
reproduces the weak interaction: each Feynman diagram is due to a strand diagram for
which only crossing switches are considered, and for which Planck size is approximated
as zero size. In particular, the strand model does not allow any other fundamental diagrams for the weak interaction.
The finite and small number of possible strand diagrams and thus of Feynman diagrams implies that the weak interaction is renormalizable. For example, the change or
‘running’ of the weak coupling with energy is reproduced by the strand model, because
the running can be determined through the appropriate Feynman diagrams.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 372
Motion Mountain – The Adventure of Physics
W
254
9 g auge interactions d ed uced from stra nds
Weak strand diagrams
electron
Weak Feynman diagrams
Z boson
time average of
crossing switches,
at lower magnification
t2
electron
Z
t2
vacuum
t1
t1
electron
time
W
neutrino
t2
W
t2
vacuum
W
electron
time
W
W
t2
W
t2
vacuum
γ
(photon)
W
W
t1
t1
γ
(photon)
time
W
t2
W
t2
t1
t1
Z
Z
time
Z
Z
F I G U R E 66 The strand model for the fundamental Feynman diagrams of the weak interaction. The
tangles for the fermions are introduced later on.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
electron
t1
t1
Motion Mountain – The Adventure of Physics
neutrino
electron
the w ea k nuclea r interaction a nd the second reidemeister m ov e
255
Fun challenges and curiosities abou t the weak interaction
Challenge 166 e
The W boson and its antiparticle are observed to annihilate through the electromagnetic
interaction, yielding two or more photons. The tangle model of the weak bosons has a
lot of advantages compared to the knot model: The annihilation is much easier to understand.
∗∗
The strand model, like the standard model of particle physics, predicts that everything
about the weak interaction is already known. Nevertheless, the most important weak
process, the decay of the neutron, is being explored by many precision experiments. The
strand model predicts that none of these experiments will yield any surprise.
∗∗
Page 329
We have deduced the main properties of the weak Lagrangian from the strand model.
We have shown that Reidemeister II moves – or pokes – in tangle cores lead to a broken
SU(2) gauge group and to massive weak bosons. We found that the deviation from
tangle core sphericity plus chirality is weak charge, and that the weak interaction is nonAbelian. We have also shown that the weak interaction naturally breaks parity maximally
and mixes with the electromagnetic interaction. In short, we have deduced the main experimental properties of the weak interaction.
Is there a difference between the strand model and the electroweak Lagrangian of the
standard model of particle physics? Before we can fully answer the question on deviations between the strand model and the standard model, we must settle the issue of the
Higgs boson. This is done later on.
In any case, the strand model predicts that the broken SU(2) gauge symmetry remains
valid at all energies. No other gauge groups appear in nature. The strand model thus
predicts again that there is no grand unification, and thus no larger gauge group, be it
SU(5), SO(10), E6, E7, E8, SO(32) or any other group. Also this result indirectly rules out
supersymmetry and supergravity.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Summary on the weak interaction and experimental predictions
Motion Mountain – The Adventure of Physics
Ref. 183
The strand model makes clear that the weak interaction and the electromagnetic interaction mix, but do not unify. There is only electroweak mixing, and no electroweak unification, despite claims to the contrary by the Nobel Prize committee and many other
physicists. In fact, Sheldon Glashow, who received the Nobel Prize in Physics for this
alleged ‘unification’, agrees with this assessment. So do Richard Feynman and, above
all, Martin Veltman, who was also involved in the result; he even makes this very point
in his Nobel Prize lecture. The incorrect habit to call electroweak mixing a ‘unification’
was one of the main reason for the failure of past unification attempts: it directed the
attention of researchers in the wrong direction.
In the strand model, the mixing of the electromagnetic and the weak interaction can
be seen as a consequence of knot geometry: the poke generators of the weak interaction
also contain twists, i.e., also contain generators of the electromagnetic interaction. In
contrast, generators of other Reidemeister moves do not mix among them or with pokes;
and indeed, no other type of interaction mixing is observed in nature.
256
Page 162
9 g auge interactions d ed uced from stra nds
Motion Mountain – The Adventure of Physics
The strand model also predicts that the combination of gravity and quantum theory
turns all Planck units into limit values, because there is a maximum density of strand
crossings in nature, due to the fundamental principle. Therefore, the strand model predicts a maximum weak field value given by the Planck force divided by the smallest weak
charge. All physical systems – including all astrophysical objects, such as neutron stars,
quark starts, gamma-ray bursters or quasars – are predicted to conform to this limit. So
far, no observed field value is near this limit, so that the prediction does not contradict
observation.
So far, our exploration of the weak interaction has left us with a few open issues: we
need to calculate the weak coupling constant and determine the tangle for each particle
of the standard model, including the Higgs boson. But we also need to explain weak fermion mixing, CP violation and the masses of all particles. Despite these open points, we
have settled another line of the millennium list: we know the origin of the weak interaction and of its main properties. Before we clarify the open issues, we explore the third
Reidemeister move.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
the strong nuclea r interaction a nd the third reid emeister m ov e
Reidemeister move III, or slide, in textbook form :
257
Slide in SU(3) form (one example) :
A slide transfer :
gluon
slide
transfer
fermion
with
different
phase
vacuum
F I G U R E 67 A gluon changes the phase of a tangle: slide transfer is the basis of the strong interaction in
the strand model. During the interaction, no strand is cut or reglued; the transfer occurs purely through
the excluded volume that results from the impenetrability of strands.
In nature, the strong interaction is the result of the absorption and the emission of massless, electrically uncharged, spin-1 gauge bosons that are called gluons. Gluons interact
with quarks, the only fermions with colour charge. Fermions can have three different colour charges, antifermions three different anticolours. Gluons form an octet, are themselves colour charged and therefore also interact among themselves. The Lagrangian
of quarks coupled to the gluon field has an unbroken SU(3) gauge symmetry. There
are three fundamental Feynman diagrams: one for quark-gluon interaction and two for
gluon-gluon interactions: a triple and a quartic gluon vertex. The strong coupling constant is about 0.5 at low energy; its energy dependence is determined by renormalization.
The previous paragraph summarizes the main observations about the strong interaction. All known observations related to the strong interaction, without any known
exception, are contained in its Lagrangian. Therefore, we need to show that the strong
interaction Lagrangian follows from the strand model.
Strands and the slide, the third R eidemeister move
Page 222
As explained above, interactions of fermions are deformations of the tangle core that
change its phase. We start directly by presenting the strand model for the strong interaction.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
the strong nuclear interaction and the third
reidemeist er move
Motion Mountain – The Adventure of Physics
fermion
258
9 g auge interactions d ed uced from stra nds
⊳ The strong interaction is the transfer of slides, i.e., the transfer of third Reidemeister moves, between a gluon and a particle. As shown in Figure 67,
strands are not cut in this process; they simply transfer deformations as a
result of their impenetrability.
Such a slide transfer will influence the phase of the affected particle tangle. Therefore,
slide transfers are indeed a type of interaction.
An introduction to SU(3)
8
? = e∑?=1 ?? ??? /2
(160)
0 1 0
cos ?/2 ? sin ?/2 0
?i?1 /2
1
0
0
?
?1 = (
) , ?1 (?) = e
= ( sin ?/2 cos ?/2 0) ,
0 0 0
0
0
1
0 ? 0
?1 = eπi?1 /2 = ( ? 0 0)
0 0 1
0 −? 0
cos ?/2 sin ?/2 0
? 2 = ( ? 0 0) , ?2 (?) = e?i?2 /2 = (− sin ?/2 cos ?/2 0) ,
0 0 0
0
0
1
πi?2 /2
?2 = e
0 1 0
= (−1 0 0)
0 0 1
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
where the eight real parameters ?? can be thought of as the eight coordinates of the group
elements on the group manifold. Since SU(3) is compact and simple, these coordinates
are best visualized as angles. Of course, ? is the imaginary unit. The generators ? ? are
complex, traceless and hermitian 3 × 3 matrices; they are used to define a basis for the
group elements. The eight generators are not group elements themselves. They describe
the structure of the group manifold near the identity matrix; for a Lie group, this local
structure defines the full group manifold. Like for any basis, also set of eight generators
? ? is not unique. Of the many possible choices for the generators, the Gell-Mann matrices
? 1 to ? 8 are the most commonly used in physics.
The Gell-Mann matrices ? ? , the corresponding group elements ?? for general angles,
and the group elements ?? for the finite angle π are given by:
Motion Mountain – The Adventure of Physics
Challenge 167 e
Before we show that slides are responsible for the strong nuclear interaction, we summarize the mathematical properties of the Lie group SU(3). This Lie group is the structure generated by the unitary 3 × 3 matrices with determinant +1. It is a group, because
matrices can be properly multiplied, because the identity matrix is included, and inverse
matrices exist. SU(3) is also a manifold; a quick check shows that it has eight dimensions.
In short, SU(3) is a Lie group: its elements behave like points on a manifold that can be
multiplied. The Lie bracket is the commutator. A general element ? of SU(3) can be
written as an exponential in the well-known way
the strong nuclea r interaction a nd the third reid emeister m ov e
259
1 0 0
cos ?/2 + ? sin ?/2
0
0
0
cos ?/2 − ? sin ?/2 0) ,
? 3 = (0 −1 0) , ?3 (?) = e?i?3 /2 = (
0 0 0
0
0
1
πi?3 /2
?3 = e
? 0 0
= (0 −? 0)
0 0 1
0 0 1
cos ?/2 0 ? sin ?/2
1
0 ),
? 4 = (0 0 0) , ?4 (?) = e?i?4 /2 = ( 0
1 0 0
? sin ?/2 0 cos ?/2
0 0 ?
?4 = eπi?4 /2 = (0 1 0)
? 0 0
0 0 1
?5 = eπi?5 /2 = ( 0 1 0)
−1 0 0
0 0 0
1
0
0
?i?6 /2
? 6 = (0 0 1) , ?6 (?) = e
= (0 cos ?/2 ? sin ?/2) ,
0 1 0
0 ? sin ?/2 cos ?/2
0 0 0
1
0
0
?i?7 /2
? 7 = (0 0 −?) , ?7 (?) = e
= (0 cos ?/2 sin ?/2 ) ,
0 ? 0
0 − sin ?/2 cos ?/2
1 0 0
?7 = eπi?7 /2 = (0 0 1)
0 −1 0
?8 =
1 0 0
1
(0 1 0 ) ,
√3 0 0 −2
√
?8 (?) = e 3 ?i?8 /2 = (
cos ?/2 + ? sin ?/2
0
0
0
cos ?/2 + ? sin ?/2
0
),
0
0
cos ? − ? sin ?
? 0 0
?8 = ?8 (π) = (0 ? 0 ) .
(161)
0 0 −1
The eight Gell-Mann matrices ? ? are hermitean, traceless and trace-orthogonal. The
corresponding group elements ?? and ?? can be thought as the unnormed and normed
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1 0 0
?6 = eπi?6 /2 = (0 0 ? )
0 ? 0
Motion Mountain – The Adventure of Physics
0 0 −?
cos ?/2 0 sin ?/2
0
1
0 ),
? 5 = (0 0 0 ) , ?5 (?) = e?i?5 /2 = (
? 0 0
− sin ?/2 0 cos ?/2
260
9 g auge interactions d ed uced from stra nds
basis vectors of the group manifold. We note that the definition of ?8 differs from that of
the other group elements ??: it contains an extra factor √3 . The fourfold concatenation
of each matrix ?? ? is the identity matrix – except for the case ?? 8 . Instead, the generator
? 8 commutes with ? 1 , ? 2 and ? 3 – though not with the other generators.
There is no ninth or tenth Gell-Mann matrix. Such a matrix would not be linearly
independent from the first eight ones. Indeed, the two matrices deduced from ? 3 using
symmetry considerations, namely
−1 0 0
cos ?/2 − ? sin ?/2 0
0
0
1
0
),
? 9 = ( 0 0 0) , ?9 (?) = e?i?9 /2 = (
0 0 1
0
0 cos ?/2 + ? sin ?/2
−? 0 0
?9 = ?9 (π) = ( 0 1 0)
0 0 ?
?10
(162)
are linear combinations of ? 3 and ? 8 ; in particular, we have ? 3 + ? 9 + ? 10 = 0 and
√3 ? 8 + ? 9 = ? 10 . Therefore, ? 9 and ? 10 are not Gell-Mann matrices. (Also two further
matrices corresponding to ? 8 in the other two triplets can be defined. The sum of these
three matrices is 0 as well.)
The multiplication properties of the Gell-Mann generators ? 1 to ? 8 are listed in
Table 10. To make the threefold symmetry more evident, the table also lists the products
containing the linearly dependent matrices ? 9 and ? 10 . Writing the table with the commutators would directly show that the generators form a Lie algebra.
The centre of SU(3), the subgroup that commutes with all other elements of the group,
is ?3 ; its threefold symmetry is useful in understanding the behaviour of the group elements and of the generators in more detail.
The group elements ?1 to ?8 listed above share the property that their fourth powers
(??)4 are the identity matrix. The first matrix triplet ?1 , ?2 , ?3 , the second triplet
?4 , ?5 , ?9 and the third triplet ?6 , ?7 , ?10 each form a SU(2) subgroup. Reflecting the
threefold symmetry of its centre, SU(3) contains three linearly independent SU(2) subgroups. The group element ?8 commutes with the first triplet ?1 , ?2 , ?3 ; therefore, these
four elements generate a U(2) subgroup of SU(3). This U(2) subgroup, often sloppily
labeled as SU(2)xU(1), is given by those 3 by 3 matrices that contain a unitary 2 by 2
matrix in the upper left, contain zeroes in the remaining four off-diagonal elements, and
contain the inverse value of the determinant of the 2 by 2 matrix in the remaining, lower
right diagonal element. In short, SU(3) contains three linearly independent U(2) subgroups.
SU(3) is characterized by the way that the SU(2) triplets are connected. In particular,
the product ?3 ?9 ?10 is the identity, reflecting the linear dependence of the three corresponding generators ? ? . We also have ?8 ?9 = ?10 . Also the product of ?8 with its
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Vol. V, page 225
1 0 0
= ?10 (π) = (0 ? 0 )
0 0 −?
Motion Mountain – The Adventure of Physics
0 0 0
1
0
0
0
? 10 = (0 1 0 ) , ?10(?) = e?i?10 /2 = (0 cos ?/2 + ? sin ?/2
),
0 0 −1
0
0
cos ?/2 − ? sin ?/2
the strong nuclea r interaction a nd the third reid emeister m ov e
261
TA B L E 10 The multiplication table for the generators ? 1 to ? 8 of SU(3), and for the additional, linearly
dependent matrices ? 9 = −? 3/2 − ? 8√3 /2 and ? 10 = −? 3/2 + ? 8√3 /2 that are not generators. Note that,
despite the appearance, ?24 = ?25 = ?29 and ?26 = ?27 = ?210.
?1
?2
?3
?4
?5
?9
?6
?7
? 10
?8
?1
2/3
+? 8 /√3
?? 3
−?? 2
? 6 /2
+?? 7 /2
−?? 6 /2
+? 7 /2
−? 1 /2
+?? 2 /2
? 4 /2
+?? 5 /2
−?? 4 /2
+? 5 /2
? 1 /2
+?? 2 /2
? 1 /√3
?2
−?? 3
2/3
+? 8 /√3
?? 1
?? 6 /2
−? 7 /2
? 6 /2
+?? 7 /2
−?? 1 /2
−? 2 /2
−?? 4 /2
+? 5 /2
−? 4 /2
−?? 5 /2
−?? 1 /2
+? 2 /2
? 2 /√3
?3
?? 2
−?? 1
2/3
+? 8 /√3
? 4 /2
+?? 5 /2
−?? 4 /2 −1/3 − ? 3 /3 −? 6 /2
+? 5 /2
+? 9 /3
−?? 7 /2
?? 6 /2 −1/3 + ? 3 /3 ? 3 /√3
−? 7 /2
+? 10 /3
?4
? 6 /2 −?? 6 /2
−?? 7 /2 −? 7 /2
? 4 /2
−?? 5 /2
2/3 + ? 3 /2
−? 8 /2√3
−?? 9
?? 5
? 1 /2
+?? 2 /2
?? 1 /2
−? 2 /2
−? 4 /2
−?? 5 /2
−? 4 /2√3
−?√3 ? 5 /2
?5
?? 6 /2 ? 6 /2
+? 7 /2 −?? 7 /2
?? 4 /2
+? 5 /2
?? 9
2/3 + ? 3 /2
−? 8 /2√3
−?? 4
−?? 1 /2
+? 2 /2
? 1 /2
+?? 2 /2
?? 4 /2
−? 5 /2
?√3 ? 4 /2
−? 5 /2√3
?9
−? 1 /2 ?? 1 /2 −1/3 − ? 3 /3
−?? 2 /2 −? 2 /2
+? 9 /3
−?? 5
?? 4
?6
+? 4 /2 ?? 4 /2
−?? 5 /2 +? 5 /2
−? 6 /2
+?? 7 /2
? 1 /2
−?? 2 /2
?? 1 /2
+? 2 /2
? 6 /2
+?? 7 /2
2/3 − ? 3 /2
−? 8 /2√3
?? 10
−?? 7
−? 6 /2√3
−?√3 ? 7 /2
?7
?? 4 /2 −? 4 /2
+? 5 /2 +?? 5 /2
−?? 6 /2
−? 7 /2
−?? 1 /2
−? 2 /2
? 1 /2
−?? 2 /2
−?? 6 /2
+? 7 /2
−?? 10
2/3 − ? 3 /2
−? 8 /2√3
?? 6
?√3 ? 6 /2
−? 7 /2√3
? 10
−? 1 /2 −?? 1 /2 −1/3 + ? 3 /3 −? 4 /2
+?? 2 /2 −? 2 /2
−? 10 /3
+?? 5 /2
?? 7
−?? 6
2/3 − ? 3 /3
+? 9 /3
1
+? 9
?8
? 1 /√3 ? 2 /√3
1
+? 9
2/3
−? 8 /√3
? 3 /√3
2/3 + 2? 3 /3 ? 6 /2
+? 9 /3
−?? 7 /2
−?? 4 /2 −1/3 − ? 9 /3
−? 5 /2
+? 10 /3
−? 4 /2√3 −?√3 ? 4 /2
+?√3 ? 5 /2 −? 5 /2√3
−1
+? 10
?? 6 /2 −1/3 − ? 9 /3
+? 7 /2
+? 10 /3
−? 6 /2√3 −?√3 ? 6 /2
+?√3 ? 7 /2 −? 7 /2√3
−1
+? 10
companions from the other two triplets is the identity.
Finally, the product (?? ?? )3 for any ? taken from the set (1, 2, 4, 5, 6, 7) and any ? from
the same set, but from a different triplet, is also the identity matrix. This property of the
third powers – taken together with the threefold symmetry of its centre – can be seen as
the essential property that distinguishes SU(3) from other Lie groups. We now return to
the strand model and show that slides indeed define an SU(3) group.
From slides to SU(3)
The slide, or third Reidemeister move, involves three pieces of strands. The textbook version of the third Reidemeister move – which is called ?0 here and is illustrated in Figure 68 – moves or ‘slides’ one strand, taken to be the horizontal blue one in the figure,
262
9 g auge interactions d ed uced from stra nds
A textbook slide, or Reidemeister III move:
E0
F I G U R E 68 The textbook version ?0 of the slide move, or third Reidemeister move, is unobservable,
because it does not involve crossing switches.
?? = eπi?? /2 .
(163)
In the strand model, the generators ? ? describe the difference between an infinitesimal
generalized slide – thus a slide-rotation with a rotation by an infinitesimal angle – and
the identity. For slides, concatenation is equivalent to group multiplication, as expected.
Slides form a group. We will now show that the slide generators obey the multiplication
table already given in Table 10.
To see how the SU(3) multiplication table follows from Figure 69, we first note that
the starting strand configuration of the Reidemeister III move contains, if all spatial configurations are considered, the same threefold symmetry as the centre of SU(3). In particular, like the generators and the basis vectors of SU(3), also the slides of the figure can
be grouped into three triplets.
We now focus on the first triplet, the one formed by the three slides ?1 , ?2 and ?3 .
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
against a crossing of the other two. Equivalently, we can say that a slide pushes two
strands against the blue strand that is kept in place. This textbook slide – we also call
it a pure slide here – does not contain any crossing switch; following the fundamental
principle of the strand model, it is therefore unobservable, or, simply said, of no physical
relevance. However, related strand moves that do involve crossing switches do exist.
We introduce eight generalized slides, or slide-rotations, for a three-strand configuration; they are shown in Figure 69. We directly call these generalized slides ?1 to ?8 , because they will turn out to correspond to the SU(3) group elements with the same name
that were introduced above. In other words, we will show that the generalized slides ??
are elements of a Lie group SU(3); in particular, they obey all the properties expected
from the correspondence with the SU(3) generators ? ? in Gell-Mann’s choice:
Motion Mountain – The Adventure of Physics
Colours and arrows are only added for clarity.
The textbook slide is unobservable, because it lacks crossing switches.
It is thus not physical and uninteresting.
the strong nuclea r interaction a nd the third reid emeister m ov e
263
The generalized slides, or Reidemeister III moves, acting on three strands, form an SU(3) group.
The 8 generalized slides are shown below, with grey background. They are
local slides and rotations by an angle π of an imaginary buckle formed
by (usually) two strands. The strands lie (mostly) in a paper plane.
λ2
For each SU(2) triplet inside SU(3), the rotation axes of the finite group
elements En are arranged at right angles to each other, as are those of
the generators λn shown on the right. The rotation axes for E3, E9 and
E10 are parallel; they are perpendicular to the paper plane. The three
imaginary belt buckles for the three SU(2) subgroups are also shown.
λ3
G
R
λ7
E1 = e i π λ 1 / 2
π
antigreen
E2 = e i π λ2 / 2
B
λ10
=
– λ3/2
+ λ8√3/2
λ4
red
green
E4 = e i π λ4 / 2
E6 = e i π λ6 / 2
π
π
antiblue
antired
E5 = e i π λ5 / 2
E7 = e i π λ7 / 2
π
antigreen
antiblue
E9 = e i π λ9 / 2
E10 = e i π λ10 / 2
π
π
π
antiblue
antired
antigreen
E3 = e i π λ3 / 2
E8 = e i π √3 λ8 / 2
π √3
F I G U R E 69 The strand deformations for the generalized slide moves ??. The corresponding generators
? ? lead to an SU(3) structure, as shown in the text. Note that the rotation vectors for the generators ? ?
and for the generalized slide moves ?? differ from each other. For clarity, the figure shows, instead of
the deformation of the strand under discussion, the complementary deformations of the other two
strands.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
π
antired
λ5
Motion Mountain – The Adventure of Physics
blue
λ9 = – λ3 / 2
– λ8 √3 / 2
λ8
λ6
Starting
position
λ1
264
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
To makes things clear, these moves can be pictured as combined deformations and slides
of the red and green strands against the horizontal blue strand. We can imagine these
moves like those of the belt trick, but acting on an imaginary buckle formed only by the
red and green strands. These generalized slides do contain crossing changes; therefore
they are observable and are of physical relevance.
We note that ‘slide’ is not a perfect term for the generalized deformations ?1 to ?8 ;
in fact, we might prefer to call them slide-rotations, because they are slide-rotations by
an angle π that are applied to an imaginary belt buckle. Despite the involved construction, these generalized, observable moves remain modelled on the textbook slide ?0 ; in
particular, they require three strand segments. The generalized, observable moves just
defined differ from the twists and pokes discussed above, in the sections on the electromagnetic and weak interactions; thus they differ from Reidemeister I and II moves. As
a result, we will usually continue to call the generalized, observable moves simply slides.
For simplicity, we assume – similarly to what we did in the discussion about the weak
interaction – that the three strand segments are (roughly) in a plane. This is an idealized situation; in fact, the arguments given in the following apply also to all other threedimensional configurations of three strands. In particular, the same results appear if all
three strands segments are assumed perpendicular to each other, instead of lying in a
plane.
We note that the rotation axes of the generalized slides ?1 and ?2 are neither aligned
nor orthogonal to the paper plane. More precisely, the rotation axes of ?1 , ?4 and ?6 are
perpendicular to the sides of a cube. ?2 , ?5 and ?7 are perpendicular to them. For the
first triplet, the rotation axes ?1 , ?2 and ?3 form an orthonormal basis; the same is valid
for the other two triplets. We now show that the slides of the first triplet define an SU(2)
group.
The observable, generalized slides in the triplet ?1 , ?2 and ?3 can be concatenated.
We distinguish two cases. The first case is the concatenation of any such slide with itself.
The result corresponds to a rotation by 2π of the chosen strand pair and its imaginary
belt buckle, and thus induces a corresponding amount of tail twisting. In fact, when
any slide of the triplet is concatenated four times with itself, the result is the identity
operation. Comparing a twofold and a fourfold concatenation, we see that they differ
only by an entangling, or algebraically, by a minus sign for the imaginary buckle. This
already realizes half of the belt trick that visualizes SU(2).
The other case to be checked is the concatenation of two different slides of the triplet.
The result is always the third slide of the triplet (up to a sign that depends on whether the
combination is cyclical or not). This behaviour realizes the other half of the belt trick. In
short, we have shown that the triplet containing the first three generalized slides defines
an SU(2) group. More precisely, the infinitesimal slide-rotations ? 1 , ? 2 and ? 3 corresponding to the finite SU(3) elements ?1 , ?2 and ?3 generate the SU(2) Lie algebra of an
SU(2) Lie group. The SU(2) subgroup just found is just one of the three linearly independent SU(2) subgroups of SU(3). The generators of the first slide triplet thus reproduce
the nine results in the upper left of Table 10. We thus retain that we can indeed visualize
the first three generalized slides with the help of the three orthogonal rotations by π of
an imaginary belt buckle formed by the red and green strands.
For the visualization of SU(3) it is essential to recall that the direction in threedimensional space of the vectors visualizing ? ? and those visualizing ?? differ from each
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Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
other. This already the case for U(1).
The remaining generalized slides that are possible in the three-strand configuration
are easily constructed using the threefold symmetry of the strand configuration; they
are illustrated in Figure 69. For each of the three strand segments there is a triplet of
observable slides; this yields a total of nine possible generalized slides for the observer
defined by the paper plane. In the second triplet, the slides corresponding to ?1 and ?2
are called ?4 and ?5 , and in the third triplet they are called ?6 and ?7 . For the three slides
corresponding to ?3 – we call the other two ?9 and ?10 – only two generators are linearly
independent. Indeed, the figure shows that ?3 ?9 ?10 – whose axes are all three parallel –
is the identity matrix; this expected from an SU(3) structure. The three operations ?3 , ?9
and ?10 also commute with all other operations; thus they form the centre of the group
defined by all ?. The second linearly independent, generalized slide of common use, ?8 ,
is also shown in the figure; it is a linear combination of ?9 and ?10 . We note that the
strand model also visualizes the factor √3 in the definition of ?8 . In total, we get eight
linearly independent generalized slides. All slides, except for ?8 , act on an imaginary
belt buckle that is formed by two strands.
We saw that the generators corresponding to the slides ?1 , ?2 and ?3 generate an
SU(2) subgroup. The same holds for the corresponding triplet ?4 , ?5 and the linear
combination ?9 = −?3 /2 − ?8 √3 /2 (corresponding to ?3 ), and for the triplet ?6 , ?7
and ?10 = −?3 /2 + ?8 √3 /2. For each of these slides, a fourfold concatenation yields the
identity; and inside each triplet, the concatenation of two different slides yields a multiple of the third slide. In short, for each triplet, the corresponding infinitesimal slides
generate an SU(2) group. These three SU(2) groups are linearly independent. We have
thus reproduced an important part of the structure of SU(3). In addition, we have found
a visualization of SU(3); since each SU(2) group can be represented by a separate imaginary buckle, the group SU(3) can be visualized – in many, but not all in aspects – with the
help of three imaginary buckles. The top right of Figure 69 illustrates this visualization.
The correspondence of the slides and the multiplication table increases further if we
change slightly the definition of the first triplet. In this first triplet we can take as imaginary buckle the set of all three central segments. Moving all three strands together simplifies the visualization, because for the first triplert, the blue strand is trapped between the
other two strands. In this way, generalized slide still consists of a rotation followed by a
slide. And we still have a SU(2) subgroup for the first triplet.
The slide ?8 differs from the other slides, as expected from SU(3). It describes a motion that rotates the red and green strands in opposite directions; this is illustrated in
Figure 69. ?8 is thus not well described with an imaginary belt buckle. It is straightforward to check that the slide ?8 commutes with ?1 , ?2 , ?3 and obviously with itself, but
not with the other generalized slides. Together, ?8 and the first triplet thus form a U(2)
Lie group, as expected. In addition, we find that ?8 commutes with ?9 and ?10 , and that
?8 ?9 = ?10 , as expected from SU(3). The strand model also implies that the product of
?8 with its two counterparts from the other triplets is the identity matrix, as expected
from SU(3).
The last step to show the equivalence of slides and SU(3) requires us to confirm the
multiplication properties – between slides ?? or between generators ? ? – from different
triplets. In fact, because of the three-fold symmetry of the centre, we only need to check
two multiplication results between slides from different triplets: one that either involves
266
The strand model for gluons
Physically, the eight slides corresponding to the Gell-Mann matrices represent the effects
of the eight gluons, the intermediate vector bosons of the strong interaction, that can act
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
? 3 or ? 8 , and one that does not.
We begin with products involving ? 3 and one of the first two elements of another
triplet. Such products yield a weighted sum of generators of the triplet. It is easier to
check these product properties by using the exemplary relation between finite group elements ?5 ?3 ?4 = ?3 . Note that only this specific permutation of 5, 3 and 4 yields this
result. Playing with the strand model confirms the relation. Similar comments apply to
?6 ?3 ?7 = ?3 – and to the corresponding products involving ?9 , such as ?1 ?9 ?2 = ?9 ,
or ?10 , such as ?1 ?10 ?2 = ?10 – as well as ?4 ?8 ?5 = ?8 and ?6 ?8 ?7 = ?8 . The strand
model allows anybody to check that these relations are satisfied.
We continue with the exemplary product ? 5 ? 7 , respectively ?5 ?7 . We note a basic difference between a product like ? 5 ? 7 and any product of two generators from the same
triplet. The product ? 5 ? 7 – like the other concatenations of generators from different
triplets – does not yield a single generator, but yields a combination, i.e., a sum of generators. The combination is not easy to visualize with strands; an easier way is to check
the SU(3) algebra using the properties of the product ?5 ?7 .
As mentioned above, in SU(3), for products involving the first two members from
different triplets, the threefold concatenation (?? ?? )3 is the identity. And indeed, Figure 69
confirms that (?2 ?4 )3 or (?5 ?7 )3 is the identity. Similarly, also the other products can be
tested with the help of three strands.
Using the visualization with three strands, we have thus confirmed all products of
generators from two different triplets that appear in Table 10. We note that Figure 69 also
illustrates that the three slides ?2 , ?5 and ?7 generate an SO(3) group, the rotation group
in three dimensions. In order to see this, we observe that the infinitesimal versions of
the three slides generate all possible rotations in three dimensions of the central triangle.
An SO(3) group also appears for the slides 1, 4 and 7, for the slides 1, 5 and 6, and for
the slides 2, 4 and 6. These are the four basic SO(3) subgroups of SU(3). The remaining
combinations of three operations from three different triplets – such as 1, 4 and 6, or the
combination 1, 5 and 7, or the combination 2, 4 and 7, or the combination 2, 5 and 6 –
do not generate any subgroup. This can be confirmed by exploring the corresponding
strand moves.
We can conclude: in a region with three strands crossing each other, the eight linearly
independent, generalized slides that can be applied to that region define the group SU(3).
In other words, the group SU(3) follows from the third Reidemeister move.
In the same way as for the other gauge groups, we find that particles whose strand
models contain configurations with three strand segments can be subject to an SU(3)
gauge interaction. In experiments, this interaction is called the strong nuclear interaction.
The strong interaction is due to the Reidemeister III move. Like for the other interactions, a particle will only interact strongly if its tangle is not too symmetric, because in
the symmetric case, averaged over time, there will be no net interaction. We will clarify
the details below, when we discuss the specific tangles and colour charges of the different
elementary matter particles.
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267
on a particle.
⊳ Given that the eight slides ?1 to ?8 represent the effects of the eight gluons,
they also represent the gluons themselves.
Interactions are transfers of a tangle process to another tangle. Therefore
⊳ The absorption of a gluon is a slide that is transferred to another particle.
⊳ The emission of a gluon is a slide that is transferred to three vacuum strands.
Motion Mountain – The Adventure of Physics
Challenge 168 e
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To visualize the concept of gluon even further, we can say that every gluon can be described as a strand structure that continuously performs an SU(3) operation, i.e., a generalized slide continuously repeating itself. We found a similar correspondence for the
other gauge interactions. In case of the electromagnetic interaction, the intermediate
vector boson, the photon, can be described as a strand that continuously performs a
U(1) operation, i.e., a rotation. In case of the weak interaction, a weak intermediate vector boson can be described as a strand that continuously performs an SU(2) operation,
i.e., an operation from the belt trick. This is most evident in the unbroken form of the
weak bosons.
Every gluon can also be seen as the deformation of a single strand that drags its surrounding with it. This single strand description of gluons implies that gluons have vanishing mass and vanishing charge. This single strand description of gluons also implies
that they have spin 1, as is observed. The strand model of the gluon also implies that free
gluons would have a huge energy.
The SU(3) multiplication table confirms that the eight gluons transform according to
the adjoint (and faithful) representation of SU(3). Therefore, each row or column in a
Gell-Mann matrix thus corresponds to one of the three colours of the strong interaction. The exploration of slide concatenation also showed that two general slides do not
commute and do not anticommute. The group SU(3) is non-Abelian. This implies that
gluons interact among themselves. Both the multiplication table and the strand model
for gluons imply that two interacting gluons can yield either one or two new gluons, but
not more. This is illustrated in Figure 70. The strand model, through its generation of
SU(3), thus implies that gluons interact among themselves, but only in triple and quartic
gluon vertices.
Slides – i.e., gluon emission or absorption – never change the topology of tangles,
and in particular, of matter tangles. Therefore, the strand model predicts that the strong
interactions conserve electric charge, baryon number, weak isospin, flavour, spin and all
parities. This is indeed observed. In particular, there is a natural lack of C, P and CP
violation by slides. This is precisely what is observed for the strong interaction.
Because gluons do not change the topology of the particle tangles they act upon, but
only change their shape, gluons are predicted to be massless in the strand model, despite interacting among themselves. And because gluons interact among themselves, free
gluons are predicted not to appear in nature. And of course, all these conclusions agree
with experiments.
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9 g auge interactions d ed uced from stra nds
A triple gluon vertex :
green
anti-green
t2
red
t2
anti-blue
vacuum
t1
red
t1
time
anti-blue
The quartic gluon vertex :
anti-green green
t2
red anti-blue
t1
anti-green red
green
t1
time
anti-blue
In summary, we have shown that in the strand model, the strong nuclear interaction
and all its properties appear automatically form slides, i.e., from Reidemeister III moves.
In particular, the strand model implies that the Lagrangian of strongly interacting fermions has a SU(3) gauge invariance that is due to generalized slide deformations.
The gluon L agrangian
Gluons are massless particles with spin 1. As a result, the field intensities and the Lagrangian are determined in the same way as for photons: energy density is the square of
crossing density, i.e., the ‘square’ of field intensity. Since there are 8 gluons, the Lagrangian density becomes
1 8
Lgluons = − ∑ ???? ???
(164)
?
4 ?=1
where the gluon field intensities, with two greek indices, are given naturally as
???? = ∂? ??? − ∂? ??? − ????? ??? ??? ,
(165)
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F I G U R E 70 The two types of self-interaction of gluons in the strand model.
Motion Mountain – The Adventure of Physics
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the strong nuclea r interaction a nd the third reid emeister m ov e
269
Strong charge, or colour :
Random slides affect
only rational tangles
with broken threefold
tail symmetry :
Random slides
do not affect
knotted tangles :
Random slides
affect gluons :
Random slides
do not affect photons :
Colour charge
Surrounded by a bath of gluons that randomly induce slides of all kinds, not all fermion
cores will change their rotation state. Generally speaking, particles have colour if a bath
of random gluons changes their phase. Only tangles which lack some symmetry will
therefore possess colour charge. Tangle that are symmetric will be neutral, or ‘white’.
Which symmetry is important here?
We see directly that the photon tangle is not sensitive to a gluon bath. The same is
valid for W and Z bosons. These tangles are too simple. The strand model predicts that
these particles are colour-neutral, i.e., that they are ‘white’, as is observed.
On the other hand, the multiplication properties given above shows that gluons interact among themselves and thus that they have colour charge. In fact, group theory
shows that their properties are best described by saying that gluons have a colour and an
anticolour; this is the simplest way to describe the representation to which they belong.
In short, the strand model of gluons automatically implies that they carry both a colour
and an anti-colour.
Fermions behave differently. In the strand model, a fermion has colour charge if the
corresponding triple belt model is affected by large numbers of random gluons. The
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Page 231
and ???? are the structure constants of SU(3) that can be deduced from the multiplication table given above. The quantities ??? , with one greek index, are the gluon vector
potentials. The last term in the definition of the field intensities corresponds to the triple
and quartic vertices in the Feynman diagrams of gluon interactions. They are shown
in Figure 70. The Lagrangian is simply the natural generalization from the U(1) case of
photons to the SU(3) case of gluons. In short, we obtain the usual free gluon Lagrangian
from the strand model.
Motion Mountain – The Adventure of Physics
F I G U R E 71 Tangles with and without colour charge. (This figure needs to be updated.)
270
9 g auge interactions d ed uced from stra nds
Strong interaction diagram :
real
quark
+
Feynman diagram :
time average
of crossing
switches
virtual
gluon
real
quark
t2
virtual
gluon
t2
core is
rotated by
2π/3 around
vertical axis
t1
t1
quark
time
correspond to the three belts.
⊳ A fermion tangle has colour charge if its three-belt model is not symmetric
for rotations by ±2π/3.
Coloured rational tangles automatically have three possible colours:
⊳ The three colour charges are the three possibilities to map a tangle to the
three belt model.* Each colour is thus a particular orientation in ordinary
space.
Page 319
Challenge 169 ny
If we want to explore more complicated types of tangles of two strands, such as prime
tangles or locally knotted tangles, we recall that such tangles are not part of the strand
* Can you define a geometric or even a topological knot invariant that reproduces colour charge?
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
first tangles that come to mind are tangles made of three strands, such as the simple
tangles shown in Figure 69. But a short investigation shows that such tangles are colourneutral, or ‘white’. We will see below that this implies that leptons are colour-neutral, or
‘white’. In contrast, a rational fermion tangle does not suffer this fate. (We recall that
a so-called rational tangle is by definition made of exactly two strands; a two-stranded
tangle is rational if the two strands can be untangled just by moving the tails around.) In a
bath of gluon strands that induce slides, i.e., third Reidemeister moves, a general rational
tangle made of two strands is expected to be influenced, and thus to be colour-charged.
Rational tangles made of two strands are the simplest possible tangles with colour. A
tangle is called rational if it can be untangled just by moving the tails around. An example
of a rational tangle is shown in Figure 72. Such tangles break the three-fold symmetry
of the three-belt structure, and are thus colour-charged. We will show below how these
tangles are related to quarks. We can thus say:
Motion Mountain – The Adventure of Physics
F I G U R E 72 The Feynman diagram of the strong interaction for a quark. The upper triplet of tails
the strong nuclea r interaction a nd the third reid emeister m ov e
271
model. The strand model thus predicts that rational tangles made of two strands are
the basic colour states. And indeed, in nature, quarks are the only fermions with colour
charge.
We can summarize that colour charge is related to orientation in space. The three
possible colours and anticolours are consequences of the possible orientations along the
three dimensions of space.
Properties of the strong interaction
The L agrangian of Q CD
/ − ?? ?2 ???? )Ψ?? −
LQCD = ∑ Ψ? (?ℏ?D
?
1 8 ?
∑ ? ? ?? ,
4 ?=1 ?? ?
(166)
where the index ? counts the coloured fermion, i.e., the quark. In this Lagrangian density,
/ is now the SU(3) gauge covariant derivative
D
/ = ∂/ − ? ?? ??? ? ? ,
D
(167)
where ? is the gauge coupling, ? ? are the generators of SU(3), i.e., the Gell-Mann
matrices given above, and the ??? are, as before, the gluon vector potentials. The last
term in the covariant derivative corresponds to the Feynman diagram and the strand
diagram of Figure 72. This is the Lagrangian density of QCD.
In summary: the strand model reproduces QCD. However, we have not yet deduced
the number and masses ?? of the quarks, nor the strong gauge coupling ?.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
We started from the idea that tangle core deformations lead to phase redefinitions. We
then found that slides imply that the strong interaction Lagrangian for matter and for
radiation fields is SU(3) gauge invariant. If we include these two gauge invariances into
the fermion Lagrangian density from the Dirac equation, we get
Motion Mountain – The Adventure of Physics
In the strand model, all interactions are deformations of the tangle core. Specifically, the
strong interaction is due to exchange of slides. Particles have strong charge, or colour, if
their tangles lack the three-belt symmetry just specified. In the case of coloured fermions, colour change is a change of the mapping to the three-belt model, i.e., a change of
orientation of the tangle in space.
If we use the strand definition of the strong interaction, visual inspection shows us
that slide exchanges, and thus gluon exchanges, are deformations that conserve topology;
therefore gluon exchange conserves colour. Since the strong interaction conserves the
topology of all involved tangles and knots, the strong interaction also conserves electric
charge, parity, and, as we shall see below, all other quantum numbers – except colour
itself, of course. All these results correspond to observation.
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R enormalization of the strong interaction
Page 382
Page 334, page 341
Curiosities and fun challenges abou t SU(3)
Vol. V, page 288
Ref. 184
∗∗
Challenge 170 ny
We have discussed the shape deformations that lead to the SU(3) group. But what are the
precise phase choices for a crossing that lead to SU(3) invariance?
∗∗
Challenge 171 ny
Do the two linear independent gluons with lined-up tails have the same properties as the
other six gluons?
∗∗
Challenge 172 s
Three strands can cross each other also in another way, such that the three strands are
interlocked. Why can we disregard the situation in this section?
∗∗
Deducing the Lie groups U(1), SU(2) and SU(3) directly from a basic principle contradicts another old dream. Many scholars hoped that the three gauge groups have something to do with the sequence complex numbers, quaternions and octonions. The strand
model quashes this hope – or at least changes it in an almost unrecognizable way.
∗∗
Challenge 173 e
The tangles for the W and Z bosons have no colour charge. Can you confirm this?
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Deducing the Lie group SU(3) from a three-dimensional model is a new result. In particular, deducing the gauge group SU(3) as a deformation gauge group is new. Frank
Wilczek, Alfred Shapere, Alden Mead, Jerry Marsden and several others have confirmed
that before this discovery, only the geometric Lie group SO(3) and its subgroups had
been found in deformations. The fundamental principle of the strand model shows its
power by overcoming this limitation. (Apparently, nobody had even realized that the
belt trick already implies the possibility of an SU(2) gauge group for deformations.)
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The slide move description of the strong interaction implies that only three Feynman
diagrams are possible: one QCD Feynman diagram is possible for quarks, and only the
triple and the quartic vertices are possible among gluons. This limited range of options
allowed us to deduce the QCD Lagrangian. The limited range of options is also essential
for the renormalization of QCD. The strand model thus automatically ensures that the
strong interaction is renormalizable.
In short, the strand model provides a new underlying picture for the Feynman diagrams of the strong interaction, but does not change the physical results at any energy
scale accessible in the laboratory. In particular, the measured running of the strong coupling constant is reproduced. Indeed, in the strand model, a flux-tube–like bond between
the quarks appears automatically, as we will see when exploring hadrons. At high kinetic energies, the bond has little effect, so that quarks behave more like free particles. In
short, we find that the strand model reproduces asymptotic freedom and also provides an
argument for quark confinement. We will return to the issue in more detail below.
the strong nuclea r interaction a nd the third reid emeister m ov e
273
∗∗
Challenge 174 ny
The Lie group SU(3) is also the symmetry group of the three-dimensional harmonic oscillator. What is the geometric relation to the Lie group SU(3) induced by slides?
∗∗
Challenge 175 e
Confirm that the strand model does not contradict the Coleman–Mandula theorem on
the possible conserved quantities in quantum field theory.
∗∗
Challenge 176 e
Confirm that the strand model does not contradict the Weinberg–Witten theorem on
the possible massless particles in quantum field theory.
∗∗
Are the Wightman axioms of quantum field theory fulfilled by the strand model with
interactions? The Haag–Kastler axioms? Is Haag’s theorem circumvented?
∗∗
Ref. 185
Challenge 178 ny
Show that the BCFW recursion relation for tree level gluon scattering follows from the
strand model.
Summary on the strong interaction and experimental predictions
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Page 353
We have deduced the Lagrangian density of QCD from the strand model with the help of
slides. Is there a difference between the strand model and QCD? No, not as long as gravity
plays no role. The strand model predicts that gravitation only comes into play near the
Planck energy √ℏ?5 /4? . And indeed, accelerator experiments have not yet found any
effect that contradicts QCD, and therefore no effect that contradicts the strand model of
the strong interaction.
The strand model also predicts that the strong interaction is naturally CP-invariant.
This means that axions – particles invented to explain the invariance – are unnecessary:
as shown below, the strand model even predicts that they do not to exist. Both predictions agree with experiment.
The strand model of the strong interaction implies that the SU(3) gauge symmetry is
valid at all energies. No other gauge group plays a role in the strong interaction. The
strand model thus predicts again that there is no grand unification in nature, and thus
no larger gauge group. Often discussed groups such as SU(5), SO(10), E6, E7, E8 or
SO(32) are predicted not to apply to nature. Also this prediction is not contradicted by
experiment.
The strand model further predicts that the combination of gravity and quantum theory turns all Planck units into limit values. The strand model thus predicts a maximum
strong field value given by the Planck force divided by the strong charge of the quark. All
physical systems – including all astrophysical objects, such as neutron stars, quark stars,
gamma-ray bursters or quasars – are predicted to conform to this field limit. So far, this
prediction is validated by experiment.
In summary, we have shown that Reidemeister III moves – or slides – in tangle cores
lead to an SU(3) gauge invariance and a Lagrangian that reproduces the strong interac-
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Challenge 177 d
274
Page 162
9 g auge interactions d ed uced from stra nds
tion. Colour charge is related to the topology of certain rational tangles. In this way, we
have deduced the origin and most observed properties of the strong interaction. We have
thus settled another issue of the millennium list. However, we still need to deduce the
tangles and the number of quarks, their masses and the strength of the strong coupling.
Motion Mountain – The Adventure of Physics
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sum m ary and pred ictions ab ou t g auge interactions
275
summary and predict ions abou t gauge interactions
Page 150
Page 162
Ref. 181
Already in 1926, Kurt Reidemeister proved an important theorem about possible deformations of knots or tangles that lead to changes of crossings. When tangles are described
with two-dimensional diagrams, all possible deformations can be reduced to exactly three
moves, nowadays called after him. In the strand model, the two-dimensional tangle diagram describes what an observer sees about a physical system. Together with the equivalence of interactions as crossing-changing deformations, Reidemeister’s theorem thus
proves that there are only three gauge interactions in nature. In particular, there is no fifth
force. Searches for additional gauge interactions are predicted to fail. And indeed, they
have all failed up to now.
Unification of interactions
Ref. 142
We can also state that there is only one Reidemeister move. This becomes especially clear
if we explore the three-dimensional shape of knots instead of their two-dimensional diagrams: all three Reidemeister moves can be deduced from the same deformation of a
single strand. Only the projection on a two-dimensional diagram creates the distinction
between the three moves. In the terms of the strand model, this means that all gauge
interactions are in fact aspects of only one basic process, a fluctuation of strand shape,
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Predicting the number of interactions in nature
Motion Mountain – The Adventure of Physics
Page 147
At this point of our adventure, we have deduced gauge theory and the three known gauge
interactions from strands. Using only the fundamental principle, we explained the dimensions of space-time, the Planck units, the principle of least action, the appearance
of the gauge groups U(1), broken SU(2) and SU(3), of renormalization, of Lorentz symmetry and of permutation symmetry. Thus we have deduced all the concepts and all the
mathematical structures that are necessary to formulate the standard model of elementary particles.
In particular, the strand model provides a description and explanation of the three
gauge interactions at Planck scales that is based on deformations of strands. The deduction of the three gauge interactions given in this text, with the help of the Reidemeister
moves, is the first and, at present, the only explanation of the three gauge forces. No other
explanation or deduction has ever been given.
We have shown that quantum field theory is an approximation of the strand model.
The approximation appears when the strand diameter is neglected; quantum field theory
is thus valid for all energies below the Planck scale. In other words, in contrast to many
other attempts at unification, the strand model is not a generalization of quantum field
theory. The strand model for the three gauge interactions is also unmodifiable. These
properties are in agreement with our list of requirements for a final theory.
We have not yet deduced the complete standard model: we still need to show which
types of particles exist, which properties they have and what couplings they produce.
However, we have found that the strand model explains all the mathematical structures
from the millennium list that occur in quantum field theory and in the standard model
of particle physics. In fact, the strand explanation for the origin of the gauge interactions
allows us to make several definite predictions.
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No divergences
Page 150
Grand unification, supersymmetry and other dimensions
Page 391
Page 348
The three gauge interactions are due to the three Reidemeister moves. Therefore, the
strand model asserts that there is no single gauge group for all interactions. In short, the
strand model asserts that there is no so-called grand unification. The absence of grand
unification implies the absence of large proton decay rates, the absence of additional, still
undiscovered gauge bosons, the absence of neutron–antineutron oscillations, and the
absence of sizeable electric dipole moments in elementary particles. All these searches
are ongoing at present; the strand model predicts that they yield null results.
Supersymmetry and approaches based on it assume gauge group unification. However, as just explained, the strand model predicts that there is no supersymmetry and
therefore no supergravity. The strand model also predicts the absence of all conjectured
‘superparticles’. In 2016 and again in 2017, the numerous experiments at CERN confirmed the prediction: there is no sign of supersymmetry in nature.
Reidemeister moves are confined to three spatial dimensions. Indeed, the strand
model is based on exactly three spatial dimensions. It predicts that there are no other,
undetected dimensions of space. The strand model also predicts the absence of noncommutative space-time, even though, with some imagination, strands can be seen as
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
The strand model implies that there are no divergences in the quantum description of
nature. This lack of divergence occurs because all measurement values appear after
strand effects have been averaged out. As mentioned above, strand effects on space-time
disappear through ‘shivering’ and strand effects on particles disappear through wavefunctions.
In summary, in the strand model, no interaction implies or contains divergences:
neither gravity nor the gauge interactions. There are neither ultraviolet nor infrared divergences. The strand model avoids divergences, infinities and singularities of any kind
from its very start.
Motion Mountain – The Adventure of Physics
and that the three gauge interactions are only distinguished by their projections. In this
way, the three gauge interactions are thus unified by the strand model.
The plane of projection used in a strand diagram defines a mapping from strand fluctuations to Reidemeister moves. The projection plane is defined by the observer, i.e.,
by the frame of reference. Depending on the projection plane, a general deformation
is mapped into different Reidemeister moves. At first sight, the nature of an interaction – whether electromagnetic, strong or weak – seems to depend on the observer. In
nature, however, this is not the case. But this contradiction is only apparent. In the
strand model, the nature of interaction of a particle results from the type of asymmetry
of its tangle core. Certain strand deformations do not lead to interactions, because their
effects are suppressed by the averaging of short-time fluctuations underlying every observation. In other words, the averaging process at the basis of observations also ensures
that interactions are effectively observer-independent at low energy.
In short, the strand model provides a natural unification of the interactions. And this
unification of the interactions differs completely from any past proposal. The final test,
of course, can only be provided by experiment.
sum m ary and pred ictions ab ou t g auge interactions
Page 147
Page 395
277
remotely related to that approach. Finally, the strand model predicts the lack of different
vacua: the vacuum is unique.
In short, the strand model differs both experimentally and theoretically from the unification proposals made in the twentieth century. In particular, the strand model predicts the absence of additional symmetries, of additional energy scales, and of additional
space-time properties at high energy. The strand model predicts that unification is not
achieved by searching for higher symmetries, nor for higher dimensions, nor for concepts that contain both. This lack of complex mathematical or symmetry concepts in
nature is disappointing; the hopes and search activities in the last fifty years are predicted
to have been misguided. In other words, the predictions of the strand model are unpopular. However, these predictions agree with our list of requirements for a final theory;
and so far, all these predictions agree with experiment.
No new observable gravit y effects in particle physics
In the ‘cube’ structure of physics shown in Figure 1, the transition from the final, unified
description to quantum field theory occurs by neglecting gravity, i.e., by assuming flat
space-time. The same transition occurs in the strand model, where neglecting gravity in
addition requires neglecting the strand diameter. In this way, the gravitational constant
? disappears completely from the description of nature.
We can summarize our findings on quantum field theory also in the following way:
Page 311
This result will be complemented below by a second, equally restrictive result that limits the observable quantum effects in the study of gravity. In short, the strand model
keeps particle physics and general relativity almost completely separated from each other.
This is a consequence of the different effects produced by tail deformations and by core
deformations. And again, the prediction of a lack of additional gravitational effects in
particle physics agrees with all experiments so far.
The status of our quest
Page 162
In this chapter, we have deduced that strands predict exactly three interactions. Interactions are deformations of tangle cores and just three classes of such core deformations
exist. The three classes of deformations are given by the three Reidemeister moves. Because of the properties of the Reidemeister moves, the three interactions are described
by a U(1), a broken SU(2) and a SU(3) gauge symmetry, respectively.
Strands also show that the three interactions are renormalizable, relativistically invariant, and that they follow the least action principle. Strands thus imply the three interaction Lagrangians of the standard model of particle physics. In addition, strands predict
the absence of other interactions, symmetries and space-time structures.
If we look at the millennium list of open issues in fundamental physics, we have now
solved all issues concerning the mathematical structures that appear in quantum field
theory and in the standard model of particle physics.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
⊳ The strand model predicts that particle masses are the only observable effect
of gravity in quantum physics and in particle physics.
Motion Mountain – The Adventure of Physics
Page 8
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9 g auge interactions d ed uced from stra nds
⊳ All mathematical structures found in quantum physics result from the fundamental principle of the strand model.
Equivalently, extension contains all quantum effects. This is an intriguing result that
induces us to continue our exploration. Only two groups of issues are still unexplained:
the theory of general relativity and the spectrum of elementary particles. We proceed in
this order.
Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
C h a p t e r 10
GENER AL R EL AT IVIT Y DEDUCED
FROM ST R ANDS
G
Page 206
Page 209
We have seen above that any observer automatically introduces a 3+1-dimensional background space-time. We have also seen that in the case of quantum theory, physical spacetime, the space-time that is formed by the fluctuations of the vacuum strands, is naturally
3+1-dimensional and flat. In the absence of gravity, physical space and background space
coincide.
Using strands, we have deduced:
⊳ ? is the invariant limit for all energy speeds.
This limit is achieved only by free massless particles, such as photons. Strands also
showed us that massive particles move more slowly than light. In short, strands reproduce special relativity.
The strand model thus predicts that pure special relativity is correct for all situations
and all energies in which gravity and quantum theory play no role. The strand model
also predicts that when gravity or quantum effects do play a role, general relativity or
quantum theory must be taken into account. This means that there is no domain of
nature in which intermediate descriptions are valid.
It is sometimes suggested that the invariant Planck energy limit for elementary
particles might lead to a ‘doubly special relativity’ that deviates from special relativity
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Fl at space, special rel ativit y and its limitations
Motion Mountain – The Adventure of Physics
eneral relativity describes the deformations of the vacuum. In everyday life,
ravitation is the only such effect that we observe. But on astronomical scale,
ravity shows more phenomena: vacuum can deflect light, producing gravitational lenses, can wobble, giving gravitational waves, and can accelerate, yielding the
darkness of the sky and the fascinating black holes. All these observations require general
relativity for their description. Therefore, general relativity must be part of any unified
description of nature.
In the following, we explain the existence of gravity as a consequence of strands. Then
we deduce the field equations of general relativity, the entropy of black holes and relativistic cosmology from the strand model. We also predict the outcome of many quantum
gravity experiments. Finally, we deduce the consequences of strands for cosmology, including a new experimental prediction. Of all Planck-scale models of space or spacetime, strands seem to be the simplest one that provides these deductions.
280
10 g eneral rel ativity d ed uced from stra nds
first mass
gravitational
interaction
~1/r2
second mass
distance r
Ref. 85
Ref. 186
Page 8
Cl assical gravitation
In nature, at low speeds and in the flat space limit, gravitation is observed to lead to
an acceleration ? of test masses that changes as the inverse square distance from the
gravitating mass;
?
?=? 2 .
(168)
?
This acceleration is called universal gravitation or classical gravitation. It is an excellent
approximation for the solar system and for many star systems throughout the universe.
In the strand model, every space-time effect, including gravitation, is due to the behaviour of tangle tails. In the strand model, every mass, i.e., every system of tangles, is
connected to the border of space by tails. The nearer a mass is to a second mass, the
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 64
at high particle energy. However, this suggestion is based on two assumptions: that at
Planck energy point masses are a viable approximation to particles, and that at Planck
energy vacuum and matter differ. In the strand model, and in nature, both assumptions
are incorrect. Nature, as general relativity shows, does not allow the existence of point
masses: the densest objects in nature are black holes, and these are not point-like for any
mass value.
In addition, quantum theory implies the fuzziness of matter and space. As a result,
near Planck energy, matter and vacuum cannot be distinguished. Put simply, no system
near Planck energy can be described without general relativity and without quantum
gravity. In short, the strand model predicts that the approach of ‘doubly special relativity’ cannot be correct. Also Figure 1 makes this point: there is no description of nature
besides the usual ones.
To sum up, the strand model reproduces special relativity when masses are approximated as point-like in flat space. But at the same time, the strand model states that a
negligibly small, light and localizable mass cannot exist – neither in flat nor in curved
space. This matches observations.
Motion Mountain – The Adventure of Physics
F I G U R E 73 Gravitational attraction as result of strands.
g enera l rel ativit y d educed from stra nds
281
more frequently the tails of the two masses cross and get tangled. Figure 73 illustrates
the situation. The strand model states:
⊳ Gravitation is due to the fluctuations of tail crossings.
Challenge 179 e
Ref. 187
Page 287
Black holes have entropy; this implies universal gravitation. There are at least two ways
to explain this connection.
An especially concise explanation was recently given by Erik Verlinde. In this view,
gravity appears because any mass ? generates an effective vacuum temperature around it.
A gravitating mass ? attracts test masses because during the fall of a test mass, the total
entropy decreases. It is not hard to describe these ideas quantitatively.
Given a spherical surface ? enclosing a gravitating mass ? at its centre, the acceleration ? of a test mass located somewhere on the surface is given by the local vacuum
temperature ?:
2π ??
?=?
,
(169)
ℏ
where ? is the Boltzmann constant. This relation is called the Fulling–Davies–Unruh
effect and relates vacuum temperature and local acceleration. Thus, an inertial or a freely
falling mass (or observer) measures a vanishing vacuum temperature.
In the strand model, the vacuum temperature at the surface of the enclosing sphere is
given by the crossing switches induced by the tails starting at the mass. We can determine
the vacuum temperature by dividing the energy ? contained inside the sphere by twice
the maximum possible entropy ? for that sphere. This maximum value is the entropy that
the sphere would have if it were a black hole horizon; it can be calculated by the strand
model, as we will see shortly.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Deducing universal gravitation from bl ack hole properties
Motion Mountain – The Adventure of Physics
Page 356
Around a mass, the tail crossings fluctuate; averaged of time, the fluctuations lead to a
crossing switch density around every mass. The resulting potential energy – where energy is action per time and thus given by the number of crossing switches per time –
changes like the inverse distance from the central mass. This is the reason for the 1/?dependence of the gravitational potential and the 1/?2 -dependence of gravitational acceleration. (This applies to all those cases where spatial curvature is negligible.) In simple
words, in the strand model, the inverse square dependence of gravitational acceleration
is due to the three-dimensionality of space combined with the one-dimensionality of
strands.
The strand model also shows that masses and energies are always positive: every
tangle contains curved strands. The model also shows qualitatively that larger masses
produce stronger attraction, as they generally produce more crossing switches. We will
show below that the number density of crossing switches is indeed given by the mass.
In the strand model, crossing switches are not only related to action and energy; they
are also related to entropy. A slightly different – but equivalent – view on gravitation
therefore appears when we put the stress on the entropic aspect.
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10 g eneral rel ativity d ed uced from stra nds
The temperature ? is thus given by the expression
?=
?
? 2?ℏ
=
.
2? ? ??
(170)
The factor 2 needs explanation: it might be due to the combination of the effects of space
and matter.
Neglecting spatial curvature, we can set ? = 4π?2 ; this gives a temperature at the
enclosing sphere given by
? ?ℏ
?= 2
.
(171)
? 2π ??
Page 281
The change of entropy d?/d? when a test mass ? moves by a distance ? can be determined from the strand model in a simple manner. When the test mass ? moves by a
(reduced) Compton wavelength, in the strand model, the mass has rotated by a full turn:
the entropy change is thus 2π? per (reduced) Compton wavelength. Thus we have
d?
2π ??
=?
.
d?
ℏ
(174)
Using the temperature ? found in expression (171), we get an expression for the gravitational force given by
??
?=? 2 .
(175)
?
Page 35
This is universal gravitation again. This time we have thus deduced universal gravitation
from the entropy and temperature generated by gravitating masses.
We note that the temperature and entropy of black holes are limit values. We can thus
state that universal gravitation is a consequence of nature’s limit values.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
This is universal gravitation, as discovered by Robert Hooke and popularized by Isaac
Newton. Since spatial curvature was neglected, and the central mass was assumed at rest,
this expression is only valid for large distances and small speeds. We have thus deduced
universal gravity from the effects of gravitating masses on vacuum temperature. Below,
we show that in the relativistic case this sequence of arguments – which was given by
Jacobson fifteen years before Verlinde – leads to the field equations of general relativity.
An alternative deduction of universal gravitation from black hole entropy is the following. The gravitational force ? on a test mass ? is given by the vacuum temperature ?
created by the central mass ? and by the change of entropy ? per length that is induced
by the motion of the test mass:
d?
?=?
.
(173)
d?
Motion Mountain – The Adventure of Physics
Page 293
Inserting this expression into the expression (169) for the Fulling–Davies–Unruh acceleration ?, we get
?
?=? 2 .
(172)
?
g enera l rel ativit y d educed from stra nds
283
The partial link :
2R
axis
other vacuum strands
Summary on universal gravitation from strands
Ref. 188
Vol. I, page 218
Curved space
In nature, observation shows that physical space is not flat around masses, i.e., in the
presence of gravity. Near mass and energy, physical space is curved. Observations also
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Page 287
Universal gravitation is due to the temperature and entropy of the (curved) vacuum
around masses. The limit case is the temperature and entropy of black holes. In the
strand model, these temperature and entropy values are a consequence of the underlying strand crossing switches; we will show this shortly. Universal gravitation thus (again)
appears as an effect of the crossing switches induced by masses.
We can say that we have several explanations of universal gravitation using strands.
We have deduced universal gravitation from the energy of strands, from the temperature
of strands and from the entropy of strands around a mass. We have also have deduced
universal gravitation from the maximum force, which strands fulfil as well. In short,
strands explain the origin of universal gravitation.
Incidentally, modelling mass as a source for strand crossing switches is remotely reminiscent of Georges-Louis Lesage’s eighteenth-century model of gravitation. Lesage
proposed that gravity appears because many tiny, usually unnoticed corpuscules push
masses together. In fact, as we will see shortly, there is a certain similarity between
these assumed tiny corpuscules and virtual gravitons. And interestingly, all criticisms
of Lesage’s model then cease to hold. First, there is no deceleration of free masses in
inertial motion, thanks to the built-in special-relativistic invariance. Secondly, there is
no heating of masses, because the entangled tails represent virtual gravitons that scatter elastically. Thirdly, and most of all, by replacing the corpuscules ultra-mondains of
Lesage by virtual gravitons – and finally by strands – we can predict an additional effect
of gravity that is not described by the inverse square dependence: space-time curvature.
Motion Mountain – The Adventure of Physics
F I G U R E 74 A schematic model of the fundamental defect, and thus the fundamental type of curvature:
the partial link.
284
10 g eneral rel ativity d ed uced from stra nds
show that curved space-time remains 3+1-dimensional. The observation of this type of
curvature was predicted long before it was measured, because curvature follows unambiguously when the observer-invariance of the speed of light ? and the observerinvariance of the gravitational constant ? are combined.
We continue directly with the strand model of spatial curvature and show that all
observations are reproduced.
⊳ Curvature (of physical space-time) is due to simple, unknotted and weakly
localized defects in the tangle of strands that make up the vacuum. An example is shown in Figure 74.
⊳ Mass is a localized defect in space and is due to tangled strands. Thus mass
curves space around it.
⊳ Energy in a volume is the number of crossing switches per unit time. As a
result, mass is equivalent to energy. As a second result, energy also curves
space.
⊳ Gravitation is the space-time curvature originating from compact regions
with mass or energy.
Vol. II, page 284
The structure of horizons and bl ack holes
In general relativity, another concept plays a fundamental role. In the strand model we
have:
⊳ A horizon is a tight, one-sided weave of strands.
Therefore, there are no strands behind the horizon. This implies that behind a horizon,
there is no matter, no light, no space and no time – just nothing. Indeed, this is the
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Page 356
These natural definitions show that curvature is due to strand configurations. In particular, curvature is built of unknotted – i.e., massless – defects. The massless defects leading
to curvature are usually dynamic: they evolve and change. Such curvature defects – virtual gravitons – originate at regions containing matter or energy. In fact, the curvature of
space around masses is a natural result of fluctuations of the strands that make up matter
tangles.
We note that curved space, being a time average, is continuous and unique. Vacuum or
curved space, more precisely, curved physical space, thus differs from background space,
which is flat (and drawn in grey in the figures).
Incidentally, the distinction between physical and background space also avoids Einstein’s hole argument; in fact, the distinction allows discussing it clearly, as only physical
space describes nature.
Motion Mountain – The Adventure of Physics
⊳ In the case of curvature, physical space-time, which is due to averaged strand
crossing switches, differs from flat background space-time, which usually
corresponds to the tangent or to the asymptotic space-time. In Figure 74,
the grey background colour can be taken as visualization of the background
space.
g enera l rel ativit y d educed from stra nds
General horizon :
(side view)
285
Black hole :
F I G U R E 75 A schematic model of a general and a spherical horizon as tight weaves, as pictured by a
distant observer. In the strand model there is nothing, no strands and thus no space, behind a horizon.
⊳ A black hole is a tight, one-sided and closed weave of strands.
Is there something behind a horizon?
A drawing of a horizon weave, such as the one of Figure 75, clearly points out the difference between the background space and the physical space. The background space is
the space we need for thinking, and is the space in which the drawing is set. The physical
space is the one that appears as a consequence of the averaging of the strand crossings.
Physical, curved space exists only on the observer side – usually outside – of the horizon.
The physical space around a black hole is curved; it agrees with the background space
only at infinite distance from the horizon. Inside the horizon, there is background space,
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In principle, closed horizons can have any shape. The simplest case is the spherical, nonrotating horizon, which defines the Schwarzschild black hole. It is illustrated on the righthand side of Figure 75.
If an observer is located outside a spherical horizon, the strand model states that there
is nothing inside the horizon: no matter, no light and no vacuum. The strand model
thus provides a simple and drastic view of black hole horizons. Figure 75 also illustrates
that the concept of radius (or size) of a black hole has to be approached with the (wellknown) care. In general, the size of a structure made of strands is the number of crossings
encountered when travelling through it. However, an observer cannot travel through a
black hole: there are no strands inside, thus there is no vacuum there! The size of a black
hole must therefore be defined indirectly. The simplest way is to take the square root
of the area, divided by 4π, as the radius. Thus the strand model, like general relativity,
requires that the size of a compact horizon be defined by travelling around it.
We note that the strand model also provides an intuitive explanation for the differences between a rotating and a non-rotating black hole.
Motion Mountain – The Adventure of Physics
experience of any observer about a horizon. A horizon is thus a structure that limits
physical space. It does not limit background space.
One particular type of horizon is well-known:
286
10 g eneral rel ativity d ed uced from stra nds
but no physical space. In short, the strand model implies that – for an observer at spatial
infinity – there is nothing, not even a singularity, inside a black hole horizon.
⊳ There is no physical space, no matter and no singularity inside a horizon.
Energy of bl ack hole horizons
?=
Challenge 181 e
?cs 4π?2 ?4
?4
=
=?
.
?
2π? 4?
2?
(176)
Strands thus imply the well-known relation between energy (or mass) and radius of
Schwarzschild black holes.
How do the crossing switches occur at a horizon of a black hole? This interesting
puzzle is left to the reader.
The tight-weave model of horizons also illustrates and confirms both the hoop conjecture and the Penrose conjecture. For a given mass, because of the minimum size of crossings, a spherical horizon has the smallest possible diameter, compared to other possible
shapes. The strand model naturally implies that, for a given mass, spherical black holes
indeed are the densest objects in nature.
The nature of bl ack holes
The strand model naturally implies the no-hair theorem. Since all strands are the same,
independently of the type of matter that formed or fell into the horizon, a black hole
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Challenge 180 e
The strand model allows us to calculate the energy content of a closed horizon. Energy is
action per unit time. In the strand model, the energy of a non-rotating spherical horizon
is thus given by the number ?cs of crossing switches per time unit. In a tight weave,
crossing switches cannot happen in parallel, but have to happen sequentially. As a result,
a crossing switch ‘propagates’ to the neighbouring Planck area on the surface. Since the
horizon weave is tight and the propagation speed is one crossing per crossing switch
time, this happens at the speed of light. In the time ? that light takes to circumnavigate
the spherical horizon, all crossings switch. We thus have:
Motion Mountain – The Adventure of Physics
Horizons are observer-dependent. Both the existence and the shape of a horizon depends on the observer. As we will see, this happens in precisely the same way as in usual
general relativity. In the strand model, there is no contradiction between the one observer at spatial infinity who says that there is nothing behind a horizon, not even physical space, and another, falling observer, who does not observe a horizon and thus states
that there is something there. In the strand model, the two statements naturally transform into each other under change of viewpoint. Indeed, the transformation between
the two viewpoints contains a deformation of the involved strands.
We note that the equivalence of viewpoints and the statement that there is nothing
behind a horizon is based on the combination of general relativity and quantum theory. If we would continue thinking that space and time is a manifold of points – thus
disregarding quantum theory – these statements would not follow.
In summary, one-sided tight weaves are a natural definition of horizons.
g enera l rel ativit y d educed from stra nds
287
has no characteristics other than mass, angular momentum and charge. Here we used a
result from the next chapter, when it will become clear that all elementary particles are
indeed made of the same featureless strands. Taking that result as given, we deduce that
flavour quantum numbers and particle number do not make sense for black holes. We
also deduce that weak and strong charge are not defined for black holes. Strands explain
naturally why neutral black holes made of antimatter and neutral black holes made of
matter do not differ, if their masses and angular momenta are the same. In short, the
strand model of nature implies the no-hair theorem: strands, not hairs.
Horizons and black holes are borderline systems between space and matter. This borderline property must be fulfilled by every final theory. The strand model fulfils this
requirement: in the strand model, black holes can either be described as curved space or
as tightly packed particles in permanent free fall.
Entropy of vacuum and mat ter
⊳ The flat and infinite vacuum has vanishing entropy, because the number of
crossing switches is zero on average.
At the same time,
The entropy of vacuum and of horizons differs from that of matter. In the absence of
gravity, the number of microstates of matter is determined – as in usual thermodynamics
(thermostatics) – by the behaviour of tangle cores.
In strong gravity, when the distinction between matter and vacuum is not so clear-cut,
the number of microstates is determined by the possible crossing switches of the strands.
In strong gravity, only tails play a role. This becomes clear when we calculate the entropy
of black holes.
Entropy of bl ack holes deduced from the strand model
Despite the tight weaving, the strands making up a horizon are fluctuating and moving:
the weave shape fluctuates and crossing switch all the time. This fluctuating motion is
the reason why horizons – in particular those of black holes – have entropy.
The weave model of a horizon, illustrated in detail in Figure 76, allows us to calculate
the corresponding entropy. Since the horizon is a tight weave, there is a crossing on each
Planck area. To a first approximation, on each (corrected) Planck area of the horizon, the
strands can cross in two different ways. The fundamental principle of the strand model
thus yields two microstates per Planck area. The number ? of Planck 2areas is given by
?2 = ??3 /4?ℏ. The resulting number of black hole microstates is 2? . The entropy is
given by the natural logarithm of the number of the possible microstates times ?. This
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⊳ Curved space and horizons have non-vanishing entropy.
Motion Mountain – The Adventure of Physics
Both vacuum and matter are made of fluctuating strands. We note directly:
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10 g eneral rel ativity d ed uced from stra nds
F I G U R E 76 The entropy of black holes results from the number of possible crossing states above a
Planck area.
?=?
Ref. 189
??3
ln 2 .
4?ℏ
(177)
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
This result is the well-known first approximation of black hole entropy: one bit per corrected Planck area. In the strand model, the proportionality of entropy and area is thus
a direct consequence of the extension of the strands. This proportionality is also well
known from studies of quantum gravity and of strings. In those approaches however,
the relation between the area proportionality and extension is less obvious.
For Schwarzschild black holes, the entropy value of expression (177) is not correct.
In the strand model, this incorrect value is explained as a consequence of neglecting
the effects of the strand tails. Indeed, additional contributions to the entropy appear at
a finite distance from the horizon, due to the crossing of the tails on their way to the
border of space, as shown in Figure 76. The actual entropy will thus be larger than the
first approximation, but still be proportional to the area ?.
The correct proportionality factor between the area and the entropy of a black hole
results when the strand tails are taken into account. (The correction factor is called the
Barbero–Immirzi parameter in the research literature on quantum gravity.) The calculation is simplest for Schwarzschild black holes. By construction, a black hole with macroscopic radius ?, being a tight weave, has ?/?Pl tails. For each given Planck area, there
are, apart from the basic, or lowest crossing, additional crossings ‘above it’, along the
radial direction, as shown in Figure 76. These additional crossings are due to the tails
from neighbouring and distant Planck areas.
Taking into effect all strand tails allows us to calculate the average number of crossings
above a given Planck area. The main point is to perform this calculation for all those
tails that start in a circular strip of Planck width centred around the Planck area under
consideration. We then add the probabilities for all possible circular strips. One such
circular strip is drawn in Figure 76.
The definition of horizons as tight weaves implies that a horizon with ?2 Planck areas
Motion Mountain – The Adventure of Physics
approximation gives an entropy of a horizon of
g enera l rel ativit y d educed from stra nds
Challenge 182 e
289
is made of ? strands. This means that for each circular strip of radius ??Pl, there is only
one strand that starts there and reaches spatial infinity as a tail. For this tail, the average
probability ? that it crosses above the central Planck area under consideration is
?=
1
.
?!
(178)
∞
Summing over all strips, i.e., over all values ?, we get a total of ∑?=0 1/?! = e = 2.71828...
microstates on and above the central Planck area under consideration. Thus the number
e replaces the number 2 of the first approximation:
the number of horizon microstates
2
2
of a Schwarzschild black hole is not 2? , but e? . As a consequence, the entropy of a
macroscopic Schwarzschild horizon becomes
?=?
(179)
This is the Bekenstein–Hawking expression for the entropy of Schwarzschild black holes.
The strand model thus reproduces this well-known result. With this explanation of the
difference between 2 and e = 2.71828..., the strand model confirms an old idea:
The above calculation, however, counts some states more than once. Topologically
identical spherical horizons can differ in the direction of their north pole and in their
state of rotation around the north–south axis. If a spherical horizon is made of ? strands,
it has ?2 possible physical orientations for the north pole and ? possible angular2 orientations around the north–south axis. The actual number of microstates is thus ?? /?3 .
Using the relation between ?2 and the surface area ?, namely ? = ?2 4?ℏ/?3 , we get
the final result
??3 3? ? ?3
?=?
−
ln
.
(180)
4?ℏ 2
4?ℏ
Ref. 190
The strand model thus makes a specific prediction for the logarithmic correction of
the entropy of a Schwarzschild black hole. This final prediction of the strand model
agrees with many (but not all) calculations using superstrings or other quantum gravity
approaches.
In summary, the entropy value (179), respectively (180), of black holes is due to the
extension of the fundamental entities in the strand model and to the three dimensions of
space. If either of these properties were not fulfilled, the entropy of black holes would
not result. This is not a surprise; also our deduction of quantum theory was based on the
same two properties. In short: like every quantum effect, also the entropy of black holes
is a result of extension and three-dimensionality. Only a three-dimensional description
of nature agrees with observation.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
⊳ The entropy of a black hole is located at and near the horizon.
Motion Mountain – The Adventure of Physics
??3
.
4?ℏ
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10 g eneral rel ativity d ed uced from stra nds
Temperature, radiation and evap oration of bl ack holes
The strands that make up a horizon fluctuate in shape. Since every horizon contains
energy, the shape fluctuations imply energy fluctuations. In other words, horizons are
predicted to have a temperature. The value of the temperature can be deduced from
the strand model by noting that the characteristic size of the fluctuations for a spherical
horizon is the radius ? of the horizon. Therefore we have
?? =
ℏ?
.
2π?
(181)
Using the definition of surface gravity as ? = ?2 /?, we get
?=
(182)
The strand model predicts that horizons have a temperature proportional to their surface
gravity. This result has been known since 1973.
All hot bodies radiate. The strand model thus predicts that Schwarzschild black holes
radiate thermal radiation of the horizon temperature, with power and wavelength
? = 2πℏ?2/?2 , ? ≈ ? .
(183)
Bl ack hole limits
In many ways, black holes are extreme physical systems. Not only are black holes the
limit systems of general relativity; black holes also realize various other limits. As such,
black holes resemble light, which realizes the speed limit. We now explore some of these
limits.
For a general physical system, not necessarily bound by a horizon, the definitions of
energy and entropy with strands allow some interesting conclusions. The entropy of a
system is the result of the number of crossing possibilities. The energy of a system is
the number of crossing changes per unit time. A large entropy is thus only possible if a
system shows many crossing changes per time. Since the typical system time is given by
the circumference of the system, the entropy of a physical system is therefore limited:
? ⩽ ?? 2π?/ℏ? .
(184)
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
This confirms a well-known consequence of the temperature of black holes.
Like all thermal systems, horizons follow thermodynamics. In the strand model, black
hole radiation and evaporation occur by reduction of the number of strands that make
up the horizon. The strand model thus predicts that black holes evaporate completely,
until only elementary particles are left over. In particular, the strand model implies that
in black hole radiation, there is no information loss.
In short, strands reproduce all aspects of black hole evaporation. The strand model
also shows that there is no information loss in this process.
Motion Mountain – The Adventure of Physics
Ref. 57, Ref. 58
ℏ?
.
2π??
g enera l rel ativit y d educed from stra nds
Ref. 191
291
This relation is known as Bekenstein’s entropy bound; the precise definitions of the quantities in the bound need some care, as Don Page explains. The bound thus also follows
from the strand model. Strands imply that the equality is realized only for black holes.
In the strand model, horizons are tight, one-sided weaves. For example, this implies
that any tangle that encounters a horizon is essentially flat. Because of tangle flatness and
the extension of the tails, at most one Planck mass can cross a horizon during a Planck
time. This yields the mass rate limit
d?/d? ⩽ ?3 /4?
(185)
that is valid in general relativity and in nature.
Black holes can rotate. The strand model states that there is a highest angular frequency possible; it appears when the equator of the black hole rotates with the speed of
light. As a result, the angular momentum ? of a black hole is limited by
Ref. 192
This limit is well known from general relativity.
The electric charge of a black hole is also limited. The force limit in nature implies
that the electrical forces between two charged black holes must be lower than their gravitational interaction. This means that
?2
??2
⩽
,
4π?0 ?2
?2
(187)
?2 ⩽ 4π?0 ??2 .
(188)
This is the well-known charge limit for (static) black holes given by the Reissner–
Nordström metric. The maximum charge of a black hole is proportional to its radius.
It follows directly from the maximum force principle.
To explain the charge limit, we deduce that the extremal charge surface density ?/?
of a black hole is proportional to 1/?. The higher the horizon curvature, the more charge
per Planck area is possible. In the strand model, a horizon is a tight weave of strands. We
are thus led to conjecture that at Planck scale, electric charge is related to and limited by
strand curvature. We will explore this connection in more detail below.
The strand model limits energy density to the Planck energy per Planck volume, or to
the value ?7 /(16?2 ℏ). This limit implies a lower size limit for black holes, particles and
any localized system. Therefore, the strand model does not allow singularities, be they
dressed or naked. And indeed, no singularity has ever been observed.
In summary, the strand model reproduces the known limit properties of horizons.
And all these results are independent of the precise fluctuation details of the strands.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
or
Page 378
(186)
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? < 2??2 /? .
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10 g eneral rel ativity d ed uced from stra nds
Curvature around bl ack holes
The tails of a black hole extend up to the border of space; the density of tails is highest at
the horizon. A black hole is therefore surrounded by partial links at any finite distance
from the horizon. In other words, the space around a black hole is curved. The value of
the space-time curvature increases as one approaches the horizon, because of the way
in which the partial links hinder each other in their motion. The nearer they are to the
horizon, the more they hinder each other. The curvature that appears is proportional to
the density of partial links and to their average strand curvature.
At the horizon, the curvature radius is the horizon radius ?. By construction, the
number of tails departing from a non-rotating black hole is proportional to ?. The spatial
curvature is given by the average crossing density gradient. Hence at a radial distance ?
from a static black hole, the spatial curvature ? is
?
.
?3
(189)
So at the horizon itself, the curvature ? is (of the order of) the inverse square of the horizon radius; further away, it decreases rapidly, with the third power of the distance. This
result is a well-known property of the Schwarzschild solution and is due to the extension
of the strands. The rapid decay with radius is the reason why in everyday situations there
is no noticeable curvature of space-time. In short, strands allow us to deduce the correct
curvature of space-time around black holes and spherical masses.
The strand model also explains and visualizes the importance of spherical horizons in
nature. First of all, strands illustrate the non-existence of (uncharged) one-dimensional
or toroidal horizons in 3 + 1 space-time dimensions. Such configurations are unstable,
in particular against transverse shear and rearrangement of the strands.
The strand model also implies that non-rotating, closed horizons are spherical. Obviously, spheres are the bodies with the smallest surface for a given volume. The minimum
horizon surface appears because the strands, through their fluctuations, effectively ‘pull’
on each Planck area of the horizon. As a result, all non-rotating macroscopic horizons
will evolve to the spherical situation in a few Planck times. (Deviations from the spherical
shape will mainly occur near Planck scales.) With the definition of gravity waves given
below, it also becomes clear that strongly deformed, macroscopic and non-spherical horizons are unstable against emission of gravity waves or of other particles. In short,
⊳ All non-rotating horizons of non-spherical shape are unstable.
The strand model thus confirms that spherical horizons are favoured and that the most
compact bodies with a given mass. The reasoning can be extended to rotating horizons,
yielding the well-known shapes.
In summary, strands reproduce all known qualitative and quantitative properties
of horizons and of black holes, and thus of general systems with strong gravitational
fields. All predictions from strands agree with observations and with other approaches
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
The shape of non-rotating bl ack holes
Motion Mountain – The Adventure of Physics
?∼
g enera l rel ativit y d educed from stra nds
293
to quantum gravity. These hints already suggest that strands imply the field equations.
The field equations of general rel ativit y
Page 280
Ref. 22
The field equations can be deduced from the fundamental principle in two different, but
related ways. Essentially, both derivations repeat the reasoning for universal gravitation
given above, but for the relativistic case. The first deduction of the field equations is based
on an old argument on the thermodynamics of space-time. Strands show that horizons
have three thermodynamic properties:
— an area–entropy relation of ? = ? ??3 /4?ℏ,
— a curvature–temperature relation of ? = ? ℏ/2π??,
— a relation between heat and entropy of ?? = ???.
Using these three properties, and using the relation
(190)
that is valid only in case of horizons, we get the first principle of horizon mechanics
?? =
(191)
From this relation, using the Raychaudhuri equation, we obtain the field equations of
general relativity. This deduction was given above.*
In other words, the field equations result from the thermodynamics of strands. It is
worth noting that the result is independent of the details of the fluctuations or of the
microscopic model of space, as long as the three thermodynamic properties just given
are valid. In fact, these properties must be fulfilled by any model of space-time; and
indeed, several competing models of space claim to fulfil them.
* Here is the argument in a few lines. The first principle of horizon mechanics can be rewritten, using the
energy–momentum tensor ???, as
?2
∫ ??? ?? dΣ? =
? ??
8π?
where dΣ? is the general surface element and ? is the Killing vector that generates the horizon. The
Raychaudhuri equation allows us to rewrite the right-hand side as
∫ ????? dΣ? =
?4
∫ ??? ?? dΣ?
8π?
where ??? is the Ricci tensor describing space-time curvature. This equality implies that
??? =
Vol. II, page 98
?4
(? − (?/2 + Λ)???)
8π? ??
where Λ is an undetermined constant of integration. These are Einstein’s field equations of general relativity.
The field equations are valid everywhere and for all times, because a suitable coordinate transformation can
put a horizon at any point and at any time. To achieve this, just change to a suitable accelerating frame, as
explained in the volume on relativity.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 32
?2
? ?? .
8π?
Motion Mountain – The Adventure of Physics
?? = ?? ,
294
Page 186
Ref. 19
Page 32
10 g eneral rel ativity d ed uced from stra nds
We can use the relation between fluctuations and strands to settle an issue mentioned
above, in the section on quantum theory. Strand fluctuations must obey the thermodynamic properties to allow us to define space-time. If they obey these properties, then
space-time exists and curves according to general relativity.
A second derivation of the field equations of general relativity follows the spirit of the
strand model most closely. It is even shorter. Strands imply that all physical quantities
are limited by the corresponding Planck limit. These limits are due to the limit to the
fundamental principle, in other words, they are due to the packing limit of strands. In
particular, the fundamental principle limits force by ? ⩽ ?4 /4? and power by ? ⩽ ?5 /4?.
We have already shown above that this limit implies the field equation.
In other words,
⊳ Given that black holes and thus horizons are thermodynamic systems, so is
curved space.
⊳ Since black holes have thermodynamic aspects, so has gravity.
And since black holes are built from microscopic degrees of freedom, so is curved space.
Or, in simple words:
⊳ Space is made of many small entities.
⊳ Space is made of strands, because strands are the simplest entities that yield
black hole entropy.
Strands are the simplest way to incorporate quantum effects into gravitation. If we take
into consideration that strands are the only way known so far to incorporate gauge interactions, we can even conclude that strands are the only way known so far to incorporate
all quantum effects into gravitation.
In summary, the strand model asserts that the field equations appear as consequences
of fluctuations of impenetrable, featureless strands. In particular, the strand model implies and confirms that a horizon and a particle gas at Planck energy do not differ. However, the value of the cosmological constant is not predicted from strand thermodynamics.
E quations from no equation
Page 147
The strand model asserts that the field equations of general relativity are not the result of another, more basic evolution equation, but result directly from the fundamental
principle. To say it bluntly, the field equations are deduced from a drawing – the fundamental principle shown in Figure 10. This strong, almost unbelievable statement is due
to a specific property of the field equations and to two properties of the strand model.
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And finally we can state:
Motion Mountain – The Adventure of Physics
The reason: both can be transformed into each other. Therefore:
g enera l rel ativit y d educed from stra nds
295
The Hilbert action of general rel ativit y
Page 208
We have just shown that the strand model implies the field equations of general relativity.
We have also shown above that, in the strand model, the least action principle is a natural
property of all motion of strands. Combining these two results, we find that a natural
way to describe the motion of space-time is the (extended) Hilbert action given by
?=
Page 283
?4
∫(? − 2Λ) d? ,
16π?
(192)
where ? is the Ricci scalar, d? = √det? d4 ? is the invariant 4-volume element of the
metric ?, and Λ is the cosmological constant, whose value we have not determined yet.
As is well known, the description of evolution with the help of an action does not add
anything to the field equations; both descriptions are equivalent.
For a curved three-dimensional space, the Ricci scalar ? is the average amount, at a
given point in space, by which the curvature deviates from the zero value of flat space.
In the strand model, this leads to a simple statement, already implied by Figure 74:
⊳ The Ricci scalar ? is the ratio of additional or missing crossings per spatial
volume, compared to flat space.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
⊳ Any principle that allows deducing the field equations cannot itself be an
evolution equation.
Motion Mountain – The Adventure of Physics
First of all, the field equations are, above all, consequences of the thermodynamics of
space-time. In the strand model, the thermodynamic properties are deduced as a consequence of the strand fluctuations. This deduction does not require underlying evolution equations; the field equations follow from the statistical behaviour of strands.
The second, essential property of the strand model is its independence from the underlying motion of the strands. In the strand model we obtain the evolution equations
of the vacuum – the field equations of general relativity – without deducing them from
another equation. We do not need an evolution equation for the strand shape; the deduction of the field equations works for any underlying behaviour of strand shapes, as
long as the thermodynamic properties of the strand fluctuations are reproduced.
The third and last essential property that allows us to deduce the field equations directly from a graph, and not from another equation, is the relation between the graph and
natural physical units. The relation with natural units, in particular with the quantum
of action ℏ and the Boltzmann constant ?, is fundamental for the success of the strand
model.
In summary, the fundamental principle of the strand model contains all the essential
properties necessary for deducing the field equations of general relativity. In fact, the discussion so far makes another important point: unique, underlying, more basic evolution
equations for the tangle shape cannot exist. There are two reasons. First, an underlying
equation would itself require a deduction, thus would not be a satisfying solution to unification. Secondly, and more importantly, evolution equations are differential equations;
they assume well-behaved, smooth space-time. At Planck scales, this is impossible.
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10 g eneral rel ativity d ed uced from stra nds
The graviton :
wavelength
F I G U R E 77 The graviton in the strand model.
As usual, the averaging is performed over all spatial orientations. A similar statement
can be made for the cosmological constant Λ. In short, we can say: the Hilbert action
follows directly from the fundamental principle of the strand model.
⊳ Space-time foam is made of fluctuating strands.
Ref. 193
At everyday scales, the foam is not noticed, because background space and physical space
are indistinguishable. At Planck scales, space-time is not fundamentally different from
everyday space-time. No unusual topology, no additional dimensions, and no new or
unusual properties appear at Planck scales. Above all, the strand model predicts that
there are no observable effects of space-time foam; for example, ‘space-time noise’ or
‘particle diffusion’ do not exist. The strand model of space-time foam is both simple and
unspectacular.
Gravitons, gravitational waves and their detection
In the strand model, gravitons can be seen as a special kind of partial links. An example
is shown in Figure 77. As a twisted pair of parallel strands, the graviton returns to itself
after rotation by π; it thus behaves like a spin-2 boson, as required.
Can single gravitons be observed? The strand model implies that the absorption of
a single graviton by an elementary particle changes its spin or position. However, such
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Quantum physics implies that at scales near the Planck length and the Planck time,
space-time fluctuates heavily. John Wheeler called the situation space-time foam; the
term quantum foam is also used. In a sense, quantum gravity can be defined, if at all, as
the description of space-time foam. This reduced view arises because no separate theory
of quantum gravity is possible in nature.
Historically, there have been many speculations on the details of space-time foam.
Apart from its fluctuations, researchers speculated about the appearance of topology
changes – such as microscopic wormholes – about the appearance of additional dimensions of space – between six and twenty-two – or about the appearance of other unusual
properties – such as microscopic regions of negative energy, networks or loop structures.
The strand model makes a simple prediction that contradicts most previous speculations:
Motion Mountain – The Adventure of Physics
Space-time foam
g enera l rel ativit y d educed from stra nds
297
A candidate
tangle for an
extended
defect :
F I G U R E 78 A speculative, highly schematic model for a cosmic string, a one-dimensional defect in
space-time
⊳ Single gravitons cannot be detected.
Ref. 194
Open challenge: Improve the argument for the graviton tangle
Challenge 184 ny
The argument that leads to the graviton tangle is too much hand-waving. Can you make
the argument more compelling? Could the four tails form a cross and thus span a plane?
O ther defects in vacuum
The strand model provides a quantum description of gravitation. The strand model does
so by explaining physical space as the average of the crossing switches induced by strand
fluctuations among untangled strands. Matter, radiation and horizons are defects in the
‘sea’ of untangled strands.
So far, we have been concerned with particles, i.e., localized, zero-dimensional defects,
and with horizons, i.e., two-dimensional defects. Now, modelling of the vacuum as a set
of untangled strands also suggests the possible existence of one-dimensional – equivalent
to dislocations and disclinations in solids – of additional two-dimensional defects, or
of three-dimensional defects. Such defects could model cosmic strings, domain walls,
wormholes, toroidal black holes, time-like loops and regions of negative energy.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Challenge 183 e
The situation changes for gravitational waves. Such waves are coherent superpositions
of large numbers of gravitons and are observable classically. In such a case, the argument against the detection of single gravitons does not apply. In short, the strand model
predicts that gravitational waves can be observed. (This prediction, made by many since
1915 and repeated in this text on the basis of the strand model in 2008, came true in
February 2016. The observations also produced the extremely low mass limit of at most
1.2 × 10−22 eV/c2 for any possible mass of the photon.)
Motion Mountain – The Adventure of Physics
a change cannot be distinguished from a quantum fluctuation, because the graviton is
predicted to be massless. Furthermore, the strand model predicts that gravitons do not
interact with photons, because they have no electric charge. In summary, the strand
model predicts:
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10 g eneral rel ativity d ed uced from stra nds
Vol. IV, page 143
Page 200
What is the gravitational field of a quantum system in a macroscopic superposition? The
issue has been raised by many scholars as an important step towards the understanding
of how to combine gravitation and quantum theory.
The strand model deflates the importance of the issue. The model shows – or predicts,
if one prefers – that the gravitational field of a superposition is the temporal and spatial
average of the evolving quantum system, possibly under inclusion of decoherence.
What is the gravitational field of a single quantum particle in a double-slit experiment? As Figure 38 shows, the gravitational field almost always appears in both slits,
and only very rarely in just one slit.
In summary, in the strand model, the combination of gravitation and quantum theory is much simpler than was expected by most researchers. For many decades it was
suggested that the combination was an almost unattainable goal. In fact, in the strand
model we can almost say that the two descriptions combine naturally.
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The gravit y of superp ositions
Motion Mountain – The Adventure of Physics
An example of such a possible new defect is illustrated in Figure 78. The illustration can be seen as the image of a one-dimensional defect or as the cross section of a
two-dimensional defect. Are such defects stable against fluctuations? The strand model
suggests that they are not. These defects are expected to decay into a mixture of gravitons, black holes, matter and radiation particles. However, this issue is still a topic of
research, and will not be covered here.
Exploring the stability of wormholes, time-like loops and toroidal black holes leads
to similar results. It seems that the strand model should not allow time-like loops of
macroscopic size, since any configuration that cannot be embedded locally into three
flat spatial dimensions is either a particle or a black hole. Alternatively, macroscopic
time-like loops would collapse or decay because of the fluctuations of the strands. In
the same way, wormholes or black holes with non-trivial topology should be unstable
against more usual strand structures, such as particles or black holes.
We also note the strand model does not allow volume defects (black holes being
surface-like defects). The most discussed types of volume defects are macroscopic regions of negative energy. Energy being action per unit time, and action being connected
to crossing changes, the model does not allow the construction of negative-energy regions. However, the strand model does allow the construction of regions with lower
energy than their environment, as in the Casimir effect, by placing restrictions on the
wavelengths of photons.
The strand model thus predicts the absence of additional defects and tangle types.
The final and general connection between tangle types and defects is shown (again) in
Table 11. The next chapter will give details of the tangles corresponding to each particle.
In summary, the strand model reproduces the results of modern quantum gravity and
predicts that the more spectacular defects conjectured in the past – linear defects such
as cosmic strings, surface defects such as wormholes, volume defects such as negativeenergy regions – do not appear in nature.
g enera l rel ativit y d educed from stra nds
299
TA B L E 11 Correspondences between physical systems and mathematical tangles.
Strands
Ta n g l e t y p e
Vacuum
Dark energy
Elementary vector boson
Quark
Lepton
Meson, baryon
many infinite unknotted strands
many fluctuating infinite strands
one infinite strand
two infinite strands
three infinite strands
three or more infinite strands
Higher-order propagating
fermion
Virtual particles
Composed systems
Graviton
Gravity wave
Horizon
Young universe
two or more infinite strands
unlinked
unlinked
a curve
rational tangle
braided tangle
composed of rational
tangles
general rational tangle
open or unlinked strands
many strands
two infinite twisted strands
many infinite twisted strands
many tightly woven infinite strands
closed strand(s)
trivial tangles
separable tangles
specific rational tangle
many graviton tangles
web-like rational tangle
knot (link)
Torsion, curiosities and challenges abou t quantum gravit y
On the one hand, the strand model denies the existence of any specific effects of torsion
on gravitation. On the other hand, the strand model of matter describes spin with the
belt trick. The belt trick is thus the strand phenomenon that is closest to the idea of
torsion. Therefore, exaggerating a bit in the other direction, it could also be argued that
in the strand model, torsion effects are quantum field theory effects.
∗∗
Ref. 196
Ref. 197
Ref. 156
Ref. 198
Ref. 157
Ref. 199
The strand model describes three-dimensional space as made of tangled strands. Several
similar models have been proposed in the past.
The model of space as a nematic world crystal stands out as the most similar. This
model was proposed by Hagen Kleinert in the 1980s. He took his inspiration from
the famous analogy by Ekkehart Kröner between the equations of solid-state elasticity
around line defects and the equations of general relativity.
Also in the 1980s, the mentioned posets have been proposed as the fundamental structure of space. Various models of quantum gravity from the 1990s, inspired by spin networks, spin foams and by similar systems, describe empty space as made of extended
constituents. These extended constituents tangle, or bifurcate, or are connected, or sometimes all of this at the same time. Depending on the model, the constituents are lines,
circles or ribbons. In some models their shapes fluctuate, in others they don’t.
Around the year 2000, another type of Planck-scale crystal model of the vacuum has
been proposed by David Finkelstein. In 2008, a specific model of space, a crystal-like
network of connected bifurcating lines, has been proposed by Gerard ’t Hooft.
All these models describe space as made of some kind of extended constituents in
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Ref. 195
Motion Mountain – The Adventure of Physics
Physical system
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10 g eneral rel ativity d ed uced from stra nds
a three-dimensional background. All these models derive general relativity from these
constituents by some averaging procedure. The lesson is clear: it is not difficult to derive
general relativity from a Planck-scale model of space. It is not difficult to unify gravity
and quantum theory. As Luca Bombelli said already in the early 1990s, the challenge for
a Planck-scale model of nature is not to derive gravity or general relativity; the challenge
is to derive the other interactions. So far, the strand model seems to be the only model
that has provided such a derivation.
∗∗
Challenge 185 e
The Planck force is the force value necessary to produce a change ℏ in a Planck time over
a Planck length. The Planck force thus appears almost exclusively at horizons.
∗∗
Already in the 1990s, Leonard Susskind speculated that black holes could be formed by
a single wound-up string. Strands differ from strings; they differ in the number of dimensions, in their intrinsic properties, in their symmetry properties, in the fields they
carry and in the ways they generate entropy. Nevertheless, the similarity with the strand
model of black holes is intriguing.
∗∗
Page 161
Ref. 201
∗∗
Challenge 186 e
The first version of the strand model assumed that space is not defined at the cosmic
horizon, and that therefore, strand impenetrability does not hold there. The same was
thought to occur at black hole horizons. The newest version of the strand model does
not seem to need this exception to impenetrability. Can you explain black hole entropy
without it?
∗∗
Page 35
The strand model also allows us to answer the question whether quantum particles are
black holes: no, they are not. Quantum particles are tangles, like black holes are, but
particles do not have horizons. As a side result, the mass of all particles is lower than a
Planck mass, or more precisely, lower than a Planck mass black hole.
Strands imply that gravity is weaker than the three gauge interactions. This consequence, like the low particle mass just mentioned, is due to the different origins of
gravity and gauge interactions. Gravity is due to the strand tails, whereas gauge interactions are due to the tangle cores. Thus gravity is the weakest interaction in everyday
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 154
In September 2010, two years after the strand model appeared, independent research
confirmed its description of physical space, as already mentioned above. In an extended article exploring the small scale structure of space from several different research
perspectives in general relativity, Steven Carlip comes to the conclusion that all these
perspectives suggest the common idea that ‘space at a fixed time is thus threaded by
rapidly fluctuating lines’.
In 2011, also independently, Marcelo Botta Cantcheff modelled space as a statistic ensemble of one-dimensional ‘strings’. He explained the main properties of space, including the thermodynamic properties of black holes.
Motion Mountain – The Adventure of Physics
Ref. 200
g enera l rel ativit y d educed from stra nds
Ref. 202
Page 8
301
life. The observation of the weakness of gravity at everyday and other energy scales is
sometimes called the weak gravity conjecture. It is naturally valid in the strand model.
The conjecture is also part of the Bronshtein cube shown in Figure 1.
∗∗
For an observer at spatial infinity, a black hole horizon is an averaged-out tight web of
strands. What does a falling observer experience? The question will still capture the
imagination in many years. Such an observer will also see strands; above all, a falling
observer will never hit any singularity. The details of the fall are so involved that they
are not discussed here, because the fall affects both the black hole appearance and the
observer.
∗∗
∗∗
Ref. 203
The strand model makes the point that entanglement and the vacuum – and thus
quantum gravity – have the same nature: both are due to crossing strands. This idea
has been explored independently by Mark van Raamsdonk.
∗∗
⊳ Quantum gravity effects cannot be distinguished from ordinary quantum
fluctuations.
Despite many attempts to disprove it, all experiments so far confirm the conjecture. Because both quantum gravity effects and quantum effects are due to tail fluctuations, the
strand model seems to imply the conjecture.
∗∗
Ref. 204
The strand model of black holes also confirms a result by Zurek and Thorne from the
1980s: the entropy of a black hole is the logarithm of the number of ways in which it
could have been made.
∗∗
Challenge 187 s
Argue that because of the strand model, no black hole can have a mass below the (corrected) Planck mass, about 11 μg, and thus that microscopic black holes do not exist. Can
you find a higher lower limit for the mass?
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
As we have seen, the strand model predicts no observable violation of Lorentz-invariance
– even though it predicts its violation at Planck scale. Strands predict the lack of dispersion, birefringence and opacity of the vacuum. Strands predict that the vacuum has three
dimensions whenever it is observed and that it is unique, without phase transitions. We
already mentioned the impossibility of detecting single gravitons.
All these negative predictions are examples of the ‘no avail’ conjecture:
Motion Mountain – The Adventure of Physics
Can black hole radiation be seen as the result of trying to tear vacuum apart? Yes and no.
The answer is no, because physical vacuum cannot be torn apart, due to the maximum
force principle. But the answer is also yes in a certain sense, because the maximum force
is the closest attempt to this idea that can be realized or imagined.
302
10 g eneral rel ativity d ed uced from stra nds
∗∗
Do atoms or the elementary fermions moving inside matter emit gravitational radiation,
and why? The question was already raised by Albert Einstein in 1916. The strand model
answers the issue in the same way as textbook physics. Elementary particles in atoms –
in the ground state – do not emit gravitational waves for the same reason that they do not
emit electromagnetic waves: for atoms in the ground state, there is no lower state into
which they could decay. Excited atomic states do not emit gravitational waves because of
the extremely low emission probability; it is due to the extremely low mass quadrupole
values.
∗∗
Ref. 205
∗∗
Page 58
It is often stated that general relativity does not allow the description of fermions if the
topology of space is kept fixed. This is wrong: the strand model shows that fermions can
be included in the case that space is seen as an average of extended fundamental entities.
∗∗
Predictions of the strand model abou t gravit y
As just presented, the strand model makes several verifiable predictions about general
relativity and quantum gravity.
— The maximum energy speed in nature is ?, at all energy scales, in all directions, at all
times, at all positions, for every physical observer. This agrees with observations.
— No deviations from special relativity appear for any measurable energy scale, as long
as gravity plays no role. No ‘double’ or ‘deformed special relativity’ holds in nature,
even though a maximum energy-momentum for elementary particles does exist in
nature. Whenever special relativity is not valid, general relativity, or quantum field
theory, or both together need to be used. This agrees with observations.
— There is a maximum power or luminosity ?5 /4?, a maximum force or momentum
flow ?4 /4?, and a maximum mass change rate ?3 /4? in nature. The limits hold for
all energy scales, in all directions, at all times, at all positions, for every physical observer. These predictions agree with observations, though only few experimental ob-
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 206
Following the fundamental principle of the strand model, ? is the fundamental constant
that describes gravitation. The strand model predicts that gravity is the same for all energy scales; in other words, the constant ? is not expected to change with energy. This
agrees with recent results from quantum gravity and distinguishes the behaviour of ?
from that of the coupling constants in the gauge interactions of particle physics.
Motion Mountain – The Adventure of Physics
Challenge 188 ny
In 2009 Mikhail Shaposhnikov and Christof Wetterich argued that if gravitation is
‘asymptotically safe’, there is no physics beyond the standard model and the Higgs mass
must be around 126 GeV – exactly the value that was found experimentally a few years
afterwards. A quantum field theory is called asymptotically safe if it has a fixed point
at extremely high energies. Does the strand model imply that gravity is – maybe only
effectively – asymptotically safe?
g enera l rel ativit y d educed from stra nds
Vol. V, page 146
All listed predictions are unspectacular; they are made also by other approaches that
contain general relativity as limiting cases. In particular, the strand model, like many
other approaches, predicts:
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
servations are close to these limit values.
— There is a minimum distance and a minimum time interval in nature. There is a maximum curvature and a maximum mass density in nature. There are no singularities in
nature. All this agrees with observations, including the newly discovered black hole
mergers.
— The usual black hole entropy expression given by Bekenstein and Hawking holds.
The value has never been measured, but is consistently found in all calculations performed so far. In fact, black hole entropy is related to the Fulling–Davies–Unruh effect, which itself is related to the Sokolov–Ternov effect. This latter effect has already
been observed in several accelerators, for the first time in 1971. However, it now seems
that this observation does not actually prove black hole entropy.
— There are no deviations from general relativity, as described by the Hilbert action, for
any measurable scale. The only deviations appear in situations with a few strands,
i.e., in situations where quantum theory is necessary. This agrees with observations,
including those of black hole mergers, but experimental data are far from sufficient;
undetected deviations could still exist.
— There is no modified Newtonian dynamics, or MOND, with evolution equations that
differ from general relativity. The rotation curves of stars in galaxies are due to dark
matter, to other conventional explanations, or both.
— There is no effect of torsion that modifies general relativity. This agrees with observations.
— There is no effect of higher derivatives of the metric on the motion of bodies. This
agrees with observations, but experimental data are far from sufficient.
— Observations are independent of the precise strand fluctuations. Mathematical consistency checks of this prediction are possible.
— No wormholes, no negative energy regions and no time-like loops exist. This agrees
with observations, but experimental data are far from covering every possible loophole.
— The Penrose conjecture and the hoop conjecture hold. Here, a mathematical consistency check is possible.
— There are no cosmic strings and no domain walls. This agrees with observations, but
experimental data are far from exhaustive.
— Gravitons have spin 2; they return to their original state after a rotation by π and are
bosons. This agrees with expectations.
— Gravitons cannot be detected, due to the indistinguishability with ordinary quantum
fluctuations of the detector. This agrees with data so far.
— Atoms emit neither gravitational waves nor gravitons.
— Gravitational waves exist and can be detected. This agrees with experiment; the final
confirmation occured in late 2015.
— The gravitational constant ? does not run with energy – as long as the strand diameter
can be neglected. In this domain, ? is not renormalized. This prediction agrees with
expectations and with data, though the available data is sparse.
Motion Mountain – The Adventure of Physics
Ref. 207
303
304
Page 307
Ref. 97
Page 8
10 g eneral rel ativity d ed uced from stra nds
⊳ With the exception of the cosmological constant and of particle mases (and
possibly the Sokolov–Ternov effect), no quantum gravity effects will be observed.
Gravity will not yield new measurable quantum effects. So far, this prediction agrees
with experiment – and with almost all proposed models of quantum foam in the research literature. In other words, we have found no unexpected experimental predictions
from the strand model in the domain of quantum gravity. This is the so-called ‘no avail’
conjecture; and it is not a surprise.
In fact, the Bronshtein cube of Figure 1 also implies:
⊳ There is no separate theory of quantum gravity that includes relativity but
does not include the other interactions.
cosmolo gy
The finiteness of the universe
In the strand model, cosmology is based on the following idea:
⊳ The universe is made of one fluctuating strand that criss-crosses from and to
the horizon. Fluctuations increase the complexity of the strand tangledness
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Cosmology is an active field of research, and new data are collected all the time. We start
with a short summary.
The sky is dark at night. This and other observations about the red shift show that
the universe is surrounded by a horizon and is of finite size and age. Precise measurements show that cosmic age is around 13 800 million years. The universe expands; the
expansion is described by the field equations of general relativity. The universe’s expansion accelerates; the acceleration is described by the cosmological constant Λ – also called
dark energy – that has a small positive value. The universe is observed to be flat, and, averaged over large scales, homogeneous and isotropic. At present, the observed average
matter density in the universe is about 18 times smaller than the energy density due to
the cosmological constant. In addition, there is a large amount of matter around galaxies
that does not radiate; the nature of this dark matter is unclear. Galaxy formation started from early density fluctuations; the typical size and amplitude of the fluctuations are
known. The topology of space is observed to be simple.
The strand model, like any unified description of nature, must reproduce and explain
these measurement results. Otherwise, the strand model is wrong.
Motion Mountain – The Adventure of Physics
There is no room for a theory of relativistic quantum gravity in nature.
In short, strands lead us to expect deviations from general relativity only in two domains: in cosmology (such as changes of the cosmological constant) and in particle physics. The rest of this chapter deals with cosmology. The subsequent chapters focus on
particle physics.
cosm olo gy
Universe’s
horizon
or
‘border
of space’
(pink)
305
Universal
tangle
(blue
lines)
Particle
tangle
(tangled
blue
lines)
Physical
space or
vacuum
(white)
Background
space
(grey)
Background
space
(grey)
Physical space (white) matches background space (grey) only inside the horizon. Physical space thus
only exists inside the cosmic horizon.
over time.
In other words, the strands of all particles are woven into the sky. The existence of finite
size and of finite age then follows automatically:
Ref. 208
Ref. 209
Page 307
Page 285
The strand model thus has a simple explanation for the finiteness of the universe and the
horizon that bounds it: The universe’s horizon is the weave that joins all strand tails. A
schematic illustration of the cosmic horizon is given in Figure 79.
The strand model predicts that the horizon of the universe is an event horizon, like
that of a black hole. Until 1998, this possibility seemed questionable; but in 1998, it was
discovered that the expansion of the universe is accelerating. This discovery implies that
the cosmic horizon is indeed an event horizon, as required by the strand model.
In fact, the strand model predicts that all horizons in nature are of the same type. This
also means that the universe is predicted to saturate Bekenstein’s entropy bound. More
precisely, the strand model predicts that the universe is a kind of inverted back hole. Like
for any situation that involves a horizon, the strand model thus does not allow us to make
statements about properties ‘before’ the big bang or ‘outside’ the horizon. As explained
above, there is nothing behind a horizon.
In particular, the strand model implies that the matter that appears at the cosmic horizon during the evolution of the universe appears through Bekenstein–Hawking radiation. This contrasts with the ‘classical’ explanation form general relativity that new matter appears simply because it existed behind the horizon beforehand and then crosses the
horizon into the ‘visible part’ of the universe.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
⊳ The universe’s horizon appears at the age or distance at which the strand
crossings cannot be embedded any more into a common three-dimensional
background space. The horizon expands over time.
Motion Mountain – The Adventure of Physics
F I G U R E 79 In the strand model, the universe is limited by a horizon, as schematically illustrated here.
306
10 g eneral rel ativity d ed uced from stra nds
‘horizon’
‘time’
F I G U R E 80 An extremely simplified view of how the universe evolved near the big bang. In this
evolution, physical time, space and the surrounding horizon are in the process of getting defined.
“
Or cette liaison ou cet accommodement de
toutes les choses créées à chacune, et de
chacune à toutes les autres, fait que chaque
substance simple a des rapports qui expriment
toutes les autres, et qu’elle est par conséquent
un miroir vivant perpétuel de l’univers.*
Gottfried Wilhelm Leibniz, Monadologie, 56.
The big bang – withou t infl ation
Ref. 210
Page 291
”
Any expanding, homogeneous and isotropic matter distribution had earlier stages of
smaller size and higher density. Also the universe has been hotter and denser in the past.
But the strand model also states that singularities do not appear in nature, because there
is a highest possible energy density. As a result, the big bang might be imagined as illustrated in Figure 80. Obviously, physical space and time are not well defined near that
* ‘Now this connexion or adaptation of all created things to each and of each to all, means that each simple
substance has relations which express all the others, and, consequently, that it is a perpetual living mirror
of the universe.’
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Challenge 189 e
Motion Mountain – The Adventure of Physics
Page 101
We note that modelling the universe as a single strand implies that it contains tangles.
In other words, the strand model makes the prediction that the universe cannot be empty,
but that it must contain particles. Strand cosmology also confirms that the question of
initial conditions for the universe does not really make sense: particles appear at the
horizon.
We also note that describing the universe as made of a single strand is a natural,
but somewhat unusual way to incorporate what particle physicists and cosmologists like
to call holography. Holography is the idea that all observables of a physical system are
defined on a boundary enclosing the system. In other words, if we would know, at Planck
scale, everything that happens on the walls of a room, we could know everything that is
and goes on inside the room. Instead of holography, we could also call it the NSA dream.
Holography is a consequence of the extension of the fundamental constituents of nature
and is a natural consequence of the strand model. As a consequence, strand cosmology
naturally reproduces holographic cosmology – though not fully, as is easy to check.
cosm olo gy
307
situation, so that the figure has to be taken with a grain of salt. Nevertheless, it shows how
the evolution of the universe can be seen as resulting from the increase in tangledness of
the strand that makes up nature.
The strand model leads to the conjecture that the evolution of the universal strand just
after the big bang automatically yields both a homogeneous and isotropic matter distribution and a flat space. Also the scale invariance of early density fluctuations seems natural in the strand model. In short, the strand model looks like a promising alternative
to inflation: the hypothesis of inflation becomes unnecessary in the strand model, because strand cosmology directly makes the predictions that seem so puzzling in classical
cosmology. This issue is still subject of research.
The cosmological constant
For a spherical system, this yields
?⩽?
Ref. 212
??3
.
4?ℏ
(195)
The application of this inequality to the universe is called the Fischler–Susskind holographic conjecture. Using the energy–entropy relation ? = ?? valid for any holographic
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 211
In particular, the strand model predicts a small positive cosmological constant, i.e., a
constant that leads to a small repulsion of masses.
The relation between the cosmological constant and the radius of the universe can
be found also with another, more precise argument, based on holography, and given by
Balázs and Szapudi. Bekenstein’s holographic entropy bound states that for all systems
of size ? and energy ? we have
2π?
? ⩽ ??
.
(194)
ℏ?
Motion Mountain – The Adventure of Physics
Page 210
The cosmological constant is due to strands. When three mutually orthogonal strands
come together at a point, they cannot be completely straight; they are slightly bent. Equivalently, because the strands of the vacuum touch each other, due to the statistics of the
fluctuations, there is a slight effective repulsion between them. This is the strand model
for the cosmological constant.
In short, in the strand model, vacuum energy, or dark energy, is due to the cosmological constant, which itself is due to strand fluctuations. As we saw above, the strand
model predicts that the cosmological constant Λ for infinitely extended flat space vanishes, because the vacuum energy density vanishes in that case. But the strand model
also predicts that for finite extension, the cosmological constant does not vanish. Indeed, in the strand model, a finite size limits the fluctuations of the strands. Fluctuations
with sizes larger than the size of space are frozen out; this leads to an effective repulsion of strands that in turn leads to a cosmological constant given by (the square of) the
extension of space:
1
Λ= 2 .
(193)
?max
308
10 g eneral rel ativity d ed uced from stra nds
system, and introducing the energy density ?E , we get the limit given by
?E ⩽
Page 290
(196)
Using the formula for temperature ? = ℏ?/2π?? for a horizon – deduced above from the
strand model – we get
1 3?4
1 3?4
?E ⩽
=
.
(197)
? 2? 4π?2 2?
The strand model predicts that the universe saturates the entropy bound. In other words,
assuming that ? is ? times the age of the universe ?0 , the strand model predicts that the
total energy density of the universe is equal to the so-called critical energy density.
The equality of the measured total energy density and the critical density is well
known. These measurements show that the present total energy density of the universe
is about
?E vac ≈ 8.5 ⋅ 10−10 J/m3 or ?m vac = 0.94(9) ⋅ 10−26 kg/m3 .
(198)
In other words, the strand model, like the holographic argument, predicts that the cosmological constant is limited by
3
Λ⩽ 2 2 .
(199)
? ?0
Ref. 214
Ref. 215
Challenge 190 s
The result confirms the result of expression (193). Modern measurements yield 74 % of
the maximum possible value.
The argument for the value of the cosmological constant can be made for any age of
the universe. Therefore,
⊳ The strand model predicts that the cosmological constant Λ(?) decreases
with increasing radius of the universe.
In particular, there is no need for a scalar field that makes the cosmological constant
decrease; the decrease is a natural result of the strand model. The strand model states
that the cosmological constant appears in the field equations as a quantum effect due to
the finite size of the universe. The strand model thus implies that there is no separate
equation of motion for the cosmological constant, but that the constant appears as a
large-scale average of quantum effects, as long as the size of the universe is limited.
In summary, the strand model predicts that not only the field equations of general relativity, but also the amount of dark energy, the expansion of the universe and its acceleration result from strand fluctuations. The cosmological constant changes roughly with
the inverse square of time. In particular, the strand model implies that the effect proposed by Wiltshire – that the cosmological constant is an artefact of the inhomogeneity
of matter distribution – is not fundamental, but may at most influence the value somewhat. (Could the difference between the maximum possible and the measured value of
the cosmological constant be due to this effect?)
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 213
Motion Mountain – The Adventure of Physics
Ref. 213
? 3??3
.
? 4ℏ?
cosm olo gy
309
The value of the mat ter densit y
The strand model predicts that horizons emit particles. As a consequence, the strand
model predicts an upper limit for the number ?b of baryons that could have been emitted
by the cosmic horizon during its expansion. For a horizon shining throughout the age
of the universe ?0 while emitting the maximum power ?5 /4?, we get
?b0 ⩽
Ref. 213
(200)
Equality would hold only if the contributions of photons, electrons, neutrinos and dark
matter could be neglected. In short, using the age ?0 = 13.8 Ga, the strand model predicts
that at most 2.6 ⋅ 1079 baryons exist in the universe at present. Modern measurements
indeed give values around this limit.
In other terms, the strand model states that the sum of all particle energies in the
universe is at most ?0 ?5 /4?, or 50 % of the critical density; this includes observable matter
as well as dark matter. The experimental value for the total matter density is about 26 %
of the critical density. In observations, 4 % of the matter density is observed, and 22 % is
dark. We will discuss the nature of dark matter later on.
The strand model also makes a statement on the change of matter density with time.
As explained, the number of baryons is predicted to increase with time ?, due to their
appearance at the horizon. Since the radius will also increase (roughly) with time, we get
This unexpected prediction contrasts with the usually assumed 1/?3 dependence in a
matter-dominated universe. The prediction has yet to be tested with observations. We
note that the strands imply that the ratio between matter density and vacuum energy
density is related to the details of the radius increase during the history of the universe.
Open challenge: What are the effects of dark mat ter?
Challenge 191 ny
Challenge 192 ny
Challenge 193 ny
In the arguments above, is there a factor of order 2 missing that induces incorrect conclusions about dark matter density? Might the prediction of dark matter increase, decrease
or even disappear after correction of this missing numerical factor?
Conventionally, it is argued that cold dark matter exists for two reasons: First, it is
necessary to grow the density fluctuations of the cosmic microwave background rapidly
enough to achieve the present-day high values. Secondly, it is needed to yield the observed amplitudes for the acoustic peaks in the cosmic background oscillations. Can the
strand model change these arguments?
Later on, it will be argued that in the strand model, dark matter is a mixture of conventional matter and black holes. How does this dark matter prediction explain the galaxy
rotation curves? This leads to a really speculative issue: Could tangle effects at the scale
of a full galaxy be related to dark matter?
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
⊳ The strand model predicts that matter density decreases roughly as 1/?2 .
Motion Mountain – The Adventure of Physics
Ref. 213
Page 353
?0 ?5 /4?
= 2.6 ⋅ 1079 .
?b ?2
310
10 g eneral rel ativity d ed uced from stra nds
The top ology of the universe
In the strand model, physical space-time, whenever it is defined, cannot be multiply connected. Also all quantum gravity approaches make this prediction, and the strand model
confirms it: because physical space-time is a result of averaging strand crossing switches,
non-trivial topologies (except black holes) do not occur as solutions. For example, the
strand model predicts that wormholes do not exist. In regions where space-time is undefined – at and beyond horizons – it does not make sense to speak of space-time topology. In these regions, the fluctuations of the universal strand determine observations.
In short, the strand model predicts that all searches for non-trivial macroscopic (and microscopic) topologies of the universe, at both high and low energies, will yield negative
results. So far, this prediction agrees with all observations.
Predictions of the strand model abou t cosmology
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
— The universe is not empty. (Agrees with observation.)
— Its integrated luminosity saturates the power limit ?5 /4?. (Agrees with observation.)
— The universe’s energy density saturates the entropy bound. (Agrees with observation.)
— The are no singularities in nature. (Agrees with observation.)
— Dark energy results from vacuum strand fluctuations. (Agrees with observation.)
— Dark energy, or vacuum energy, is completely described by a cosmological constant Λ(?)
that is positive and changes with the radius ?(?) of the universe as 1/?(?)2. (This prediction differs from the usual cosmological models, which assume that Λ is constant
or changes with time in other ways. The strand prediction might be checked in the
near future by testing whether the minimum acceleration around galaxies changes
with distance – if this minimum is related to Λ.)
— The number of baryons in nature is limited by the maximum luminosity times the
age of the universe, thus by 2.6 ⋅ 1079 baryons. (Agrees with observation.)
— The matter density of the universe decreases with age, roughly as ?? ∼ 1/?2 . (Checks
are under way. This prediction differs from the usual cosmological models.)
— There is nothing behind the cosmic horizon. Matter, energy and space appear at the
horizon. (Agrees with observations and requirements of logic.)
— Early density fluctuations are scale-invariant. (Agrees with observation.)
— The universe is flat and homogeneous. (Agrees with observation.)
— Apart from the cosmological constant Λ(?), all other fundamental constants of nature
are constant over time and space. (Agrees with observation, despite claims of the
contrary.)
— Inflation is unnecessary.
— The universe’s topology is trivial. There are no wormholes, no time-like loops, no
cosmic strings, no toroidal black holes, no domain walls and no regions of negative
energy. (Agrees with observation.)
— The above statements are independent of the precise fluctuation details. (Can be
tested with mathematical investigations.)
Motion Mountain – The Adventure of Physics
In the domain of cosmology, the strand model makes the following testable predictions.
sum m ary on m illennium issues a b ou t rel ativity a nd cosm olo gy
311
All these predictions can and will be tested in the coming years, either by observation or
by computer calculations.
summary on millennium issues abou t rel ativit y and
cosmolo gy
Page 8
Page 147
Page 162
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 277
Motion Mountain – The Adventure of Physics
Page 353
We have deduced special relativity, general relativity and cosmology from the strand
model. The fundamental principle of the strand model implies the invariant Planck units,
the Lagrangian and action of general relativity, the finiteness of the universe and, above
all, it explains in simple terms the entropy of black holes.
Space-time foam is replaced by the strand model of the vacuum: empty space is the
time-average of untangled strands. More precisely, space is the thermodynamic average
of crossing switches that are due to shape fluctuations of untangled strands.
The strand model – and in particular, the strand model of the vacuum – explains the
number of space-time dimensions, the vacuum energy density, the matter density and
the finiteness of the universe. The cosmological constant is a consequence of the finite
size of the universe. The issue of the initial conditions of the universe has been defused.
The macroscopic and microscopic topology of the universe is simple. And dark matter
is predicted to be, as shown in the next chapter, a combination of conventional matter
and black holes.
The most important predictions of the strand model are the decrease of the cosmological constant with time and the absence of inflation. Various experiments will test these
predictions with increased precision in the coming years. So far, measurements do not
contradict these predictions.
The strand model confirms that the speed of light ? and the corrected Planck force
?4 /4? are limit values. The strand model also predicts that no variation in space and
time of ?, ?, ℏ and ? can be detected, because they define all measurement units.
The strand model predicts that the cosmological constant and the masses of the elementary particles are the only quantum effects that will be observed in the study of
gravitation. Strands strongly suggest that additional effects of quantum gravity cannot
be measured. In particular, no effects of space-time foam will be observed.
The strand model is, at present, the simplest – but not the only – known model of
quantum gravity that allows deducing all these results. In particular, the strands’ explanation of black hole entropy is by far the simplest one known.
General relativity is an approximation of the strand model. The approximation appears when the quantum of action and, in particular, the strand diameter are neglected.
General relativity and cosmology thus appear by approximating ℏ as 0 in the strand
model – as required by the Bronshtein cube of physics that is shown in Figure 1. Strands
imply that general relativity is valid for all energies below the Planck energy. In other
words, the strand model is not a generalization of general relativity. This conforms to
the list of requirements for the final theory.
If we look at the millennium list of open issues in physics, we see that – except for the
issue of dark matter – all issues about general relativity and cosmology have been settled.
The strand model explains the mathematical description of curved space-time and of
312
Page 275
10 g eneral rel ativity d ed uced from stra nds
general relativity. The strand model also provides a simple model of quantum gravity –
maybe the simplest known one. Above, we had already shown that the strand model explains all mathematical structures that appear in quantum theory and in particle physics.
Together with the results from this chapter we can now say: the strand model explains all
concepts, i.e., all mathematical structures that appear in physical theories. In particular,
strands explain the metric, curvature, wave functions, field intensities – and the probabilistic behaviour of all of them. They all result from averaging crossing switches.
In summary, starting from the fundamental principle of the strand model, we have
understood that strands are the origin of gravitation, general relativity, quantum gravity
and cosmology. We have also understood the mathematical description of gravitation –
and, before, that of quantum physics – found in all textbooks. These results encourage us
to continue our quest. Indeed, we are not done yet: we still need to deduce the possible
elementary particles and to explain their properties.
Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
C h a p t e r 11
T HE PART ICLE SPECT RUM DEDUCED
FROM ST R ANDS
Ref. 216
Page 161
”
Particles and quantum numbers from tangles
In nature, we observe three entities: vacuum, horizons and particles. Of these,
(quantum) particles are localized entities with specific intrinsic properties, i.e., properties that do not depend on their motion.
In nature, all the intrinsic properties of every particle, every object and every image
are completely described by three types of basic properties: (1) the elementary particles
they contain, (2) their behaviour under space-time transformations, (3) their interactions. The full list of these basic intrinsic properties of particles is given in Table 12.
Given the basic intrinsic properties for each elementary particle, physicists can deduce
all those intrinsic particle properties that are not listed; examples are the half life, decay
modes, branching ratios, electric dipole moment, T-parity, gyromagnetic ratio or electric
polarizability. Of course, the basic intrinsic properties also allow physicists to deduce
every property of every object and image, such as size, shape, colour, brightness, density,
** Voltaire (b. 1694 Paris, d. 1778 Paris) was an influential philosopher, politician and often satirical writer.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
trands describe quantum theory, gauge interactions and general relativity. But do
trands also settle all issues left open by twentieth-century physics? Do they
ettle the origin of all the elementary particles, their quantum numbers, their masses
and their mixing angles? How does the infinite number of possible tangles lead to a finite
number of elementary particles? And finally, do strands explain the coupling constants?
In the millennium list of open issues in fundamental physics, these are the issues that
remain. The strand model is correct only if these issues are resolved.
In this chapter, we show that the strand model indeed explains the known spectrum of
elementary particles, including the three generations of quarks and leptons. The strand
model is the first approach of modern physics that can provide such an explanation.
It should be stressed that from this point onwards, the ideas are particularly speculative. In the chapters so far, the agreement of the strand model with quantum field
theory and general relativity has been remarkable. The following chapters assign specific
tangles to specific particles. Such assignments are, by nature, not completely certain. The
speculative nature of the ideas now becomes particularly apparent.
Motion Mountain – The Adventure of Physics
S
“
No problem can withstand the assault of
sustained thinking.
Voltaire**
314
11 the pa rticle spectrum d ed uced from stra nds
Tangles made of one strand :
1a
curve
1b
prime long
knot
1c
composed
long knot
1a’
unknot
1b’
prime
knot
not in the
strand model
found in the early universe and at horizons
1c’
composed
knot
Correspondence :
not in the
strand model
F I G U R E 81 Examples for each class of tangles made of one strand.
particles made of one strand
Page 174
In the strand model, all particles made of one strand have spin 1, are elementary, and are
bosons. Conversely, all massless elementary spin-1 bosons can only have two tails, and
thus must be made of a single strand. Such one-stranded tangles return to the original
strand after a core rotation by 2π. Massive elementary spin-1 bosons can have one or
more strands. Tangles of more than one strand can only have spin 1 if they represent
massive elementary or composed particles. In short, classifying one-stranded tangles
allows classifying all elementary gauge bosons.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 173
elasticity, brittleness, magnetism or conductance.
In short, understanding all properties of matter and images thus only requires understanding the basic properties of quantum particles; and understanding the basic properties of quantum particles only requires understanding the basic properties of the elementary particles.
The strand model states that all elementary (and all composed) particles are tangles
of strands. This leads us to ask: Which tangle is associated to each elementary particle?
What kinds of elementary particles are possible? Do these tangles reproduce, for each
elementary particle, the observed values of the basic properties listed in Table 12?
It turns out that the strand model only allows a limited number of elementary particles.
In addition, the tangles of these elementary particle have intrinsic properties that match
the observed properties. To prove these strong statements, we first recall that all massive
elementary particles are represented by an infinite sequence of tangles. We now explore
tangles according to the number of strands they are made of.
Motion Mountain – The Adventure of Physics
photon,
W and Z,
gluon,
vacuum
pa rticles m a d e of one stra nd
315
TA B L E 12 The full list of basic intrinsic properties of quantum particles, from which all other observed
intrinsic properties of particles, objects and images can be deduced.
P roperty
P o s s i b l e Va l u e
Determines
Quantum numbers due to space-time symmetries:
Spin ? or ?
P parity
C parity
integer or half-integer
multiple of ℏ
even (+1) or odd (−1)
even (+1) or odd (−1)
statistics, rotation behaviour, conservation
behaviour under reflection, conservation
behaviour under charge conjugation,
conservation
Interaction properties:
Mass ?
Weak charge
Mixing angles
CP-violating phases between 0 and π/2
rational multiple of strong
coupling constant
Lorentz force, coupling to photons,
conservation
weak scattering and decays, coupling to W
and Z, partial conservation
mixing of quarks and neutrinos, flavour
change
degree of CP violation in quarks and
neutrinos
confinement, coupling to gluons,
conservation
Flavour quantum numbers, describing elementary particle content:
Lepton number(s) ?? integer(s)
Baryon number ?
integer times 1/3
Isospin ?? or ?3
+1/2 or −1/2
Strangeness ??
integer
Charmness ??
integer
Bottomness ??
integer
Topness ??
integer
conservation in strong and e.m.
interactions
conservation in all three gauge
interactions
up and down quark content, conservation
in strong and e.m. interactions
strange quark content, conservation in
strong and e.m. interactions
charm quark content, conservation in
strong and e.m. interactions
bottom quark content, conservation in
strong and e.m. interactions
top quark content, conservation in strong
and e.m. interactions
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Strong charge, i.e.,
colour
gravitation, inertia
Motion Mountain – The Adventure of Physics
Electric charge ?
between 0 and the Planck
mass
integer multiples of one
third of electron or proton
charge
rational multiple of weak
coupling constant
between 0 and π/2
316
11 the pa rticle spectrum d ed uced from stra nds
Photon γ :
Weak vector bosons W – before SU(2) breaking :
Wx
Wy
W0
Gluons g :
Motion Mountain – The Adventure of Physics
though, for clarity, the gluons are shown here using their complementary two-strand moves.
Mathematicians have already classified one-stranded tangles; they are usually called
open knots or long knots. To get an overview, we list an example for each class of onestranded tangles on the left-hand side of Figure 81. For completeness, closed curves are
shown on the right-hand side of the figure. We now explore each of these classes.
Unknot ted curves
Page 250
The simplest type of tangle made of one strand is an unknotted curve, shown as example
1a in Figure 81. The study of gauge interactions has shown that unknotted strands are,
depending on their precise average shape, either vacuum strands or gauge bosons.
The time-average of a vacuum strand is straight. A single strand represents a particle
if the time-averaged strand shape is not a straight line.
In the strand model, vacuum strands in flat space are, on average, straight. In this
property, vacuum strands differ from gauge bosons, which, on average, have curved
strands, and thus carry energy.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
F I G U R E 82 The gauge bosons in the strand model. All differ from vacuum by one curved strand –
pa rticles m a d e of one stra nd
317
Gauge bosons – and R eidemeister moves
Page 222
Gauge bosons are the carrier particles of the interactions. In the strand model, the gauge
interactions are due to the three Reidemeister moves. The electromagnetic, the weak and
the strong interaction correspond to respectively the first, second and third Reidemeister
move. As we have seen above, when the three Reidemeister moves deform fermion tangle
cores they generate U(1), SU(2) and SU(3) gauge symmetries. The detailed exploration
of the correspondence between tangle deformation and gauge theory led us to the gauge
boson tangles shown in Figure 82.
⊳ All gauge bosons – before symmetry breaking when applicable – are single,
curved strands.
Page 257
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 244
Motion Mountain – The Adventure of Physics
Page 226
A single strand represents a particle if the time-averaged strand shape is not a straight
line. The lack of straightness implies non-vanishing energy. A single-strand particle can
thus be either a strand with a bulge or a strand whose tails are not aligned along a straight
line.
As explained above, the first Reidemeister move, the twist, leads to the modelling of
photons as helical strands. Therefore, photons have vanishing mass and two possible polarizations. Photons do not have tangled, localized family members; photons are massless. Their specific unknotted and twisted strand shapes also imply that photons generate
an Abelian gauge theory and that photons do not interact among themselves. Automatically, photons have no weak and no strong charge. The strand model further implies
that photons have negative P-parity and C-parity, as is observed.
The study of the second Reidemeister move, the poke, showed that deformations induced by pokes can also involve braiding of tangle tails; this leads to the symmetry breaking of the weak interaction. As a result, the observed W and the Z boson strands become
massive. The tangle of the W is chiral, and thus it is electrically charged; the tangle of the
Z is achiral and thus electrically neutral. Being tangled, the W and the Z also carry weak
charge and thus interact among themselves, generating a non-Abelian gauge theory. The
strand model also implies that the W and the Z have no P-parity, no C-parity and no
colour charge, as is observed.
The study of the third Reidemeister move, the slide, led us to the existence of eight
gluons. The eight gluons are unknotted, thus they carry no mass, no electric charge and
no weak charge. Each gluon tangle has two possible polarizations. The strand model
of gluons also implies that they have negative P-parity and no C-parity, as is observed.
Gluons tangles carry colour and interact among themselves, thus they generate a nonAbelian gauge theory. In contrast to the other two interactions, free, single gluons are
short-lived, because their structure induces rapid hadronization: when gluons act on
the vacuum, quark–antiquark pairs are produced. Gluons do not have tangled family
members; they are massless in the high energy limit, when their tails are aligned.
For completeness we mention that by assignment, all gauge bosons differ from vacuum by a single curved strand, have vanishing lepton and baryon numbers, and thus
also lack all flavour quantum numbers. All this is as observed.
The strand model explains the lack of classical SU(2) field waves as a consequence
of the breaking of the SU(2) symmetry and the consequent mass of the weak bosons.
318
11 the pa rticle spectrum d ed uced from stra nds
Strands explain the lack of classical SU(3) waves, also called gluonic waves, as a consequence of the topological impossibility to produce such waves, which is related to the
infinite mass of single free gluons.
In somewhat sloppy language we can say that the shape and the effects of photons are
one-dimensional, those of the unbroken weak bosons are two-dimensional, and those
of the gluons are three-dimensional. This is the essential reason that they reproduce the
U(1), SU(2) and SU(3) groups, and that no higher gauge groups exist in nature.
In summary, Reidemeister’s theorem implies that the list of known gauge bosons with
spin 1 is complete. But the list of possible tangles made of a single strand is much more
extensive; we are not done yet.
Open or long knots
Page 251
Closed tangles: knots
Summary on tangles made of one strand
Page 275
In summary, a single strand represents a particle if the strand shape is, on average, not a
straight line. This distinguishes a vacuum strand from a particle strand. A particle strand
can thus be a strand with a bulge or a strand whose tails are not aligned along a straight
line. All tangles made of one open strand represent elementary particles of spin 1, thus
elementary vector bosons.
Massless elementary spin-1 particles are made of one open strand also because other
tangles cannot reproduce both zero mass and the spin-1 behaviour under rotations: only
one-stranded tangles return to the original strand after a core rotation by 2π and allow
vanishing mass at the same time.
In the strand model, all tangles made of one open curved strand are assigned to the
known gauge bosons. The strand model correctly reproduces and thus explains the gauge
boson spectrum and the quantum numbers for each gauge boson. In short, there is no
room for additional elementary gauge bosons.
In other words, the strand model predicts that all gauge bosons and thus all interactions are already known. We have thus a second argument – after the non-existence of
other gauge groups – stating that no other gauge interaction exists in nature. (Both arguments against the existence of other gauge interactions are related; in particular, both
are due to the three-dimensionality of space.) In particular, we find again that grand
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Figure 81 shows, on the right-hand side, examples for all classes of closed tangles of one
strand, i.e., of tangles without tails. They are usually just called knots in mathematics. In
the strand model knots may appear only in the early universe, and maybe near horizons.
They do not seem to have physical relevance and we do not explore them here.
Motion Mountain – The Adventure of Physics
Single strands could also contain knotted regions. We have explained earlier on that all
such possibilities – mathematically speaking, all so-called open knots or long knots – have
no relation to particles. In the strand model, they cannot appear and thus play no role.
The original strand model from 2008 did include such configurations as particles (for
example as W and Z bosons), but it now – i.e., after 2014 – seems that this inclusion is
an unnecessary complication.
pa rticles m a d e of t wo stra nds
319
Tangles made of two strands :
2a
trivial
tangle
2b
simple
crossing
2c
rational
tangle
(locally
unknotted)
2d
prime
tangle
(locally
unknotted)
2e
locally
knotted
tangle
Particle correspondence :
elementary:
quark or
graviton,
with higher
orders
not in
the strand
model
not in
the strand
model
2a’
unlink
2b’
Hopf
link
2c’
rational
link
2d’
prime
link
2e’
composed
link
2f
mixed
open-closed
tangle
Particle correspondence : none - only found in the early universe or near horizons.
F I G U R E 83 Possible tangles made of two strands.
unification and supersymmetry are not allowed in nature.
particles made of t wo strands
In the strand model, particle tangles can also be made of two strands. Examples for all the
classes of two-stranded tangles are given in Figure 83. Each class has a physical particle
assignment.
— The simplest tangle made of two strands is the trivial tangle, shown as example 2a
in Figure 83. In the strand model, the trivial tangle, like all separable tangles, is a
composite system. Each of the two strands can represent either the vacuum or a gauge
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
composed
of vacuum
or gauge
bosons; see
also 2c
Motion Mountain – The Adventure of Physics
composed
of vacuum
or gauge
bosons
320
11 the pa rticle spectrum d ed uced from stra nds
boson. Simply stated, the trivial tangle of two strands is not an elementary particle.
— The simplest non-trivial tangle made of two strands is the crossing, shown as 2b in
Figure 83. In the strand model, the crossing appears as part of the W and Z bosons;
in addition, for certain tail configurations, it can represent a graviton or the simplest
state of a down quark, as we will see below.
— A new class of tangles are the rational tangles, represented by example 2c in the figure.
A rational tangle is a tangle that can be untangled by moving its tails around. (Also
example 2b is a rational tangle.) Rational tangles are distinct from prime and from
locally knotted tangles, shown as examples 2d and 2e, which require pulling the tail
through the tangle to untangle it. Rational tangles are thus weakly tangled. As we will
see,
⊳ Rational tangles of two strands represent the graviton and the quarks.
Q uarks
Page 269
Page 378
Page 334, page 341
The exploration of the strand model and of the strong interaction showed: the tangle of
a coloured fermion, thus of a quark, must be rational, must reproduce the three possible
colour options, and must break the three-belt symmetry.
The simplest tangles that realize these requirements are shown in Figure 84: quark
tangles are rational tangles made of two strands. Higher quark generations have larger
crossing numbers. The four tails form the skeleton of a tetrahedron. A particle with two
strands tangled in this way automatically has spin 1/2. The electric charges of the quarks
are 1/3 and −2/3, an assignment that is especially obvious for up and down quarks and
that will become clearer later on, in the study of hadrons. Parity is naturally assigned as
done in Figure 84. Baryon number and the other flavour quantum numbers – isospin,
strangeness, charm, bottomness, topness – are naturally assigned as usual. The flavour
quantum numbers simply ‘count’ the number of corresponding quark tangles. Like all
localized tangles, quarks have weak charge. We will explore weak charge in more detail
below. Antiquarks are mirror tangles and have opposite quantum numbers. We will see
below that these assignments reproduce the observed quantum numbers of all mesons
and baryons, as well as all their other properties.
We note that the simplest version of the down quark is a simple crossing; nevertheless, it differs from its antiparticle, because the simple crossing mixes with the braid with
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
In short, the only two-stranded tangles of interest in the strand model are the rational
tangles. We now explore them in more detail.
Motion Mountain – The Adventure of Physics
We will discuss them in detail in the next two sections. More complicated rational
tangles are higher-order propagating states of the simpler ones.
— Another class of tangles are prime tangles, for which the tangle 2d is an example.
Like knotted one-stranded tangles, we conclude that prime tangles are not part of
the strand model.
— Still another class of tangles are locally knotted tangles, shown as example 2e. Also
this class is not part of the strand model.
— Finally, closed tangles, links and mixed tangles, shown in the lower row of Figure 83,
have no role in the strand model – except for the one closed strand that makes up the
whole universe. horizons.
pa rticles m a d e of t wo stra nds
321
Antiquarks :
Parity P = +1, B = 1/3, spin S = 1/2
Q = –1/3
Q = +2/3
P = –1, B = –1/3, S = 1/2
Q = +1/3
Q = –2/3
d
d
u
5.0 ± 1.6 MeV
2.5 ± 1.1 MeV
s
c
105 ± 35 MeV
1.27 ± 0.11 GeV
b
t
4.20 ± 0.17 GeV
171.3 ± 2.3 GeV
s
b
u
c
t
Seen from a larger distance, the tails follow (on average) the skeleton of a tetrahedron :
s
s
Page 323
Page 265
Page 334
seven crossings, 13 crossings, etc.; this mixing is due to the leather trick, as shown below.
And for every quark type, these more complicated braids differ from those of their antiparticles.
For each quark, the four tails form the skeleton of a tetrahedron. In Figure 84 and
Figure 85, the tetrahedral skeletons are drawn with one tail in the paper plane; of the
other three tails, the middle one is assumed to be above the paper plane, and the outer two
tails to be below the paper plane. This is important for the drawing of quark compounds
later on. The three tails allow us to reproduce the strong interaction and the colour charge
of the quarks: each colour is one of three possible orientations in space; more precisely,
the three colours result from the three possible ways to map a quark tangle to the three
belt structure. Each colour corresponds to a different choice for the tail that lies above
the paper plane, as shown in Figure 85. The colour interaction of quarks will be clarified
in the section on mesons.
In the strand model, the quark tangles thus carry colour. In nature, no free coloured
particle has been observed. The strand model reproduces this observation in several
ways. First of all, all leptons and baryons are colour-neutral, as we will see shortly.
Secondly, only free quark tangles, as shown in Figure 84, have a definite colour state, because they have a fixed orientation in space. Thirdly, free quark states, thus quark states
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
F I G U R E 84 The simplest tangles assigned to the quarks and antiquarks. For reference, the experimental
mass values are also given.
Motion Mountain – The Adventure of Physics
Quarks :
322
11 the pa rticle spectrum d ed uced from stra nds
The three colour states for a strange quark :
red
green
F I G U R E 85 The three colour charges correspond to the three possible spatial orientations; the centre
tail on the right is always above the paper plane, the other two tails on the right are below the paper
plane.
Page 338
Page 326
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 334
in the tetrahedral configuration of Figure 84, do not fit into vacuum even at large distances from the core; thus free quarks carry infinitely high energy. In practice, this means
that free quark states do not occur in nature. Indeed, a free, coloured quark tangle can
reduce its energy by interacting with one or several other quarks. The result is a strong
colour attraction between quarks that leads to colourless composites.
In short, also in the strand model, only colourless composites of quarks exist as stable
free particles. We will explore quark composites and the issue of confinement of quarks
in more detail shortly.
In nature, quarks are weakly charged and interact with W bosons. In the strand
model, the absorption or the emission of a W boson is the operation that takes a quark
tangle and adds or subtracts a braiding step. This process is illustrated in Figure 86, which
shows that a braiding (unbraiding) operation corresponds to the emission (absorption)
of an W boson before symmetry breaking. It is straightforward to check that this operation fulfils all conservation laws and properties that are observed for these so-called
flavour-changing charged currents. The absorption or emission of an (unbroken) Z boson
has no braiding effect. The strand model thus reproduces the result that only the charged
weak bosons can change quark flavours, as is observed.
For completeness, we mention that quarks, being tangles of two strands, have vanishing lepton number. Indeed, as we will see below, lepton tangles are made of three strands.
In summary, all quantum numbers of quarks are reproduced by the strand model, as
long as quarks are modelled as braids of two strands with ends directed along the corners
of a tetrahedron.
Motion Mountain – The Adventure of Physics
blue
pa rticles m a d e of t wo stra nds
323
An example for a flavour-changing charged current :
Strand model :
Observation :
s quark
s quark
W boson
W boson (unbroken)
t2
t2
t1
c quark
vacuum
t1
c quark
time
The leather trick :
F I G U R E 87 The leather trick is the deformation process that changes these two structures into each
Q uark generations
Page 363
Page 331
We stress that the quark tangles shown Figure 84 represent only the simplest tangle for
each quark. First of all, longer braids are mapped to each of the six quarks. This might
seem related to the leather trick shown in Figure 87. This trick is well-known to all people
in the leather trade: if a braid of three strands has ? ⩾ 6 crossings, it can be deformed
into a braid with ? − 6 crossings. We might conjecture that, due to the leather trick, there
is no way to introduce more than 6 quarks in the strand model.
In fact, the leather trick argument assumes that the braid end – and thus the ends
of the strands – can be moved through the braids. In the strand model, this can only
happen at the horizon, the only region where space (and time) are not well-defined, and
where such manipulations become possible. The low probability of such a process will
be important in the determination of quark masses.
Instead of resting on the leather trick, it is simpler to assume that braids with large
numbers of crossings are mapped modulo 6 to the braids with the smallest number of
crossings. This is consistent, because in the strand model, a braid with six additional
crossings is mapped to a particle together with a virtual Higgs boson. The modulo 6 rule
thus represents the Yukawa mass generation mechanism in the strand model.
In summary, in the strand model, each quark is not only represented by the tangles
shown in Figure 84, but also by tangles with 6 additional crossings, with 12 additional
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
other. The leather trick limits structures made of three-stranded braids to six basic types.
Motion Mountain – The Adventure of Physics
F I G U R E 86 Absorption or emission of a W boson changes quark flavour.
324
11 the pa rticle spectrum d ed uced from stra nds
The graviton :
F I G U R E 88 The graviton in the strand model.
The graviton
Page 296
Glueballs
Ref. 217, Ref. 218
Challenge 195 ny
Ref. 219
Ref. 220
There is no observational evidence for glueballs yet, even though simulations of QCD on
the lattice predict the existence of several such states in the 1.5 GeV/?2 mass range. The
lack of experimental confirmation is usually explained by the strong background noise
in the reaction that produces glueballs, and by the expected strong mixing with mesons
of similar quantum numbers. The experimental search for glueballs is still ongoing.
The lowest-mass glueball is usually expected to be made of two gluons. In the strand
model, a glueball made of two gluons would be made of two curved strands. However,
the strand model of gluons does not seem to allow such a tangle.
Could a situation in which two gluons are linked in such a way that the four tails are
perpendicular and span a plane lead, through averaging, to a zero spin value? The issue
of glueballs needs a more precise investigation.
Whatever the situation for glueballs might be, the strand model of gluons seems in
contrast with the models of glueballs as knots that were proposed by Buniy and Kephart
or by Niemi. These models are based on closed knots, not on tangles with tails. The strand
model does not seem to allow real particles of zero spin that are composed of gluons. On
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 279
One rational tangle made of two strands is special. This special tangle is shown (again)
in Figure 88. It differs from a quark tangle in one property: the tails are parallel (and
near) to each other, and thus lie (almost completely) in a plane. Its tangle core returns
to its original state after rotation by π, and therefore models a spin-2 particle. The tangle
is not localized along its propagation direction; thus it has no mass, no electric and no
weak charge. It also has no colour charge. The tangle represents the graviton. Similar
tangles with higher winding numbers represent higher orders in the perturbation theory
of gravitation.
The chapter on gravitation has already shown how gravitons lead to curvature, horizons and the field equations of general relativity.
Motion Mountain – The Adventure of Physics
Challenge 194 e
crossings, etc.
As a mathematical check, we can also ask whether all other rational tangles are
mapped to quarks. Rational tangles of higher complexity arise by repeatedly twisting
any pair of tails of a quark tangle. This process produces an infinite number of complex
two-stranded tangles. In the strand model, these tangles are quarks surrounded by virtual particles. Equivalently, we can say that all the more complex rational tangles that do
not appear in Figure 84 are higher-order propagators of quarks.
pa rticles m a d e of t wo stra nds
325
the other hand, if closed knots were somehow possible in the strand model, they would
imply the existence of glueballs.
In summary, the issue of glueballs is not settled; a definitive solution might even lead
to additional checks of the strand model.
The mass gap problem and the Cl ay Mathematics Institute
Ref. 221
Challenge 196 s
The topic of two-stranded tangles also requires to solve the puzzle of Figure 89. To which
physical states do the three pictured tangles correspond?
Summary on t wo-stranded tangles
In summary, the strand model predicts that apart from the six quarks and the graviton, no other two-stranded elementary particle exists in nature. Concerning composite
particles, the two-stranded glueball issue is not completely settled, but points towards
non-existence.
Quarks and the graviton, the elementary particles made of two strands, are rational
tangles. Their strand models are thus not tangled in a complicated way, but tangled in
the least complicated way possible. This connection will be of importance in our search
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
A puzzle
Motion Mountain – The Adventure of Physics
Page 275
The Clay Mathematics Institute offers a large prize to anybody who proves the following
statement: For any compact simple non-Abelian gauge group, quantum gauge theory exists
in continuous, four-dimensional space-time and produces a mass gap. This is one of their
so-called millennium problems.
The strand model does not allow arbitrary gauge groups in quantum field theory. According to the strand model, the only compact simple non-Abelian gauge group of interest is SU(3), the gauge group of the strong nuclear interaction. And since the strand
model does not seem to allow for glueballs, for SU(3) an effective mass gap of the order
of the Planck mass is predicted. (If glueballs would exist in the strand model, the mass
gap would still exist but be smaller.) Indeed, the strand model explains the short range
of the strong interaction as a consequence of the details of Reidemeister III moves and
the quark tangle topology.
The strand model further states that space-time and gauge groups are low-energy approximations, because neither points nor fields exist at a fundamental level; points and
fields are approximations to strands. According to the strand model, the quantum properties of nature result from the extension of strands. As a consequence, the strand model
denies the existence of any quantum gauge theory as a separate, exact theory on continuous space-time.
In summary, the strand model does predict a mass gap for SU(3); but the strand model
also denies the existence of quantum gauge theory for any other compact simple nonAbelian gauge group. And even in the case of SU(3) it denies – like for any other gauge
groups – the existence of a quantum gauge theory on continuous space-time. As deduced
above, the strand model allows only the three known gauge groups, and allows their existence only in the non-continuous strand model of space-time. In short, it is impossible
to realize the wish of the Clay Mathematics Institute.
326
11 the pa rticle spectrum d ed uced from stra nds
achiral 6-fold
crossing tangle
chiral 6-fold
crossing tangle
chiral 8-fold
crossing tangle
Motion Mountain – The Adventure of Physics
F I G U R E 89 Which particle states are described by these tangles?
Tangles made of three strands :
3b
simple
crossings
3c
braided
tangle
3d
rational
tangle
3e
prime
tangle
3f
locally
knotted
tangle
3g
closed or
mixed
open-closed
tangles
not part of
the strand
model
near the
big bang
or
horizons
Particle correspondence :
composed
of vacuum
and gauge
bosons
composed
of vacuum
and gauge
bosons
elementary: composed:
not part of
leptons and mesons of
the strand
model
Higgs boson spin 0 and
other particles
F I G U R E 90 Examples for all the classes of tangles made of three strands.
for elementary particles that are still undiscovered.
particles made of three strands
In the strand model, the next group are particles made of three strands. Examples for
all classes of three-stranded tangles are given in Figure 90. Several classes of three-
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
3a
trivial
tangle
pa rticles m a d e of three stra nds
327
Lepton tangles, all with spin S = 1/2 , parity P = +1 , lepton number L’ = 1 and baryon number B = 0 :
e
0.5 MeV
105 MeV
νμ
τ
all three with
Q = –1
1.77 GeV
ντ
all three with
Q=0
1±1 eV
1±1 eV
Seen from a larger distance, the tails follow (on average) the x, y and z axes of a coordinate system.
F I G U R E 91 The simplest tangles of the leptons, with the experimental mass values. Antileptons are
mirror tangles.
stranded tangles turn out to be composites of two-stranded particles. However, a number
of tangles are new and represent elementary particles.
L eptons
The candidate tangles from 2008 for the leptons shown in Figure 91 are the simplest
possible non-trivial tangles with three strands. These lepton tangles are simple braids
with tails reaching the border of space. The six tails probably point along the coordinate
axes. These braided tangles have the following properties.
— Each lepton is localized. Each lepton has mass: its three tails con be braided, thus
have non-vanishing Yukawa coupling, thus generate mass. And each lepton has spin
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
1±1 eV
Motion Mountain – The Adventure of Physics
νe
μ
328
11 the pa rticle spectrum d ed uced from stra nds
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 222
In summary, tangles of three strands have precisely the quantum numbers and most
properties of leptons. In particular, the strand model predicts exactly three generations
of leptons, and predicts that all leptons have mass.
This implies that searches for the neutrino-less double beta decay should yield negative results, that the magnetic moments of the neutrinos should have the exceedingly
Motion Mountain – The Adventure of Physics
1/2. Each lepton thus follows the Dirac equation.
— Each lepton has weak charge.
— Charged leptons and antileptons differ. Each has two possible chiralities.
— Three of the tangles are topologically chiral, thus electrically charged, and three other
tangles are topologically achiral, thus uncharged.
— The spatial parity ? of the charged lepton tangles is opposite to that of their antiparticles.
— Being made of three strands, lepton tangles have vanishing colour charge and vanishing baryon number.
— In contrast to quarks, lepton tangles can be inserted in the vacuum using a localized,
i.e., finite amount of energy and are thus predicted to exist as free particles.
— The three types of lepton (flavour) numbers can be assigned as usual; the lepton numbers are conserved in reactions, apart for neutrino mixing effects, as we will see below.
— The strand model predicts that the electron, the charged tangle with the lowest mass,
is stable, as there is no way for it to decay and conserve charge and spin. The other
two generations are predicted to be unstable, due to weak decays that simplify their
topology.
— The three generations are reproduced by the strand model, as every more complicated
braid can be seen as equivalent to one of the first six braids, with the same braiding
argument that limits the number of quarks.
— There is a natural mapping between the six quarks and the six leptons. It appears
when the final bend of the ‘longer’ quark strand is extended to the border of space,
thus transforming a two-stranded quark braid into a three-stranded lepton braid.
Thus we get three common generations for quarks and leptons.
— The neutrino strands differ by tail braiding; the strand model thus predicts that the
weak interaction mixes neutrinos.
— All lepton tangles differ from each other. Thus the mass values are different for each
lepton.
— Due to the small amount of tangling, the strand model predicts that the masses of
the leptons are much smaller than those of the W and Z boson. This is indeed observed. (This also suggests a relation between the mass and the total curvature of a
tight tangle.)
— The simplest tangle for the electron neutrino also suggests that the mass values for
the electron neutrino is naturally small, as its tangle is almost not tangled.
— The strand model predicts that lepton masses increase with the generation number.
Since the neutrino masses are not precisely known, this prediction cannot yet be
checked.
— Neutrinos and antineutrinos are both massive and differ from each other. If the tangle
of the electron neutrino is correct, the electron neutrino of opposite chirality is expected to be seen only rarely – as is observed.
pa rticles m a d e of three stra nds
329
A candidate tangle for the Higgs boson :
or
F I G U R E 92 A candidate tangle for the Higgs boson in the strand model: the open version (left) and the
corresponding closed version (right). For the left version, the tails approach the six coordinate axes at
infinity.
Ref. 223
Open issue: are the lepton tangles correct?
Page 386
The Higgs boson – the mistaken section from 2009
Page 250
The existence of the Higgs boson is predicted from the standard model of elementary
particle physics using two arguments. First of all, the Higgs boson prevents unitarity
violation in longitudinal W–W and Z–Z boson scattering. Secondly, the Higgs boson
confirms the symmetry breaking mechanism of SU(2) and the related mass generation
mechanism of fermions. Quantum field theory predicts that the Higgs boson has spin
0, has no electric or strong charge, and has positive C and P parity. In other words, the
Higgs boson is predicted to have, apart from its weak charge, the same quantum numbers
as the vacuum.
In the strand model, there seems to be only one possible candidate tangle for the Higgs
boson, shown on the left of Figure 92. The tangle has positive C and P parity, and has
vanishing electric and strong charge. The tangle also corresponds to the tangle added by
the leather trick; it thus could be seen to visualize how the Higgs boson gives mass to the
quarks and leptons. However, there are two issues with this candidate. First, the tangle
is a deformed, higher-order version of the electron neutrino tangle. Secondly, the spin
value is not 0. In fact, there is no way at all to construct a spin-0 tangle in the strand
model. These issues lead us to reconsider the arguments for the existence of the Higgs
boson altogether.
We have seen that the strand model proposes a clear mechanism for mass generation:
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Challenge 197 ny
The argument that leads to the lepton tangles is vague. The tangle assignments might
need corrections. There are two issues.
First, there is an aesthetic issue: in most particle tangles, the electric charge unit is
given by three crossings of the same sign. It seems odd that leptons should form an exception.
Secondly, the candidate tangles suggest that the muon neutrino is more massive than
the electron. Most probably therefore, the tangles need amends. Can you improve the
situation, either by finding better tangles or by finding better arguments?
Motion Mountain – The Adventure of Physics
small values predicted by the standard model of particle physics, and that rare muon and
other decays should occur at the small rates predicted by the standard model.
330
11 the pa rticle spectrum d ed uced from stra nds
Strand model for a longitudinal massive spin 1 boson :
Observation :
spin
precession
orthogonal
spin
time average
of crossing
switches
precession
motion
motion
time
Strand model for a transversal massive spin 1 boson :
time average
of crossing
switches
spin
parallel
spin
motion
motion
F I G U R E 93 In the strand model, transverse and longitudinal W and Z bosons differ. (Note added in
2012: this statement is mistaken.)
⊳ Mass is due to strand braiding.
Page 207
Ref. 224
Ref. 225
This mechanism, due to the weak interaction, explains the W and Z boson mass ratio,
as we will see below. The leather trick that explains fermion masses can be seen as the
addition of a sixfold tail braiding. In particular, the rarity of the braiding process explains
why particle masses are so much smaller than the Planck mass. In short, the strand model
explains mass without a Higgs boson.
If the Higgs boson does not exist, how is the unitarity of longitudinal W and Z boson
scattering maintained? The strand model states that interactions of tangles in particle
collisions are described by deformations of tangles. Tangle deformations in turn are described by unitary operators. Therefore, the strand model predicts that unitarity is never
violated in nature. In particular, the strand model automatically predicts that the scattering of longitudinal W or Z bosons does not violate unitarity.
In other terms, the strand model predicts that the conventional argument about unitarity violation, which requires a Higgs boson, must be wrong. How can this be? There are
at least two loopholes available in the research literature, and the strand model realizes
them both.
The first known loophole is the appearance of non-perturbative effects. It is known
for a long time that non-perturbative effects can mimic the existence of a Higgs boson in
usual, perturbative approximations. In this case, the standard model could remain valid
at high energy without the Higgs sector. This type of electroweak symmetry breaking
would lead to longitudinal W and Z scattering that does not violate unitarity.
The other loophole in the unitarity argument appears when we explore the details of
the longitudinal scattering process. In the strand model, longitudinal and transverse W
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 358
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core
pa rticles m a d e of three stra nds
Ref. 226
331
Vol. V, page 253
Vol. V, page 258
In July 2012, CERN researchers from two different experiments announced the observation of a new neutral boson with a mass of 125 GeV. Additional data analysis showed
that the boson has spin 0 and positive parity. All experimental checks confirm that the
boson behaves like the Higgs boson predicted in 1963 by Peter Higgs and a number of
other researchers.
The results lead to question several statements made in 2009 in the previous section.
— Is the tangle on the left-hand side of Figure 92 really a higher order version of the
electron neutrino? It turns out that this statement is wrong: in contrast to the tangle
* If the arguments against the Higgs boson turn out to be wrong, then the strand model might be saved with
a dirty trick: we could argue that the tangle on the left-hand side of Figure 92 might effectively have spin 0.
In this case, the ropelength of the Borromean rings, 29.03, together with the ropelengths of the weak bosons,
lead to a Higgs mass prediction, to first order, in the range from (29.03/10.1)1/3 ⋅ 80.4 GeV = 114 GeV to
(29.03/13.7)1/3 ⋅ 91.2 GeV = 117 GeV, plus or minus a few per cent.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
The Higgs boson – the corrected section of 2012
Motion Mountain – The Adventure of Physics
or Z bosons are modelled as shown in Figure 93. For longitudinal bosons, spin and its
precession leads to a different situation than transversal bosons: longitudinal bosons are
more delocalized than transversal bosons. This is not the case for fermions, where the
belt trick leads to the same delocalization for longitudinal and transverse polarization.
Interestingly, it is also known for a long time that different delocalization for longitudinal
and transversal bosons maintains scattering unitarity, and that in the case of delocalization the conventional argument for the necessity of the Higgs boson is wrong. These
are well-known consequences of the so-called non-local regularization in quantum field
theory. The strand model thus provides a specific model for this non-locality, and at the
same time explains why it only appears for longitudinal W and Z bosons.
The issue of different scattering behaviour for longitudinal and transverse weak bosons also raises the question whether the mass of the longitudinal and the transversal
bosons are precisely equal. The possibility, triggered by Figure 93, might seem appealing
at first sight in order to solve the unitarity problem. However, the strand model forbids
such a mass difference. In the strand model, mass is due to tangle fluctuations, but does
not depend on spin direction.
In other words, the strand model predicts that the scattering of longitudinal W and Z
bosons is the first system that will show effects specific to the strand model. Such precision scattering experiments might be possible at the Large Hadron Collider in Geneva.
These experiments will allow checking the non-perturbative effects and the regularization
effects predicted by the strand model. For example, the strand model predicts that the
wave function of a longitudinal and a transversally polarized W or Z boson of the same
energy differ in cross section.
In summary, the strand model predicts well-behaved scattering amplitudes for longitudinal W and Z boson scattering in the TeV region, together with the absence of the
Higgs boson.* The strand model explains mass generation and lack of unitarity violations in longitudinal W or Z boson scattering as consequences of tail braiding, i.e., as
non-perturbative and non-local effects, and not as consequences of an elementary spin0 Higgs boson. The forthcoming experiments at the Large Hadron Collider in Geneva
will test this prediction.
332
11 the pa rticle spectrum d ed uced from stra nds
Higgs boson :
of the neutrino, the tangle of Figure 92 is not twisted.
— Does the tangle of Figure 94 have spin 1/2 or spin 0? As mentioned already in 2009,
an effective spin 0 might be possible, in a similar way that it is possible for spin-0
mesons. Spin 0 behaviour might appear because the tangle can be oriented in different directions or because of the Borromean property: no two strands have more
crossings than two vacuum strands; the time average of these situations has the same
symmetry as the vacuum, and thus implies spin 0.
— Does the tangle of Figure 94 have the correct, positive, C and P values expected for a
Higgs boson? It seems so.
— Is the mentioned non-locality effect for W and Z bosons real? If the effect were real, it
should also appear for other spin-1 particles. In the strand model, mass values should
not depend on spin orientation, but only on tangle core topology. The statements
made in 2009 on delocalization and longitudinal scattering seem wrong in retrospect.
— Would the Higgs boson tangle assignment of Figure 94 be testable? Yes; any tangle
assignment must yield the observed mass value and the observed branching ratios
and decay rates. This is a subject of research. But already at the qualitative level, the
proposed tangle structure of the Higgs boson suggests decays into leptons that are
similar to those observed at CERN.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
be oriented in different directions, so that the time average has spherical symmetry. The tangle has 9
crossings: 3 crossings appear already in the vacuum configuration of three strands, and the additional 6
crossings (see Figure 92) are due to the Higgs boson.
Motion Mountain – The Adventure of Physics
F I G U R E 94 The tangle of the Higgs boson in the strand model. Spin 0 appears because the braid can
pa rticles m a d e of three stra nds
Page 366
— Is the tangle of Figure 94 elementary? Yes.
— Are there other possible Higgs boson tangles? This issue is open. The braid structure
seems the most appealing structure, as it embodies the effect of tail braiding, an effect
that is important for the appearance of mass.
— Are knots and links, i.e., closed tangles, really forbidden? The discussion about the
Higgs boson concerns the open tangle shown in Figure 94, not the Borromean link
shown on the right-hand side of Figure 92. So far, there is no evidence for closed
tangles in the strand model. Such evidence would mean a departure from the idea
that nature is a single strand.
— Does the Higgs boson issue put into question the strand model as a whole? First of all,
SU(2) breaking is unaffected. Secondly, a mistaken tangle–particle assignment can be
accommodated in the strand model; new forces or symmetries cannot. Therefore the
strand model is not put into question.
— Could several, possibly charged, Higgs bosons exist? No such tangles seem possible
– as long as a tangle with two Figure 94 Higgs cores in sequence is not a separate
particle.
— Has some other strand model effect been overlooked? Could other elementary or
composed particles exist? For example, the structure of the Higgs boson might be
seen to suggest that lepton families reappear (roughly) every 125 GeV. Is that the
case? The issue is not completely settled. It seems more probable that those higher
tangles simply yield corrections to the Higgs mass.
2012 predictions abou t the Higgs
— The Higgs tangle implies a Higgs boson with vanishing charge, positive parity, being
elementary – as is observed.
— The Higgs tangle allows us to estimate the Higgs/Z mass ratio. Using the new, unknotted, tangle model for the W and Z bosons, the estimates are in the region of the
observed values. Improving the estimates is still subject of research.
— The Higgs tangle and the strand model imply that the standard model is correct up to
Planck energy, and that the Higgs mass value should reflect this. The observed Higgs
mass of 125 GeV complies also with this expectation.
— Therefore, the strand model suggests that no deviations between the standard model
and data should ever be observed in any experiment.
— The strand model again and consistently predicts the lack of supersymmetry.
— In the case that several Higgs bosons exist or that the braided Higgs tangle does not
apply, the strand model is in trouble.
— In the case that effects, particles or interactions beyond the standard model are observed, the strand model is in trouble.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
In short, the existence of the standard model Higgs boson seems compatible with the
strand model. The 2009 mistake about the Higgs also shows that the exploration of the
strand model is not yet complete. In any case, the strand model has not been falsified by
the discovery of the Higgs boson.
Assuming that the Higgs tangle shown in Figure 94 is correct, we have an intuitive
proposal for the mechanism that produces mass, namely tail braiding. The proposed
Higgs tangle also allows a number of experimental predictions.
Motion Mountain – The Adventure of Physics
Page 332
333
334
11 the pa rticle spectrum d ed uced from stra nds
Pseudoscalar and vector mesons made of up and down quarks :
Spin S = 0, L = 0
Parity P = –1
η : 548 MeV
C = +1
π+ : 140 MeV
π– : 140 MeV
η,
π0
ω0, ρ0
uu
uu
π+
ρ+
ud
ud
π–
ρ–
ud
ud
ω0
: 783 MeV
C = –1
ρ+ : 775 MeV
ρ– : 775 MeV
π0, η
ρ0, ω0
ρ0 : 775 MeV
dd
dd
C = –1
Circles indicate
pairs of tails to the
border of space
crossed tail pairs to the border of space), with the observed mass values.
Q uark-antiquark mesons
Ref. 227
Ref. 228
Ref. 229
In the strand model, all three-stranded tangles apart from the leptons, as well as all fourstranded tangles represent composite particles. The first example are mesons.
In the strand model, rational tangles of three strands are quark-antiquark mesons with
spin 0. The quark tangles yield a simple model of these pseudoscalar mesons, shown on
the left-hand sides of Figure 95, Figure 97 and Figure 98. The right-hand sides of the
figures show vector mesons, thus with spin 1, that consist of four strands. All tangles are
rational. Inside mesons, quarks and antiquarks ‘bond’ at three spots that form a triangle
oriented perpendicularly to the bond direction and to the paper plane. To increase clarity, the ‘bonds’ are drawn as circles in the figures; however, they consist of two crossed
(linked) tails of the involved strands that reach the border of space, as shown in Figure 96.
With this construction, mesons made of two quarks are only possible for the type ? ?.
Other combinations, such as ? ? or ? ?, turn out to be unlinked. We note directly that
this model of mesons resembles the original string model of hadrons from 1973, but also
the Lund string model and the recent QCD string model.
To compare the meson structures with experimental data, we explore the resulting
quantum numbers. As in quantum field theory, also in the strand model the parity of
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
F I G U R E 95 The simplest strand models for the light pseudoscalar and vector mesons (circles indicate
Motion Mountain – The Adventure of Physics
π0 : 135 MeV
C = +1
Spin S = 1, L = 0
Parity P = –1
Note: the larger circle
is above the paper plane.
pa rticles m a d e of three stra nds
335
Simplified drawing :
Spin S = 0, L = 0
Parity P = –1
π0 : 135 MeV
C = +1
Spin S = 1, L = 0
Parity P = –1
Note: the large circle
is above the paper plane.
π0, η
ρ0, ω0
ρ0 : 775 MeV
dd
dd
C = –1
Complete drawing :
Simplification used :
F I G U R E 96 The meaning of the circles used in the tangle graphs of mesons and baryons.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
a particle is the product of the intrinsic parities and of wave function parity. The states
with orbital angular momentum ? = 0 are the lowest states. Experimentally, the lightest
mesons have quantum numbers ??? = 0−+ , and thus are pseudoscalars, or have ??? =
1−− , and thus are vector mesons. The strand model reproduces these observed quantum
numbers. (We note that the spin of any composite particle, such as a meson, is lowenergy quantity; to determine it from the composite tangle, the tails producing the bonds
– drawn as circles in the figures – must be neglected. As a result, the low-energy spin of
mesons and of baryons is correctly reproduced by the strand model.)
In the strand model, the meson states are colour-neutral, or ‘white’, by construction,
because the quark and the antiquark, in all orientations, always have opposite colours
that add up to white.
In the strand model, the electric charge is an integer for all mesons. Chiral tangles are
charged, achiral tangles uncharged. The charge values deduced from the strand model
thus reproduce the observed ones.
In experiments, no mesons with quantum numbers 0−− , 0+− , or 1−+ are observed.
Also this observation is reproduced by the quark tangles, as is easily checked by direct
inspection. The strand model thus reproduces the very argument that once was central
to the acceptance of the quark model itself.
It is important to realize that in the strand model, each meson is represented by a
tangle family consisting of several tangle structures. This has three reasons. First, the
‘circles’ can be combined in different ways. For example, both the ? ? and the ? ? have
as alternate structure a line plus a ring. This common structure is seen as the underlying
Motion Mountain – The Adventure of Physics
Circles indicate
pairs of tails to the
border of space
336
11 the pa rticle spectrum d ed uced from stra nds
Pseudoscalar and vector mesons containing
strange and charm quarks :
K–
su
494 MeV
Note: the large circle
is above the paper plane.
K*–
su
892 MeV
K0
sd
498 MeV
K*0
sd
899 MeV
K0
sd
498 MeV
K*0
sd
899 MeV
η'
ss
958 MeV
ϕ'
ss
1020 MeV
D0
cu
1864 MeV
D*0
cu
2007 MeV
D0
cu
1864 MeV
D*0
cu
2007 MeV
D+
cd
1870 MeV
D*+
cd
2010 MeV
D–
cd
1870 MeV
D*–
cd
2010 MeV
Ds+
cs
1970 MeV
Ds*+
cs
2112 MeV
Ds–
cs
1968 MeV
Ds*–
cs
2112 MeV
ηc
cc
2981 MeV
J/ψ
cc
3097 MeV
F I G U R E 97 The simplest strand models for strange and charmed mesons with vanishing orbital angular
momentum. Mesons on the left side have spin 0 and negative parity; mesons on the right side have
spin 1 and also negative parity. Circles indicate crossed tail pairs to the border of space; grey boxes
indicate tangles that mix with their antiparticles and which are thus predicted to show CP violation.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
K*+
su
892 MeV
Motion Mountain – The Adventure of Physics
K+
su
494 MeV
pa rticles m a d e of three stra nds
337
Pseudoscalar and vector mesons containing a bottom quark :
Spin S = 0, L = 0
Parity P = –1
Spin S = 1, L = 0
Parity P = –1
Note: the large circle
is above the paper plane.
B*–
bu
5325 MeV
B0
bd
5279 MeV
B*0
bd
5325 MeV
B0s
bs
5366 MeV
B*s0
bs
5412 MeV
Bc–
bc
6286 MeV
B*c–
bc
not yet
discovered
ηb (C=+1)
bb
9300 MeV
Y (C=–1)
bb
9460 MeV
F I G U R E 98 The simplest strand models for some heavy pseudoscalar and vector mesons, together with
their experimental mass values. Antiparticles are not drawn; their tangles are mirrors of the particle
tangles. Circles indicate crossed tail pairs to the border of space; grey boxes indicate tangles that mix
with their antiparticles and which are thus predicted to show CP violation.
reason that these two quark structures mix, as is indeed observed. (The same structure is
also possible for ? ?, and indeed, a full description of these mesons must include mixing
with this state as well.) The second reason that mesons have several structures are the
mentioned, more complicated braid structures possible for each quark, namely with 6,
12, etc. additional braid crossings. The third reason for additional tangle structures is the
occurrence of higher-order Feynman diagrams of the weak interaction, which add yet
another group of more complicated topologies that also belong to each meson.
In short, the mesons structures of Figure 95, Figure 97 and Figure 98 are only the
simplest tangles for each meson. Nevertheless, all tangles, both the simplest and the more
complicated meson tangles, reproduce spin values, parities, and all the other quantum
numbers of mesons. Indeed, in the strand model, the more complicated tangles automatically share the quantum numbers of the simplest one.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Circles indicate
pairs of tails to the
border of space
Motion Mountain – The Adventure of Physics
B–
bu
5279 MeV
338
11 the pa rticle spectrum d ed uced from stra nds
Meson form factors
Ref. 230
The strand model also predicts directly that all mesons from Figure 95, Figure 97 and
Figure 98, in fact all mesons with vanishing orbital momentum, are prolate. This (unsurprising) result is agreement with observations. Mesons with non-vanishing orbital
momentum are also predicted to be prolate. This latter prediction about meson shapes
is made also by all other meson models, but has not yet been checked by experiment.
There is another way to put what we have found so far. The strand model makes
the following prediction: When the meson tangles are averaged over time, the crossing densities reproduce the measured spatial, quark flavour, spin and colour part of the
meson wave functions. This prediction can be checked against measured form factors
and against lattice QCD calculations.
Meson masses, excited mesons and quark confinement
? = ?0 + ?1 ?2
Ref. 232
(201)
with an (almost) constant factor ?1 for all mesons, about 0.9 GeV/f m. These relations,
the famous Regge trajectories, are explained in quantum chromodynamics as deriving
from the linear increase with distance of the effective potential between quarks, thus
from the properties of the relativistic harmonic oscillator. The linear potential itself is
usually seen as a consequence of a fluxtube-like bond between quarks.
In the strand model, the fluxtube-like bond between the quarks is built-in automatically, as shown in Figure 99. All mesons have three connecting ‘bonds’ and these three
bonds can be seen as forming one common string tube. In the simplified drawings, the
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 356
Motion Mountain – The Adventure of Physics
Ref. 231
The strand model also allows us to understand meson masses. We recall that a topologically complicated tangle implies a large mass. With this relation, Figure 95 predicts that
the π0 , ? and π+/− have different masses and follow the observed meson mass sequence
?(π0 ) < ?(π+/− ) < ?(?). The other mass sequences can be checked with the help of Figure 95, Figure 97 and Figure 98; there are no contradictions with observations. However,
there is one limit case: the strand model predicts different masses for the ?0 , ?, and ?+/− .
So far, observations only partly confirm the prediction. Recent precision experiments
seem to suggest that ?0 and ?+/− have different mass; this result has not been confirmed
yet.
More precise mass determinations will be possible with numerical calculations. This
will be explored in more detail later on. In any case, the strand model for mesons suggests that the quark masses are not so important for the determination of meson masses,
whereas the details of the quark-antiquark bond are. Indeed, the light meson and baryon
masses are much higher than the masses of the constituent quarks.
The relative unimportance of quark masses for many meson masses is also confirmed
for the case of excited mesons, i.e., for mesons with orbital angular momentum ?. It is
well known that mesons of non-vanishing orbital angular momentum can be grouped
into sets which have the same quark content, but different total angular momentum ? =
? + ?. These families are observed to follow a well-known relation between total angular
momentum ? and mass ?, called Regge trajectories:
pa rticles m a d e of three stra nds
339
Vanishing orbital angular momentum
Simplified drawing :
Spin S = 0, L = 0
Parity P = –1
π0 : 135 MeV π0, η
C = +1
dd
Spin S = 1, L = 0
Parity P = –1
Note: the large circle
is above the paper plane.
ρ0, ω0
dd
ρ0 : 775 MeV
C = –1
Complete drawing :
Simplified drawing :
Complete drawing :
bond or string tube is the region containing the circles. In orbitally excited mesons, the
three bonds are expected to lengthen and thus to produce additional crossing changes,
thus additional effective mass. The strand model also suggests a linear relation. Since the
mechanism is expected to be similar for all mesons, which all have three bonding circles,
the strand model predicts the same slope for all meson (and baryon) Regge trajectories.
This is indeed observed.
In summary, the strand model reproduces meson mass sequences and quark confinement in its general properties.
CP viol ation in mesons
Ref. 233
Ref. 231
In the weak interaction, the product CP of C and P parity is usually conserved. However,
rare exceptions are observed for the decay of the ?0 meson and in various processes
that involve the ?0 and ?0s mesons. In each of these exceptions, the meson is found to
mix with its own antiparticle. CP violation is essential to explain the matter–antimatter
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
F I G U R E 99 The strand model for mesons without (top) and with (bottom) orbital angular momentum.
Motion Mountain – The Adventure of Physics
With orbital angular momentum
340
O ther three-stranded tangles and glueballs
Page 324
Challenge 198 s
In the strand model, the aditted complicated tangles made of three strands are either
higher-order propagating versions of the tangles just presented or composites of onestranded or two-stranded particles.
The often conjectured glueball could also be made of three gluons. In the strand
model, such a structure would be a simple tangle made of three strands. However, the
masslessness of gluons does not seem to allow such a tangle. The argument is not watertight, however, and the issue is, as mentioned above, still subject of research.
We note that the three-strand analog of the graviton – three parallel, but twisted
strands – is not an elementary particle, but a composed structure.
Spin and three-stranded particles
Why do three strands sometimes form a spin 0 particle, such as the elementary Higgs
boson, sometimes a spin 1/2 particle, such as the elementary electron, and sometimes a
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 353
asymmetry of the universe.
The strand model allows us to deduce whether the mixing of a meson with its own
antiparticle is possible or not. As expected, only neutral mesons are candidates for such
mixing, because of charge conservation. In the strand model, particle–antiparticle mixing is possible whenever the transition from a neutral meson to its antiparticle is possible
in two ways: by taking the mirror of the meson tangle or by shifting the position of the
binding strands. All mesons for which this is possible are shown in grey boxes in Figure 95, Figure 97 and Figure 98. The strand model also makes it clear that such mixing
requires shifting of the bonds; this is a low-probability process that is due to the weak
interaction. The strand model thus predicts that the weak interaction violates CP invariance in mesons that mix with their antiparticles.
Since the spin 1 mesons decay strongly and thus do not live long enough, the small
effect of CP violation is de facto only observed in pseudoscalar, spin-0 mesons. The
strand model thus predicts observable mixings and CP violation for the mesons pairs
?0 − ?0 , ?0 − ?0 , ?0 − ?0 , ?0s − ?0s . The prediction by the strand model corresponds
precisely to those systems for which CP violation is actually observed. (CP violation in
? mesons was finally discovered at CERN in 2011, after it was predicted both by the
standard model and the strand model, in earlier editions of this volume.)
In the strand model, meson–antimeson mixing is possible because the various quarks
are braided strands. Because of this braid structure, the existence of meson–antimeson
mixing is a consequence of the existence of three quark generations. The meson structures also make it clear that such mixings would not be possible if there were no third
quark generation. The strand model thus reproduces the usual explanation of CP violation as the result of three quark generations.
For the strong and the electromagnetic interaction, the strand model predicts that
there is no mixing and no CP violation, because gluons and photons do not change
particle topology. Therefore, the strand model suggests the absence of axions. The lack of
a suitable tangle for axions, shown later on, then turns this suggestions into a prediction.
In summary, the existence of CP violation in the weak interactions and the lack of CP
violation in the strong interaction are natural consequences of the strand model.
Motion Mountain – The Adventure of Physics
Ref. 231
11 the pa rticle spectrum d ed uced from stra nds
ta ng les of four a nd m ore stra nds
341
spin 1 particle, such as a composed meson? The answer depends on how the strands are
free to move against each other.
The Higgs tangle appears through tangling of vacuum strands, and inherits the zero
spin of vacuum. The W and Z tangles have a special property: two strands can rotate
around the third; this makes them bosons as well, but of spin 1. Fermion tangles have
neither property; their core can only rotate through the belt trick; thus they are fermions.
Summary on three-stranded tangles
tangles of four and more strands
Baryons
In the strand model, rational tangles made of five or six strands are baryons. The quark
tangles of the strand model yield the tangles for baryons in a natural way, as Figure 100
shows. Again, not all quark combinations are possible. First of all, quark tangles do not
allow mixed ? ? ? or ? ? ? structures, but only ? ? ? or ? ? ? structures. In addition, the
tangles do not allow (fully symmetric) spin 1/2 states for ? ? ? or ? ? ?, but only spin 3/2
states. The model also naturally predicts that there are only two spin 1/2 baryons made of
? and ? quarks. All this corresponds to observation. The tangles for the simplest baryons
are shown in Figure 100.
The electric charges of the baryons are reproduced. In particular, the tangle topologies imply that the proton has the same charge as the positron. Neutral baryons have
topologically achiral structures; nevertheless, the neutron differs from its antiparticle, as
can be deduced from Figure 100, through its three-dimensional shape. The Δ baryons
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
If we add one or more strand to a three-strand tangle, no additional class of tangles
appears. The tangle classes remain the same as in the three-strand case. In other words,
no additional elementary particles arise in the strand model. To show this, we start our
exploration with the rational tangles.
We saw above that the rational tangles made of four strands represent the vector
mesons. We have already explored them together with the scalar mesons. But certain
more complicated rational tangles are also important in nature, as we consist of them.
Motion Mountain – The Adventure of Physics
Page 331
Compared to two-stranded tangles, one new class of elementary particles appears for
three strands; the new class is somewhat less tangled than general rational tangles but still
more tangled than the trivial vacuum tangle: the braided tangles. Braided tangles represent the Higgs boson and the leptons; the tangles reproduce all their observed quantum
numbers. The braided tangles also imply that neutrinos and anti-neutrinos differ, are
massive, and are Dirac particles.
The strand model (corrected in 2012) also predicts that, apart from the six leptons and
one Higgs boson, no other elementary particle made of three strands exist in nature.
In the case of composite particles made of three strands, the strand model proposes
tangles for all pseudoscalar mesons; the resulting quantum numbers and mass sequences
match the observed values. In the spectrum of composite particles, the glueball issue is
not completely settled.
342
11 the pa rticle spectrum d ed uced from stra nds
The proton has two basic graphs,
corresponding to u↑ u↓ d↑ and u↑ u↑ d↓ :
Spin S = 1/2, L = 0,
d parity P = +1
u
u
u
uud = p
938 MeV
u
d
The neutron has two basic graphs,
corresponding to d↑ d↓ u↑ and d↑ d↑ u↓ :
d
d
u
d
u
d
d
uud=Δ+
1232 MeV
Circles indicate
pairs of tails to the
border of space
u
u
uuu=Δ++
1232 MeV
u
u
u
d
udd=Δ0
1232 MeV
d
d
ddd=Δ–
1232 MeV
d
d
F I G U R E 100 The simplest strand models for the lightest baryons made of up and down quarks (circles
indicate linked tail pairs to the border of space), together with the measured mass values.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
The four Δ baryons have one graph each, corresponding
to u↑ u↑ u↑, u↑ u↑ d↑, u↑ d↑ d↑ and d↑ d↑ d↑ :
Spin S = 3/2, L = 0,
parity P = +1
u
Motion Mountain – The Adventure of Physics
udd = n
940 MeV
ta ng les of four a nd m ore stra nds
343
Baryons with spin S = 1/2 and angular momentum L = 0 made of up, down and strange quarks :
s
s
s
dss=Ξ−
1322 MeV
s
uss=Ξ0
1315 MeV
d
d
dds=Σ−
1197
MeV
d
s
u
s
u
uds=Λ0
1116
MeV
Parity P = +1 expected
for both Ξ baryons, but
not yet measured.
d
u
uds=Σ0
1192 MeV
u
u
uus=Σ+
1189 MeV
d
s
s
d
d
Circles indicate
pairs of tails to the
border of space
udd=n
940 MeV
u
u
P = +1 for both
neutron and proton.
uud=p
938 MeV
u
d
Page 385
Ref. 230
have different electric charges, depending on their writhe.
Baryons are naturally colour-neutral, as observed. The model also shows that the baryon wave function usually cannot be factorized into a spin and quark part: the nucleons
need two graphs to describe them, and tangle shapes play a role. Baryon parities are reproduced; the neutron and the antineutron differ. All this corresponds to known baryon
behaviour. Also the observed baryon shapes (in other words, the baryon quadrupole
moments) are reproduced by the tangle model.
The particle masses of proton and neutron differ, because their topologies differ. However, the topological difference is ‘small’, as seen in Figure 100, so the mass difference is
small. The topological difference between the various Δ baryons is even smaller, and
indeed, their mass difference is barely discernible in experiments.
The strand model naturally yields the baryon octet and decuplet, as shown in Figure 101 and Figure 102. In general, complicated baryon tangles have higher mass than
simpler ones, as shown in the figures; this is also the case for the baryons, not illustrated
here, that include other quarks. And like for mesons, baryon Regge trajectories are due
to ‘stretching’ and tangling of the binding strands. Since the bonds to each quark are
again (at most) three, the model qualitatively reproduces the observation that the Regge
slope for all baryons is the same and is equal to that for mesons. We note that this also
implies that the quark masses play only a minor role in the generation of hadron masses;
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
F I G U R E 101 One tangle (only) for each baryon in the lowest J=L+S=1/2 baryon octet (circles indicate
linked tail pairs to the border of space), together with the measured mass values.
Motion Mountain – The Adventure of Physics
P = +1 for all four.
344
11 the pa rticle spectrum d ed uced from stra nds
Baryons with spin S = 3/2 and angular momentum L = 0 made of up, down and strange quarks :
s
s
sss=Ω−
1672 MeV
d
s
dss=Ξ∗–
1535 MeV
dds=Σ∗–
1387
MeV
d
d
u
uds=Σ∗0
1384
MeV
d
u
udd=Δ0
1232
MeV
d
s
s
u
uus=Σ∗+
1383
MeV
d
d
u
uud=Δ+
1232
MeV
u
s
u
u
uuu=Δ++
1232
MeV
u
u
F I G U R E 102 One tangle for each baryon in the lowest J=3/2 baryon decuplet (circles indicate linked
tail pairs to the border of space), together with the measured mass values.
this old result from QCD is thus reproduced by the strand model.
The arguments presented so far only reproduce mass sequences, not mass values. Actual hadron mass calculations are possible with the strand model: it is necessary to compute the number of crossing changes each tangle produces. There is a chance, but no
certainty, that such calculations might be simpler to implement than those of lattice QCD.
Tetraquarks and exotic mesons
Ref. 218
Ref. 234
Among the exotic mesons, tetraquarks are the most explored cases. It is now widely believed that the low-mass scalar mesons are tetraquarks. In the strand model, tetraquarks
are possible; an example is given in Figure 103. This is a six-stranded rational tangle. Spin,
parities and mass sequences from the strand model seem to agree with observations. If
the arrangement of Figure 103 would turn out to be typical, the tetraquark looks more
like a bound pair of two mesons and not like a state in which all four quarks are bound in
equal way to each other. On the other hand, a tetrahedral arrangement of quarks might
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
ddd=Δ–
1232
MeV
d
d
s
uss=Ξ∗0
1532 MeV
s
s
u
Motion Mountain – The Adventure of Physics
d
Circles indicate
pairs of tails to the
border of space
s
ta ng les of four a nd m ore stra nds
345
The scalar σ meson as a tetraquark (ud)(ud), c. 0.5 GeV :
d
u
d
u
Circles indicate
pairs of tails to the
border of space
F I G U R E 103 The strand model for a specific tetraquark (circles indicate linked tail pairs to the border of
space).
Page 324, page 340
O ther tangles made of four or more strands
Summary on tangles made of four or more strands
By exploring all possible tangle classes in detail, we have shown that every localized structure made of strands has an interpretation in the strand model. In particular, the strand
model makes a simple statement on any tangle made of four or more strands: such a
tangle is composite of the elementary tangles made of one, two or three strands. In other
terms, there are no elementary particles made of four or more strands in nature.
The strand model states that each possible tangle represents a physical particle sys-
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
We do not need to explore other prime tangles or locally knotted tangles made of four
or more strands. They are either not allowed or are higher-order versions of rational
tangles, as explained already in the case of two and three strands. We also do not need to
explore separable tangles. Separable tangles are composite of tangles with fewer strands.
One class of tangles remains to be discussed: braided tangles of four or more strands.
Now, a higher-order perturbation of the weak interaction can always lead to the topological entanglement of some vacuum strand with a tangle of fewer strands. Braided
tangles of four or more strands are thus higher-order propagating states of three-stranded
leptons or hadrons.
We can also state this in another way. There are no tangles of four or more strands
that are more tangled than the trivial tangle but less tangled than the lepton tangles.
Therefore, no additional elementary particles are possible. In short, the tangle model
does not allow elementary particles with four or more strands.
Motion Mountain – The Adventure of Physics
Ref. 220
also be possible. The details of this topic are left for future exploration.
The strand model makes an additional statement: knotted (hadronic) strings in
quark–antiquark states are impossible. Such states have been proposed by Niemi. In the
strand model, such states would not be separate mesons, but usual mesons with one or
several added virtual weak vector bosons. This type of exotic mesons is therefore predicted not to exist.
The situation for glueballs, which are another type of exotic mesons, has already been
discussed above.
346
11 the pa rticle spectrum d ed uced from stra nds
TA B L E 13 The match between tangles and particles in the strand model.
Strands
Ta n g l e
Pa r t i c l e
Ty pe
1
1
unknotted
knotted
elementary
–
vacuum, (unbroken) gauge boson
not in the strand model
2
2
2
unknotted
rational
prime, knotted
composed
elementary
–
composed of simpler tangles
quark or graviton
not in the strand model
3
3
3
unknotted
braided
rational
composed of simpler tangles
lepton
leptons
3
prime, knotted
composed
elementary
elementary or
composed
–
4 & more
like for 3 strands
all composed
composed of simpler tangles
not in the strand model
Challenge 199 s
In the strand model, mass appears due to tail braiding. But mass is also due to tangle
rotation and fluctuation. How do the two definitions come together?
∗∗
The following statement seems absurd, but is correct:
⊳ The tangle model implies that all elementary particles are point-like, without
internal structure.
Indeed, if at all, the strand model implies deviations from point-like behaviour only at
Planck scale; particles are point-like for all practical purposes.
∗∗
Challenge 200 e
In the strand model, only crossing switches are observable. How then can the specific tangle structure of a particle have any observable effects? In particular, how can
quantum numbers be related to tangle structure, if the only observables are due to crossing changes?
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
fun challenges and curiosit ies abou t particle
tangles
Motion Mountain – The Adventure of Physics
tem: an overview is given in Table 13. The mapping between tangles and particles is
only possible because (infinitely) many tangles are assigned to each massive elementary
particle.
The result of this exploration is that the strand model limits the number of elementary
particles to those contained in the standard model of particle physics.
fun cha llenges a nd curi osities a b ou t pa rticle ta ng les
347
∗∗
No neutral weak currents that change strangeness or other flavours are observed. In the
strand model this observation is a consequence of the tangle shape of the Z boson.
∗∗
Challenge
Ref.201
235r
In 2014, Marek Karliner predicted the existence of six-quark states. Can the strand model
reproduce them? Can it settle whether they are molecules of three mesons or genuine
six-quark states?
∗∗
Challenge 202 e
Can you use the strand model to show that pentaquarks do not exist?
∗∗
Ref. 236
What is the relation of the model shown here to the ideas of Viro and Viro on skew lines?
Ref. 237
The most prominent proponent of the idea that particles might be knots was, in 1868,
William Thomson–Kelvin. He proposed the idea that different atoms might be differently
‘knotted vortices’ in the ‘ether’. The proposal was ignored – and rightly so – because
it did not explain anything: neither the properties nor the interactions of atoms were
explained. The proposal simply had no relation to reality. In retrospect, the main reason
for this failure was that elementary particles and quantum theory were unknown at the
time.
Purely topological models for elementary particles have been proposed and explored by
various scholars in the past. But only a few researchers ever proposed specific topological
structures for each elementary particle. Such proposals are easily criticized, so that it is
easy to make a fool of oneself; any such proposal thus needs a certain amount of courage.
Ref. 238
Ref. 143
Ref. 239
Ref. 139
Ref. 141
Ref. 140
— Herbert Jehle modelled elementary particles as closed knots already in the 1970s.
However, his model did not reproduce quantum theory, nor does it reproduce all
particles known today.
— Ng Sze Kui has modelled mesons as knots. There is however, no model for quarks,
leptons or bosons, nor a description for the gauge interactions.
— Tom Mongan has modelled elementary particles as made of three strands that each
carry electric charge. However, there is no connection with quantum field theory or
general relativity.
— Jack Avrin has modelled hadrons and leptons as Moebius bands, and interactions
as cut-and-glue processes. The model however, does not explain the masses of the
particles or the coupling constants.
— Robert Finkelstein has modelled fermions as knots. This approach, however, does not
explain the gauge properties of the interactions, nor most properties of elementary
particles.
— Sundance Bilson-Thompson, later together with his coworkers, modelled elementary
fermions and bosons as structures of triple ribbons. The leather trick is used, like in
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
∗∗
Motion Mountain – The Adventure of Physics
∗∗
348
11 the pa rticle spectrum d ed uced from stra nds
the strand model, to explain the three generations of quarks and leptons. This is by
far the most complete model from this list. However, the origin of particle mass, of
particle mixing and, most of all, of the gauge interactions is not explained.
∗∗
Ref. 240
Strands do not require higher dimensions. On the other hand, it can be argued that
strands do produce an additional non-commutative structure at each point in space. In
a sense, when strands are averaged over time, a non-commutative inner space is created
at each point in space. As a result, when we focus at a specific spatial position over
somewhat longer times scales than the Planck time, we can argue that, at that point of
space, nature is described by a product of three-dimensional space with an internal, noncommutative space. Since many years, Alain Connes and his colleagues have explored
such product spaces in detail. They have discovered that with an appropriately chosen
non-commutative inner space, it is possible to reproduce many, but not all, aspects of the
standard model of particle physics. Among others, choosing a suitable non-commutative
space, they can reproduce the three gauge interactions; on the other hand, they cannot
reproduce the three particle generations.
Connes’ approach and the strand model do not agree completely. One way to describe the differences is to focus on the relation of the inner spaces at different points of
space. Connes’ approach assumes that each point has its own inner space, and that these
spaces are not related. The strand model, instead, implies that the inner spaces of neigh-
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
∗∗
Motion Mountain – The Adventure of Physics
Ref. 144
Strands are not superstrings. In contrast to superstrings, strands have a fundamental
principle. (This is the biggest conceptual difference.) The fundamental principle for
strands is not fulfilled by superstrings. In contrast to superstrings, strands have no tension, no supersymmetry and no own Lagrangian. (This is the biggest physical difference.)
Because strands have no tension, they cannot oscillate. Because strands have no supersymmetry, general relativity follows directly. Because strands have no own Lagrangian,
particles are tangles, not oscillating superstrings, and quantum theory follows directly.
In fact, the definitions of particles, wave functions, fields, vacuum, mass and horizons
differ completely in the two approaches.
In contrast to superstrings, strands describe the number of gauge interactions and of
particle generations. In contrast to superstrings, strands describe quarks, hadrons, confinement, Regge behaviour, asymptotic freedom, particle masses, particle mixing and
coupling constants. In the strand model, in contrast to ‘open superstrings’, no important configuration has ends. In contrast to open or closed superstrings, strands move in
three spatial dimensions, not in nine or ten; strands resolve the anomaly issue without
higher dimensions or supersymmetry, because unitarity is automatically maintained, by
construction; strands are not related to membranes or supermembranes. In the strand
model, no strand is ‘bosonic’ or ‘heterotic’, there is no E(8) or SO(32) gauge group, there
are no general ‘pants diagrams’ for all gauge interactions, there is no AdS/CFT duality,
there is no ‘landscape’ with numerous vacuum states, and there is no ‘multiverse’. In
contrast to superstrings, strands are based on Planck units. And in contrast to superstrings, strands yield the standard model of elementary particles without any alternative.
In fact, not a single statement about superstrings is applicable to strands.
fun cha llenges a nd curi osities a b ou t pa rticle ta ng les
Hanging
situation
349
A
B
bouring points are related; they are related by the specific topology and entanglement of
the involved strands. For this very reason the strand model does allow to understand the
origin of the three particle generations and the details of the particle spectrum.
There are further differences between the two approaches. Connes’ approach assumes
that quantum theory and general relativity, in particular, the Hilbert space and the spatial manifold, are given from the outset. The strand model, instead, deduces these structures from the fundamental principle. And, as just mentioned, Connes’ approach is not
unique or complete, whereas the strand model seems to be. Of the two, only the strand
model seems to be unmodifiable, or ‘hard to vary’.
∗∗
The strand model implies that there is nothing new at small distances. At small distances,
or high energies, nature consists only of strands. Thus there are no new phenomena
there. Quantum theory states that at small scales, nothing new appears: at small scales,
there are no new degrees of freedom. For example, quantum theory states that there is no
kingdom Lilliput in nature. The strand model thus confirms the essence of quantum theory. And indeed, the strand model predicts that between the energy scale of the heaviest
elementary particle, the top quark, 173 GeV, and the Planck energy, 1019 GeV, nothing
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
motion along the chain, when holding ring B
while dropping ring A.
Motion Mountain – The Adventure of Physics
F I G U R E 104 A ring chain gives an impression of
350
F I G U R E 105 The ring chain trick produces an illusion of motion (mp4 film © Franz Aichinger). Can
more rings be added in horizontal directions?
Translational motion of a photon :
t2
Translational motion of an electron :
t1
t2
F I G U R E 106 Motion of photons and electrons through strand hopping.
is to be found. There is a so-called energy desert – empty of interesting features, particles
or phenomena – in nature.
∗∗
Most ropes used in sailing, climbing or other domains of everyday life are produced
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
t1
Motion Mountain – The Adventure of Physics
Challenge 203 e
11 the pa rticle spectrum d ed uced from stra nds
fun cha llenges a nd curi osities a b ou t pa rticle ta ng les
351
F I G U R E 107 A discarded candidate tangle for the W boson.
∗∗
Challenge 204 e
Not all tangle assignments are self-evident at first sight. Figure 107 shows a tangle whose
status in the strand model is not clear. Can you explain what the tangle represents?
∗∗
Page 150
Challenge 205 ny
What is the effect of shivering on braiding, and thus on weak particle mixing, on particle
tangle families and on the number of generations?
Challenge 206 e
Are all bosons made of strands whose ends are exactly opposite to each other at spatial
infinity? Photon, graviton, gluon, W, Z and the virtual Higgs comply. The unbroken ones
are axial, the broken ones are flat. Is there a reason or a sense for this issue?
CPT invariance
CPT invariance is a fundamental property of quantum field theory. In the strand model,
charge conjugation C is modelled as a mirror transformation of the tangle; parity P is
modelled as the change of sign of the belt trick of the tangle core; and motion inversion
T is modelled as the inverse motion of the core of a particle tangle.
In other words, CPT invariance is natural in the strand model. Therefore, the strand
model predicts that particles and antiparticles have the same g-factor, the same dipole
moment, the same mass, the same spin, exactly opposite charge value, etc. All this is also
predicted by quantum field theory, and is confirmed by experiment.
Motion through the vacuum – and the speed of light
Ref. 241
Up to now, one problem was left open: How can a particle, being a tangle of infinite
extension, move through the web of strands that makes up the vacuum? An old trick,
known already in France in the nineteenth century, can help preparing for the idea of
particle motion in space. Figure 104 shows a special chain that is most easily made with a
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
∗∗
Motion Mountain – The Adventure of Physics
by braiding. Searching for ‘braiding machine’ on the internet yields a large amount of
videos. Searching for ‘LEGO braiding machine’ shows the most simple and beautiful
examples and allows you to see how they work.
352
11 the pa rticle spectrum d ed uced from stra nds
few dozen key rings. If the ring B is grabbed and the ring A released, this latter ring seems
to fall down along the whole chain in a helical path, as shown in the film of Figure 203.
If you have never seen the trick, try it yourself; the effect is astonishing. In reality, this is
an optical illusion. No ring is actually falling, but the sequence of rings moves in a way
that creates the impression of ring motion. And this old trick helps us to solve a number
of issues about particle motion that we swept under the carpet so far.
The main idea on particle motion in the strand model is the following:
⊳ Translational particle motion is also due to strand substitution, or ‘strand
hopping’.
”
summary on millennium issues abou t particles and
the vacuum
We have discovered that the strand model makes a strong statement: elementary particles
can only be made of one, two or three tangled strands. Each elementary particle is represented by an infinite family of rational tangles of fixed strand number. The family members are related through various degrees of tangling, such as tail braiding or the leather
trick.
For one-stranded particles, the strand model shows that the photon, the W, the Z and
the gluons form the full list of spin-1 bosons. For two-stranded particles, the strand model
shows that there are precisely three generations of quarks. For three-stranded elementary
particles, the strand model shows that there is a Higgs boson and three generations of
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
“
The ground of science was littered with the
corpses of dead unified theories.
Freeman Dyson
Motion Mountain – The Adventure of Physics
A schematic illustration of translational motion is given in Figure 106. In the strand
model, contrary to the impression given so far, a tangle does not always need to move
as a whole along the strand. This is seen most easily in the case of a photon. It is easy to
picture that the tangle structure corresponding to a photon can also hop from strand to
strand. At any stage, the structure is a photon; but the involved strand is never the same.
The idea of motion through strand hopping also works for massive particles. The
motion of a massive particle, such as an electron, is shown schematically in Figure 106.
The figure shows that through a tail unbraiding, the structure that describes an electron
can get rid of one strand and grab a new one. This process has a low probability, of
course. In the strand model, this is one reason that massive particles move more slowly
than light, even if the first approximation yields a zero mass value.
We note that this explanation of motion is important also for the mapping from strand
diagrams to Feynman diagrams. For many such diagrams, for example for the annihilation of particles and antiparticles in QED, strand hopping and tail unbraiding play a role.
Without them, the mapping from strands to quantum field theory would not be possible.
In summary, tangles of massive particles can move through the vacuum using hopping – via tail unbraiding – and this naturally happens more slowly than the motion of
photons, which do not need any process at the border of space to hop. The speed of
photons is thus a limit speed for massive particles; special relativity is thus recovered.
sum m ary on m illennium issues a b ou t pa rticles a nd the vacuum
Page 153
Page 162
leptons. Neutrinos and antineutrinos differ and are massive Dirac particles. The strand
model thus predicts that the neutrino-less double-beta decay will not be observed. Glueballs probably do not exist.
The strand model explains the origin of all quantum numbers of the observed elementary particles. Also all predicted quantum numbers for composed particles agree
with observations. Therefore, we have also completed the proof that all observables in
nature are due to crossing switches.
The strand model reproduces the quark model, including all the allowed and all the
forbidden hadron states. For mesons and baryons, the strand model predicts the correct
mass sequences and quantum numbers. Tetraquarks are predicted to exist. A way to
calculate hadron form factors is proposed.
In the strand model, all tangles are mapped to known particles. The strand model
predicts that no elementary particles outside the standard model exist, because no tangles
are left over. For example, there are no axions, no leptoquarks and no supersymmetric
particles in nature. The strand model also predicts the lack of other gauge bosons and
other interactions. In particular, the strand model – corrected in 2012 – reproduces the
existence of the Higgs boson. In fact, any new elementary particle found in the future
would contradict and invalidate the strand model.
In simple words, the strand model explains why the known elementary particles exist
and why others do not. We have thus settled two further items from the millennium list
of open issues. In fact, the deduction of the elementary particle spectrum given here is,
the first and, at present, also the only such deduction in the research literature.
The omnipresent number 3
Predictions abou t dark mat ter, the LHC and the vacuum
Astrophysical observations show that galaxies and galaxy clusters are surrounded by
large amounts of matter that does not radiate. This unknown type of matter is called
dark matter.
In the strand model, the known elementary particles are the only possible ones.
Therefore, the galactic clouds made of dark matter must consist of those particles mentioned up to now, or of black holes.
⊳ The strand model thus predicts that dark matter is a mixture of particles of
the standard model and black holes.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 206
The strand model shows that the number 3 that appears so regularly in the standard
model of particle physics – 3 generations, 3 interactions, charge values ?/3 and 2?/3
of quarks (as shown below), 3 colours and SU(3) – is, in each case, a consequence of
the three-dimensionality of space. In fact, the strand model adds a further, but related
number 3 to this list, namely the maximum number of strands that make up elementary
particles.
The three-dimensionality of space is, as we saw already above, a result of the existence
of strands: only three dimensions allow tangles of strands. In short, all numbers 3 that
appear in fundamental physics are explained by strands.
Motion Mountain – The Adventure of Physics
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353
354
Page 162
11 the pa rticle spectrum d ed uced from stra nds
This statement settles a further item from the millennium list of open issues.
The prediction from 2008 of a lack of new elementary particles in dark matter is at
odds with the most favoured present measurement interpretations, but cannot yet be
ruled out. The detection of black hole mergers in 2015 can even be seen as a partial
confirmation. However, the issue is obviously not yet settled. In fact, the prediction
provides another hard test of the model: if dark matter is found to be made of yet unknown particles, the strand model is in trouble.
We can condense all the results on particle physics found so far in the following statement:
⊳ There is nothing to be discovered about nature outside general relativity and
the standard model of particle physics.
Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Strands predict that there is no hidden aspect of nature left. In particular, the strand
model predicts a so-called high-energy desert: it predicts the lack of any additional elementary particle. Equivalently, the strand model predicts that apart from the Planck
scale, there is no further energy scale in particle physics. Researchers blinded by beliefs
sometimes call this the nightmare scenario.
In other words, there is no room for discoveries beyond the Higgs boson at the Large
Hadron Collider in Geneva, nor at the various dark matter searches across the world. If
any new elementary particle is discovered, the strand model is wrong. More precisely, if
any new elementary particle that contradicts the strand model is discovered, the strand
model is wrong. That some unknown elementary particle has been missed in the present
exploration of tangle classes is still a logical possibility.
Because the strand model confirms the standard model and general relativity, a further prediction can be made: the vacuum is unique and stable. There is no room for other
options. For example, there are no domains walls between different vacuum states and
the universe will not decay or change in any drastic manner.
In summary, the strand model predicts a lack of any kind of science fiction in modern
physics.
C h a p t e r 12
PART ICLE PR OPERT IES DEDUCED
FROM ST R ANDS
”
** ‘Everything you see, I owe it to spaghetti.’ Sofia Villani Scicolone is an Italian actress and Hollywood
star.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
he Planck units, via strands and the fundamental principle, explain almost all
hat is known about motion: strands explain what moves and how it moves. But
he strand model is only correct if it also explains every measured property of every
elementary particle. So far, we only deduced the quantum numbers of the elementary
particles. Three kinds of particle properties from the millennium list remain open: the
masses, the mixing angles and the couplings. These measured particle properties are important, because they determine the amount of change – or physical action – induced by
the motion of each elementary particle.
So far, the strand model has answered all open questions on motion that we explored.
In particular, the strand model has explained why quantum field theory, the interactions,
the particle spectrum, general relativity and cosmology are what they are. But as long
as we do not understand the measured properties of elementary particles, we do not
understand motion completely.
In short, the next step is to find a way to calculate these particle properties – and
obviously, to show that the calculations agree with the measurements. The step is particularly interesting; so far, no other unified model in the research literature has ever
achieved such calculations – not even calculations that disagree with measurements.
Because the strand model makes no experimental predictions that go beyond general
relativity and the standard model of particle physics, explaining the properties of elementary particles is the only way to confirm the strand model. Many ways to test and
to refute the strand model are possible; but only a calculation of the measured particle
properties can confirm it.
The ideas in this chapter are more speculative than those of the past chapters, because
the reasoning depends on the way that specific tangles are assigned to specific particles.
Such assignments are never completely certain. We continue keeping this in mind.
Motion Mountain – The Adventure of Physics
T
“
Tutto quel che vedete, lo devo agli spaghetti.**
Sophia Loren
356
12 pa rticle properties d ed uced from stra nds
TA B L E 14 The measured elementary particle masses, as given by the Particle Data Group in 2016.
Electron neutrino
Muon neutrino quark
Tau neutrino
< 2 eV/c2
< 2 eV/c2
< 2 eV/c2
Electron
Muon
Tau
0.510 998 9461(31) MeV/c2
105.658 3745(24) MeV/c2
1776.86(12) MeV/c2
Up quark (? = 2/3 ?)
Down quark (? = −1/3 ?)
Strange quark (? = −1/3 ?)
Charm quark (? = 2/3 ?)
Bottom quark (? = −1/3 ?)
Top quark (? = 2/3 ?)
2.2(6) MeV/c2
4.7(5) MeV/c2
96(8) MeV/c2
1.27(3) GeV/c2
4.18(4) GeV/c2
173.21(1.22) GeV/c2
W boson
Z boson
Higgs boson
80.385(15) GeV/c2
91.1876(21) GeV/c2
125.09(24) GeV/c2
Photon
Gluons
Graviton
not detectable
not detectable
not detectable
For comparison:
the corrected Planck mass
0.61 ⋅ 1019 GeV/c2
the masses of the elementary particles
The mass describes the inertial and gravitational effects of a body. The strand model
must reproduce all mass values observed in nature; if it doesn’t, it is wrong.
To reproduce the masses of all bodies, it is sufficient that the strand model reproduces
the measured masses, the mixing angles and the coupling strengths of the elementary
particles. We start with their masses. The measured values are given in Table 14; all these
values are unexplained. They are part of the millennium list of open issues in fundamental physics.
In nature, the gravitational mass of a particle is determined by the space curvature
that it induces around it. In the strand model, this curvature is due to the modified
fluctuations that result from the presence of the tangle core; in particular, the curvature
is due to the modified fluctuations of the particle tails and to the modified vacuum strand
fluctuations just around the particle position. The modified strand fluctuations produce
a crossing switch distribution around the tangle core; the crossing switch distribution
leads to spatial curvature; at sufficiently large distances, this curvature distribution is
detected as a gravitational mass.
In contrast, inertial mass appears in the Dirac equation. In the strand model, iner-
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
M a s s va l u e
Motion Mountain – The Adventure of Physics
E l e m e n tary pa rt i c l e
the m asses of the elem enta ry pa rticles
Page 280
357
tial mass is determined by the frequency and the wavelength of the helix drawn by the
rotating phase vector. These quantities in turn are influenced by the type of tangle, by
the fluctuations induced by the particle charges, by the topology changes induced by the
weak interaction, and, in the case of fermions, by the average frequency and size of the
belt and leather tricks. All these processes are due to strand fluctuations.
In short, both gravitational and inertial particle mass are due to strand fluctuations.
More specifically, the mass seems mainly due to the fluctuations of the tails of the particle
tangle. The strand model thus suggests that gravitational and inertial mass are automatically equal. In particular, the strand model suggests that every mass is surrounded by
fluctuating crossing switches whose density decreases with distance and is proportional
to the mass itself. As discussed above, this idea leads to universal gravity.
General properties of particle mass values
The general properties of particle masses are thus reproduced by the strand model.
Therefore, continuing our exploration makes sense. We start by looking for ways to
determine the mass values from the tangle structures. We discuss each particle class
separately, first looking at mass ratios, then at absolute mass values.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 370
— The strand model implies that the masses of elementary particles are not free parameters, but that they are determined by the topology, or tangledness, of the underlying
tangles and their tangle families. Particle masses are thus fixed and discrete in the
strand model – as is observed. Of course, we have to take into account the many
members in each tangle family.
— The strand model implies that masses are always positive numbers.
— The strand model implies that the more complex a tangle is, the higher its mass value
is. This follows from the behaviour of tangle tail fluctuations around the tangle core.
— Because particle masses are due to strand fluctuations, the strand model also implies
that all elementary particle masses are much smaller than the Planck mass, as is observed. Also this result follows from the behaviour of tangle tail fluctuations around
the tangle core.
— Because particle masses are due to strand fluctuations, particle and antiparticle
masses – their tangles are mirrors of each other – are always equal, as is observed.
— Because particle masses are due to strand fluctuations, particle masses do not depend
on the age of the universe, nor on their position in the universe, nor on any other state
variable: The strand model predicts that particle masses are constant and invariant,
as is observed.
— Because particle masses are due to strand fluctuations, and the fluctuations differ
somewhat for tight and loose tangles of the same shape, the strand model predicts
that particle masses change – or run – with energy, as is observed.
Motion Mountain – The Adventure of Physics
So far, our adventure allows us to deduce several results on the mass values of elementary
particles:
358
12 pa rticle properties d ed uced from stra nds
Tight W boson tangle candidate
Tight Z boson tangle candidate
B oson masses
Three elementary particles of integer spin have non-vanishing mass: the W boson, the Z
boson and the Higgs boson. Mass calculations are especially simple for bosons, because
in the strand model, they are clean systems: each boson is described by a relatively simple
tangle family; furthermore, bosons do not need the belt trick to rotate continuously.
We expect that the induced curvature, and thus the gravitational mass, of an elementary boson is due to the disturbance it introduces into the vacuum. At Planck energy,
this disturbance will be, to a large extent, a function of the ropelength introduced by the
corresponding tight tangle. Let us clarify these concepts.
Tight or ideal tangles or knots are those tangles or knots that appear if we imagine
strands as being made of a rope of constant diameter that is infinitely flexible, infinitely
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
F I G U R E 108 Tight tangle candidates (of 2015/2016) for the simplest tangles of the W, the Z and the
Higgs bosons.
Motion Mountain – The Adventure of Physics
Tight Higgs tangle candidate
the m asses of the elem enta ry pa rticles
359
slippery and pulled as tight as possible. Examples of tight tangles are shown in Figure 108.
With physical ropes from everyday life, tight knots and tangles can only be approximated, because they are not infinitely flexible and slippery; tight tangles are mathematical
idealizations. But tight tangles of strands are of special interest: if we recall that each
strand has an effective diameter of one Planck length, tight tangles realize the Planck
limit of the strand model.
— The ropelength of a tight closed knot is the length of a perfectly flexible and slippery rope of constant diameter required to tie the tight knot. In other words, the
ropelength is the smallest amount of idealized rope needed to tie a knot.
— The ropelength of a tight open knot is the length by which a very long rope tied into a
tight knot is shortened.
— With a bit of care, the concept of ropelength can be also be defined for tangles of
several strands.
W/Z boson mass ratio and mixing angle (in the 2016 tangle model)
Page 358
Ref. 244
Candidates for the simplest tangles of the W boson and of the Z boson families are shown
in Figure 108. The corresponding ropelength values for tight tangles, determined numerically, are ? W = 4.28 and ? Z = 7.25 rope diameters. The strand model estimates the W/Z
mass ratio by the cube root of the ropelength ratio:
1/3
?W
?
≈ ( W ) = 0.84 .
?Z
?Z
Ref. 231
(202)
This value has to be compared with the experimental ratio of 80.4 GeV/91.2 GeV=0.88.
The agreement between experiment and strand model is not good, but acceptable, for
two reasons.
On the one hand, the strand model reproduces the higher value of the neutral Z bo-
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 242
Motion Mountain – The Adventure of Physics
Page 357
In the following, the ropelength is assumed to be measured in units of the rope diameter.
Measuring ropelength in units of the rope radius is less common.
In the strand model, the ropelength measures, to a large extent, the amount by which
a tight knot or tangle disturbs the vacuum around it. The ropelength fulfils all the properties of particle mass mentioned above: the ropelength is discrete, positive, increases with
tangle complexity, is equal for particles and antiparticles, and is a constant and invariant
quantity. The ropelength will thus play an important role in any estimate of a particle
mass.
It is known from quantum field theory that the masses of W and Z bosons do not
change much between Planck energy and everyday energy, whatever renormalization
scheme is used. This allows us, with a good approximation, to approximate the weak
boson masses at low, everyday energy with their mass values at Planck energy. Thus we
can use tight tangles to estimate boson masses.
In the strand model, the gravitational mass of a spin 1 boson is proportional to the
radius of the disturbance that it induces in the vacuum. For a boson, this radius, and
thus the mass, scales as the third root of the ropelength of the corresponding tight tangle.
360
Page 250
Ref. 162
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 194, page 212
son’s mass: a tangle with spatial symmetry is more complex than one without. On the
other hand, is also clear why the calculated mass ratio does not match the experimental
result. First of all, the simple tangles represent and approximate W and Z bosons only to
the first order. As mentioned above, in the strand model, every massive particle is represented by an infinite family of tangles. The strand model thus also predicts that the match
between the calculated and the measured ratio ?W /?Z should improve when higherorder Feynman diagrams, and thus more complicated tangle topologies, are taken into
account. Improving the calculation is still a subject of research. Secondly, approximating
the tight knot effects with an effective radius, thus just using the ropelength to determine
the mass, implies neglecting the actual shape, and effectively approximating their shape
by a sphere. Thirdly, as already mentioned, this calculation assumes that the low energy
mass ratio and the mass ratio at Planck energy are equal.
Despite the used approximations, the tight tangle estimate for the W/Z mass ratio
gives an acceptable agreement with experiment. The main reason is that we expect the
strand fluctuations from the various family members to be similar for particles with the
same number of strands. For these mass ratios, the tail braiding processes cancel out.
Also the other two approximations are expected to be roughly similar for the two weak
bosons. This similarity explains why determining the W/Z boson mass ratio is possible
with acceptable accuracy.
The W/Z mass ratio also determines the weak mixing angle ?w of the weak interaction
Lagrangian, through the relation cos ?w = ?W /?Z. The strand model thus predicts the
value of the weak mixing angle to the same accuracy as it predicts the W/Z mass ratio.
This argument leads to a puzzle: Can you deduce from the strand model how the W/Z
mass ratio changes with energy?
Also the inertial masses of the W and Z bosons can be compared. In quantum theory, the inertial mass relates the wavelength and the frequency of the wave function. In
the strand model, a quantum particle that moves through vacuum is a tangle core that
rotates while advancing. The frequency and the wavelength of the helix thus generated
determine the inertial mass. The process is analogous to the motion of a body moving
at constant speed in a viscous fluid at small Reynolds numbers. Despite the appearance
of friction, the analogy is possible. If a small body of general shape is pulled through a
viscous fluid by a constant force, such as gravity, it follows a helical path. This analogy
implies that, for spin 1 particles, the frequency and the wavelength are above all determined by the effective radius of the small body. The strand model thus suggests that the
inertial mass – inversely proportional to the path frequency and the path wavelength
squared – of the W or the Z boson is approximately proportional to its tight knot radius.
This yields again a cube root of the ropelength and thus gives the same result as for the
gravitational mass.
Also the inertial mass is not exactly proportional to the average tight knot radius;
the precise shape of the tight knot and the other tangle family members play a role. The
strand model thus predicts that a more accurate mass calculation has to take into account
these effects.
In summary, the strand model predicts a W/Z mass ratio and thus a weak mixing
angle close to the observed ratio, and explains the deviation of the approximation from
the measured value – provided that the tangle assignments are correct.
Motion Mountain – The Adventure of Physics
Challenge 207 ny
12 pa rticle properties d ed uced from stra nds
the m asses of the elem enta ry pa rticles
361
The g-factor of the W boson
Ref. 231
Ref. 243
Experiments show that the W boson has a g-factor with the value ?? = 2.2(2). This result
– whose limited accuracy does not allow to detect any anomalous magnetic moment yet
– can be compared to the prediction of the strand model. In particular, the observation
might be used to eliminate certain tangle candidates for the W boson.
The strand model makes a simple prediction for charged elementary particles: because mass rotation and charge rotation are both due to the rotation of the particle core,
the g-factor of all such particles is 2 – in the approximation that neglects Feynman diagrams of higher order. In particular, the g-factor of the W boson is predicted to be 2 in
this approximation. Also this prediction thus agrees both with experiment and with the
standard model of particle physics.
The Higgs/Z boson mass ratio
Page 358
Ref. 244
(203)
Starting with the W boson yields an estimate for the Higgs mass of 128 GeV. Both estimates are not good but acceptable, given that the non-sphericity of the W, Z and Higgs
boson tangles have not been taken into account. (The strand model suggests that for a
strongly non-spherical shape – such as the shape of the W, Z and Higgs tangle – the effective mass is higher than the value deduced from ropelength alone.) Deducing better
mass ratio estimates for the W, Z and Higgs tangles is still a subject of research.
In summary, the strand model predicts a Higgs/Z, a Higgs/W and a W/Z mass ratio
close to the observed values; and the model suggests explanations for the deviations of
the approximation from the observed value – provided that the tangle assignments for
the three bosons are correct.
A first approximation for absolute boson mass values
The tangles for the W, Z and Higgs bosons also provide a first approximation for their
absolute mass values. The tangles are rational; in particular, each tangle is made of strands
that can be pulled straight. This implies, for each strand separately, that a configuration
with no extra strand length and no net core rotation is possible. As a result, in the first
approximation, the gravitational mass and the inertial mass of the elementary bosons
both vanish.
A better approximation for mass values requires to determine, for each boson, the
probability of crossing switches in and around its tangle core. This probability depends
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
?Higgs ≈ 121 GeV .
Motion Mountain – The Adventure of Physics
Page 332
The observed mass value of the Higgs boson is 125(1) GeV. The observed mass value for
the Z boson is 91.2(1) GeV. Like for the other bosons, the strand model suggests using
the ropelength to estimate the mass of the Higgs boson tangle. The candidate tangle
for the Higgs boson was illustrated above in Figure 94, and its tight version is shown in
Figure 108.
The ropelength of the tight Higgs tangle turns out to be 17.1 diameters, again determined by numerical approximation. This value yields a naive mass estimate for the Higgs
boson of (17.1/7.25)1/3 ⋅ 91.2 GeV, i.e.,
362
12 pa rticle properties d ed uced from stra nds
Antiquarks :
Parity P = +1, B = 1/3, spin S = 1/2
Q = –1/3
Q = +2/3
P = –1, B = –1/3, S = 1/2
Q = +1/3
Q = –2/3
d
u
d
u
5.0 ± 1.6 MeV
2.5 ± 1.1 MeV
5.0 ± 1.6 MeV
2.5 ± 1.1 MeV
s
c
s
c
105 ± 35 MeV
1.27 ± 0.11 GeV
105 ± 35 MeV
1.27 ± 0.11 GeV
b
t
b
t
4.20 ± 0.17 GeV
171.3 ± 2.3 GeV
4.20 ± 0.17 GeV
171.3 ± 2.3 GeV
Seen from a larger distance, the tails follow (on average) the skeleton of a tetrahedron :
s
s
are also given.
on the probabilities for tail braiding and for core rotation. These probabilities are low,
because, sloppily speaking, the corresponding strand fluctuations are rare. The rarity is
a due to the specific tangle type: tangles whose strands can be pulled straight have low
crossing switch probabilities at their core or at their tails when they propagate.
The strand model thus predicts that elementary boson masses, like all other elementary particle masses, are much smaller than the Planck mass, though not exactly zero. This
prediction agrees with observation: experimentally, the three elementary boson mass
values are of the order of 10−17 Planck masses. We will search for more precise mass
estimates below.
Q uark mass ratios
Quarks are fermions. In the strand model, mass estimates for fermions are more difficult
than for bosons, because their propagation involves the belt trick. Still, using Figure 109,
the strand model allows several predictions about the relations between quark masses.
— The quark masses are predicted to be the same for every possible colour charge. This
is observed.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
F I G U R E 109 The simplest tangles assigned to the quarks and antiquarks. The experimental mass values
Motion Mountain – The Adventure of Physics
Quarks :
the m asses of the elem enta ry pa rticles
Page 320
Page 372
363
— Furthermore, the progression in ropelength of the tight basic tangles for the six
quarks suggests a progression in their masses. This is observed, though with the exception of the up quark mass. For this exceptional case, effects due to tail braiding
and to quark mixing are expected to play a role, as argued below.
? ∼ e??
Ref. 245
Ref. 247
where ? is an unknown number of order 1. We note directly that such a relation promises
general agreement with the observed quark mass ratios.
Actual ropelength calculations by Eric Rawdon and Maria Fisher show that the
ropelength of quark tangles increases roughly linearly with ?, as expected from general
knot theoretic arguments. Their results are given in Table 15. Comparing these calculated ropelength differences with the known low-energy quark masses confirms that the
number ? has an effective value in the range between 0.4 and 0.9, and thus indeed is of
order one.
The results of Table 15 suggest that the top quark should be particularly heavy – as
is observed. The results of Table 15 also suggest that something special is going on for
the ?-? quark pair, which is out of sequence with the other quarks. Indeed, the strand
model predicts a very small mass, – at the Planck scale – for the down quark. However,
in nature, the down mass is observed to be larger than the up mass. (We note that despite
this issue, meson mass sequences are predicted correctly.) It could well be that the mirror
symmetry of the simplest down quark tangle is the reason that the braiding, i.e., the
mixing with more massive the family members with six and more additional crossings,
is higher than that for the up quark.
The experimental values for the quark masses are given in Table 16; the table also
includes the values extrapolated to Planck energy for the pure standard model. The calculation of the strand model does not agree with the data. The only encouraging aspect
is that the ropelength approximation provides an approximation for older speculations
on approximately fixed mass ratios between the up-type quarks u, c, t and fixed mass ratios between down-type quarks d, s, b. The attempted strand model estimate shows that
ropelength alone is not sufficient to understand quark mass ratios. Research has yet to
show which effect has to be included to improve the correspondence with experiment.
In fact, the strand model predicts that everyday quark masses result from a combin-
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 246
(204)
Motion Mountain – The Adventure of Physics
Let us try to extract numerical values for the quark mass ratios. We start by exploring the
tight quark tangles, thus Planck-scale mass values. For each quark number ?, the quark
mass will be the weighted average over the mass of its family tangles with ?, ? + 6, ? + 12,
... crossings, where the period 6 is due to the leather trick. Each tight tangle has a certain
ropelength. The mass of each tangle will be determined by the frequency of crossing
changes at the core, including those due to the belt trick. The quark mass then is the
average over all family tangles; it will be determined by the frequency of tail braiding
and of all other fluctuations that generate crossing switches.
For determining mass ratios, the frequency of the crossing switches in the core are
the most important. Given that the particles are fermions, not bosons, this frequency
is expected to be an exponential of the ropelength ?. Among quarks, we thus expect a
general mass dependence of the type
364
12 pa rticle properties d ed uced from stra nds
TA B L E 15 Calculated ropelengths, in units of the rope diameter, of tight quark tangles of Figure 84
(Page 321) with tails oriented along the skeleton of a tetrahedron.
Ta n g l e
Length
Ropelength
Difference
skeleton (vacuum)
simplest d
simplest u
simplest s
simplest c
simplest b
simplest t
138.564065
139.919533
142.627837
146.175507
149.695643
153.250364
157.163826
base value
1.355468
4.063773
7.611443
11.131578
14.686299
18.599761
1.355468
2.708305
3.547670
3.520136
3.554721
3.913462
TA B L E 16 For comparison: the quark masses at Planck energy,, calculated from the measured quark
Low energy
mass
Planck energy
mass
u (? = 2/3?)
d (? = −1/3?)
s (? = −1/3?)
c (? = 2/3?)
b (? = −1/3?)
t (? = 2/3?)
2.5(1.1) MeV
5.0(1.6) MeV
105(35) MeV
1270(110) MeV
4200(170) MeV
171300(2300) MeV
0.45(0.16) MeV
0.97(0.10) MeV
19.4(1.2) MeV
213(8) MeV
883(10) MeV
66993(880) MeV
ation of three effects: the effect of ropelength and of tangle core shape on rotation and
the belt trick, the effect of sixfold tail braiding, and the effect of the energy dependence
of mass between Planck energy and everyday energy, due to core loosening.
Even though an analytic calculation for quark masses seems difficult, better approximations are possible. With sufficient computer power, it is possible to calculate the effects of the core shape rotations, including the belt trick, and of the energy dependence
of the quark masses. The most difficult point remains the calculation of the probabilities
for tail braiding. More research is needed on all these points.
L epton mass ratios
Mass calculations for leptons are as involved as for quarks. Each lepton, being a fermion,
has a large family of associated tangles: there is a simplest tangle and there are the tangles
that appear through repeated application of tail braiding. Despite this large tangle family,
some results can be deduced from the simplest lepton tangles alone, disregarding the
higher-order family members.
Both for neutrinos and for charged leptons, the progression in ropelength of the tight
versions of the basic tangles predicts a progression in their masses. This is indeed observed. This is valid for all candidate set of lepton tangles.
For each lepton tangle with ? crossings, knot theory predicts a ropelength ? that in-
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Q ua rk
Motion Mountain – The Adventure of Physics
masses using the standard model of particle physics – assuming that it is correct up to Planck energy.
the m asses of the elem enta ry pa rticles
365
Lepton tangles, all with spin S = 1/2 , parity P = +1 , lepton number L’ = 1 and baryon number B = 0 :
e
0.5 MeV
105 MeV
νμ
τ
all three with
Q = –1
1.77 GeV
ντ
all three with
Q=0
1±1 eV
1±1 eV
Seen from a larger distance, the tails follow (on average) the x, y and z axes of a coordinate system.
F I G U R E 110 Simple candidate tangles for the leptons. Antileptons are mirror tangles. The experimental
mass values are also given.
creases roughly proportionally to the crossing number: ? ∼ ?. Each lepton mass value
will again be given by the frequency of crossing switches due to rotations, including the
belt trick, and of tail braiding. We thus expect a general relation of the type
?? ∼ e?? ?
(205)
where ? is a number of order 1 that takes into account the shape of the tangle core. Such
a relation is in general agreement with the observed ratios between lepton masses. Research on these issues is ongoing; calculations of ropelengths and other geometric properties of the lepton tangles will allow a more detailed analysis. The most important challenges are, first, to deduce the correct mass sequence among the muon neutrino and the
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
1±1 eV
Motion Mountain – The Adventure of Physics
νe
μ
366
Page 331
Ref. 248
12 pa rticle properties d ed uced from stra nds
electron, and second, to estimate the neutrino masses.
We note that the lepton mass generation mechanism of the strand model differs from
other proposals in the research literature. It agrees with the Higgs mechanism but goes
beyond it. For neutrinos, the mechanism contradicts the see-saw mechanism but confirms the Yukawa mechanism directly. From a distance, the mass mechanism of the
strand model also somewhat resembles conformal symmetry breaking.
On the absolu te values of particle masses
Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
In nature, the masses of elementary particles are observed to be much lower than the
Planck mass: the observed values lie between about 10−30 for neutrinos and 10−17 for the
top quark. Particle masses are constant over space and time. Antiparticles have the same
mass as particles. Gravitational and inertial masses are the same. Following the standard
model, particle masses are due to the Higgs mechanism. Finally, elementary particles
masses run with energy.
All qualitative observations about mass are reproduced by the strand model. The however, the explanation of the numerical values is still lacking.
In the strand model, the gravitational mass of elementary particles is due to disturbance of the vacuum, in particular to the disturbance of the vacuum fluctuations. Larger
masses are due to more complex tangles. Since rest mass is localized energy, rest mass is
due to crossing switches per time. Larger masses have more crossing switches per time
than lower masses.
In the strand model, the inertial mass of elementary particles is their reluctance to
rotate. Inertial mass describes the relation between rotation frequency and wavelength;
in other terms, inertial mass described the steepness of the helix drawn by the rotating
phase arrow of a propagating particle. Larger masses low steepness, smaller masses have
higher steepness. Larger masses are due to more complex tangles.
As we just saw, the strand model predicts mass sequences and ratios of elementary
particle masses that corroborate or at least do not contradict observations too much.
The next step is to determine absolute mass values from the strand model. So far we
only found that elementary particle masses are much smaller than a Planck mass. But to
validate the strand model, we need more precise statements. To determine gravitational
mass values, we need to count those crossing switches that occur at rest; to determine
inertial mass values, we need to look for crossing switches in the case of a moving particle
– or, if we prefer, to understand the origin of the steepness of the helix drawn by the phase
arrow. All these methods should first lead to mass value estimates and then to mass value
calculations.
In general, the strand model reduces mass determination to the calculation of the
details of a process: How often do the fluctuations of strands lead to crossing switches?
There are various candidates for the crossing switches that lead to particle mass.
The first candidate for mass-producing crossings is tail switching. In general however,
tail switching leads to different particle types. Only for the Higgs process, i.e., the addition of a full Higgs braid to a particle, is this process expected to be relevant. We can also
say that tail braid addition are the strand model’s version of the Yukawa coupling terms.
The next candidate for mass-producing crossings is the belt trick. However, the belt
trick cannot be the full explanation, as there are also massive bosons, and the belt trick
the m asses of the elem enta ry pa rticles
Page 331
⊳ Mass is crossing switch rate.
* What is a Planck mass? In the strand model, a Planck mass corresponds to a structure that produces one
crossing switch for every Planck time, constantly, without interruption. But the strand model predicts that
such structures do not appear as localized particles, because every localized particle – i.e., every tangle –
has, by construction, a much smaller number of induced crossing switches per time. Following the strand
model, elementary particles with Planck mass do not exist. This conclusion agrees with observation. But
the strand model also implies that black holes with a Planck mass do not exist. Indeed, such Planck-scale
black holes, apart from being extremely short-lived, have no simple strand structure. We can state that a
Planck mass is never localized. Given these results, we cannot use a model of a localized Planck mass as a
unit or a benchmark to determine particle masses.
The impossibility of using Planck mass as a unit is also encountered in everyday life: no mass measurement in any laboratory is performed by using this unit as a standard.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
only applies to fermions.
The third candidate for mass-producing crossings could appear when particles shed
one strand and grap a new one. The influence of this process is not clear yet.
A fourth candidate for mass-producing crossings is the leather trick. However, the
leather trick cannot be realized for strands that reach spatial infinity; therefore it is expected that it play no role.
A fifth candidate for mass-producing crossings are those crossings that occur above
the core, similar to the crossing that occur above the horizon of a black hole. This candidate includes the belt trick in the case of fermions, includes the Higgs mechanism, and
thus seems most promising.
It might be that some other mass-producing switching processes are being overlooked. Nevertheless, in the following we explore the fifth candidate in more detail, the
crossings around a given tangle core.
Before looking for estimates, we note that in the past, various researchers have reached
the conclusion that all elementary particle masses should be due to a common process
or energy scale. Among theoretical physicists, the breaking of conformal symmetry has
always been a candidate for the associated process. Among experimental physicists, the
Higgs mechanism – now confirmed by experiment – is the favourite explanation of all
elementary particle masses. In the strand model, crossing switches around tangles are
related to the Higgs boson. At the same time we can also argue that tangles break the
conformal symmetry of vacuum. With a bit of distance, we can thus say that the strand
model agrees with both research expectations.
We now continue with the quest for absolute mass estimates. In the strand model,
absolute mass values are not purely geometric quantities that can be deduced directly
from the shapes of tangle knots. Particle masses are due to dynamical processes. Absolute
mass values are due to strand fluctuations; and these fluctuations are influenced by the
core topology, the core shape, the core ropelength and core tightness.
To determine absolute particle mass values, we need to determine the ratio between
the particle mass and the Planck mass. This means to determine the ratio between the
crossing switch probability for a given particle and the crossing switch probability for a
Planck mass, namely one switch per Planck time.*
Energy is action per time. Mass is localized energy. In other words, the absolute mass
of a particle is given by the average number of crossing switches it induces per time:
Motion Mountain – The Adventure of Physics
Ref. 248, Ref. 249
367
368
12 pa rticle properties d ed uced from stra nds
three polymer strands
monomers
with
Planck length
diameter
tangled
core
region
R
F I G U R E 111 Determining lepton
liquid solution
More precisely, the crossing switch rate of a particle at rest is its gravitational mass, and
the crossing switch rate induced by propagation is its inertial mass. Let us explore the
relations.
Given that mass is determined by the crossing switch rate, we deduce that particle
mass values are determined by tangle topology, are fixed, are discrete, are positive, increase with tangle core complexity, are identical for particle and antiparticles, are constant over time, and are much smaller than the Planck mass. Because all these properties
match observations, the local crossing switch rate indeed realizes all qualitative requirements for absolute particle mass values. We can thus proceed with the hope to learn
more. In order to calculate absolute particle masses, we just need to determine the number of crossing switches per time that every particle tangle induces.
One general way to perform a particle mass calculation is to use a computer, insert
a strand model of the fluctuating vacuum plus the strand model of the particle under
investigation, and count the number of crossing switches per time. The basis for one such
approach, using the analogy of the evolution of a polymer in liquid solution, is shown in
Figure 111. In contrast to polymers, also the change of strand length has to be taken into
account. By determining, for a given core topology, the average frequency with which
crossing switches appear for a tethered core, we can estimate the masses of the leptons,
quarks and bosons. In such a mass calculation, the mass scale is set indirectly, through
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 357
mass values with the help of a
polymer analogy of strands. After
rescaling, the probability of
crossing switches around the
tangle core yields an estimate for
the mass of the elementary
particle with that tangle.
Motion Mountain – The Adventure of Physics
tails
the m asses of the elem enta ry pa rticles
369
the time scale of the fluctuation spectrum. This is tricky but feasible. One would first
need to find the parameter space and the fluctuation spectrum for which the polymer
tangle follows the Schrödinger equation. Calculations with different tangles should then
yield the different mass values. Such a simulation would also of interest for exploring the
strand model of quantum mechanics.
A more precise computer simulation would also model the vacuum itself with strands.
This approach would even allow to explore gravitational and inertial mass separately. In
such a simulation, the particle mass appears when the helical motion of a tangle moving
through a strand vacuum is observed. The required effort can be reduced by using the
most appropriate computer libraries.
A further general way to determine particle masses is to search for analytical approximations. This is a fascinating conceptual and mathematical challenge. The main issue is
to clarify which crossing switches contribute most to particle mass.
A first analytical attempt is the following. We assume that the inertial mass for a moving
fermion is proportional to the fluctuation-induced appearance of the belt trick. If the
tight core has a diameter of, say, three Planck lengths – and thus a circumference of
around 9 Planck lengths – then the probability ? of the belt trick for a particle with six
tails will be in the range
? ≈ (e−9 )6 ≈ 10−24 .
(206)
Challenge 208 ny
Challenge 209 r
Challenge 210 ny
⊳ What is the numerical probability of the belt trick for a tethered core of given
topology with fluctuating tails?
So far, several experts on polymer evolution have failed to provide even the crudest estimate for the probability of the belt trick in a polymer-tethered ball. Can you provide
one?
A second analytical approach starts from the following question:
⊳ How often does a tail cross above the tangle core?
This question is loosely related to the previous one; in addition, this approach illustrates
why complex cores have larger mass. The probability of such crossings, when squared,
would be an estimate for the crossing switch rate, and thus for the particle mass. (There
are additional details to the calculation.) We note directly that the number of tails will
have a smaller impact on mass than the complexity of the tangle. So far, a reliable estimate for the crossing number, as a function of the tangle core properties, is still missing
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
This value would be the order of magnitude for the mass estimate, in Planck units. Such
an estimate is only very rough, and the exponent can be quite different. Nevertheless, we
do get an explanation for the large difference between the Planck mass and the typical
fermion mass. A more precise analytical approximation for the belt trick probability –
not an impossible feat – will therefore solve the so-called mass hierarchy problem. We
thus want to know:
Motion Mountain – The Adventure of Physics
Analytical estimates for particle masses
370
Challenge 211 ny
12 pa rticle properties d ed uced from stra nds
– even a crude one. Can you find one?
Open issues abou t mass calcul ations
Challenge 212 e
Calculating absolute particle masses from tangle fluctuations, either numerically or with
an analytical approximation, will allow the final check of the statements in this section.
The strand model predicts that the resulting values will match experiments. For these
calculations, it is essential that the tangle assignment for each elementary particle is correct. In 2017, the tangles for the W tangle and for the lepton tangles are still doubtful.
∗∗
Because the strand model predicts a lack of new physics beyond the standard model of
particle physics, the calculation of neutrino masses, and thus their mass sequence, is one
of the few possible predictions – in contrast to retrodictions – that are left over in the
strand model.
Challenge 213 s
Is the mass of a tangle related to the vacuum density of strands?
∗∗
Challenge 214 s
Do particle masses depend on the cosmological constant?
∗∗
∗∗
Challenge 215 s
Can the concept of total curvature of a tangle help to calculate particle masses?
∗∗
Challenge 216 d
Does the effect of tail braiding confirm the conjecture that every experiment is described
by a small energy scale, determining the resolution or precision, and a large energy scale,
less obvious, that determines the accuracy?
∗∗
If tail braiding is due to the weak interaction, and if the Higgs is a tail-braided vacuum,
can we deduce that the Higgs interaction is a higher order effect of the weak interaction?
Can we deduce a concrete experimental prediction from this relation?
On fine-tuning and naturalness
It has become fashionable, since about a decade, to state that the standard model of elementary particle physics is ‘fine-tuned’. The term expresses several ideas. First of all,
the extremely low value of the vacuum energy is not obvious when all the zero-point
field contributions from the various elementary particles of the standard model are included. A low vacuum energy seems only possible if the masses and the particle types
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
The mass of an elementary particle does not depend on the spin direction. In particular,
the W and Z bosons have equal longitudinal and transversal mass. The strand model
does not allow an influence of spin orientation on mass.
Motion Mountain – The Adventure of Physics
∗∗
the m asses of the elem enta ry pa rticles
of the standard model are somehow interrelated. In other words, the term ‘fine tuning’
expresses, above all, the lack of understanding of the origin of the masses, mixings and
coupling constants of elementary particles.
The term ‘fine tuning’ is also used to state that the universe would be very different
if the fundamental constants would be different. But this statement lacks deep truth.
In this usage, the term ‘fine tuning’ states that particle masses are not parameters that
can be varied at will. In common usage, ‘parameters’ are variable constants; but the low
value of the vacuum energy – as well as many other observations – shows that the masses
of elementary particles cannot be varied without destroying the validity of the standard
model of particle physics.
Some people suggest that ‘fine-tuning’ implies that the standard model of particle
physics is ‘unnatural’, whatever this might mean in detail. Some even suggest that the
parameters of the standard model lack any explanation. The strand model – but also
common sense – show that this suggestion is false.
The strand model naturally has a low vacuum energy, because the unknotted strands
of flat space naturally have a zero energy density, and the particle masses, mixings and
coupling constants are not variable or random, but naturally unique and fixed in value.
Any correct description of nature must be ‘fine-tuned’. If the standard model would not
be ‘fine-tuned’, it would not describe nature.
In short, the fashionable term ‘fine-tuned’ is equivalent to the terms ‘unmodifiable’
and ‘hard to vary’ that were discussed above. All these terms highlight the lack of alternatives to the world as we observe it, the existence of explanations for the processes
around us, and our ability to discover and grasp them. This is part of the wonders of
nature. And the strand model makes those wonders apparent at the Planck scale.
Page 369
Page 162
The strand model implies that masses are dynamic quantities fixed by processes due to
the geometric and topological properties of specific tangle families. As a result, strands
explain why the masses of elementary particles are not free parameters, but fixed and
unique constants, and why they are much smaller than the Planck mass by many orders
of magnitude. Strands also reproduce all known qualitative properties of particle masses.
Strands provide estimates for a number of elementary particle mass ratios, such as
?W /?Z and ?Higgs /?W . Most quark and lepton mass sequences and first rough estimates of mass ratios agree with the experimental data. All hadron mass sequences are
predicted correctly. The strand model also promises to calculate absolute mass values, including their change or ‘running’ with energy. Such future calculations will allow either
improving the match with observations or refuting the strand model.
The results are encouraging for two reasons. First of all, no other unified model that
agrees with experiment explains the qualitative properties of mass and mass sequences.
Secondly, no research on statistical tangles exists; an understanding of the parameters of
nature might be lacking because results in this research field are still few.
In the millennium list of open issues we have thus seen how to settle the origin of
particle masses – though we have not calculated them yet. Because a few even more
interesting challenges are awaiting us, we continue nevertheless. In the next leg, we investigate how elementary particle states mix.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Summary on elementary particle masses and millennium issues
Motion Mountain – The Adventure of Physics
Page 163
371
372
12 pa rticle properties d ed uced from stra nds
A quark mass eigenstate :
A quark weak eigenstate :
s
s‘
two tails
above, one tail
below paper plane
all three tails
in paper plane
magnified core
F I G U R E 112 Tail shifting leads to quark mixing: mass eigenstates and weak eigenstates differ.
mixing angles
Q uark mixing – the experimental data
Vol. V, page 250
??
?
( ?? ) = (??? ) ( ? ) .
?
??
Ref. 231
(207)
where, by convention, the states of the +2/3 quarks ?, ? and ? are unmixed. Unprimed
quarks names represent strong (and electromagnetic) eigenstates, primed quark names
represent weak eigenstates. In its standard parametrization, the mixing matrix reads
?12 ?13
?13 e−??
?12 ?13
? = (−?12 ?23 − ?12 ?23 ?13 e??
?12 ?23 − ?12 ?23 ?13 e?? ?23 ?13 )
??
?12 ?23 − ?12?23 ?13 e
−?12 ?23 − ?12 ?23 ?13 e?? ?23 ?13
(208)
where ??? = cos ???, ??? = sin ??? and ? and ? label the generation (1 ⩽ ?, ? ⩽ 3). The mixing
matrix thus contains three mixing angles, ?12 , ?23 and ?13 , and one phase, ?. In the limit
?23 = ?13 = 0, i.e., when only two generations mix, the only remaining parameter is
the angle ?12 , called the Cabibbo angle; this angle is Cabibbo’s original discovery. The
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 231
In nature, the quark mass eigenstates and their weak eigenstates differ. This difference
was discovered in 1963 by Nicola Cabibbo and is called quark mixing. The values of the
elements of the quark mixing matrix have been measured in many experiments, and
more experiments aiming to increase the measurement precision are under way.
The quark mixing matrix, also called CKM mixing matrix, is defined by
Motion Mountain – The Adventure of Physics
In nature, the mass eigenstates for fermions differ from their weak eigenstates: quarks
mix among themselves, and so do neutrinos. Quarks also show CP violation; for neutrinos, the issue is still open. These effects are described by two so-called mixing matrices.
The two mixing matrices contain fundamental constants of nature. For the strand model
to be correct, it must allow calculating the measured values of all components of the two
mixing matrices.
m ixing a ng les
Challenge 217 e
Ref. 231
373
last parameter, the so-called CP-violating phase ?, by definition between 0 and 2π, is
measured to be different from zero; it expresses the observation that CP invariance is
violated in the case of the weak interactions. The CP-violating phase only appears in
the third column of the matrix; therefore CP violation requires the existence of (at least)
three generations.
The present 90 % confidence values for the measured magnitude of the complex quark
mixing matrix elements are
0.97427(14) 0.22536(61) 0.00355(15)
|?| = (0.22522(61) 0.97343(15) 0.0414(12) ) .
0.00886(33) 0.0405(12) 0.99914(5)
(209)
Q uark mixing – expl anations
Page 250
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Page 331
In the standard model of particle physics, the quark mixing matrix is usually seen as due
to the coupling between the vacuum expectation value of the Higgs field and the lefthanded quark doublets or the right handed quark singlets. However, this description
does not lead to a numerical prediction.
A slightly different description of quark mixing is given in the strand model. In the
strand model, the Higgs field and its role as mass generator and unitarity maintainer is
a special case of the process of tail braiding. And braiding is related to the weak interaction. Because the various quarks are differently tangled rational tangles, tail braiding
can reduce or increase the crossings in a quark tangle, and thus change quark flavours.
We thus deduce from the strand model that quark mixing is an automatic result of the
strand model and related to the weak interaction. We also deduce that quark mixing is
due to the same process that generates quark masses, as expected. But we can say more.
In the strand model, the mass eigenstate – and colour eigenstate – is the tangle shape
in which colour symmetry is manifest and in which particle position is defined. The
mass eigenstates of quarks correspond to tangles whose three colour-tails point in three
directions that are equally distributed in space. The shape in which the tails point in
three, equally spaced directions is the shape that makes the SU(3) representation under
core slides manifest.
In contrast, the weak eigenstates are those shapes that makes the SU(2) behaviour
of core pokes manifest. For a quark, the weak eigenstate appears to be that shape of a
tangle for which all tails lie in a plane; for such plane configuration, the tails and the core
mimic a belt and its buckle, the structure that generates SU(2) behaviour. The two types
of eigenstates are illustrated in Figure 112.
In the strand model, masses are dynamical effects related to tangle shape. In the case
of quarks, the two configurations just mentioned will thus behave differently. We call
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All these numbers are unexplained constants of nature, like the particle masses. Within
experimental errors, the matrix ? is unitary.
A huge amount of experimental work lies behind this short summary. The data have
been collected over many years, in numerous scattering and decay experiments, by thousands of researchers. Nevertheless, this short summary represents all the data that any
unified description has to reproduce about quark mixing.
374
12 pa rticle properties d ed uced from stra nds
the transformation from a mass eigenstate to a weak eigenstate or back tail shifting. Tail
shifting is a deformation: the tails as a whole are rotated and shifted. On the other hand,
tail shifting can also lead to untangling of a quark tangle; in other words, tail shifting
can lead to tail braiding and thus can transform quark flavours. The process of tail shifting can thus explain quark mixing. (Tail shifting also explains the existence of neutrino
mixing, and the lack of mixing for the weak bosons.)
Tail shifting can thus be seen as a partial tail braiding; as such, it is due to the weak
interaction. This connection yields the following predictions:
Page 207
Ref. 231
A challenge
Ref. 251
Can you deduce the approximate expression
tan ?u mix = √
Challenge 218 r
?u
?c
(210)
for the mixing of the up quark from the strand model?
CP viol ation in quarks
Ref. 231
Page 339
Page 336
The CP violating phase ? for quarks is usually expressed with the Jarlskog invariant,
2
defined as ? = sin ?12 sin ?13 sin ?23
cos ?12 cos ?13 cos ?23 sin ?. This involved expression
is independent of the definition of the phase angles and was discovered by Cecilia Jarlskog, an important Swedish particle physicist. Its measured value is ? = 3.06(21) ⋅ 10−5 .
Because the strand model predicts three quark generations, the quark model implies
the possibility of CP violation. In the section on mesons we have seen that the strand
model actually predicts the existence CP violation. In particular, Figure 97 shows that
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Performing a precise calculation of mixing angles and their running with energy is still
a subject of research.
Motion Mountain – The Adventure of Physics
— Tail shifting, both with or without tail braiding at the border of space, is a generalized
deformation. Therefore, it is described by a unitary operator. The first result from the
strand model is thus that the quark mixing matrix is unitary. This is indeed observed.
— For quarks, tail braiding is a process with small probability. As a consequence, the
quark mixing matrix will have its highest elements on the diagonal. This is indeed
observed.
— Tail shifting also naturally predicts that quark mixing will be higher between neighbouring generations, such as 1 and 2, than between distant generations, such as 1 and
3. This is also observed.
— The connection between mixing and mass also implies that the 1–2 mixing is stronger
than the 2–3 mixing, as is observed.
— Finally, tail shifting predicts that the numerical values in the quark mixing matrix
can be deduced from the difference between the shapes of the two kinds of tangles
shown in Figure 112. In particular, tail shifting also predicts that the quark mixing
angles change, or run, with energy. In addition, the effect is predicted to be small.
On the other hand, so far there is no reliable experimental data on the effect.
m ixing a ng les
375
A neutrino mass eigenstate :
A neutrino weak eigenstate :
νe’
νe
each set of three tails
follows three mutually
perpendicular directions
all tails lie in one plane
F I G U R E 113 Tail shifting leads to neutrino mixing: mass eigenstates and weak eigenstates differ.
Ref. 252
Ref. 231
The observation, in 1998, of neutrino mixing is comparably recent in the history of
particle physics, even though the important physicist Bruno Pontecorvo predicted the
effect already in 1957. Again, the observation of neutrino mixing implies that also for
neutrinos the mass eigenstates and the weak eigenstates differ. The values of the mixing matrix elements are only known with limited accuracy so far, because the extremely
small neutrino mass makes experiments very difficult. Experimental progress across the
world is summarized on the website www.nu-fit.org. The absolute value of the so-called
PMNS mixing matrix ? is
0.82(1) 0.54(2) −0.15(3)
−0.35(6)
0.70(6) 0.62(6) ) .
|?| = (
0.44(6) −0.45(6) 0.77(6)
(211)
Again, these numbers are unexplained fundamental constants of nature. Within experimental errors, the matrix ? is unitary. The mixing among the three neutrino states is
strong, in contrast to the situation for quarks. Neutrino masses are known to be positive; however, present measurements are not precise and only yield values of the order of
1 ± 1 eV.
In the strand model, the lepton mass eigenstates correspond to tangles whose tails
point along the three coordinate axes. In contrast, the weak eigenstates again correspond to tangles whose tails lie in a plane. The two kinds of eigenstates are illustrated
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Neu trino mixing
Motion Mountain – The Adventure of Physics
with the help of tail shifting, ?0 and ?0 mesons mix, and that the same happens with
certain other neutral mesons. Figure 98 shows a further example. As just mentioned,
the possibility of tail shifting implies that CP violation is small, but non-negligible – as
is observed.
The strand model thus predicts that the quark mixing matrix has a non-vanishing CPviolating phase. The value of this phase is predicted to follow from the geometry of the
quark tangles, as soon as their shape fluctuations are properly accounted for. This topic
is still a subject of research.
376
12 pa rticle properties d ed uced from stra nds
in Figure 113. Again, the transition between the two eigenstates is due to tail shifting, a
special kind of strand deformation.
We thus deduce that neutrino mixing, like quark mixing, is an automatic result of the
strand model and is related to the weak interaction. Given that the neutrino masses are
small and similar, and that neutrinos do not form composites, the strand model predicts
that the mixing values are large. This is a direct consequence of tail shifting, which in
the case of similar masses, mixes neutrino tangles leads to large mixings between all
generations, and not only between neighbouring generations. In the strand model, the
large degree of neutrino mixing is thus seen as a consequence of their low and similar
masses, of their tangle structure, and of their existence as free particles.
Like for quarks, the strand model predicts a unitary mixing matrix for neutrinos. The
strand model also predicts that the geometry of the neutrino tangles and their fluctuations will allow us to calculate the mixing angles. More precise predictions are still
subject of research.
Ref. 254
Open challenge: calcul ate mixing angles and phases ab initio
Challenge 219 ny
Calculating the mixing angles and phases ab initio, using the statistical distribution of
strand fluctuations, is possible in various ways. In particular, it is interesting to find the
relation between the probability for a tail shift and for a tail braiding. This will allow
checking the statements of this section.
Because the strand model predicts a lack of new physics beyond the standard model
of particle physics, the calculation of neutrino mixing angles is one of the few possible
predictions that are left over in fundamental physics. Since the lepton tangles are still
tentative, a careful investigation is necessary.
One possibility is that only the electron neutrino tangle given above is correct, and
that the other two neutrinos are similar to it, just with more built-in torsion. Figure 114 illustrates this possibility. If this assignment were correct, two of the mixing angles should
be large (and maybe have the zero-order value of 120°/3=40°). In addition, very low mass
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 253
The strand model predicts that the three neutrinos are massive Dirac particles, not Majorana particles. This has not yet been confirmed by experiment. The strand model thus
predicts that the neutrino mixing matrix has only one CP-violating phase. (It would have
three such phases if neutrinos were Majorana particles.) The value of this phase is predicted to follow from the neutrino tangles and a proper accounting of their fluctuations.
Also this calculation is still a subject of research.
On the one hand, the strand model suggests the appearance of CP violation in neutrinos. On the other hand, it is unclear when the value of the CP-violating phase will
ever be measured with sufficient precision. This is one of the hardest open challenge of
experimental particle physics.
The mechanism of CP violation has important consequences in cosmology, in particular for the matter–antimatter asymmetry. Since the strand model predicts the absence of the see-saw mechanism, the strand model rules out leptogenesis, an idea invented to explain the lack of antimatter in the universe. The strand model is more on the
line with electroweak baryogenesis.
Motion Mountain – The Adventure of Physics
CP viol ation in neu trinos
m ixing a ng les
377
F I G U R E 114 An alternative candidate assignment for the three neutrino tangles that generates large
mixing between neighbouring generations and strong preference for one handedness.
Page 162
The strand model implies that mixing angles for quarks and neutrinos are properties of
their tangle families. The existence of mixing is due to the shape of tangles and their
fluctuations. As a result, strands explain why mixing angles are not free parameters, but
discrete and unique constants of nature. The strand model also predicts that mixing
angles are constant during the evolution of the universe.
We have shown that tangles of strands predict non-zero mixing angles for quarks and
neutrinos, as well as CP-violation in both cases. The strand model also predicts that
the mixing angles of quarks and neutrinos can be calculated from strand fluctuations.
Strands predict that mixing matrices are unitary and that they run with energy. Strands
also predict a specific sequence of magnitudes among matrix elements; the few predictions so far agree with the experimental data. Finally, the strand model rules out leptogenesis.
We have thus partly settled four further items from the millennium list of open issues.
All qualitative aspects and some sequences are reproduced correctly, but no hard quantities were deduced yet. The result is somewhat disappointing, but it is also encouraging.
At present, no other explanation for quark and neutrino mixing is known. Future calculations will allow either improving the checks or refuting the strand model. We leave this
topic unfinished and proceed to the most interesting topic that is left: understanding the
coupling constants.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Summary on mixing angles and the millennium list
Motion Mountain – The Adventure of Physics
values would arise naturally, in normal ordering. This tangle assignment would also explain the difficulty of observing neutrinos of opposite handedness: neutrinos would have
an extremely high preference for one handedness (belt-trick); the mirror images of the
tangles in Figure 114 would correspond to antineutrinos. Despite these appealing aspects,
this tentative assignment has one unclear issue: explaining the lack of a fourth neutrino
generation is not straightforward.
378
12 pa rticle properties d ed uced from stra nds
1/α i
weaker
60
50
Data
Prediction
by the standard
model
1/α1
40
electromagnetism
1/α2
30
20
stronger
θw
1/α3
√3 α1/5 weak
hypercharge
√α2 weak
interaction
10
10
5
10
10
15
10
19
10
Q / GeV
coupling constant s and unificat ion
Ref. 5
In nature, electric, weak and strong charge are quantized. No experiment has ever found
even the smallest deviation from charge quantization. All charges in nature are integer
multiples of a smallest charge unit. Specifically, the electric charge of every free particle
is observed to be an integer multiple of the positron electric charge. We call the integer
the electric charge quantum number.
In nature, the strength of a gauge interaction for a unit charge is described by its coupling constant. The coupling constant gives the probability with which a unit charge emits
a virtual gauge boson, or, equivalently, the average phase change produced by the absorption of a gauge boson. There are three charge types and three coupling constants: for the
electromagnetic, for the weak and for the strong interaction. All particles with a given
charge type and value share the same coupling constant, even if their masses differ. The
three coupling constants depend on energy. The known data and the change with energy
predicted by the standard model of particle physics are shown in Figure 115.
In nature, the fine structure constant ?, i.e., the electromagnetic coupling constant, at
the lowest possible energy, 0.511 MeV, has the well-known measured value
? = 1/137.035 999 139(31) .
(212)
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
F I G U R E 115 Left: How the three coupling constants (squared) change with energy, as predicted by the
standard model of particle physics; the graph shows the constant ?1 = 53 ?/ cos2 ?W for the weak
hypercharge coupling (related to the electromagnetic fine structure constant ? through the weak
mixing angle ?W and a historical factor 5/3 that is useful in grand unification), the coupling constant
?2 = ?? = ?/ sin2 ?W for the weak interaction, and the coupling constant ?3 = ?s for the strong
interaction. The three black points are measurement points; at lower and slightly higher energies, data
and calculation match within experimental errors. (Courtesy Wim de Boer) Right: The relation between
the coupling constants ? for the electromagnetic U(1)EM , ?2 = ?? for the weak SU(2), ?1 for the weak
hypercharge U(1)Y gauge groups and the weak mixing angle ?W .
Motion Mountain – The Adventure of Physics
0
0
10
√α
coupling consta nts a nd unification
379
Equivalently, the electromagnetic coupling of the positron can also be described by the
equivalent number
√? = 1/11.706 237 6167(13) = 0.085424543114(10) ,
(213)
Interaction strengths and strands
In the strand model, all three gauge interactions are due to shape changes of tangle cores.
We first classify the possible shape changes. Given a tangle core, the following shape
changes can occur:
In the strand model, the fluctuation probabilities for each Reidemeister move – twist,
poke or slide – determine the coupling constants. We thus need to determine these
probability values. We can directly deduce a number of conclusions, without any detailed calculation:
— The coupling constants are not free parameters, but are specified by the geometric,
three-dimensional shape of the particle tangles.
— By relating coupling constants to shape fluctuation probabilities, the strand model
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
— Small changes of core shape do not produce any crossing switch. Small shape changes
thus have no physical significance: for a given observer, they leave all observables
unchanged.
— Twist shape changes of a strand segment in the core produce an electric field, if the
particle is charged. More precisely, the electric field around a particle is the difference
between the average number ?tr of right twists and the average number ?tl of inverse,
left twists that a particle tangle produces per unit time.
— Poke shape changes of a strand segment in the core produce a weak interaction field.
More precisely, the weak field is the asymmetry among the probabilities ?p? , ?p? and
?p? for the three fundamental poke types and their inverses.
— Slide shape changes of a strand segment in the core produce a colour field, if the
particle has colour. More precisely, the colour field is the asymmetry among the probabilities ?s1 to ?s8 for the eight fundamental slide types and their inverses.
— A combination of these moves can also appear.
Motion Mountain – The Adventure of Physics
which is also called the electric charge unit (at low energy). Quantum electrodynamics
predicts the precise change with energy of this charge unit; the experiments performed so
far, up to over 100 GeV, agree with this prediction. Quantum electrodynamics also predicts that the charge unit, when extrapolated right up to the Planck energy, would have a
value of 1/10.2(1). These predictions are shown, in a common, but somewhat scrambled
way, in Figure 115.
Explaining the value of ?, which determines all colours and all material properties
in nature, is the most famous millennium issue. If the strand model cannot reproduce
every observation about ? and other coupling constants, it is wrong. In particular, we
thus need to understand, using the strand model, the quantization of charges on the one
hand, and the mysterious value of the charge unit – either at low energy or at Planck
energy – on the other hand.
380
12 pa rticle properties d ed uced from stra nds
predicts that coupling constants are positive numbers and smaller than 1 for all energies. This is indeed observed.
— A still stricter bound for coupling constants can also be deduced. The sum of all
possible fluctuations for a particular tangle has unit probability. We thus have
8
1 = ?small + ?tr + ?tl + ∑ (?p? + ?p−? ) + ∑ (?s? + ?s−? ) + ?combination . (214)
?=?,?,?
In summary, the strand model implies, like quantum field theory, that coupling constants
are probabilities. The obvious consequences are
⊳
⊳
⊳
⊳
? < 1,
?w < 1,
?s < 1
? + ?w + ?s < 1 and √? + √?w + √?s < 1.
These properties are valid both in quantum field theory and in the strand model. Despite
the agreement with experiment, we have not deduced any new result yet – except one.
Strands imply unification
Page 275
In fact, one new point is made by the strand model. Each gauge interaction is due to a
different Reidemeister move. However, given a specific tangle core deformation, different
observers will classify the deformation as a different Reidemeister move. Indeed, every
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
The strand model thus predicts that the sum of the three charge units must be strictly
smaller than 1, for every energy value. This is easily checked, both with the data
and with the prediction of quantum field theory. In quantum field theory, the three
(modified) coupling constants are given, as a function of energy, in the popular graph
shown in Figure 115. The values are a combination of experimental data – for low energies – and theoretical extrapolations – for high energies. In this popular graph, the
electromagnetic coupling is traditionally multiplied by 5/(3 cos2 ?W ). (This is done
in order to test grand unification; we keep the traditional factor, even though grand
unification is shown by experiment and predicted by the strand model not to apply to
nature.) The graph allows us to confirm that the sum of the three unmodified charge
units is indeed smaller than 1 for all energy values, as predicted by the strand model.
— The strand model also predicts that the three coupling constants are related by small
numbers, as the corresponding fluctuations differ only in the number of involved
strands. This is also observed, as Figure 115 shows – especially if we remember that
the couplings are the square roots of the values shown in the graph, corrected for the
traditional factor.
— The strand model further predicts that the coupling constants are independent of
time and space, and that in particular, they do not depend on the age of the universe.
This is also observed, despite occasional claims to the contrary.
— Finally, strand model predicts that the coupling constants are the same for particles
and antiparticles, as is observed.
Motion Mountain – The Adventure of Physics
Ref. 255
?=1
coupling consta nts a nd unification
Reidemeister move can be realized by the same deformation of a single strand: for each
Reidemeister move, it is sufficient to add a curved section to a straight strand segment.
Such a deformation can look like a type I Reidemeister move for one observer, like a type
II move for another, and like a type III move for a third observer.
Because all interactions follow from the same kind of strand deformation of tangle
cores, the strand model thus provides unification of the interactions. This result is new:
in fact, this kind of strand unification of the interactions differs completely from any
other approach ever proposed. In contrast to other approaches, strand unification does
not require that the three coupling constants have the same value at high energy.
A given shape deformation thus has five probabilities associated to it: the probabilities describe what percentage of observers sees this deformation as a type I move, as a
type II move, as a type III move, as a combination of such moves, or as no move at all,
i.e., as a small move without any crossing switch. On the other hand, at energies measurable in the laboratory, the moves can almost always be distinguished, because for a given
reaction, usually all probabilities but one practically vanish, due to the time averaging
and spatial scales involved.* In short, at energies measurable in the laboratory, the three
gauge interactions almost always differ.
Calcul ating coupling constants
⊳ The strand model predicts that the fine structure constant can be calculated
by determining the probability of twists, i.e., Reidemeister I moves, in the
fluctuating tangle shapes of a given particle with nonzero electrical charge.
In other words, the strand model must show that the probability of the first Reidemeister
move in chiral particle tangles is quantized. This probability must be an integer multiple
of a unit that is common to all tangles; and this coupling unit must be the fine structure
constant. Any check for the existence of a coupling unit requires the calculation of twist
appearance probabilities for each chiral particle tangle. The strand model is only correct
if all particles with the same electric charge yield the same twist emission probability.
Instead of emission, also absorption can be used to calculate the fine structure constant:
* The strand model thus predicts that at extremely high energy, meaning near the Planck energy, for each
gauge interaction, also particles with zero charge can interact. At Planck energy, when horizons form, the
time averaging is not perfect, and interactions become possible even with zero charge.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
The strand model predicts that the calculation of the three coupling constants is a problem of tangle geometry and fluctuation statistics. Thus it can be approached, at each
energy scale, either analytically or with computer calculations. The calculations need to
determine the probabilities of the corresponding Reidemeister moves. If the results do
not agree with the experimental values, the strand model is false. We note that there is
no freedom to tweak the calculations towards the known experimental results.
In particular, in the strand model, one way to proceed is the following. The (square
root of the) fine structure constant is the probability for the emission of twists by a fluctuating chiral tangle.
Motion Mountain – The Adventure of Physics
Challenge 220 e
381
382
12 pa rticle properties d ed uced from stra nds
⊳ The strand model predicts that the fine structure constant can be calculated
from the average angle that a tangle core rotates when absorbing a photon.
Challenge 221 r
First hint: the energy dependence of physical quantities
Page 378
⊳ Coupling constants, masses and mixing angles change with energy because
they are quantities that depend on the average geometrical details, and in
particular, on the scale of the underlying particle tangles.
More precisely, the running quantities depend on the fluctuations of the geometric tangle
shapes, and these fluctuations depend somewhat on the spatial and thus the energy scale
under consideration. We note that the strand model predicts a running only for these
three types of observables; all the other observables – spin, parities or other quantum
numbers – are predicted to depend on the topology of the particle tangles, and thus to be
independent of energy. This prediction agrees with observation. Therefore, we can now
Ref. 227
Page 331
* In the standard model of particle physics, the running of the electromagnetic and weak coupling constants
– the slope in Figure 115 – depends on the number of existing Higgs boson types. The (corrected) strand
model predicts that this number is one. Measuring the running of the constants thus allows checking the
number of Higgs bosons. Unfortunately, the difference is small; for the electromagnetic coupling, the slope
changes by around 2 % if the Higgs number changes by one. But in future, such a measurement accuracy
might be possible.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
In nature, all effective charges, i.e., the coupling constants, change with energy. One also
says that they run with energy. Figure 115 shows the details. Running also occurs for
masses and mixing angles. All other intrinsic particle properties, such as spin, parities
and all other quantum numbers, are found not to change with energy. For the coupling constants, the measured changes between everyday energy and about 100 GeV agree
with the prediction from quantum field theory.*
The strand model predicts
Motion Mountain – The Adventure of Physics
We will pursue this alternative shortly.
So far, there do not seem to exist any analytical tool that permits the calculation of
shape deformation probabilities. Thus, at present, computer calculations seem to be the
only possible choice. Of all existing software programs, the most adapted to calculating
fluctuation probabilities are the programs that simulate the dynamics of tangled polymers; but also the programs that simulate the dynamics of cosmic strings or the dynamics of helium vortices are candidates. The main issue, apart from a large computer time, is
the correct and self-consistent specification of the shape fluctuation distribution at each
energy scale.
In summary, using the strand model we expect to be able to calculate the electromagnetic coupling constant and to understand its validity across all elementary particles. The
same expectation obviously also holds for the two nuclear interactions. If any of the expectations on tangle interactions are found to be incorrect, the strand model is false. The
strand model must yield quantized tangle equivalence classes for the electromagnetic, weak
and colour charge. Even though the calculation issues are still subject of research, there
are encouraging hints that these expectations will be validated.
coupling consta nts a nd unification
383
explore the details of the running.
Second hint: the running of the coupling constants at low
energy
Page 378
Page 334
d?
>0.
d?
Also for the two nuclear interactions, the washing out effect for loose tangle cores
by the vacuum does occur as predicted by quantum field theory. In the weak interaction, the antiscreening of the weak charge appears in this way. In the strong interaction,
both virtual quark–antiquark pairs and virtual gluon pairs can appear from the strands
that make up the vacuum. Virtual quark–antiquark pairs lead to screening, as virtual
electron–antielectron pairs do for the electromagnetic interaction. In addition, however,
we have seen that the strand model of mesons implies that virtual gluon pairs lead to
antiscreening. (In contrast, virtual photon pairs do not lead to such an effect.) Because
the strand model fixes the number of quark and gluons, the strand model is consistent
with the result that the screening of the colour charge by quark pairs is overcompensated
by the antiscreening of the virtual gluon pairs.
In other words, the strand model reproduces the observed signs for the slopes of the
coupling constants in Figure 115, for the same reason that it reproduces the quantum field
theoretic description of the three gauge interactions. The predicted running could also
be checked quantitatively, by taking statistical averages of tangle fluctuations of varying
dimension. This is a challenge for future research.
Third hint: further predictions at low energy
As we just saw, the complete explanation of the running of the couplings depends on
the explicit boson and fermion content of nature and on the fact that the strand model
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⊳
Motion Mountain – The Adventure of Physics
The strand model proposes a new view on the screening and antiscreening effects that
are part of quantum field theory. In the strand model, screening effects are consequences
of the statistics of shape deformations for loose tangle cores that are embedded into the
strands that form the vacuum. Since these statistical effects can in principle be calculated,
it is expected that such calculations can be compared with the predictions of quantum
field theory shown in Figure 115. This check is in progress. A few results, however, can
be deduced without any calculations at all.
In the strand model, the electromagnetic interaction is due to the first Reidemeister
move, the twist. For a charged particle – thus one with a chiral tangle core – the average
difference in the occurrence of right and left twists determines the effective charge. It is
expected that this difference decreases when the strand core is loose, because the loose
strands are more similar to those of the surrounding vacuum, so that the differences
due to the chirality of the tangle will be washed out. In the language of quantum field
theory, the virtual particle-antiparticle pairs – created by the fluctuations of the vacuum
strands – screen the central, naked charge. The screening is reduced when the energy
is increased, and thus when the scales are reduced. In other words, the strand model
predicts that the electromagnetic coupling increases with energy, as is observed:
384
12 pa rticle properties d ed uced from stra nds
⊳ ?em < ?w < ?s .
The running of the coupling constants up to Pl anck energy
At energies near the Planck energy, quantum field theory is modified: effects due to the
strand diameter start to play a role. Near Planck energy, tangles get tighter and tighter
and fluctuations get weaker, because there is less room for them. In other words, near
Planck energy tangles tend to approach the structure of horizons. Therefore, near the
Planck energy, the strand model predicts deviations from the energy dependence of the
coupling constants that is predicted by quantum field theory. So far, estimating such
deviations has not been possible.
Another calculation might seem more promising: to calculate the coupling constants
near Planck energy. It could be argued that the approach to calculate the low-energy
coupling constants from Planck-energy values seems unsatisfactory, due to the approximations and extrapolations involved. But it is possible if we are convinced that quantum
field theory is correct up to Planck energy. And this is just what the strand model predicts. Such a Planck-scale calculation might then allow us to estimate the low-energy
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 378
The prediction matches observations. Unfortunately, this argument is not reliable. If
the strand number were the only cause of the running, the argument would imply that
the three slopes for the running of the three coupling constants should behave like 3:2:1.
However, the graph of Figure 115 shows otherwise, even if the difference between the
electromagnetic coupling and the weak hypercharge coupling is taken into account. Indeed, the running of the coupling constants is not due to strand number only, but also
to the explicit boson and fermion content of nature, as we just saw.
Motion Mountain – The Adventure of Physics
reproduces quantum field theory. Interestingly, the strand model also proposes a simpler,
though less precise explanation of the running.
At energies much smaller than the Planck energy, such as everyday energies, the
strand model implies that the average size of the tangle core is of the order of the position
uncertainty of a particle. In other words, any thickness of the strands – real or effective
– can be neglected at low energies. Therefore, at low energies, the average strand length
within a particle tangle core is also of the order of the de Broglie wavelength. Low, everyday energy thus implies large, loose and spherical/ellipsoidal tangle cores.
At low energies, shape fluctuations can lead to any of the three Reidemeister moves.
The probabilities of such shape deformations will scale with some power of the average strand length within the tangle core. In other words, coupling constants depend on
energy. But how exactly?
We note directly that higher Reidemeister moves, which involve larger numbers of
strand segments, will scale with larger power values. In particular, the longer the strand
in the core – i.e., the lower the energy – the more the relative probability for the higher
Reidemeister moves will increase.
In summary, the strand model predicts that when a tangle is loose and long, i.e., when
energies are low, the strong nuclear interaction, due to the third Reidemeister move,
is the strongest gauge interaction, followed by the weak nuclear interaction, due to the
second Reidemeister move, in turn followed by the electromagnetic interaction:
coupling consta nts a nd unification
385
coupling constants from their Planck energy values. However, so far, also this approach
has not led to success, despite a number of attempts. The challenge seems to be to understand core deformation for case of tight tangle cores. We keep this option in mind.
L imits for the fine structure constant d o not provide
expl anations
Ref. 260
When searching for ways to determine the fine structure constant, we need to be careful.
Here is an example that explains why.
Numerous observations of nature imply a limit on the fine structure constant. A pretty
one appeared in a post on the internet in 2017. The electrostatic repulsion between two
electrons at a given distance must be larger than the radiation force between to small
neutral black holes at that same distance. In other words,
Vol. V, page 150
Here it is assumed that thermal radiation from one black hole acts on the cross section
of the other black hole by pushing it away. Multiplying both sides by ?2 /ℏ? and inserting
the expressions for the black hole luminosity ? bh and the black hole radius ?bh gives
?>
1
.
15 320 π
(216)
The bound is not tight, but is obviously correct.
Various researchers are looking for observations that give the best possible bound for
?. Such a search can indeed yield much better bounds. However, such a search cannot
explain the value of ?. We can indeed use thermodynamics, gravity or other observed
properties to deduce observational limits on ?. Many formulae of physics contain ? in a
more or less obvious way. Maybe, one day, known physics will be able to yield very tight
upper and lower bounds for ?. Still, the explanation of the value of ? would still lack.
To explain the fine structure constant ?, we need an approach based on the final theory, not one based on known, millennium physics, such as expression (215). Millennium
physics can measure ?, but cannot explain it. To explain the fine structure constant, a
final theory is needed. In our case, we need to check whether we can calculate ? with
strands. Therefore, we now explore tangle topology, tangle shapes and tangle motion
with this aim in mind.
Charge quantization and top ological writhe
In the strand model, electric charge is related to the chirality of a tangle. Only chiral
tangles are electrically charged. The strand model thus implies that a topological quantity for tangles – defined for each tangle in the tangle family corresponding to a specific
elementary particle – must represent electric charge. Which quantity could this be?
The first candidate for charge in the strand model is provided by knot theory:
⊳ The usual topological quantity to determine chirality of knots and tangles is
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 261
(215)
Motion Mountain – The Adventure of Physics
2
? bh π ?bh
?2 1
>
.
4π?0 ?2
? 4π?2
386
12 pa rticle properties d ed uced from stra nds
the topological writhe.
To determine its value, we draw a minimal projection, i.e., a two-dimensional knot or
tangle diagram with the smallest number of crossings possible. We then count the righthanded crossings and subtract the number of left-handed crossings. This difference, an
integer, is the topological writhe. Topological writhe is thus a two-dimensional concept
and does not depend on the shape of a knot or tangle. We note:
Page 327
⊳ The electric charge quantum number behaves similarly to topological writhe
(times one third or times one): it is quantized, has two possible signs, vanishes for achiral tangles, is a topological invariant – and thus is conserved.
Challenge 222 e
Challenge 223 ny
Ref. 5
In short, a topological quantity, namely topological writhe, reproduces the electric charge
quantum number in the strand model. Three issues remain. First, given that every
particle is described by a tangle family with an infinite number of members, how is
the electric charge, i.e., the topological writhe of the other tangle family members accounted for? It is not hard to see that family members do not change topological
writhe. The second issue is more thorny: why is the charge definition different for
leptons? We skip this problem for the time being. The third issue is the central one:
What is the origin of the peculiar value of the charge unit, whose square has the value
? = 1/137.035 999 139(31) at low energy?
Charge quantization and linking number
An alternative conjecture for charge quantization is the following:
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
In other terms, the electric charge quantum number can be reproduced with the help of
topological writhe. And indeed, the electric charge of massless bosons, i.e., photons and
gravitons, vanishes.
Let us sum up. In nature, electric charge is quantized. The strand model describes
charged particles with the help of fluctuating alternating tangles, and charge quantization
is a topological effect that results because all particles are made of strands. In particular,
Motion Mountain – The Adventure of Physics
Page 321
— The topological writhe of the W boson tangles is +3 or −3, depending on which mirror image we look at; the topological writhe of the Z boson and Higgs boson tangles
vanishes. The topological writhe of any unknotted strand also vanishes. In this way, if
we define the electric charge quantum number as one third of the topological writhe,
we recover the correct electric charge quantum number of the weak and all other
gauge bosons. We note that the leather trick does not change this result, so that all
family members of a particle share the same topological writhe.
— The tangles of the quarks show that if we define the electric charge quantum number
as one third of the topological writhe, we recover the correct electric charge quantum
number of all quarks. The leather trick has no effect on this definition.
— The tangles of the leptons show that if we define the electric charge quantum number as the topological writhe of the centre region only, we recover the correct electric
charge quantum number of all leptons. Again, the leather trick does not change this
result.
coupling consta nts a nd unification
387
⊳ Electric charge, i.e., twist emission probability, might be proportional to the
linking number of ribbons formed by strand pairs.
The following arguments speak in favour of this conjecture.
How to calcul ate coupling constants
The strand model suggests that crossing number and linking number somehow define
electric and weak charge. In simple words, the model suggests that quantization of all
charge types is a topological effect; quantization is due to the multiple ways in which
strands cross inside tangles.
Coupling constants describe the probability of interaction with gauge bosons. Experiments show that these quantities are slightly scale-dependent, since they run with energy.
But in the strand model, coupling constants are not really shape-dependent: electrons,
muons and antiprotons have the same electric charge and fine structure constant values
despite being described by different tangles. Coupling constants do not depend on the
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Challenge 224 e
In short, linking number, an integer, might be a better topological quantity to explain
electric charge quantization than topological writhe. On the other hand, it might well be
that linking number, being a quantity that depends on two strands, is related to the weak
charge rather than to the electric charge.
If the conjectured relation between linking number and electric or weak charge is
correct, it might lead to a calculation of the corresponding coupling constant, once the
tangle shape or, better, once the tangle dynamics is included in the proper way. For example, the photon emission probability could depend on the writhe or on the twist of
the (averaged) ribbons. Both these properties might lead to virtual photon emission.
(The sum of writhe and twist of a ribbon is given by the linking number, as explained by
Calugareanu’s theorem.)
In this and any topological definition of electric charge, we face two slight hurdles:
First, we have to watch out for the graviton: it is uncharged. Secondly, we have to explain
why the strand model for the simplest family member of the d quark is not chiral. Both
hurdles can be overcome.
If the linking of two strands is connected to weak charge, it might well be that a similar
quantity defined for three strands is related to colour charge. All these possibilities are
topic of research.
Motion Mountain – The Adventure of Physics
— In knot theory, a ribbon is the strip associated to and limited by two strands.
— The linking number of a ribbon is the number of times that the two edges of a ribbon
wind around each other. The linking number is a topological invariant and an integer.
— In particle tangles, only wound up, i.e., linked ribbons should lead to (net) boson
emission. For tangles made of three strands, we define a total linking number as the
sum of all three possible linking numbers.
— The linking number of the Higgs boson strand pairs is zero; that of the Z boson strand
pairs is the sum of 1, 0 and -1, thus also zero. The linking number for the W boson is
3 or -3, that of the quarks is 1, -1, 2 or -2. We thus conjecture that the charge quantum
number is one third of the total linking number.
— Massless bosons, i.e., photons, gluons and gravitons, have no electric charge.
388
12 pa rticle properties d ed uced from stra nds
kind of tangle. Experiments show that they just depend somewhat on its size. In short,
⊳ We need a definition of each coupling constant that is tangle-independent
and shape-independent, and only depends on a topological invariant of
tangles.
In fact, this conclusion eliminates many speculations, including a number of calculation
approaches that were included in this chapter in previous editions. We are left with just
a few options. To explore them, we start with an overview.
C oupling constants in the strand model
In the strand model, neutral particles are those that cannot receive Reidemeister moves
or that receive them all in equal way:
1. Electromagnetism: Neutral ‘tangles’ are made of one strand (e.g., the photon) or are
achiral (e.g., the Z and the neutrinos).
2. Weak interaction: Neutral tangles are made of one strand (e.g., the photon) or of two
straight or unpokeable strand pairs (e.g., the Z, the right-handed leptons and quarks).
3. Strong interaction: Neutral tangles are made of one strand or of three strands.
In the strand model, charged particles are specific tangles:
1. Electric charge is due to the observability of crossings during photon emission or
absorption, i.e., when twists are applied. Particles with electric charge, i.e., with preferred twist transfer, have a global asymmetry, global twistedness, namely topological
chirality. Locally, electrically charged particles have crossings; electric charge is positive or negative. Charge is 1/3 of the signed crossing number. Examples are the charged
leptons, the quarks and the W boson.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
In the strand model, the gauge interactions are modelled as transfers of Reidemeister
moves:
1. The electromagnetic interaction is twist transfer and the electric charge is preferred
twist transfer to or from a massive particle. Twists can be added abd form a circle:
they form a U(1) Lie group. They change the tangle phase by exchanging one observable crossing.
2. The weak interaction is poke transfer and the weak isospin is preferred poke transfer
to or from a massive particle. Pokes exist in three linearly independent directions
and their generators behave like the belt trick: they generate an SU(2) Lie group.
They change the tangle phase by exchanging two observable crossings.
3. The strong interaction is slide transfer and the colour charge is preferred slide transfer
to or from a massive particle. Slides can be added, its generators have a ?3 symmetry
and they form an SU(3) Lie group. They change the tangle phase by exchanging two
or three? crossings.
Motion Mountain – The Adventure of Physics
In experiments, there are the following gauge interactions with their charges:
1. The electromagnetic interaction with electric charge and U(1) symmetry.
2. The weak interaction with weak isospin and SU(2) symmetry.
3. The strong interaction with colour and SU(3) symmetry.
coupling consta nts a nd unification
389
Coupling strength is the ease of crossing rotation, of poke creation, and of slide induction. These connections allow calculating the coupling strength values.
Deducing ? from precession
Page 226
In nature, magnetic fields rotate charged particles. In the strand model, as shown in
Figure 52, magnetic fields are made of moving twists. In fact, from the strand definition of the electromagnetic interaction and the electric charge and from the drawing in
Figure 49, we deduce:
⊳ Moving twists rotate crossings.
We note that this description differs slightly from a pure twist transfer. But this formulation is the key to calculating ?.
We assume that the typical, average crossing is lying in the paper plane, as in the drawing of the fundamental principle. For an average crossing, the two strands lie along the x
and y axes. When a photon, i.e., a twist, arrives along the diagonal in the first quadrant,
it rotates the crossing completely, by one turn. If the twist arrives at a different angle, its
effect is lowered. We approximate this angle effect with simple trigonometry: we assume
that the angular projection describes the reduction of the effect with the incoming angle
of the twist.
For the incident photon, we call ? the angle from the y-axis and ? the angle out of
the paper plane. The average rotation angle induced by an absorbed photon or twist on
a charged particle with three crossings, corresponding to one elementary charge, can be
calculated. We include sin ? for the volume element in spherical coordinates and average
over the possible angle values ? between the strands at the crossing. Further terms arise
from the trigonometric approximation. In particular, a second power arises from the two
tails, and a further squaring is required to get probabilities. Nevertheless, the expression
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 231
Motion Mountain – The Adventure of Physics
2. Weak charge is thus due to the observability of crossings during W or Z emission or
absorption, i.e., when pokes are applied. Particles with weak isospin, i.e., with preferred poke transfer, have a global asymmetry that prevents all pokes to act equally
effectively: For fermions, such an asymmetry arises when tangle twistedness and the
belt trick have the same sign; thus all left-handed fermions and right-handed antifermions have weak isospin. Locally, weakly charged fermions behave like a belt buckle
that rotates in the appropriate direction. Due to their tangle topology, some fermions
have positive, others negative weak isospin. For the W boson, the asymmetry is built
into the tangle; due to the tangle structure, the W and its antiparticle have plus or
minus twice the weak isospin of fermions.
3. Colour, strong charge, is due to the observability of crossings during gluon emission
or absorption, i.e., when slides are applied. Particles with colour charge, i.e., with preferred slide transfer, have a global asymmetry that prevents all slides to act equally effectively: Coloured particles are made of exactly two strands with tails in tetrahedron
skeleton directions. Only two-stranded tangles allow certain slides and prevent others. Therefore only quarks have colour charge. Locally, red, blue and green colours
correspond to three directions in one plane that differ by an angle of 2π/3.
390
12 pa rticle properties d ed uced from stra nds
remains open to dispute:
π/2
3 π ?/2
∫
∫
∫
cos ? (sin ?)2 (cos(?π/?) cos ?)4 d? d? d? = 0.15 .
2π2 ?=0 ?=−?/2 ?=−π/2
(217)
The resulting value of 0.15 is not an acceptable approximation to reality, in which √? =
0.08542454311(1) at low energy and √? = 0.10(1) at Planck energy. Neither is the value
a good approximation to the hypercharge coupling, which changes from √?1 = 0.10(1)
at 100 GeV to √?1 = 0.13(1) at Planck energy. We need a better approximation for the
value of the electromagnetic coupling strength.
√?calc =
Deducing the weak coupling
⊳ A moving poke rotates a pair of strands.
π/2
√?w calc = ∫
?=0
2π
∫
sin ? (cos2 ? cos2 ?)4 d? d? ≈ 0.19 .
(218)
?=0
If we need to average over the different angles between the strands that make up the pair
experiencing the poke, we get a different value.
The calculated value of the weak coupling is not an acceptable approximation to reality, in which √?w = 0.18 at the (low) energy of 100 GeV and √?w = 0.14 at Planck
energy. We need a better approximation.
Deducing the strong coupling
Strong fields deform specific three-strand configurations by adding generalized slides.
The generalized slides are due to gluons. Strong colour is related to the order and orient-
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
This process is the key to calculating ?w . We note that there is a certain similarity to the
setting used for calculating the electromagnetic coupling: in both cases, the incoming
boson acts on a target consisting of two strands. This similarity is the reason for electroweak mixing.
We calculate the coupling constant for a single belt buckle, assuming parallel strands.
The average rotation angle induced by one incoming weak (unbroken) boson (out of
the three possible cases) is one full turn when the impact is perpendicular to the two
strands and to the plane defined by them. For a general incidence angle the induced
rotation angle is lower. We again use trigonometrical projection to approximate the induced crossing rotation angle in the general case, with the same issues as in the previous
case. We call ? the angle from ideal incidence, and ? the longitude. The average angle is
then given by
Motion Mountain – The Adventure of Physics
Weak fields deform strand (crossing) pairs by adding or transferring generalized pokes.
Weak fields are collections of pokes; pokes represent virtual weak bosons. The weak
isospin, the weak charge, is related to the orientation of the strand pairs. The weak interaction occurs through an incoming poke that deforms a strand pair:
coupling consta nts a nd unification
391
ation of the strands in these specific three-strand configurations. In short:
⊳ Incoming, moving slides deform three-strand configurations.
This is the key to calculating ?s.
We assume that one of eight possible gluons is incident. In an average triple strand
configuration, the three strands are oriented in a way that in the paper plane they look
like three symmetrically arranged rays. One ray lies along the y axis. When a gluon
arrives, it performs a slide. For an incident gluon, we call ? the angle from the y-axis to
the next strand and ? the angle out of the paper. In the trigonometric approximation,
the average slide angle induced on a coloured particle is given by
π/2
√?s calc = ∫
2π
∫
(219)
?=0
This is not an acceptable approximation to reality, in which √?s = 0.7(1) at the (low)
energy of 1 GeV and √?s = 0.13(1) at Planck energy. We need a better approximation
for the strong coupling.
Open challenge: calcul ate coupling constants with precision
Page 352
Ref. 262
Electric dip ole moments
Experimental physicists are searching for electric dipole moments of elementary
particles. No non-zero value has been detected yet. The idea of electric dipole moment is based on a non-spherical distribution of electric charge in space.
In the strand model, particles are tangles. As a consequence, the electric charge dis-
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Challenge 225 r
The approximations used above for estimating the coupling constants can be dismissed
as mere educated guesses. Despite this objection, these guesses show that a determination of the coupling constants from the strand model is within reach, and that it can be
realized with limited effort. It is sufficient to improve the three approximations; this is
can be realized by using computer simulations for the transfer of Reidemeister moves or
by finding an improved analytical model.
Calculating all three coupling constants ab initio with high precision will allow checking the statements of this section in an independent manner and, above all, will allow
testing the strand model. The calculations should be performed at different energies,
to confirm the energy dependence of the couplings. Also the influence of the effective
strand diameter on the fine structure constant should be explored.
In order to reach highest precision, the effects of the various tangle family members
might have to be taken into account, because in the strand model, each particle is described by a family of tangles. On the other hand, the strand model predicts that family
members have a small effect on the coupling constant, so that the family issue can be
neglected in the beginning.
In the case of the nuclear coupling constants, Arnold’s results on plane curves may
help in the estimations and calculations.
Motion Mountain – The Adventure of Physics
?=0
sin ? (cos3 ? cos3 ?)4 d? d? ≈ 0.11 .
392
12 pa rticle properties d ed uced from stra nds
tribution – the distribution of the crossings in a tangle – is intrinsically a slightly nonspherical quantity, thus a quantity unequally distributed in space. However, it is only
non-local on a scale of the order of a Planck length. In other terms, the electric dipole
moment ? of elementary particles is predicted to be
⊳ ? = ? ? ?Pl ,
Ref. 263
Five key challenges abou t coupling strengths
There are many ways to evaluate candidates for unified models. A concrete evaluation
focuses on four key challenges about coupling constants. These challenges must be resolved by any candidate model in order to be of interest.
⊳ The strand model explains why the fine structure constant, or equivalently,
the electric charge, is the same for electrons and protons.
Deducing this equality is a key challenge for any unified model. In fact, all coupling
constants must be independent of particle type. This is the case in the strand model.
2. The second key challenge was the energy-dependence of the coupling constants.
The strand model predicts that coupling constants run with energy in exactly the way that
is predicted by QED, QCD and electroweak theory. We could also argue that this is not a
real challenge for any unified model that reproduces these theories. In the strand model,
the running of the electromagnetic coupling constant can be seen as a consequence of
the gradual tightening of tangles with energy. For a typical electrically charged particle
at low energy, the tangle is very loose; therefore:
⊳ The Planck scale number of crossings is shielded by an additional cloud of
crossings created by the loose strands of the tangle.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
1. So far, we explained particle charges with topological properties of the tangle models of the particles, and we explained coupling strengths with the transfer of crossings,
pokes and slides. This allowed deducing a rough approximation of coupling constants.
By doing so, we have settled a first key challenge:
Motion Mountain – The Adventure of Physics
where the factor ? arises from averaging the tangle and is of order one. Similar values are
predicted by the standard model in the absence of supersymmetry and grand unification.
However, the sensitivity of measurements has not reached these values yet, by several
orders of magnitude.
We note that the strand model predicts that the dipole moment changes, or ‘runs’,
with energy. This follows from the shape-dependence of the dipole moment. Such a
dependence is also predicted by quantum field theory.
In summary, we expect that up to a region close to a Planck length, the strand model
should not yield dipole moments that differ in order of magnitude from those predicted
by the standard model of particle physics. In the future, more precise calculations and
measurements could allow testing the strand model using dipole moments.
coupling consta nts a nd unification
393
In this way, the strand model explains the running of the fine structure constant in exactly the same way as QED.
3. The third key challenge has only been touched upon very briefly:
⊳ Any unified model needs to clarify the relation between the hypercharge,
the electric charge and the weak isospin (the ‘weak charge’).
4. The fourth key challenge, related to the previous one, still needs to be explored in
more detail:
⊳ Any unified model must explain why the mass ratio of the intermediate
weak vector bosons is related to the coupling ratio of the weak and the electromagnetic interaction as
?? 2 ?
) +
=1.
??
?w
(220)
The strand model strongly suggests that it can explain the relation, but the detailed argument must yet be provided. Using more drastic language, we can repeat what many
have said already in the past: explaining the electroweak mixing expression (220) is the
key challenge for any unified model.
In the strand model, the two electroweak coupling constants are measures for interaction probabilities of crossings with twists and with pokes. In contrast, masses are interaction probabilities of crossings with spatial curvature. Why are they related by expression
(220)? Here is a short brainstorm on the issue.
In the strand model, mass appears by tail braiding. Tail braiding adds crossings, and
in this way adds mass. Added crossings also imply added weak and sometimes electric
charges. The Z boson arises from vacuum by different tail braidings than the W. The W
arises by the braiding of two tail pairs at 90 degrees; the Z arises by braiding one tail pair
at 90 degrees.
In case of the W and the Z bosons, the Z tangle produces a larger disturbance of the
vacuum than the W; therefore it is more massive than the W.
At which angle does a clasp start to form a ‘‘enclosed space in between’’? How does
this space change with scale, given that scale might change the clasp angle? This question might be related to the running of masses, mixing angles or coupling constants. In
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
(
Motion Mountain – The Adventure of Physics
The strand model explains electromagnetism as acting on crossings and the weak interaction as acting on parallel strands. This general statement contains the required explanation; but the details still need to be worked out. It is expected that in electromagnetism,
a single crossing is rotated, mainly by rotating one strand around the other. In contrast,
in the weak interaction, two strands are rotated together, producing ar switching two
crossings. The number of crossings differs between electromagnetism and the weak interaction, but the total number of involved strands is tow in both cases. As a result of this
similarity, the two interactions mix. The final explanation of electroweak mixing might
even allow to deduce a intuitive geometric meaning of ?w , the weak mixing angle or
Weinberg angle.
394
12 pa rticle properties d ed uced from stra nds
particular, we should answer the following question: Which physical observable does
this enclosed space influence? Mass, couplings, or mixings? Is mass more related to
ropelength or more related to the enclosed space?
5. The fifth key challenge is, of course, the precise calculation of the coupling constants.
Summary on coupling constants
Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
The strand model implies that coupling constants are geometric properties of tangle families that correspond to charged particles. As a consequence, strands explain why the
coupling constants are not free parameters in nature, but fixed constants. Strands predict
that coupling constants are the same for particles with the same charge, and that coupling
constants are constant during the macroscopic evolution of the universe. Strands predict
small electric dipole moments for elementary particles, compatible with and lower than
present measurement limits. Strands also predict the correct sequence of the coupling
constants at low energy and the correct sign of their running with energy. Strands thus
reproduce all observed qualitative properties of coupling constants. No other unified
model achieves this yet.
Using tangle shapes, the strand model proposes several ways to calculate coupling
constants ab initio. First estimates of the fine structure constant, based on the knotted particle models of 2010, deviated from experiment by 40 %. However, those particle
models turned out to be mistaken. New estimates based on the new tangled particle
models yield a much better accuracy; nevertheless, the errors due to the approximations
are still larger that the measurement errors. Improved calculations are ongoing and will
allow to confirm or to refute the strand model.
C h a p t e r 13
EXPER IMENTA L PR EDICT IONS OF
T HE ST R AND MODEL
“
Es gibt viele Theorien,
die sich jedem Test entziehen.
Diese aber kann man checken,
elend wird sie drum verrecken.**
A
round the world, numerous researchers are involved in experiments that
re searching for new effects. They are searching for new observations that
re unexplained by the standard model of particle physics or by the conventional view of cosmology. At the same time, all these experiments are testing the strand
model presented here. In fact, most people working on these experiments have not
heard about the strand model, so that there is not even the danger of unconscious bias.
To simplify the check with experiments, the most important predictions of the strand
model that we deduced in our adventure are listed in Table 17.
typeface distinguishes predictions that are unsurprising, that are unconfirmed or unique to the strand
model, and those that are both unconfirmed and unique.
Page 36
Page 329
Experiment
Prediction (most
from 2008/2009)
S tat us ( 2 0 1 7)
Planck units (?, ℏ, ?, ?4 /4?)
are limit values.
Higgs boson
2008/9: does not exist.
2012: does exist.
2008/9: implies no Higgs.
2012: implies one Higgs.
2008/9: show non-local
effects at the Large Hadron
Collider.
2012: show no non-local
effects at the Large Hadron
Collider.
None has been
exceeded, but more
checks are possible.
Falsified.
Verified.
No data yet.
No data yet.
No data yet.
Page 331
Page 382
Running of the coupling constants
Page 382
Page 329
Page 331
Longitudinal W and Z boson scattering
None found yet.
** No adequate translation is possible of this rhyme, inspired by Wilhelm Busch, claiming that any theory
that can be tested is bound to die miserably.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
TA B L E 17 The main predictions of the strand model that follow from the fundamental principle. The
Motion Mountain – The Adventure of Physics
”
Anonymous
396
13 experimental predictions
TA B L E 17 (Continued) The main predictions of the strand model that follow from the fundamental
principle. The typeface distinguishes predictions that are unsurprising, that are unconfirmed or unique to
the strand model, and those that are both unconfirmed and unique.
Page 329
Page 352
Page 314
Page 276, page 318
Page 313
Page 375
Page 341
Page 376
Page 328
Page 276, page 328
Page 344
Page 324, page 340
Page 276, page 328
Page 341
Page 341
Page 338
S tat us ( 2 0 1 7)
Longitudinal W and Z boson scattering
W boson g-factor
Unknown fermions (supersymmetric
particles, magnetic monopoles, dyons,
heavy neutrinos etc.)
Unknown bosons (other gauge bosons,
supersymmetric particles, axions etc.)
Unknown interactions, energy scales and
symmetries (grand unification,
supersymmetry, quantum groups,
technicolour etc.)
Particle masses, mixing angles and
coupling constants
is unitary at the LHC.
is near to 2.
do not exist.
Obvious.
Is observed.
None found yet.
do not exist.
None found yet.
do not exist.
None found yet.
Particle masses, mixing angles and
coupling constants
Particle masses, mixing angles, coupling
constants and g-factors
Mixing matrix for quarks
Mixing matrix for neutrinos
Neutrinos
Neutrinos
Neutrino-less double beta decay
Electric dipole moments of elementary
particles, magnetic dipole moment of
neutrinos
Tetraquarks
Glueballs
Proton decay and other rare decays,
neutron-antineutron oscillations
Neutron decay
Neutron charge
Hadron masses and form factors
are calculable by modifying Most not yet
existing software packages. calculated;
approximations very
encouraging.
are constant in time.
Is observed.
are identical for antimatter.
Is observed.
is unitary.
is unitary.
are Dirac particles.
violate CP symmetry.
does not exist.
have extremely small,
calculable values.
Is observed.
No data yet.
No data yet.
No data yet.
Not yet found.
No data yet.
exist.
probably do not exist; if they
do, the spectrum can be
compared to the strand
model.
occur at extremely small,
standard model rates.
follows the standard model.
vanishes.
can be calculated ab initio.
Likely.
Not yet observed.
Not yet observed.
No deviations found.
None observed.
Not yet calculated;
value sequences and
signs correct.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 372
Prediction (most
from 2008/2009)
Motion Mountain – The Adventure of Physics
Page 313
Experiment
of the stra nd m od el
397
TA B L E 17 (Continued) The main predictions of the strand model that follow from the fundamental
principle. The typeface distinguishes predictions that are unsurprising, that are unconfirmed or unique to
the strand model, and those that are both unconfirmed and unique.
Page 353
Page 314
Page 146
Page 293
Page 293
Page 291
Page 304
Page 307
Page 307
Page 309
Page 307
Page 376
Page 310
Page 354
Dark matter
is conventional matter plus
black holes.
S tat us ( 2 0 1 7)
Partly confirmed by
black hole mergers
and lack of other
results.
Standard model of particle physics
2008/9: is essentially correct, Not yet falsified, but
with deviations for the
deviations not yet
scattering of longitudinal
observed.
vector bosons at LHC energy.
2012: is correct for all
All data agrees.
measurable energies.
Additional dimensions
do not exist.
Not observed.
Non-commutative space-time
does not exist.
Not observed.
General relativity
is correct at all accessible
No deviation found.
energies.
Short-distance deviations from universal do not exist.
All data agrees.
gravitation and modified gravity
Space-time singularities, cosmic strings, do not exist.
None observed.
wormholes, time-like loops, negative
energy regions, domain walls
Quantum gravity effects
will not be found.
None observed yet.
Behind a horizon
nothing exists.
Nothing observed.
Cosmological constant (dark energy)
is small and positive.
Is observed.
Cosmological constant (dark energy)
decreases with time squared. Data are inconclusive.
Cosmic matter density
decreases with time squared. Data are inconclusive.
Cosmic inflation
did not occur.
Data not in contrast.
Leptogenesis
did not occur.
Data are inconclusive.
Cosmic topology
is trivial.
As observed.
Vacuum
is stable and unique.
As observed.
In summary: all motion
results from strands.
Not yet falsified.
In this list, the most interesting predictions of the strand model are the numerical
predictions on the decay of the cosmological constant, the various mass ratios and mass
sequences – including the Z/W and Higgs/W mass ratios – and the relative strength
of the three gauge interactions. There is the clear option to calculate all fundamental
constants in the foreseeable future.
In addition, the strand model reproduces the quark model, gauge theory, wave functions and general relativity; at the same time, the model predicts the lack of measurable
deviations. The strand model solves conceptual problems such as the dark matter problem, inflation, confinement, the strong CP problem and the anomaly issue; by doing so,
the strand model predicts the lack of unknown effects in these domains.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 297, page 277
Prediction (most
from 2008/2009)
Motion Mountain – The Adventure of Physics
Page 146
Experiment
398
13 experimental predictions
The fundamental principle of the strand model
Strand model :
?1
Observation :
Some
deformation,
but no
passing
through
? = ℏ/2
Δ? = ?Pl
Δ? = ?Pl
? = ?/2
?2
F I G U R E 116 The fundamental principle of the strand model: Planck units are defined by a crossing
switch in three spatial dimensions. With this principle, as shown in the previous chapters, the
fundamental principle implies general relativity and the standard model of particle physics.
In our adventure, we have argued that Planck’s natural units should be modelled with the
fundamental principle for strands, which is shown again in Figure 116. As we discovered,
the fundamental principle explains the following measured properties of nature:
— Strands explain the principle of least action and the invariance of ?, ℏ, ? and ?.
— Strands explain the three dimensions of space, the existence of gravitation, curvature
and horizons, the equations of general relativity, the value of black hole entropy and
the observations of modern cosmology.
— Strands explain all the concepts used in the Lagrangian of the standard model of
particle physics, including wave functions, the Dirac equation and the finite, discrete
and small mass of elementary particles.
— Strands explain the existence of electromagnetism and of the two nuclear interactions, with their gauge groups and all their other observed properties.
— Strands describe the observed gauge and Higgs bosons, their charges, their quantum
numbers and their mass mass ranges.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
final summary abou t the millennium issues
Motion Mountain – The Adventure of Physics
The strand model deduces all its experimental predictions from a single and simple
fundamental principle: events and Planck units are due to crossing switches of strands.
Provided there are no errors of reasoning, there is no way to change the predictions
summarized here. The strand model is both simple and unmodifiable.
Naturally, errors of reasoning in the preceding chapters are well possible. A few have
occurred in the past. The exploration was performed at high speed – possibly too high. If
any experiment ever contradicts a prediction of the strand model, the model is doomed.
When the above experimental predictions were first deduced in 2008 and 2009, they were
quite unpopular. Practically all other attempts at unification predicted the existence of
yet undiscovered particles and effects. However, so far, experiment does not confirm
these other attempts; in fact, no prediction of the strand model has been falsified yet.
fina l sum mary a b ou t the m illennium issues
399
All these results translate to specific statements on experimental observations. So far,
there is no contradiction between the strand model and experiments. These results allow
us to sum up our adventure in three statements:
Page 21
We have not yet literally reached the top of Motion Mountain – because certain numerical predictions of the fundamental constants are not yet precise enough – but if no cloud
has played a trick on us, we have seen the top from nearby. In particular, we finally know
the origin of colours.
The last leg, the accurate calculation of the constants of the standard model of particle
physics, is still under way. The drive for simplicity and the spirit of playfulness that we
invoked at the start have been good guides.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 19
1. Strands solve all open issues. With one simple fundamental principle, the strand
model solves or at least proposes a way to solve all issues from the millennium list of
open issues in fundamental physics. All fundamental constants can be calculated.
2. Strands agree with all observations. In particular, the strand model implies that general relativity, quantum theory and the standard model of elementary particles are a
precise description of motion for all practical purposes.
3. Nothing new will be discovered in fundamental physics. Unexpectedly but convincingly, strands predict that general relativity, quantum theory and the standard model
of elementary particles are a complete description of motion for all practical purposes.
Motion Mountain – The Adventure of Physics
— Strands explain the three generations of quarks and leptons, their charges and
quantum numbers, their mixing, their mass sequences, as well as their confinement
properties.
— Strands explain the quark model of hadrons, including CP violation, mass sequences,
signs of quadrupole moments, the lack of unobserved hadrons, common Regge
slopes and the existence of tetraquarks.
— Strands do not allow arbitrary values for masses, coupling constants, mixing angles
and CP violating phases.
— Strands enable calculations of particle masses, their coupling constants, their mixing
angles and the CP violating phases. First rough estimates of these values agree with
the (much more precise) experimental data. Computer calculations will allow us to
improve these checks in the near future.
— Strands predict the lack of unknown dark matter and of unknown inflation mechanisms.
— Finally, strands predict that nature does not hide any unknown elementary particle,
fundamental interaction, fundamental symmetry or additional dimension. In particular, strands predict that no additional mathematical or physical concepts are required for a final theory.
C h a p t e r 14
T HE TOP OF MOT ION MOUNTAIN
”
“
The labour we delight in physics pain.
William Shakespeare, Macbeth.
Thales**
our path to the top
”
Our walk had a simple aim: to talk accurately about all motion. This 2500 year old quest
drove us to the top of this mountain. We can summarize our path in three legs: everyday
life, general relativity plus quantum theory, and unification.
** Thales of Miletus (c. 624 – c. 546 bce) was the first known philosopher, mathematician and scientist.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
ho am I? Where do I come from? What shall I do? Where does the
orld come from? Can the whole world really come to a sudden end? What
ill happen in the future? What is beauty? All these questions have a common
aspect: they are questions about motion. But what is motion? Our search for an answer
led us to study motion in all its details. In this quest, every increase in the precision of
our description of motion was a step towards the peak of Motion Mountain. Now that
we arrived there, we can savour what we have achieved and recall the emotions that we
have experienced.
In our ascent, we have learned how we move, how we experience our environment,
how we grow, what parts we are made of, and how our actions and our convictions about
them can be understood. We have learned a lot about the history and a bit about the
future of matter, of radiation and of space. We have experienced and understood the
many ways in which beauty appears in nature: as colours, as shapes, as rhythms and
most of all: as simplicity.
Savouring our achievement means that first of all, we now can look back to where we
came from. Then we enjoy the view we are offered and look out for what we could not
see before. After that, we search for what is still hidden from our sight. And finally, we
take a different path back down to where we live.
Motion Mountain – The Adventure of Physics
W
“
All things are full of gods.
our path to the top
401
Everyday life: the rule of infinit y
Ref. 1, Ref. 3
“
Ref. 2, Ref. 4
Ref. 264
Vorhin haben wir gesehen, daß in der Wirklichkeit
das Unendliche nirgends zu finden ist, was für
Erfahrungen und Beobachtungen und welcherlei
Wissenschaft wir auch heranziehen.*
David Hilbert
”
The idea that nature offers an infinite range of possibilities is often voiced with deep
personal conviction. However, the results of relativity and quantum theory show the
opposite. In nature, speeds, forces, sizes, ages and actions are limited. No quantity in
nature is infinitely large or infinitely small. No quantity in nature is defined with infinite
precision. There never are infinitely many examples of a situation; the number of possibilities is always finite. The world around us is not infinite; neither its size, nor its age,
nor its content. Nature is not infinite. This is general relativity and quantum theory in
one statement.
Relativity and quantum theory show that the idea of infinity appears only in approximate descriptions of nature; it disappears when talking with precision. Nothing in nature
* ‘Above we have seen that in the real world, the infinite is nowhere to be found, whatever experiences and
observations and whatever knowledge we appeal to.’
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
R el ativit y and quantum theory: the absence of infinit y
Motion Mountain – The Adventure of Physics
Galilean physics is the description of everyday life. We all learned Galilean physics before
secondary school. Galilean physics is the exploration and description of the motion of
stones, water, trees, heat, the weather, electricity and light. To achieve this description of
our environment, our first and main act in life is to partition experience into experiences.
In other words, our first intellectual act is the invention of parts; we invented the plural.
The act of partitioning allows us to define sequences among our experiences, and thus
to define the concept of time. The concept of space arises similarly by our possibility to
distinguish observations that occur at the same time. By comparing parts with other
parts, we define measurement. Using all of this, we become able to define velocity, mass
and electric charge, among others. These allow us to introduce action, the quantity that
quantifies change.
For a simple description of observations, we assume that division is possible without
end: thus we introduce the infinitely small. We also assume that widening our scope
of observation is possible without end. Thus we introduce the infinitely large. Defining
parts thus leads us to introduce infinity.
Using parts and, with them, the infinitely small and the infinitely large, we found,
in volumes I and III, that everyday motion has six main properties: it is continuous,
conserved, relative, reversible, mirror-invariant and lazy. Motion is lazy – or efficient –
because it produces as little change as possible.
Nature minimizes change. This is Galilean physics, the description of everyday motion,
in one statement. It allows us to describe all our everyday experiences with stones, fluids,
stars, electric current, heat and light. The idea of change-minimizing motion is based on
a concept of motion that is continuous and predictable, and a concept of nature that
contains the infinitely small and the infinitely large.
402
Ref. 4
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
is infinite. For example, we found in volume II that the sky is dark at night (also) because
space is not infinite. And we found, in volumes IV and V, that quantum theory contains
probabilities because there is a smallest action value in nature. In fact, the statement that
a quantity is infinitely large or infinitely small cannot be confirmed or reproduced by
any experiment. Worse, such a statement is falsified by every measurement. In short, we
found that infinity is a fantasy of the human mind. In nature, it does not appear. Infinity
about nature is always a lie.
The number of particles, their possible positions, the states they can have, our brain,
our creativity, our possible thoughts: all this is not infinite. Nevertheless, quantum theory
and relativity changed the world: they allowed building ultrasound imaging, magnetic
resonance imaging, lasers, satellite navigation systems, music players and the internet.
Despite the vast progress due to modern physics and the related technologies, one
result remains: nothing in our environment is infinite – neither our life, nor our experiences, nor our memories, not even our dreams or our fantasies. Neither the information
necessary to describe the universe, nor the paper to write down the formulae, nor the
necessary ink, nor the time necessary to understand the formulae is infinite. Nature is
not infinite. On the other hand, we also know that the illusion of the existence of infinity
in nature is one the most persistent prejudices and myths ever conceived. Why did we
use it in the first place?
The habit to use infinity to describe the world has many emotional reasons. For some,
it reflects the deep-rooted experience of smallness that we carry within us as a remnant
our personal history, when the world seemed so large and powerful. For others, the idea
of our smallness allows us to deny somehow the responsibility for our actions or the
existence of death. For others again, the idea of a finite universe often, at a first glance,
produces deception, disbelief and discouragement. The absence of infinity means that
we cannot achieve everything we want, and that our dreams and our possibilities are
limited. Clinging to the idea of infinity is a way to avoid confronting this reality.
However, once we face and accept the absence of infinity, we make a powerful experience. We gain in strength. We are freed from the power of those who use this myth
to put themselves above others. It is an illuminating experience to reread all those sentences on nature, on the world and on the universe containing the term ‘infinite’, knowing that they are incorrect, and then clearly experience the manipulations behind them.
The desire to make others bow to what is called the infinite is a common type of human
violence.
At first, the demise of infinity might also bring panic fear, because it can appear as a
lack of guidance. But at closer inspection, the absence of infinity brings strength. Indeed,
the elimination of infinity takes from people one of the deepest fears: the fear of being
weak and insignificant.
Moreover, once we face the limits of nature, we react like in all those situations in
which we encounter a boundary: the limit becomes a challenge. For example, the experience that all bodies unavoidably fall makes parachuting so thrilling. The recognition
that our life is finite produces the fire to live it to the full. The knowledge of death gives
meaning to our actions. In an infinite life, every act could be postponed without any
consequence. The disappearance of infinity generates creativity. A world without limits is discouraging and depressing. Infinity is empty; limits are a source of strength and
pour passion into our life. Only the limits of the world ensure that every additional step
Motion Mountain – The Adventure of Physics
Challenge 226 e
14 the top of m otion m ountain
our path to the top
403
in life brings us forward. Only in a limited universe is progress possible and sensible.
Who is wiser, the one who denies limits, or the one who accepts them? And who lives
more intensely?
Unification: the absence of finitude
“
Pray be always in motion. Early in the morning go
and see things; and the rest of the day go and see
people. If you stay but a week at a place, and that an
insignificant one, see, however, all that is to be seen
there; know as many people, and get into as many
houses as ever you can.
Philip Stanhope,* Letters to his Son on the Fine Art of
Becoming a Man of the World and a Gentleman.
”
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
* Philip D. Stanhope (b. 1694 London, d. 1773 London) was a statesman and writer.
Motion Mountain – The Adventure of Physics
Page 127
The last part of our adventure, described in this volume, produced an unexpected result.
Not only is nature not infinite; nature is not finite either. None of the quantities which
were supposed to be finite turn out to be so. Finitude turns out to be an approximation,
or better, an illusion, though a subtle one. Nature is not finite. This is the unification of
physics in one statement.
Precise observation shows that nothing in nature can be counted. If nature were finite
it would have to be (described by) a set. However, the exploration of Planck scales shows
that such a description is intrinsically incomplete and inaccurate. Indeed, a description
of nature by a set can never explain the number of its elements, and thus cannot explain
finitude itself. In other words, any approach that tries to describe nature as finite is a
belief, and is never correct. Finitude is a lie.
We thus lost our security of thought a second time. Nature is neither infinite nor finite.
We explored the possibilities left over and found that only one option is left: Nature is
indivisible. In other words, all parts that we experience are approximations. Both finitude
and infinity are approximation of nature. All distinctions are approximate. This central
conclusion solved the remaining open issues about motion. Nature has no parts.
The impossibility to count and the lack of parts imply that nature is not a computer,
not an automaton, nor a physical system. Nature is not discrete.
Recognizing all distinctions as being approximate abolishes the distinction between
the permanent aspects of nature (‘objects’, described by mass, charge, spin, etc.) and the
changing aspects (‘states’, described by position, momentum, energy). Taking all distinctions as approximate introduces extended constituents: fluctuating strands. Looking
even closer, these extended constituents are all the same one. Space, formally only used to
describe states, also acquires changing aspects: it is made from fluctuating strands. Also
properties like mass or charge, which formally were seen as static, become aspects of the
ever changing interplay between these fundamental constituents. Describing nature as
one fluctuating strand allows us to avoid finitude and to answer all questions left open
by quantum theory and general relativity.
In a sense, the merging of objects and states is a resolution of the contrasting views
on motion of the Greek thinkers Parmenides – ‘there is no motion’, i.e., in physical
language, ‘there are no states, there is only permanence’ – and Heraclitus – ‘everything
404
14 the top of m otion m ountain
moves’, i.e., in physical language ‘there is no permanence, there are only states’. Both
turn out to be right.
We can thus sum up the progress during our adventure of physics in the following
table:
TA B L E 18 The progress of physics.
Step 1
Step 2
Galilean Physics
Relativity
Step 3
Quantum field theory
Step 4
Unification
Nature is continuous.
Nature has no infinitely
large.
Nature has no infinitely
small.
Nature is not finite.
Nature has no parts.
We live in Galilean space.
We live in Riemannian
space.
We live in a Hilbert/Fock
space.
We do not live in any space;
we are space.
new sights
sustanze e accidenti e lor costume
quasi conflati insieme, per tal modo
che ciò ch’i’ dico è un semplice lume.
La forma universal di questo nodo
credo ch’i’ vidi, perché più di largo,
dicendo questo, mi sento ch’i’ godo.*
Dante, La (Divina) Commedia, Paradiso,
XXXIII, 85-93.
”
Modelling nature as a complicated web of fluctuating strands allowed us to describe at
the same time empty space, matter, radiation, horizons, kefir, stars, children and all our
other observations. All everyday experiences are consequence of everything in nature
being made of one connected strand. This result literally widens our horizon.
* ‘In its depth I saw gathered, bound with love into one volume, that which unfolds throughout the universe:
substances and accidents and their relations almost joined together, in such a manner that what I say is only
a simple image. The universal form of that knot, I think I saw, because, while I am telling about it, I feel deep
joy.’ This is, in nine lines, Dante’s poetic description of his deepest mystical experience: the vision of god.
For Dante, god, at the depth of the light it emanates, is a knot. That knot spreads throughout the universe,
and substances and accidents – physicists would say: particles and states – are aspects of that knot. Dante
Alighieri (b. 1265 Florence, d. 1321 Ravenna) was one of the founders and the most important poet of the
Italian language. Most of the Divine Comedy, his magnum opus, was written in exile, after 1302, the year
when he had been condemned to death in Florence.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
“
Nel suo profondo vidi che s’interna,
legato con amore in un volume,
ciò che per l’universo si squaderna:
Motion Mountain – The Adventure of Physics
In summary, we are made of space. More precisely, we are made of the same constituents as space. In fact, the fascination of this result goes further than that.
new sig hts
405
The beau t y of strands
“
Someday, surely, we will see the principle underlying
existence itself as so simple, so beautiful, so obvious,
that we will all say to each other, “Oh, how could we
all have been so blind, so long.”
John Wheeler, A Journey Into Gravity And Spacetime.
”
Page 163
“
”
Die Natur kann besser Physik als der beste Physiker.*
Carl Ramsauer
As mentioned above, mathematical physicists are fond of generalizing models. Despite
this fondness, we required that any final, unified description must be unique: any final,
unified description must be impossible to reduce, to modify or to generalize. In par* ‘Nature knows physics better than the best physicist.’ Carl Ramsauer (b. 1879 Oldenburg, d. 1955 Berlin),
influential physicist, discovered that electrons behave as waves.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Can the strand model be generalized?
Motion Mountain – The Adventure of Physics
Describing everything as connected does not come natural to us humans. After all, in
our life, we perform only one act: to partition. We define pluralities. There is no way we
can avoid doing this. To observe, to think, to talk, to take a decision, to move, to suffer,
to love or to enjoy life is impossible without partitioning.
Our walk showed us that there are limits to the ability to distinguish. Any kind of
partitioning is always approximate. In fact, most people can summarize their personal
experience by saying that they learned to make finer and finer distinctions. However,
talking with highest precision about a part of the world inevitably leads to talk about the
whole universe. The situation resembles a person who gets a piece of rope in his hand,
and by following it, discovers a large net. He continues to pull and finally discovers that
everything, including himself, is part of the net.
For the strand model, the term ‘theory of everything’ is therefore not acceptable.
Nature cannot be divided into ‘things’. In nature, things are never separable. There is no
way to speak of ‘every’ thing; there are no sets, no elements and no parts in nature. A
theory describing all of nature cannot be one of ‘everything’, as ‘things’ are only approximate entities: properly speaking, they do not exist. The strand model is not a theory of
everything; it is a final theory.
The strand model shows that nature is not made of related parts. Nature is made of
relations only. Parts only exist approximately. The strand model also shows: being in
motion is intrinsic to being a part. Parts, being approximate, are always in motion. As
soon as we divide, we observe motion. The act of dividing, of partitioning, of defining
parts is the very one which produces order out of chaos. Strands force us to rethink this
habit.
Despite being so tough to grasp, strands yield a precise description of motion that
unifies quantum field theory and general relativity. The strand model for the unification
of motion is both simple and powerful. There are no free parameters. There are no
questions left. Our view from the top of the mountain is thus complete. No uncertainty,
no darkness, no fear and no insecurity are left over. Only wonder remains.
406
Challenge 227 r
What is nature?
Ref. 265
“
”
Nature is what is whole in each of its parts.
Hermes Trismegistos, Book of Twenty-four
Philosophers.
At the end of our long adventure, we discovered that nature is not a set: everything is
connected. Nature is only approximately a set. The universe has no topology, because
space-time is not a manifold. Nevertheless, the approximate topology of the universe is
that of an open Riemannian space. The universe has no definite particle number, because
the universe is not a container; the universe is made of the same stuff of which particles
are made. Nevertheless, the approximate particle density in the universe can be deduced.
In nature, everything is connected. This observation is reflected in the conjecture that
all of nature is described by a single strand.
We thus arrive at the (slightly edited) summary given around the year 1200 by the
author who wrote under the pen name Hermes Trismegistos: Nature is what is whole in
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 155
ticular, a final theory must neither be a generalization of particle physics nor of general
relativity. Let us check this.
The strand model is not a generalization of general relativity: the definitions of
curvature, of gravitons and of horizons differ radically from general relativity’s approach.
The strand model is also not a generalization of particle physics: the definitions of
particle and of interactions differ radically from the concepts of quantum field theory.
Indeed, we have shown that quantum field theory and general relativity are approximations to the strand model; they are neither special cases nor reductions of the strand
model.
But what about the other requirements for a unified theory? Can the strand model
be modified or generalized? We have seen that the model does not work in more spatial dimensions, does not work with more families of quarks, does not work with more
interactions, and does not work with other evolution equations in general relativity or
particle physics. The strand model does not work with other fundamental constituents,
such as bifurcating entities, membranes, bands, or networks. (Though it does work with
the equivalent funnels, as explained earlier on, but that description is equivalent to the
one with strands.) The strand model does not work with any modified fundamental principle. Obviously, exploring all possible variations and modifications remains a challenge
for the years to come. If an actual modification of the strand model can be found, the
strand model instantly loses its value: in that case, it would need to be shelved as a failure.
Only a unique unified model can be correct.
In summary, one of the beautiful aspects of the strand model is its radical departure
from twentieth-century physics in its basic concepts, combined with its almost incredible uniqueness. No generalization, no specialization and no modification of the strand
model seems possible. In short, the strand model qualifies as a unified, final theory.
What is a requirement to one person, is a criticism to another. A number of researchers deeply dislike the strand model precisely because it doesn’t generalize previous theories and because it cannot be generalized. This attitude deserves respect, as it is born
from the admiration for several ancient masters of physics. However, the strand model
points into a different direction.
Motion Mountain – The Adventure of Physics
Page 169
14 the top of m otion m ountain
new sig hts
407
each of its parts. But in contrast to that author, we now also know how to draw testable
conclusions from the statement.
Q uantum theory and the nature of mat ter and vacuum
”
C osmology
Page 8
”
The strand model also showed us how to deduce general relativity. The strand model
clarified the fabric of horizons and explained the three dimensions of space. Most fascinating is the idea of a universe as the product of a single strand. A single strand implies
that there was nothing before the big bang, and that there is nothing outside the night
sky. For example, the strand model implies that there is no ‘multiverse’ and that there
are no hidden worlds of any kind. And the fluctuating strand explains all observations
of our universe.
The cosmological constant is not constant; it only measures the present age and size of
the universe. Therefore, the constant does not need to appear in Figure 1. In other words,
the cosmological constant simply measures the time from the big bang to the present.
* ‘Modern physics is not far from the question whether everything that exists could possibly be made from
aether. These things are the extreme goals of our science, physics.’ Hertz said this in a well-known speech
he gave in 1889. If we recall that ‘aether’ was the term of the time for ‘vacuum’, the citation is particularly
striking.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
“
Der heutigen Physik liegt die Frage nicht mehr
ferne, ob nicht etwa alles, was ist, aus dem Äther
geschaffen sei. Diese Dinge sind die äußersten
Ziele unserer Wissenschaft, der Physik.*
Heinrich Hertz
The strand model shows that as soon as we separate the universe into space-time and
the rest, i.e., as soon as we introduce the coordinates ? and ?, quantum mechanics appears automatically. More precisely, quantum effects are effects of extension. Quantum
theory appears when we realize that observations are composed of smallest events due
to crossing switches, each with a change given by the quantum of action. All events and
observations appear through the fluctuations of the strand that composes nature.
We found that matter is made of tangled strands. In fact, the correct way would be to
say: matter is made of tangled strand segments. This connection leads to Schrödinger’s
equation and to Dirac’s equation.
Insofar as matter is of the same fabric as the vacuum, we can rightly say that everything
is made of vacuum and that matter is made of nothing. But the most appropriate definition arises when we realize that matter is not made from something, but that matter is
a certain aspect of the whole of nature. Unification showed that every single elementary particle results from an arrangement of strands that involves the whole of nature, or,
if we prefer, the entire universe. In other words, we can equally say: matter is made of
everything.
We can also turn the equivalence of matter and vacuum around. Doing so, we arrive
at the almost absurd statement: vacuum is made of everything.
Motion Mountain – The Adventure of Physics
“
In everything there is something of everything.
Anaxagoras of Clazimenes (500
–428 bce Lampsacus)
408
14 the top of m otion m ountain
The ‘big bang’ is the name for what we observe if we try to make observations approaching the limits of nature. The ‘big bang’ appears automatically from the strand
model whenever we observe nature at the most distant times, the largest distances or at
the largest energies: ‘big bang’ is the name for Planck scale physics.
The universe consists of a single strand. There are many particles in nature, because
the strand is tangled up in complicated ways. What we call the ‘horizon’ of the universe
is the place where new tangles appear.
The belief that the big bang or the horizon are examples of creation is incorrect. What
happened at the big bang still happens at the horizon today. Both the black sky at night
and the big bang are nature’s way to tell us: ‘Galilean physics is approximate! Quantum
theory is approximate! General relativity is approximate!’
Musings abou t unification and strands
”
All is made from one sort of thing: all is one substance. This idea, monism, sounds
a lot like what the influential philosopher Baruch Spinoza (b. 1632 Amsterdam,
d. 1677 The Hague) held as conviction. Monism, though mixed up with the idea of
god, is also the basis of the philosophical ideas that Gottfried Wilhelm Leibniz (b. 1646
Leipzig, d. 1716 Hannover) presents in his text La Monadologie.
Ref. 266
Any complete theory of motion, also the strand model, is built on a single statement
about nature: The many exists only approximately. Nature is approximately multiple.
The etymological meaning of the term ‘multiple’ is ‘it has many folds’; in a very specific
sense, nature thus has many folds.
∗∗
Any precise description of nature is free of arbitrary choices, because the divisions that
we have to make in order to think are all common to everybody, and logically inescapable. Because physics is a consequence of this division, it is also ‘theory-free’ and
‘interpretation-free’. This consequence of the final theory will drive most philosophers
up the wall.
∗∗
For over a century, physics students have been bombarded with the statement:
‘Symmetries are beautiful.’ Every expert on beauty, be it a painter, an architect, a
sculptor, a musician, a photographer or a designer, fully and completely disagrees, and
rightly so. Beauty has no relation to symmetry. Whoever says the contrary is blocking
out his experiences of a beautiful landscape, of a beautiful human figure or of a beautiful
work of art.
* Lao Tse (sixth century bce) was an influential philosopher and sage.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
∗∗
Motion Mountain – The Adventure of Physics
“
Continuing motion masters coldness.
Continuing rest masters heat.
Motion based on rest:
Measure of the all-happening for the single one.
Lao Tse,* Tao Te King, XXXXV.
new sig hts
Ref. 267
409
The correct statement is: ‘Symmetries simplify descriptions.’ Symmetries simplify
physical theories. That is the background for the statement of Werner Heisenberg: ‘In
the beginning there was symmetry.’ On the other hand, the strand model shows that
even this statement is incorrect. In fact, neither the search for beauty nor the search
for symmetry were the right paths to advance towards unification. Such statements have
always been empty marketing phrases. In reality, the progress of fundamental theoretical
physics was always driven by the search for simplicity.
∗∗
Strands unify physics. In particular, strands extend our views on quantum theory and
mathematical physics, on particle physics and field theory, on axiomatic physics and algebraic physics, on polymer physics and gauge theory, on general relativity and cosmology. It will take several years before all these extensions will have been explored.
∗∗
Page 22
Almost all discoveries in physics were made at least 30 years too late. The same is true for
the strand model. If we compare the strand model with what many physicists believed
in the twentieth century, we can see why: researchers had too many wrong ideas about
unification. All these wrong ideas can be summarized in the following statement:
— ‘Unification requires generalization of existing theories.’
This statement is subtle: it was rarely expressed explicitly but widely believed. But the
statement is wrong, and it led many astray. On the other hand, the development of the
strand model also followed a specific guiding idea, namely:
— ‘Unification requires simplification.’
Hopefully this guiding idea will not become a dogma itself: in many domains of life,
simplification means not to pay attention to the details. This attitude does a lot of harm.
∗∗
The strand model shows that achieving unification is not a feat requiring difficult abstraction. Unification was not hidden in some almost inaccessible place that can reached only
by a few select, well-trained research scientists. No, unification is accessible to everyone
who has a basic knowledge of nature and of physics. No Ph.D. in theoretical physics is
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
∗∗
Motion Mountain – The Adventure of Physics
Vol. III, page 279
The description of nature with strands is surprisingly simple, mainly because it uses so
few basic concepts. Is this result astonishing? In our daily life, we describe our experiences with the help of a few thousand words, e.g. taking them from the roughly 350 000
words which make up the English language, or from a similar number from another
language. This set is sufficient to talk about everything, from love to suffering, from
beauty to happiness. And these terms are constructed from no more than about 35 basic
concepts, as we have seen already. We should not be too surprised that we can in fact
talk about the whole universe using only a few basic concepts: the act and the results of
(approximate) distinction, or more specifically, a basic event – the crossing switch – and
its observation.
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14 the top of m otion m ountain
∗∗
Page 84
The strand model settles all questions about determinism. Quantum theory and general
relativity are deterministic. Nevertheless, when both descriptions are combined, time
turns out to be an approximate, low-energy concept. The same applies to determinism.
Even though nature is deterministic for all practical purposes and shows no surprises,
determinism shares the fate of all its conceivable opposites, such as fundamental randomness, indeterminism of all kinds, existence of wonders, creation out of nothing, or
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
needed to understand or to enjoy it. The knowledge presented in the previous volumes
of this series is sufficient.
When Andrew Wiles first proved Fermat’s last theorem after three centuries of attempts by the brightest and the best mathematicians, he explained that his search for a
proof was like the exploration of a dark mansion. And seen the conceptual difficulties he
had to overcome, the analogy was fitting. Recalling how many more people have already
searched for unification without success, the first reaction is to compare the search for
unification to the exploration of something even bigger, such as a complex dark cave
system. But that analogy was not helpful. In contrast to the proof of Fermat’s theorem,
the goal of the quest for unification turned out to be simple and lying out in the open.
Researchers had simply overlooked it, because they were convinced that the goal was
complex, hidden in the dark and hard to reach. It was not.
The adventure of climbing Motion Mountain is thus not comparable to climbing
Cerro Torre, which might be the toughest and most spectacular challenge that nature
offers to mountain climbers. Figure 117 gives an impression of the peak. Motion Mountain does not resemble this peak at all. Neither does Motion Mountain resemble the
Langtang Lirung peak in the Nepalese Himalayas shown on the cover of this volume.
Climbing Motion Mountain is more like walking up a gentle green hill, alone, with a
serene mind, on a sunny day, while enjoying the surrounding beauty of nature.
Motion Mountain – The Adventure of Physics
F I G U R E 117 Motion Mountain does not resemble Cerro Torre, but a gentle hill (© Davide Brighenti,
Myriam70)
new sig hts
411
divine intervention: determinism is an incorrect description of nature at the Planck scale
– like all its alternatives.
∗∗
Challenge 228 e
The strand model also settles most so-called really big questions that John Wheeler used
to ask: Why the quantum? How come existence? It from bit? A "participatory universe"?
What makes "meaning"? Enjoy the exploration.
∗∗
Any unified model of nature encompasses a lot of ideas, issues and knowledge. Due to
the sheer amount of material, publishing it in a journal will be challenging.
∗∗
Ref. 268
∗∗
Page 331
∗∗
Many researchers believed during all their life that the final theory is something useful,
important and valuable. This common belief about the importance and seriousness of
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Page 8
Historically, the strand model evolved from an exploration, started in the 1990s, of the
maximum force in nature, the belt trick and the entropy of black holes. After the first six
chapters of the present volume were completed in 2002, meditating on their implications
led to the strand model and its fundamental principle.
Above all, it was the description of general relativity with the help of the maximum
force that triggered the search for a unified description that was purely based on Planck
units. Another essential point was the drive to search for a final theory directly, from its
requirements (‘top down’ in Figure 1), and not from the unification of quantum theory
and general relativity (‘bottom up’). In the years from 2002 to 2007, most of the ideas of
the strand model took shape, mainly in Munich’s underground trains, while commuting
between home and work. In those years, it appeared that strands could explain the Dirac
equation, the entropy of black holes, general relativity and the particle spectrum with the
three particle generations. While walking in the woods and fields around Munich during 2008 and 2009, it appeared that strands explain the three gauge interactions, predict
(with almost complete certainty) the lack of a Higgs boson – a big mistake due to faulty
reasoning, as turned out in 2012 – and of any new physical effects beyond the standard
model, and allow calculating the unexplained constants of particle physics. The model
thus yielded all its main predictions before the accelerator experiments at the Large Hadron Collider at CERN in Geneva were switched on in autumn 2010. Thus much of the
work was done in a haste – future will show what is of lasting value.
In 2012, the discovery of the Higgs boson, and in 2014, the comments by Sergei Fadeev
led to an improvement and simplification of the strand model, eliminating knotted
strands. In 2016 and 2017, the experimental results at the LHC, of dark matter searches,
and of the LIGO observatory confirmed the lack of deviations from the standard model
of particle physics and from general relativity, as predicted by the strand model.
Motion Mountain – The Adventure of Physics
The strand model is so simple that it fits on a tombstone - or on a T-shirt. This would
surely be god’s favourite T-shirt. It is available at www.motionmountain.net/gfts.html.
412
14 the top of m otion m ountain
the quest has led, over the past decades, to an increasingly aggressive atmosphere among
these researchers. This unprofessional atmosphere, combined with the dependence of
researchers on funding, has delayed the discovery of the final theory by several decades.
In fact, the final theory is not useful: it adds nothing of practical relevance to the
combination of the standard model and general relativity. The final theory is also not
important: it has no application in everyday life or in industry and does not substantially
change our view of the world; it just influences teaching – somewhat. Finally, the final
theory is not valuable: it does not help people in their life or make them happier. In
short, the final theory is what all fundamental theoretical research is: entertaining ideas.
Even if the strand model were to be replaced by another model, the conclusion remains: the final theory is not useful, not important and not valuable. But it is enjoyable.
∗∗
“
“
”
”
Anonymous
Cum iam profeceris tantum, ut sit tibi etiam tui
reverentia, licebit dimittas pedagogum.*
Seneca
The final theory of motion has a consequence worth mentioning in detail: its lack of
infinity and its lack of finitude eliminate the necessity of induction. This conclusion is of
* ‘When you have profited so much that you respect yourself you may let go your tutor.’ Seneca, the influential Roman poet and philosopher, writes this in his Epistulae morales ad Lucilium, XXV, 6.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
The elimination of induction
Only boring people get bored.
Motion Mountain – The Adventure of Physics
Page 86
The strand model will take a long time to get accepted. The first reason is obvious: The
strand model contradicts thinking habits in many research fields. Researchers working on
the foundations of quantum theory, on general relativity, on cosmic strings, on mathematical physics, on classical and quantum field theory, on polymer physics, on shape
deformations, on quantum gravity, on strings, on the visualization of quantum mechanics, on knot theory, on higher dimensions, on supersymmetry, on the axiomatization
of physics, on group theory, on the foundation of physics, on quantum optics and on
particle physics have to give up many life-long thinking habits. So do all other physicists. Strands supersede particles and points.
There is also a second reason for the slow acceptance of the model presented here:
The strand model, in its simplicity, is only a small step away from present research. Many
researchers are finding out how close they have been to the ideas of the strand model,
and for how long they were overlooking or ignoring such a simple option. The simplicity
of the fundamental principle contrasts with the expectation of most researchers, namely
that the final theory is complicated, difficult and hard to discover. In fact, the opposite is
true. Strands are based on Planck units and provide a simple, almost algebraic description
of nature.
In summary, for many researchers and for many physicists, there is a mixture of confusion, anger and disappointment. It will take time before these feelings subside and are
replaced by the fascination provided by the strand model.
new sig hts
Page 162
413
“
That which eludes curiosity can be grasped in action.
Traditional saying.
”
Where do we come from? Where does the world come from? What will future bring?
What is death? All these questions are questions about motion – and its meaning. To
all such questions, the strand model does not provide answers. We are a collection of
tangled strands. We are everything and nothing. The strand(s) we are made of will con-
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
What is still hidden?
Motion Mountain – The Adventure of Physics
importance for general discussions on man’s grasp of nature.
In physics, as in the other natural sciences, there is a tradition to state that a certain
description of nature – once confusingly called a ‘law’ – is valid in all cases. In these
statements, ‘all’ means ‘for all values of the quantities appearing’. As a concrete example,
the ‘law’ of universal gravitation is always claimed to be the same here and today, as well
as at all other places and times, such as on the other end of the universe and in a few
thousand years. The full list of such all-claims is part of the millennium list of open
issues in twentieth-century physics. For many decades, the habit of claiming general
validity from a limited and finite number of experiences, also called induction, has been
seen, and rightly so, as a logically dubious manoeuvre, tolerated only because it works.
But the developments described in this text show that this method is indeed justified.
First of all, a claim of generality is not that enormous as it may seem, because the
number of events that can be distinguished in nature is finite, not infinite. The preceding
sections showed that the maximal number ? of events that can be distinguished in the
universe is of the order of ? = (?0 /?Pl)4 = 10244±2, ?0 being the age of the universe and
?Pl the Planck time. This is a big, but certainly finite number.
The unified description of nature has thus first reduced the various all-claims from
an apparently infinite to a finite number of cases, though still involving astronomically
large numbers. This reduction results from the recognition that infinities do not appear
in the description of nature. We now know that when talking about nature, ‘all’ cases
never means an infinite number.
A second, important result is achieved by the description of nature with strands. In
any all-claim about fundamental motion, the checking of each of the large number of
possibilities is not necessary any more, because all events result from a single entity, in
which we introduce distinctions with our senses and our brain. And the distinctions
we introduce imply automatically that the symmetries of nature – the ‘all-claims’ or
‘inductions’ – that are used in the description of motion are correct. Nature does not
contain separate parts. Therefore, there is no way that separate parts can behave differently. Induction is a consequence of the unity of nature.
Ultimately, the possibility to verify statements of nature is due to the fact that all the
aspects of our experience are related. Complete separation is impossible in nature. The
verification of all-claims is possible because the strand model achieves the full description of how all ‘parts’ of nature are related.
The strand model shows that we can talk and think about nature because we are a part
of it. The strand model also shows that induction works because everything in nature is
related to everything else: nature is one.
414
Challenge 229 s
Vol. I, page 16
14 the top of m otion m ountain
tinue to fluctuate. Birth, life and death are aspects of tangled strands. The universe is a
folded strand that grows in complexity.
Obviously, abstract statements about tangles do not help in any human quest. Indeed, we aimed at achieving a precise description of moving particles and bending space.
Studying them was a sequence of riddles; but solving these riddles does not provide
meaning, not even at the top of Motion Mountain. From the top we cannot see the
evolution of complicated systems; in particular, we cannot see or describe the evolution
of life, the biological evolution of species, or the growth of a human beings. Nor can we
understand why we are climbing at all.
In short, from the top of Motion Mountain we cannot see the details down in the
valleys of human relations or experience; strands do not provide advice or meaning. Remaining too long on the top is of no use. To find meaning, we have to descend back
down to real life.
“
Ref. 270
”
Enjoying life and giving it meaning requires to descend from the top of Motion Mountain. The return path can take various different directions. From a mountain, the most
beautiful and direct descent might be the use of a paraglider. After our adventure, we
take an equally beautiful way: we leave reality.
The usual trail to study motion, also the one of this text, starts from our ability to talk
about nature to somebody else. From this ability we deduced our description of nature,
starting from Galilean physics and ending with the strand model. The same results can
be found by requiring to be able to talk about nature to ourselves. Talking to oneself
is an example of thinking. We should therefore be able to derive all physics from René
Descartes’ sentence ‘je pense, donc je suis’ – which he translated into Latin as ‘cogito
ergo sum’. Descartes stressed that this is the only statement of which he is completely
sure, in opposition to his observations, of which he is not. He had collected numerous
examples in which the senses provide unreliable information.
However, when talking to ourselves, we can make more mistakes than when asking
for checks from others. Let us approach this issue in a radically different way. We directly
proceed to that situation in which the highest freedom is available and the largest number
of mistakes are possible: the world of dreams. If nature would only be a dream, could we
deduce from it the complete set of physical knowledge? Let us explore the issue.
— Dreaming implies the use of distinctions, of memory and of sight. Dreams contain
parts and motion.
— Independently on whether dreams are due to previous observations or to fantasies,
through memory we can define a sequence among them. The order relation is called
time. The dream aspects being ordered are called events. The set of all (dream) events
forms the (dream) world.
— In a dream we can have several independent experiences at the same time, e.g. about
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 269
I hate reality. But it is the only place where one can
get a good steak.
Woody Allen
Motion Mountain – The Adventure of Physics
a return path: je rêve, donc je suis
a return path: j e rêv e, d onc j e suis
415
thirst and about hunger. Sequences thus do not provide a complete classification of
experiences. We call the necessary additional distinction space. Dream space has
three dimensions.* Dreaming thus means to use space and time.
— We can distinguish between dream contents. Distinguishing means that we can
count items in dreams. Counting means that we have a way to define measurements.
Dreams are thus characterized by something which we can call ‘observables’. Dream
experiences at a given instant of time are characterized by a state.
— Because we can describe dreams, the dream contents exist independently of dream
time. We can also imagine the same dream contents at different places and different
times in the dream space. There is thus an invariance of dream concepts in space and
time. There are thus symmetries in dream space.
— Dream contents can interact. Dreams appear to vary without end. Dreams seem to
be infinite.
Challenge 230 s
— Dreams show that space can warp.
— Dream motion, as you may want to check, shows a maximum speed.
— Dreams show a strange limit in distance. There is a boundary to our field of vision,
even though we do not manage to see it.
— Both the number of items we can dream of at the same time and the memory of
previous dreams is finite.
— Dreams have colours.
— There are pixels in dreams, though we do not experience them directly. But we can
do so indirectly: The existence of a highest number of things we can dream of at the
same time implies that dream space has a smallest scale.
In summary, the world of dreams has something similar to a minimum change. The
world of dreams and that of films is described by a simple form of quantum theory. The
difference with nature is that in dreams and films, space is discrete from the outset. But
there is still more to say about dreams.
— There is no way to say that dream images are made of mathematical points, as there
is nothing smaller than pixels.
— In dreams, we cannot clearly distinguish objects (‘matter’) and environment
(‘space’); they often mix.
* Though a few mathematicians state that they can think in more than three spatial dimensions, all of them
dream in three dimensions.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Pondering these issues shows that there are limits to dreams. In summary, the world of
dreams has a maximum size, a maximum speed and three dimensions that can warp.
The world of dreams and of films is described by a simple form of general relativity.
Motion Mountain – The Adventure of Physics
In other words, a large part of the world of dreams is described by a modified form
of Galilean physics. We note that the biggest difference between dreams and nature is
the lack of conservation. In dreams, observations can appear, disappear, start and stop.
We also note that instead of dreams, we could equally explore cinema films. Films, like
dreams, are described by a modified form of Galilean physics. And films, like dreams,
do not follow conservation laws. But dreams teach us much more.
416
—
—
—
—
Challenge 231 s
14 the top of m otion m ountain
In dreams, fluctuations appear both for images as well as for the background.
In dreams, sharp distinctions are impossible. Dream space-time cannot be a set.
Dream motion appears when approximate conservation (over time) is observed.
In dreams, dimensionality at small distances is not clear; two and three dimensions
are mixed up there.
In summary, the world of dreams seems to behave as if points and point particles do not
exist; and since quantum theory and general relativity hold, the world of dreams seems
to be described by extended constituents! We thus conclude this short exploration of
the physics of dreams with a fascinating conjecture: even if nature would be a dream, an
illusion or a fantasy, we might still get most of the results that we discovered in our ascent
of Motion Mountain. (What differences with modern physics would be left?) Speaking
with tongue in cheek, the fear of our own faults of judgement, so rightly underlined by
Descartes and many others after him, might not apply to fundamental physics.
Ref. 5
⊳ The fine structure constant describes the probability that a fluctuation adds
a twist to the chiral tangles of electrically charged particles.
We have not yet deduced an accurate value for the fine structure constant, but we seem
to have found out how to do so.
In short, we seem to glimpse the origin of all colours – and thus of all beauty around
us. Strands provide a beautiful explanation for beauty.
summary: what is motion?
“
”
Deep rest is motion in itself. Its motion rests in itself.
Lao Tse, Tao Te King, VI, as translated by Walter
Jerven.
We can now answer the question that drove us through our adventure:
⊳ Motion is the observation of crossing switches of the one, unobservable,
tangled and fluctuating strand that describes all of nature.
Nature’s strand forms particles, horizons and space-time: these are the parts of nature.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
All colours around us are determined by the fine structure constant ? – the coupling
constant for the electromagnetic interaction at low energy – with its measured value
1/137.035 999 139(31). The fine structure constant is also essential to describe most
everyday devices and machines, as well as all human thoughts and movements. The
constant is an aspect of every electric charge in nature.
The strand model showed us that electrical charge is a property of tangles of strands.
In particular, the strand model showed:
Motion Mountain – The Adventure of Physics
what is the origin of colours?
sum m ary: w hat is m otion?
“
All the great things that have happened in the world first took place in a
person’s imagination, and how tomorrow’s world will look like will largely
depend on the power of imagination of those who are just learning to read right
now.
Astrid Lindgren*
* Astrid Lindgren (b. 1907 Näs, d. 2002 Stockholm) was a beloved writer of children books.
”
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Ref. 271
Particles are tangles of strands; horizons and space-time are weaves of strands. The parts
of nature move. The parts move because their strands fluctuate.
Motion appears because all parts in nature are approximate. Indeed, the observation
of crossing switches and the description of strand segments fluctuating in a background
space result and are possible because we approximate from the one strand that makes up
nature to the many parts inside nature. The one strand (approximately) forms the many
elementary particles inside us. Strand segments and particles (approximately) lead us
to introduce background space, matter and radiation. Introducing background space
implies observing motion. Motion thus appears automatically when approximate parts
of nature, such as humans, animals or machines, describe other approximate parts of
nature, such as other bodies or systems.
The observation of motion is due to our introduction of the plural. Motion results from
of our forced use of many (approximate) parts to describe the unity of nature. The observation of motion results from approximations. All these approximate distinctions are
unavoidable and are due to the limitations of our experience.
Motion appears as soon as we divide the world into parts and then follow these parts.
Dividing nature into parts is not a conscious act; our human nature – our senses and
our brain – force us to perform it. And whenever we experience or talk about parts of
the universe, we find motion. Our senses and our brain are made to distinguish and to
divide – and cannot do otherwise. We need to distinguish in order to survive, to think
and to enjoy life. In a sense, we can say that motion appears as a logical consequence
of our limitations; the fundamental limitation is the one that makes us distinguish and
introduce parts, including points and sets.
Motion is an ‘artefact’ of locality. Locality is an approximation and is due to our human nature. Distinction, localization and motion are inextricably linked.
Motion is low energy concept. Motion does not exist at Planck scales, i.e., at the limits
of nature.
Motion is an artefact due to our limitations. This conclusion resembles what Zeno of
Elea stated 2500 years ago, that motion is an illusion. But in contrast to Zeno’s pessimistic view, we now have a fascinating spectrum of results and tools at our disposition:
they allow us to describe motion and nature with high precision. Most of all, these tools
allow us to change ourselves and our environment for the better.
Motion Mountain – The Adventure of Physics
Vol. I, page 15
417
PO ST FACE
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Plato’s Phaedrus, written around 380 b ce, is available in many pocket editions. Do not waste
your time learning ancient Greek to read it; the translated versions are as beautiful as the original.
Plato’s lifelong avoidance of the natural sciences had two reasons. First of all, he was jealous
of Democritus. Plato never even cites Democritus in his texts. Democritus was the most prolific,
daring, admired and successful philosopher of his time (and maybe of all times). Democritus
was a keen student of nature. His written works did not survive, because his studies were not
congenial to the followers of christianity, and thus they were not copied by the monks in the
Middle Ages. The loss of these texts is related to the second reason that kept Plato away from
the natural sciences: he wanted to save his life. Plato had learned one thing from men in the
town: talking about nature is dangerous. Starting around his lifetime, for over 2000 years people
practising the natural sciences were regularly condemned to exile or to death for impiety. Fortunately, this is only rarely the case today. But such violence still occurs, and we can honour the
dangers that those preceding us had to overcome in order to allow us enjoying this adventure.
Motion Mountain – The Adventure of Physics
Perhaps once you will read Plato’s Phaedrus, one of the beautiful philosophical Greek
texts. In it, Socrates is made to say that he almost never left the city walls because to him,
as a ‘lover of learning, trees and the open country do not teach anything, whereas men
in the town do.’ This is a veiled critique of Democritus, the most important and famous
philosopher in Greece during Plato’s time. Democritus was the natural philosopher par
excellence, and arguably had learned from nature – with its trees and open country –
more than anybody else after him.
After this adventure you can decide for yourself which of these two approaches is more
congenial to you. It might be useful to know that Aristotle refused to choose and cultivated them both. There is no alternative in life to following one’s own mind, and to enjoy
doing so. If you enjoyed this particular trip, show it to your friends. For yourself, after
this walk, sense intensively the pleasure of having accomplished something important.
Many before you did not have the occasion. Enjoy the beauty of the view offered. Enjoy
the vastness of horizon it provides. Enjoy the impressions that it creates inside you. Collect them and rest. You will have a treasure that will be useful in many occasions. Then,
when you feel the desire of going further, get ready for another of the adventures life has
to offer.
Appendix A
KNOT AND TANGLE GEOMET RY
The following table provides a terse summary of the mathematics of knot shapes.
TA B L E 19 Important properties of knot, links and tangles.
Defining property
Normal vector or
curvature vector
local vector normal to the curve, in
direction of the centre of the
‘touching’ circle, with length given by
the curvature
Binormal vector
local unit vector normal to the
tangent and to the normal/curvature
vector
local speed of rotation of the
binormal vector; positive (negative)
for right-handed (left-handed) helix
Torsion
Frenet frame at a
curve point
ropelength is integral of
arclength; ropelength is
shape-dependent.
at present, all non-trivial ideal
shapes are only known
approximately; most ideal knots
(almost surely) have kinks.
measures departure from
straightness, i.e., local bending of
a curve.
is given by the second and first
derivatives of the curve.
measures departure from
flatness, i.e., local twisting or
local handedness of a curve;
essentially a third derivative of
the curve.
‘natural’ local orthogonal frame of
the Frenet frame differs at each
reference defined by unit tangent, unit curve point, the Frenet frame is
normal/curvature and binormal
not uniquely defined if the curve
vector
is locally straight.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Knot / link / tangle one closed / several closed / one or
several open curves, all in 3d and
without intersections
Ideal knot, link,
tightest possible knot, link or tangle
tangle (shape)
(shape) assuming a rope of constant
diameter that is infinitely flexible and
infinitely slippery
Ribbon or framing short perpendicular (or non-tangent)
vector attached at each point of a
curve
Curvature of a
inverse curvature radius of ‘touching’
curve
circle
O t h e r p r o pe rt i es
Motion Mountain – The Adventure of Physics
Concept
420
a knot g eom etry
TA B L E 19 (Continued) Important properties of knot, links and tangles.
Concept
Defining property
O t h e r p r o pe rt i es
‘Natural’ framing
or Frenet ribbon
defined by the local normal, i.e., local
curvature vector
for a closed curve, it is always
closed and two-sided, and thus
never a Moebius band.
Linking number
sloppily, number of times that two
topological invariant, i.e.,
between two closed curves wind around each other, or,
shape-independent;
curves
equivalently, half the number of times Lk(?1, ?2) =
? (d? ×d? )
1
that the curves ‘swap’ position
∮ ∮ 12 ?31 2 .
4π ?2 ?1
12
Twist of a ribbon,
open or closed
Tw(?) is the total angle, in units of 2π,
by which the ribbon rotates around
the central axis of the ribbon; sloppily
said, it measures the local helicity; this
type of twist has no relation to the
first Reidemeister move
Tw(?) is the total angle, in units of
2π, by which the Frenet frame rotates
around the tangent direction, or
equivalently, (total) twist of the Frenet
ribbon, also called the total torsion of
the curve; this type of twist has no
relation to the first Reidemeister move
Twist of a curve or
knot
Signed crossing
number
number of times that the edges of the
natural/Frenet ribbon wind around
each other
generalization of the linking number
for knots to open curves
sum of positive minus sum of
negative crossings in a given oriented
2d projection of a curve or knot
(sometimes called ‘2d-writhe’)
2d-writhe of a knot, signed crossing number for a minimal
or topological
crossing number diagram/projection
(sometimes the term ‘2d-writhe’ is
writhe, or Tait
number
used for the signed crossing number
of any configuration)
topological invariant, i.e.,
shape-independent; always an
integer.
not a topological invariant,
because of existence of inflection
points.
usually not an integer.
vanishes for ribbons that are
everywhere flat.
not an integer even in case of
knots; depends on curve/knot
shape; is different from zero for
chiral curves/knots; is zero for
achiral curves/knots that have a
rigid reflective symmetry; twist
and torsion are only equal if the
twist is defined with the Frenet
ribbon – with other framings
they differ.
always an integer; depends on
shape.
is shape-invariant; is always an
integer; differs from 0 for all
chiral knots; has the value 3 for
the trefoil, 0 for the figure-eight
knot, 5 for the 51 and 52 knots, 2
for the 61 knot, 7 for the 71 and
72 knots, 4 for the 81 knot, and 9
for the 92 knot.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
number of times that the edges wind
around each other
Motion Mountain – The Adventure of Physics
Linking number
for a closed
two-sided ribbon
Self-linking number
or ‘natural’ linking
number for a knot
Link integral for an
open curve
knot g eom etry
421
TA B L E 19 (Continued) Important properties of knot, links and tangles.
Concept
Defining property
Writhing number or Wr(?) is the average, over all
3d-writhe of a knot projection directions, of the signed
crossing number; sloppily said, it
measures how wrapped, coiled and
chiral a knot is, i.e., it measures the
global helicity
Writhe of a ribbon
the value is quasi-quantized for
alternating knots with small crossing
numbers (< 11) in values that differ
from ?4/7 by only a few per cent
the value is quasi-quantized for
alternating links with small crossing
numbers (< 11) in values that differ
from 2/7 + ?4/7 by only a few per
cent
sloppily said, measures how wrapped,
coiled and chiral a ribbon is, i.e.,
measures its global helicity
vanishes for plane curves.
for any knot ? and any ribbon ?
attached to it,
Lk(?, ?) = Tw(?, ?) + Wr(?)
for applying the theorem to open
curves, a (standardized) closing
of curves is required.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Writhe of an open
curve
Calugareanu’s
theorem
depends on knot shape; usually
is not an integer; is different
from zero for chiral knots; is
zero for achiral knots that have a
rigid reflective symmetry;
1
∮? ∮? ?12 (d??31 ×d?2) ;
Wr(?) = 4π
12
uses no ribbon and thus is
independent of the ribbon shape
attached to the knot.
is additive under knot addition
for knots with small crossing
numbers (< 11) within less than
1 %.
Motion Mountain – The Adventure of Physics
Writhe of ideal,
alternating knots
and of
odd-component
links
Writhe of ideal,
alternating
even-component
links
O t h e r p r o pe rt i es
CHALLENGE HINT S AND SOLUT IONS
Challenge 2, page 28: Take Δ? Δ? ⩾ 1 and substitute Δ? = ?/Δ? and Δ? = ?/Δ?.
Challenge 16, page 43: Yes. But we can also argue its opposite, namely that matter appears when
space is compressed too much. Both viewpoints are correct.
Challenge 22, page 46: The strictest upper limits are those with the smallest exponent for length,
Challenge 24, page 48: To my knowledge, no such limits have been published. Do it yourself!
Challenge 25, page 48: The system limits cannot be chosen in other ways; after the limits have
been corrected, the limits given here should still apply.
Challenge 28, page 49: Just insert numbers to check this.
Challenge 30, page 50: No.
Challenge 31, page 51: This is a trick question due to two issues. First, is the cosmological con-
Challenge 33, page 52: If you ever write such a table, publish it and send me a copy. I will include
it in the text.
Challenge 36, page 65: Sloppily speaking, such a clock is not able to move its hands in a way that
guarantees precise time reading.
Challenge 40, page 82: The final energy ? produced by a proton accelerator increases with its
radius ? roughly as ? ∼ ?1.2 ; as an example, CERN’s LHC achieved about 13 TeV for a radius of
4.3 km. Thus we would get a radius of more than 100 light years for a Planck energy accelerator.
Building an accelerator achieving Planck energy is impossible.
Nature has no accelerator of this power, but gets near it. The maximum measured value of
cosmic rays, 1022 eV, is about one millionth of the Planck energy. The mechanism of acceleration
is still obscure. Neither black holes nor the cosmic horizon seem to be sources, for some yet
unclear reasons. This issue is still a topic of research.
Challenge 41, page 83: The Planck energy is ?Pl = √ℏ?5 /? = 2.0 GJ. Car fuel delivers about
43 MJ/kg. Thus the Planck energy corresponds to the energy of 47 kg of car fuel, about a tankful.
Challenge 42, page 83: Not really, as the mass error is equal to the mass only in the Planck case.
Challenge 43, page 83: It is improbable that such deviations can be found, as they are masked by
Page 279
the appearance of quantum gravity effects. However, if you do think that you have a prediction
for a deviation, publish it, and send the author an email.
Challenge 44, page 83: The minimum measurable distance is the same for single particles and
systems of particles.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
stant the same for all observers in the universe that are, like ourselves, more or less at rest with
respect to the background radiation? Most researchers would agree that this is the case. Secondly,
is the cosmological constant the same for extremely rapid observers, observers that move at extremely high energy with respect to the background radiation? Enjoy finding out.
Motion Mountain – The Adventure of Physics
and the strictest lower limits are those with the largest exponent of length.
challenge hints and solu tions
423
Challenge 45, page 84: There is no gravitation at those energies and there are no particles. There
Vol. I, page 259
Challenge 70, page 104: You will not find one.
Challenge 72, page 105: If you find one, publish it, and send the author an email.
Challenge 74, page 106: For the description of nature this is a contradiction. Nevertheless, the
Vol. III, page 323
Challenge 84, page 110: Plotinus in the Enneads has defined ‘god’ in exactly this way. Later,
Augustine in De Trinitate and in several other texts, and many subsequent theologians have taken
up this view. (See also Thomas Aquinas, Summa contra gentiles, 1, 30.) The idea they propose is
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Vol. II, page 258
term ‘universe’, ‘set of all sets’ and other mathematical terms, as well as many religious concepts
are of this type.
Challenge 75, page 107: No, for the reasons mentioned earlier on: fundamental measurement
errors for horizon measurements, as well as many other effects, prevent this. The speculation is
another example of misguided fantasy about extremal identity.
Challenge 76, page 108: The physical concepts most related to ‘monad’ are ‘strand’ and
‘universe’, as shown in the second half of this text.
Challenge 77, page 108: The macroscopic content of the universe may be observer-dependent.
But to speak about many universes (Many ‘everythings’?) or a ‘multiverse’ (What is more than
everything? Why only one multiverse?) is pure nonsense.
Challenge 80, page 108: True only if it were possible to do this. Because particles and space are
indistinguishable, removing particles means to remove everything. (The strand model visualizes
this connection most clearly.)
Challenge 82, page 108: True. Existence is the ability to interact. If the ability disappears, existence disappears. In other words, ‘existence’ is a low-energy concept.
Challenge 83, page 110: If you find a sensible statement about the universe, publish it! And send
it to the author as well. The next challenge shows one reason why this issue is interesting. In
addition, such a statement would contradict the conclusions on the combined effects of general
relativity and quantum theory.
Motion Mountain – The Adventure of Physics
is thus no paradox.
Challenge 46, page 84: The issue is still being debated; a good candidate for a minimum momentum of a single particle is given by ℏ/?, where ? is the radius of the universe. Is this answer
satisfying?
Challenge 47, page 85: All mentioned options could be valid at the same time. The issue is not
closed and clear thinking about it is not easy.
Challenge 48, page 85: The precise energy scale is not clear. The scale is either the Planck energy or within a few orders of magnitude from it; the lowest possible energy is thus around a
thousandth of the Planck energy.
Challenge 50, page 87: If you can think of an experiment, publish the proposal, and send the
author an email.
Challenge 51, page 90: The table of aggregates shows this clearly.
Challenge 52, page 91: The cosmic background radiation is a clock in the widest sense of the
term.
Challenge 53, page 92: The way to deduce cosmological limits is presented in detail in the section starting on page 45.
Challenge 64, page 100: Also measurement errors at Planck scales prevent the determination of
topology at those scales.
Challenge 66, page 102: The measurement error is as large as the measurement result.
424
Challenge 232 e
simple: it is possible to clearly say what ‘god’ is not, but it is impossible to say what ‘god’ is.
This statement is also part of the official Roman Catholic Catechism: see part one, section one,
chapter one, IV, 43, found at www.vatican.va/archive/ENG0015/__PC.HTM. Similar statements
are found in Judaism, Hinduism and Buddhism.
In other terms, theologians admit that ‘god’ cannot be defined, that the term has no properties or content, and that therefore the term cannot be used in any positive sentence. The aspects
common to ‘universe’ and to ‘god’ suggest the conclusion that both are the same. Indeed, the
analogy between the two concepts can be expanded to a proof: both concepts have the same
content, the same boundary, and the same domain of application. (This is an intriguing and
fascinating exercise.) In fact, this might be the most interesting of all proofs of the existence of
‘god’, as it lacks all the problems that the more common ‘proofs’ have. Despite its interest, this
proof of equivalence is not found in any book on the topic yet. The reason is twofold. First, the
results of modern physics – showing that the concept of universe has all these strange properties
– are not common knowledge yet. Secondly, the result of the proof, the identity of ‘god’ and
‘universe’ – also called pantheism – is a heresy for most religions. It is an irony that the catholic
catechism, together with modern physics, can be used to show that pantheism is correct, because
any catholic who defends pantheism (or other heresies following from modern physics) incurs
automatic excommunication, latae sententiae, without any need for a formal procedure.
If one is ready to explore the identity of universe and ‘god’, one finds that a statement like
‘god created the universe’ translates as ‘the universe implies the universe’. The original statement
is thus not a lie any more, but is promoted to a tautology. Similar changes appear for many
other – but not all – statements using the term ‘god’. (The problems with the expression ‘in the
beginning’ remain, though.) In fact, one can argue that statements about ‘god’ are only sensible
and true if they remain sensible and true after the term has been exchanged with ‘universe’.
Enjoy the exploration of such statements.
no such effects exist.
Challenge 88, page 113: In fact, no length below the Planck length itself plays any role in nature.
Challenge 90, page 114: You need quantum humour, because the result obviously contradicts a
previous one given on page 93 that includes general relativity.
Challenge 93, page 122: The number of spatial dimensions must be given first, in order to talk
about spheres.
Challenge 94, page 126: This is a challenge to you to find out. It is fun, it may yield a result in
contradiction with the arguments given so far (publish it in this case), or it may yield an independent check of the results of the section.
Challenge 96, page 130: This issue is open and still a subject of research. The conjecture of the
author is that the answer is negative. If you find an alternative, publish it, and send the author
an email.
Challenge 98, page 135: The lid of a box must obey the indeterminacy relation. It cannot be at
perfect rest with respect to the rest of the box.
Challenge 100, page 136: No, because the cosmic background is not a Planck scale effect, but an
effect of much lower energy.
Challenge 101, page 136: Yes, at Planck scales all interactions are strand deformations; therefore
collisions and gravity are indistinguishable there.
Challenge 102, page 136: No. Time is continuous only if either quantum theory and point
particles or general relativity and point masses are assumed. The argument shows that only the
combination of both theories with continuity is impossible.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Challenge 86, page 112: If you find one, publish it and send it also to me. The conjecture is that
Motion Mountain – The Adventure of Physics
Challenge 233 e
challenge hints and solu tions
challenge hints and solu tions
425
Challenge 103, page 136: You should, because at Planck scales nature’s inherent measurement
errors cannot clearly distinguish between different measurement results.
Challenge 104, page 136: We still have the chance to find the best approximate concepts pos-
sible. There is no reason to give up.
Challenge 105, page 136: Here are a few thoughts. A beginning of the big bang does not exist;
something similar is given by that piece of continuous entity which is encountered when going
backwards in time as much as possible. This has several implications.
In summary, no starting point of the big bang exists, because time does not exist there. For the
same reason, no initial conditions for particles or space-time exist. In addition, this shows that
the big bang involved no creation, because without time and without possibility of choice, the
term ‘creation’ makes no sense.
Challenge 106, page 136: The equivalence follows from the fact that all these processes require
Planck energy, Planck measurement precision, Planck curvature, and Planck shutter time.
Page 366
Challenge 107, page 136: No, as explained later on in the text.
Challenge 108, page 137: Probably there is nothing wrong with the argument. For example, in
Challenge 109, page 137: If not, force yourself. Brainstorming is important in life, as is the sub-
sequent step: the checking of the speculations.
Challenge 114, page 149: The author would like to receive a mail on your reasons for disagree-
ment.
Challenge 115, page 151: Let the author know if you succeed. And publish the results.
Challenge 116, page 151: Energy is action per time. Now, the Planck constant is the unit of ac-
tion, and is defined by a crossing switch. A system that continuously produces a crossing switch
for every Planck time running by thus has Planck energy. An example would be a tangle that is
rotating extremely rapidly, once per Planck time, producing a crossing switch for every turn.
Momentum is action per length. A system that continuously produces a crossing switch
whenever it advances by a Planck length has Planck momentum. An example would be a tangle
configuration that lets a switch hop from one strand to the next under tight strand packing.
Force is action per length and time. A system that continuously produces a crossing switch
for every Planck time that passes by and for every Planck length it advances through exerts a
Planck force. A tangle with the structure of a screw that rotates and advances with sufficient
speed would be an example.
Challenge 120, page 160: Yes; the appearance of a crossing does not depend on distance or on
the number of strands in between.
Challenge 121, page 160: No; more than three dimensions do not allow us to define a crossing
switch.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
the strand model, all observables are composed of fundamental events, and so, in some way, all
observables are fundamentally indistinguishable.
Motion Mountain – The Adventure of Physics
— Going backwards in time as far as possible – towards the ‘beginning’ of time – is the same as
zooming to smallest distances: we find a single strand of the amoeba.
— In other words, we speculate that the whole world is one single piece, fluctuating, and possibly
tangled, knotted or branched.
— Going far away into space – to the border of the universe – is like taking a snapshot with a
short shutter time: strands everywhere.
— Whenever we sloppily say that extended entities are ‘infinite’ in size, we only mean that they
reach the horizon of the universe.
426
challenge hints and solu tions
Challenge 122, page 160: If so, let the author know. If the generalization is genuine, the strand
model is not correct.
Challenge 134, page 188: The magnitude at a point should be related to the vectorial sum of all
inverse shortest crossing distances at that point.
Challenge 140, page 199: This algebraic transformation is shown in all textbooks that treat the
Pauli equation. It can also be checked by writing the two equations out component by component.
Challenge 143, page 218: Yes, as can easily be checked by rereading the definitions with the
spinor tangle description in mind.
Challenge 146, page 219: No contradiction is known.
Challenge 147, page 219: In the relativistic case, local space curvature is also taken into account.
Challenge 149, page 219: Find out, publish the result, and let the author know.
Challenge 150, page 219: If the strand interpenetration is allowed generally, quantum theory is
Challenge 152, page 219: The belt trick would imply that a wheel rolls over its own blood supply
at every second rotation.
Challenge 161, page 242: If you find one, publish it!
Challenge 172, page 272: No slide is possible, thus no crossing change appears; thus the situ-
ation has no observable effects. If we deform one slide before the slide – which is possible – we
get back the situation already discussed above.
proof is still missing. The same is expected for the Haag–Kastler axioms.
Challenge 187, page 301: A black hole has at least one crossing, thus at least a Planck mass.
Challenge 190, page 308: The present consensus is no.
Challenge 196, page 325: These tangles are not rational. In the renewed strand model of 2015,
they cannot form; they are not allowed and do not represent any particle.
Challenge 198, page 340: Such a tangle is composed of several gravitons.
Challenge 199, page 346: Tail braiding leads to tangledness, which in turn is the basis for core
rotation. And core rotation is kinetic energy, not rest mass.
Challenge 201, page 347: The issue is topic of research; for symmetry reasons it seems that a
state in which each of the six quarks has the same bound to the other five quarks cannot exist.
Challenge 209, page 369: If you find such an estimate, publish it and send it to the author. A
really good estimate also answers the following question: why does particle mass increase with
core complexity? A tangle with a complex core, i.e., with a core of large ropelength, has a large
mass value. Any correct estimate of the mass must yield this property. But a more complex knot
will have a smaller probability for the belt trick. We seem to be forced to conclude that particle
mass is not due to the belt trick alone.
Challenge 213, page 370: Probably not.
Challenge 214, page 370: Probably not.
Challenge 215, page 370: Probably not.
Challenge 216, page 370: Find out – and let the author know.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Challenge 177, page 273: For the Wightman axioms, this seems to be the case; however, a formal
Motion Mountain – The Adventure of Physics
impossible to derive, as the spinor behaviour would not be possible. If strand interpenetration
were allowed only under certain conditions (such as only for a strand with itself, but not among
two different strands), quantum theory might still possible. A similar process lies at the basis of
mass generation, as shown in the section on the weak interaction.
challenge hints and solu tions
427
Challenge 218, page 374: This would be an interesting result worth a publication.
Challenge 221, page 382: If you plan such a calculation, the author would be delighted to help.
Challenge 225, page 391: Take up the challenge!
Challenge 227, page 406: There is a good chance, however, that such alternatives can be elimin-
ated rather quickly. If you cannot do so, do publish the argument, and let the author know.
Challenge 229, page 414: Nobody can really answer ‘why’-questions about human actions.
Climbing, like every other passion, is also a symbolic activity. Climbing can be a search for
adventure, for meaning, for our mother or father, for ourselves, for happiness, or for peace.
Challenge 230, page 415: Also in dreams, speeds can be compared; and also in dreams, a kind
of causality holds (though not a trivial one). Thus there is an invariant and therefore a maximum
speed.
Challenge 231, page 416: Probably none. The answer depends on whether the existence of
Motion Mountain – The Adventure of Physics
strands can be deduced from dreams. If strands can be deduced from dreams, all of physics
follows. The conjecture is that this deduction is possible. If you find an argument against or in
favour of this conjecture, let the author know.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
BIBLIO GR APHY
“
1
”
See the first volume of the Motion Mountain series, Fall, Flow and Heat, available as free
download at www.motionmountain.net. Cited on pages 17 and 401.
See the second volume of the Motion Mountain series, Relativity, available as free download
at www.motionmountain.net. Cited on pages 17, 18, 401, and 430.
See the third volume of the Motion Mountain series, Light, Charges and Brains, available
as free download at www.motionmountain.net, as well as the mentioned fourth and fifth
volumes. Cited on pages 17 and 401.
4
See the fourth and fifth volumes of the Motion Mountain series, The Quantum of
Change and Pleasure, Technology and the Stars, available as free download at www.
motionmountain.net. Cited on pages 18, 401, 402, and 429.
5
The most precise value of the fine structure constant is determined from a weighted
world average of high-precision measurements by a special international scientific
committee called CODATA. Its website is www.codata.org/committees-and-groups/
fundamental-physical-constants. The site also provides the latest official publication
with the values of the fundamental constants. The most recent value of the fine structure
constant is published at physics.nist.gov/cgi-bin/cuu/Value?alphinv and physics.nist.gov/
cgi-bin/cuu/Value?alph. Cited on pages 18, 226, 378, 386, and 416.
6
See for example, the book by Rob ert L aug hlin, A Different Universe: Reinventing Physics from the Botton Down Basic Books, 2005. Of the numerous books that discuss the idea of
a final theory, this is the only one worth reading, and the only one cited in this bibliography.
The opinions of Laughlin are worth pondering. Cited on page 21.
7
Many physicists, including Steven Weinberg, regularly – and incorrectly – claim in interviews that the measurement problem is not solved yet. Cited on page 21.
8
Undocumented sentences to this effect are regularly attributed to Albert Einstein. Because
Einstein was a pantheist, as he often explained, his statements on the ‘mind of god’ are not
really to be taken seriously. They were all made – if at all – in a humorous tone. Cited on
page 21.
9
For an example for the inappropriate fear of unification, see the theatre play Die Physiker
by the Swiss author Friedrich D ürrenmatt. Several other plays and novels took over
this type of disinformation. Cited on page 21.
* This is a statement from the brilliant essay by the influential writer Samuel Johnson, Review of Soame
Jenyns’ “A Free Enquiry Into the Nature and Origin of Evil”, 1757. See www.samueljohnson.com.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
3
Motion Mountain – The Adventure of Physics
2
The only end of writing is to enable the readers
better to enjoy life, or better to endure it.
Samuel Johnson*
b ib liography
11
See e.g. the 1922 lectures by Lorentz at Caltech, published as H. A . L orentz, Problems of
Modern Physics, edited by H. Bateman, Ginn and Company, 1927, page 99. Cited on page
27.
12
Bohr explained the indivisibilty of the quantum of action in his famous Como lecture, printed in N . B ohr, Atomtheorie und Naturbeschreibung, Springer, 1931. It was translated into
English language as N . B ohr, Atomic Theory and the Description of Nature, Cambridge
University Press, 1934. More statements about the indivisibility of the quantum of action
can be found in N . B ohr, Atomic Physics and Human Knowledge, Science Editions, New
York, 1961. For summaries of Bohr’s ideas by others see Max Ja m mer, The Philosophy
of Quantum Mechanics, Wiley, first edition, 1974, pp. 90–91, and John Honner, The Description of Nature – Niels Bohr and the Philosophy of Quantum Physics, Clarendon Press,
1987, p. 104. Cited on page 28.
13
For an overview of the quantum of action as a basis of quantum theory, see the first chapter
of the fourth volume of the Motion Mountain series, Ref. 4. Cited on page 29.
14
An overview of EBK quantization can be found in the volume on quantum theory. Cited
on page 29.
15
Minimal entropy is discussed by L. Sz il ard, Über die Entropieverminderung in einem
thermodynamischen System bei Eingriffen intelligenter Wesen, Zeitschrift für Physik 53,
pp. 840–856, 1929. This classic paper can also be found in English translation in his collected works. Cited on page 30.
16
See for example A . E . Sha lyt-Margolin & A . Ya . Treg ubovich, Generalized
uncertainty relation in thermodynamics, preprint at arxiv.org/abs/gr-qc/0307018, or
J. Uffink & J. va n L ith- van D is, Thermodynamic uncertainty relations, Foundations
of Physics 29, pp. 655–692, 1999. Cited on page 30.
17
See also the fundamental paper by A . D iSessa, Momentum flow as an alternative perspective in elementary mechanics, 48, p. 365, 1980, and A . D iSessa, Erratum: “Momentum
flow as an alternative perspective in elementary mechanics” [Am. J. Phys. 48, 365 (1980)], 48,
p. 784, 1980. Cited on page 31.
18
The observations of black holes at the centre of galaxies and elsewhere are summarised by
R . B l a ndford & N . G ehrels, Revisiting the black hole, Physics Today 52, June 1999.
Their existence is now well established. Cited on page 31.
19
It seems that the first published statement of the maximum force as a fundamental principle was around the year 2000, in this text, in the chapter on gravitation and relativity. The
author discovered the maximum force principle, not knowing the work of others, when
searching for a way to derive the results of the last part of this adventure that would be so
simple that it would convince even a secondary-school student. In the year 2000, the author
told his friends in Berlin about his didactic approach for general relativity.
The concept of a maximum force was first proposed, most probably, by Venzo de Sabbata
and C. Sivaram in 1993. Also this physics discovery was thus made much too late. In 1995,
Corrado Massa took up the idea. Independently, Ludwik Kostro in 1999, Christoph Schiller
just before 2000 and Gary Gibbons in the years before 2002 arrived at the same concept.
Gary Gibbons was inspired by a book by Oliver Lodge; he explains that the maximum force
value follows from general relativity; he does not make a statement about the converse, nor
do the other authors. The statement of maximum force as a fundamental principle seems
original to Christoph Schiller.
Page 56
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Exploring the spirit of play is the subject of research of the famous National Institute for
Play, founded by Stuart Brown, and found at www.nifplay.org. Cited on page 21.
Motion Mountain – The Adventure of Physics
10
Vol. IV, page 15
Vol. IV, page 182
429
430
b ib lio graphy
Vol. II, page 107
Maximal luminosity is often mentioned in connection with gravitational wave detection;
nevertheless, the general power maximum has never been mentioned before. See for
example L. Ju, D. G . B l a ir & C . Z hao, Detection of gravitational waves, Reports on
Progress in Physics 63, pp. 1317–1427, 2000. See also C . W. Misner, K. S. Thorne &
J. A . Wheeler, Gravitation, Freeman, 1973, page 980. Cited on page 32.
21
See for example Wolf g ang R indler, Relativity – Special, General and Cosmological,
Oxford University Press, 2001, p. 70 ss, or R ay d ’ Inverno, Introducing Einstein’s Relativity, Clarendon Press, 1992, p. 36 ss. Cited on page 33.
22
T. Jacob son, Thermodynamics of spacetime: the Einstein equation of state, Physical Review Letters 75, pp. 1260–1263, 1995, preprint at arxiv.org/abs/gr-qc/9504004; this deep
article remains fascinating to this day. Even the author was scared to draw all the possible
conclusions. The general concepts are explained, almost without formulae, in L. Sm olin,
On the nature of quantum fluctuations and their relation to gravitation and the principle of
inertia, Classical and Quantum Gravity 3, pp. 347–359, 1986. Cited on pages 33 and 293.
23
This relation was pointed out by Achim Kempf. The story is told in A . D. Sa kha rov,
General Relativity and Gravitation 32, pp. 365–367, 2000, a reprint of his paper Doklady
Akademii Nauk SSSR 177, pp. 70–71, 1967. Cited on pages 34 and 43.
24
Indeterminacy relations in general relativity are discussed in C . A . Mea d, Possible connection between gravitation and fundamental length, Physical Review B 135, pp. 849–862, 1964.
The generalized indeterminacy relation is implicit on page 852, but the issue is explained
rather unclearly. Probably the author considered the result too simple to be mentioned explicitly. (That paper took 5 years to get published; comments on the story, written 37 years
later, are found at C . A . Mea d, Walking the Planck length through history, Physics Today
54, p. 15 and p. 81, 2001, with a reply by Frank Wilczek.) See also P. K. Tow nsend, Smallscale structure of space-time as the origin of the gravitational constant, Physical Review D 15,
pp. 2795–2801, 1977, or the paper by M. -T. Ja ekel & S. R enaud, Gravitational quantum
limit for length measurement, Physics Letters A 185, pp. 143–148, 1994. Cited on pages 35,
66, 67, 68, 71, and 118.
25
M. Kramer & al., Tests of general relativity from timing the double pulsar, preprint at arxiv.
org/abs/astro-ph/060941. Cited on page 35.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
20
Motion Mountain – The Adventure of Physics
The temporal order of the first papers on maximum force seems to be V. d e Sa b bata
& C . Sivaram, On limiting field strengths in gravitation, Foundations of Physics Letters 6,
pp. 561–570, 1993, followed by C . Massa, Does the gravitational constant increase?, Astrophysics and Space Science 232, pp. 143–148, 1995, and by L. Kostro & B. L a ng e, Is ?4 /?
the greatest possible force in nature?, Physics Essays 12, pp. 182–189, 1999. The next references
are the paper by G . W. G ibb ons, The maximum tension principle in general relativity,
Foundations of Physics 32, pp. 1891–1901, 2002, preprint at arxiv.org/abs/hep-th/0210109 –
though he developed the ideas before that date – and the older versions of the present text,
i.e., C hristoph S chiller, Motion Mountain – The Adventure of Physics, a free pdf available at www.motionmountain.net. Then came C . S chiller, Maximum force and minimum distance: physics in limit statements, preprint at arxiv.org/abs/physics/0309118, and
C . S chiller, General relativity and cosmology derived from principle of maximum power
or force, International Journal of Theoretical Physics 44, pp. 1629–1647, 2005, preprint
at arxiv.org/abs/physics/0607090. See also R . B eig, G . W. G ib b ons & R . M. S choen,
Gravitating opposites attract, Classical and Quantum Gravity 26, p. 225013, 2009. preprint
at arxiv.org/abs/09071103.
A detailed discussion of maximum force and power is given in the volume on general
relativity, Ref. 2. Cited on pages 32, 42, 294, and 440.
b ib liography
431
27
Maximal curvature, as well as area and volume quantization, are discussed in
A . A shtekar, Quantum geometry and gravity: recent advances, preprint at arxiv.org/
abs/gr-qc/0112038 and in A . A shtekar, Quantum geometry in action: big bang and black
holes, preprint at arxiv.org/abs/math-ph/0202008. Cited on pages 37, 75, and 439.
28
Maximons, elementary particles of Planck mass, are discussed by A . D. Sa kha rov, Vacuum quantum fluctuations in curved space and the theory of gravitation, Soviet Physics –
Doklady 12, pp. 1040–1041, 1968. Cited on pages 39, 78, and 121.
29
Wolf g a ng R indler, Relativity – Special, General and Cosmological, Oxford University
Press, 2001, p. 230. Cited on page 41.
30
Several incorrect counterclaims to the entropy limit were made in R . B ousso, The holographic principle, Review of Modern Physics 74, pp. 825–874, 2002, preprint at arxiv.org/
abs/hep-th/0203101. However, this otherwise good review has some errors in its arguments,
as explained on page 146 in volume V. Bousso has changed his position in the meantime;
he now accepts the entropy limit. Cited on pages 43, 47, 437, and 439.
31
Gamma-ray bursts are discussed by G . P reparata, R . Ruffini & S. - S. Xue, The dyadosphere of black holes and gamma-ray bursts, Astronomy and Astrophysics 338, pp. L87–
L90, 1998, and C . L. B ia nco, R . Ruffini & S. - S. Xue, The elementary spike produced
by a pure e+ e− pair-electromagnetic pulse from a black hole: the PEM pulse, Astronomy and
Astrophysics 368, pp. 377–390, 2001. Cited on page 44.
32
See for example the review in C . W.J. B eena kker & al., Quantum transport in semiconductor nanostructures, pp. 1–228, in H. E hrenreich & D. Turnbull editors, Solid
State Physics, volume 44, Academic Press, 1991. Cited on page 44.
33
A discussion of a different electrical indeterminacy relation, between current and charge,
can be found in Y- Q. L i & B. C hen, Quantum theory for mesoscopic electronic circuits and
its applications, preprint at arxiv.org/abs/cond-mat/9907171. Cited on page 44.
34
Ha ns C . Oha nian & R em o Ruffini, Gravitation and Spacetime, W.W. Norton & Co.,
1994. Cited on pages 45 and 444.
35
The entropy limit for black holes is discussed by J. D. B ekenstein, Entropy
bounds and black hole remnants, Physical Review D 49, pp. 1912–1921, 1994. See also
J. D. B ekenstein, Universal upper bound on the entropy-to-energy ratio for bounded
systems, Physical Review D 23, pp. 287–298, 1981. Cited on pages 47 and 131.
36
The statement is also called the Kovtun-Son-Starinets conjecture. It was published as
P. Kov tun, D. T. S on & A . O. Sta rinets, A viscosity bound conjecture, preprint at
arxiv.org/abs/hep-th/0405231. See also P. Kov tun, D. T. S on & A . O. Sta rinets, Viscosity in strongly interacting quantum field theories from black hole physics, Physical Review
Letters 44, p. 111601, 2005. For an experimental verification, see U. Hohm, On the ratio
of the shear viscosity to the density of entropy of the rare gases and H2, N2, CH4, and CF4,
Chemical Physics 444, pp. 39–42, 2014. Cited on page 48.
37
B ria n G reene, The Elegant Universe – Superstrings, Hidden Dimensions, and the Quest
for the Ultimate Theory, Jonathan Cape 1999. Cited on page 52.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Minimal length and minimal time intervals are discussed, for example, by G . A m elinoC a m elia, Limits on the measurability of space-time distances in (the semiclassical approximation of) quantum gravity, Modern Physics Letters A 9, pp. 3415–3422, 1994, preprint at
arxiv.org/abs/gr-qc/9603014, and by Y. J. Ng & H. va n Da m, Limit to space-time measurement, Modern Physics Letters A 9, pp. 335–340, 1994. Many other authors have explored
the topic. Cited on pages 37 and 67.
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26
432
38
39
40
41
42
43
b ib lio graphy
J. L. Friedman & R . D. S orkin, Spin 1/2 from gravity, Physical Review Letters 44,
pp. 1100–1103, 1980. Cited on page 58.
45
A . P. Ba l achandran, G . B imonte, G . Ma rm o & A . Simoni, Topology change and
quantum physics, Nuclear Physics B 446, pp. 299–314, 1995, preprint at arxiv.org/abs/
hep-th/9503046. Cited on page 58.
46
J. E hlers, Introduction – Survey of Problems, pp. 1–10, in J. E hlers, editor, Sistemi
gravitazionali isolati in relatività generale, Rendiconti della scuola internazionale di fisica
“Enrico Fermi”, LXVIIo corso, Società Italiana di Fisica/North Holland, 1979. Cited on
page 58.
See C . S chiller, Le vide diffère-t-il de la matière? in E . G unz ig & S. D iner editors,
Le Vide – Univers du tout et du rien – Des physiciens et des philosophes s’interrogent, Les
Éditions de l’Université de Bruxelles, 1998. An older, English-language version is available
as C . S chiller, Does matter differ from vacuum? preprint at arxiv.org/abs/gr-qc/9610066.
Cited on pages 58, 118, 120, 121, 122, 133, and 134.
See for example R ichard P. Fey nman, Rob ert B. L eig hton & Matthew Sa nd s, The Feynman Lectures on Physics, Addison Wesley, 1977. Cited on page
59.
47
48
49
Stev en Weinb erg, Gravitation and Cosmology, Wiley, 1972. Cited on pages 59, 65,
and 67.
50
The argument is given e.g. in E . P. Wig ner, Relativistic invariance and quantum phenomena, Reviews of Modern Physics 29, pp. 255–258, 1957. Cited on page 64.
The starting point for the following arguments is taken from M. S chön, Operative time
definition and principal indeterminacy, preprint at arxiv.org/abs/gr-qc/9304024, and from
T. Pa d manabhan, Limitations on the operational definition of space-time events and
quantum gravity, Classical and Quantum Gravity 4, pp. L107–L113, 1987; see also Padmanabhan’s earlier papers referenced there. Cited on page 64.
W. Heisenberg, Über den anschaulichen Inhalt der quantentheoretischen Kinematik und
Mechanik, Zeitschrift für Physik 43, pp. 172–198, 1927. Cited on page 64.
E . H. Kenna rd, Zur Quantenmechanik einfacher Bewegungstypen, Zeitschrift für Physik
44, pp. 326–352, 1927. Cited on page 64.
51
52
53
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
44
Motion Mountain – The Adventure of Physics
S. Weinberg, The cosmological constant problem, Reviews of Modern Physics 61, pp. 1–
23, 1989. Cited on page 57.
Stev en Weinb erg, The Quantum Theory of Fields, Cambridge University Press, volumes
I, 1995, and II, 1996. Cited on page 57.
See the excellent presentation on the cosmological constant in general relativity by
E . B ia nchi & C . Rov elli, Why all these prejudices against a constant?, preprint at
arxiv.org/abs/1002.3966 Cited on page 57.
The difficulties are summarised by B. S. D eWit t, Quantum field theory in curved spacetime, Physics Reports 19, pp. 295–357, 1975. Cited on page 57.
C . W. Misner, K. S. Thorne & J. A . Wheeler, Gravitation, Freeman, 1973. Cited on
pages 58, 59, and 67.
J. A . Wheeler, in Relativity, Groups and Topology, edited by C . D eWit t &
B. S. D eWit t, Gordon and Breach, 1994. See also J. A . Wheeler, Physics at the Planck
length, International Journal of Modern Physics A 8, pp. 4013–4018, 1993. However, his
claim that spin 1/2 requires topology change is refuted by the strand model of the vacuum.
Cited on page 58.
b ib liography
433
55
H. Sa lecker, & E . P. Wig ner, Quantum limitations of the measurement of space-time
distances, Physical Review 109, pp. 571–577, 1958. Cited on pages 65, 93, and 114.
56
E . J. Z im merman, The macroscopic nature of space-time, American Journal of Physics 30,
pp. 97–105, 1962. Cited on pages 65, 93, and 114.
57
J. D. B ekenstein, Black holes and entropy, Physical Review D 7, pp. 2333–2346, 1973.
Cited on pages 65, 131, and 290.
58
S. W. Hawking, Particle creation by black holes, Communications in Mathematical Physics 43, pp. 199–220, 1975; see also S. W. Haw king, Black hole thermodynamics, Physical
Review D 13, pp. 191–197, 1976. Cited on pages 65, 131, and 290.
59
P. G ib bs, The small scale structure of space-time: a bibliographical review, preprint at arxiv.
org/abs/hep-th/9506171. Cited on pages 65 and 84.
60
The impossibility of determining temporal ordering in quantum theory is discussed by
J. Oppenheimer, B. R ez nik & W. G . Unruh, Temporal ordering in quantum mechanics, Journal of Physics A 35, pp. 7641–7652, 2001, preprint at arxiv.org/abs/quant-ph/
0003130. Cited on page 66.
61
M. -T. Jaekel & S. R enaud, Gravitational quantum limit for length measurement, Physics Letters A 185, pp. 143–148, 1994. Cited on page 67.
62
D. V. A hluwalia, Quantum measurement, gravitation and locality, Physics Letters B 339,
pp. 301–303, 1994, preprint at arxiv.org/abs/gr-qc/9308007. Cited on page 67.
63
L. G a ray, Quantum gravity and minimum length, International Journal of Modern Physics A 10, pp. 145–165, 1995, preprint at arxiv.org/abs/gr-qc/9403008. This paper also includes an extensive bibliography. See also R . J. A d ler & D. I. Sa ntiago, On gravity and
the uncertainty principle, Modern Physics Letters A 14, pp. 1371–1381, 1999, preprint at arxiv.
org/abs/gr-qc/9904026. Cited on page 67.
64
C . Rov elli & L. Sm olin, Discreteness of area and volume in quantum gravity, Nuclear
Physics B 442, pp. 593–619, 1995. R . L oll, The volume operator in discretized quantum
gravity, preprint at arxiv.org/abs/gr-qc/9506014. See also C . Rov elli, Notes for a brief history of quantum gravity, preprint at arxiv.org/abs/gr-qc/0006061. Cited on page 68.
65
D. A m ati, M . C ia fa loni & G . Venez ia no, Superstring collisions at Planckian energies, Physics Letters B 197, pp. 81–88, 1987. D. J. G ross & P. F. Mend e, The high energy behavior of string scattering amplitudes, Physics Letters B 197, pp. 129–134, 1987.
K. Konishi, G . Pa ffuti & P. P rov ero, Minimum physical length and the generalized
uncertainty principle, Physics Letters B 234, pp. 276–284, 1990. P. A spinwa ll, Minimum
distances in non-trivial string target spaces, Nuclear Physics B 431, pp. 78–96, 1994, preprint
at arxiv.org/abs/hep-th/9404060. Cited on page 68.
66
M. Mag giore, A generalised uncertainty principle in quantum mechanics, Physics Letters
B 304, pp. 65–69, 1993. Cited on page 68.
67
A simple approach is S. D oplicher, K. Fred enhagen & J. E . Rob erts, Space-time
quantization induced by classical gravity, Physics Letters B 331, pp. 39–44, 1994. Cited on
pages 68 and 83.
68
A . Kem pf, Uncertainty relation in quantum mechanics with quantum group symmetry,
Journal of Mathematical Physics 35, pp. 4483–4496, 1994. A . Kem pf, Quantum groups
and quantum field theory with nonzero minimal uncertainties in positions and momenta,
Czechoslovak Journal of Physics 44, pp. 1041–1048, 1994. Cited on page 68.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
M. G. R ay mer, Uncertainty principle for joint measurement of noncommuting variables,
American Journal of Physics 62, pp. 986–993, 1994. Cited on page 64.
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434
69
70
71
72
74
75
77
78
79
80
N . F. R a m sey & A . Weis, Suche nach permanenten elektrischen Dipolmomenten: ein
Test der Zeitumkehrinvarianz, Physikalische Blätter 52, pp. 859–863, 1996. See also
W. B ernreuther & M. Suz uki, The electric dipole moment of the electron, Reviews of
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
76
E . J. Hellund & K. Ta na ka, Quantized space-time, Physical Review 94, pp. 192–195,
1954. Cited on page 69.
This intriguing extract from a letter by Einstein was made widely known by
John J. Stachel, in his paper The other Einstein: Einstein contra field theory, that is
best found in his book Einstein from ‘B’ to ‘Z’, Birkhäuser, 2002. The German original
of the letter is found in Rob ert S chulmann, A . J. Knox, Michel Ja nssen &
Józ sef Illy, The Collected Papers of Albert Einstein, Volume 8A – The Berlin Years: Correspondence, 1914–1917, letter 299, Princeton University Press, 1998. Barbara Wolff helped
in clarifying several details in the German original. The letter is now available online, at
einsteinpapers.press.princeton.edu/vol8a-doc/463. Cited on page 69.
A . P eres & N . Rosen, Quantum limitations on the measurement of gravitational fields,
Physical Review 118, pp. 335–336, 1960. Cited on page 71.
It is the first definition in Euclid’s Elements, c. 300 b ce. For an English translation see
T. Heath, The Thirteen Books of the Elements, Dover, 1969. Cited on page 72.
A beautiful description of the Banach–Tarski paradox is the one by Ia n Stewart, Paradox of the spheres, New Scientist, 14 January 1995, pp. 28–31. Cited on page 72.
H. S. Sny der, Quantized space-time, Physical Review 71, pp. 38–41, 1947. H. S. Sny der,
The electromagnetic field in quantized space-time, Physical Review 72, pp. 68–74, 1947.
A . S child, Discrete space-time and integral Lorentz transformations, Physical Review 73,
pp. 414–415, 1948. E . L. Hill, Relativistic theory of discrete momentum space and discrete space-time, Physical Review 100, pp. 1780–1783, 1950. H. T. Flint, The quantization of space-time, Physical Review 74, pp. 209–210, 1948. A . Das, Cellular space-time and
quantum field theory, Il Nuovo Cimento 18, pp. 482–504, 1960. Cited on page 74.
D. Finkelstein, ‘Superconducting’ causal nets, International Journal of Theoretical
Physics 27, pp. 473–519, 1985. Cited on page 74.
N . H. C hrist, R . Friedberg & T. D. L ee, Random lattice field theory: general formulation, Nuclear Physics B 202, pp. 89–125, 1982. G . ’ t Ho oft, Quantum field theory for
elementary particles – is quantum field theory a theory?, Physics Reports 104, pp. 129–142,
1984. Cited on page 74.
For a discussion, see R . S ora bji, Time, Creation and the Continuum: Theories in Antiquity
and the Early Middle Ages, Duckworth, 1983. Cited on page 74.
See, for example, L. B om b elli, J. L ee, D. Mey er & R . D. S orkin, Space-time
as a causal set, Physical Review Letters 59, pp. 521–524, 1987. G . B right well &
R . G reg ory, Structure of random space-time, Physical Review Letters 66, pp. 260–
263, 1991. Cited on page 74.
The false belief that particles like quarks or electrons are composite is slow to die out. See for
example: S. Fredriksson, Preon prophecies by the standard model, preprint at arxiv.org/
abs/hep-ph/0309213. Preon models gained popularity in the 1970s and 1980s, in particular
through the papers by J. C . Pati & A. Sa l a m, Lepton number as the fourth “color”, Physical Review D 10, pp. 275–289, 1974, H. Ha ra ri, A schematic model of quarks and leptons,
Physics Letters B 86, pp. 83–86, 1979, M. A . Shupe, A composite model of leptons and
quarks, Physics Letters B 86, pp. 87–92, 1979, and H. Fritzsch & G . Ma nd elbaum,
Weak interactions as manifestations of the substructure of leptons and quarks, Physics Letters B 102, pp. 319–322, 1981. Cited on page 76.
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b ib liography
81
82
83
84
85
435
?=
?2 ??
1+
and
?=
???
1+
?2 ??
?Pl
.
(221)
Another, similar approach of recent years, with a different proposal, is called ‘doubly
special relativity’. A recent summary is G . A m elino-Camelia, Doubly-special relativity: first results and key open problems, International Journal of Modern Physics 11,
pp. 1643–1669, 2002, preprint at arxiv.org/abs/gr-qc/0210063. The paper shows how conceptual problems hinder the advance of the field. Another such discussion R . A loisio,
A . G al a nte, A . F. G ri llo, E . Luz i o & F. Ménd ez, Approaching space-time through
velocity in doubly special relativity, preprint at arxiv.org/abs/gr-qc/0410020. The lesson from
these attempts is simple: special relativity cannot be modified to include a limit energy
without also including general relativity and quantum theory. Cited on pages 83 and 280.
W. Jauch, Heisenberg’s uncertainty relation and thermal vibrations in crystals, American
Journal of Physics 61, pp. 929–932, 1993. Cited on page 83.
87
H. D. Z eh, On the interpretation of measurement in quantum theory, Foundations of Physics 1, pp. 69–76, 1970. Cited on page 84.
88
See Y. J. Ng, W. A . C hristiansen & H. va n Da m, Probing Planck-scale physics with
extragalactic sources?, Astrophysical Journal 591, pp. L87–L90, 2003, preprint at arxiv.org/
abs/astro-ph/0302372; D. H. C oule, Planck scale still safe from stellar images, Classical
and Quantum Gravity 20, pp. 3107–3112, 2003, preprint at arxiv.org/abs/astro-ph/0302333.
Negative experimental results (and not always correct calculations) are found in R . L ieu
& L. Hillman, The phase coherence of light from extragalactic sources – direct evidence
against first order Planck scale fluctuations in time and space, Astrophysical Journal 585,
pp. L77–L80, 2003, and R . R ag azz oni, M. Turat to & W. G a essler, The lack of observational evidence for the quantum structure of spacetime at Planck scales, Astrophysical
Journal 587, pp. L1–L4, 2003. Cited on page 87.
B. E . S cha efer, Severe limits on variations of the speed of light with frequency, Physical
Review Letters 82, pp. 4964–4966, 21 June 1999. Cited on page 87.
89
90
A . A . A b d o & al., (Fermi GBM/LAT collaborations) Testing Einstein’s special relativity
with Fermi’s short hard gamma-ray burst GRB090510, preprint at arxiv.org/0908.1832. Cited
on page 87.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
86
?2 ??
?Pl
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Modern Physics 63, pp. 313–340, 1991, and the musings in Ha ns D ehm elt, Is the electron
a composite particle?, Hyperfine Interactions 81, pp. 1–3, 1993. Cited on page 77.
K. A ka m a, T. Hat tori & K. Katsuura, Naturalness bounds on dipole moments from
new physics, preprint at arxiv.org/abs/hep-ph/0111238. Cited on page 77.
The paper by J. Ba ron & al., Order of magnitude smaller limit on the electric dipole moment
of the electron, preprint at arxiv.org/abs/1310.7534 gives an upper experimental limit to the
dipole moment of the electron of 8.7 ⋅ 10−31 e m. Cited on page 77.
C . Wolf, Upper limit for the mass of an elementary particle due to discrete time quantum
mechanics, Il Nuovo Cimento B 109, pp. 213–218, 1994. Cited on page 79.
W. G . Unruh, Notes on black hole evaporation, Physical Review D 14, pp. 870–875, 1976.
W. G . Unruh & R. M. Wa ld, What happens when an accelerating observer detects a Rindler particle, Physical Review D 29, pp. 1047–1056, 1984. Cited on page 81.
The first example was J. Mag ueijo & L. Sm olin, Lorentz invariance with an invariant
energy scale, Physical Review Letters 88, p. 190403, 2002, preprint at arxiv.org/abs/hep-th/
0112090. They propose a modification of the mass energy relation of the kind
436
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92
G . A m elino-C amelia, Phenomenological description of space-time foam, preprint at
arxiv.org/abs/gr-qc/0104005. The paper includes a clearly written overview of present
experimental approaches to detecting quantum gravity effects. See also his update
G . A m elino-C amelia, Quantum-gravity phenomenology: status and prospects, preprint at arxiv.org/abs/gr-qc/0204051. Cited on pages 87 and 88.
93
G . A m elino-C amelia, An interferometric gravitational wave detector as a quantum
gravity apparatus, Nature 398, pp. 216–218, 1999, preprint at arxiv.org/abs/gr-qc/9808029.
Cited on page 87.
94
F. Ka rolyhazy, Gravitation and quantum mechanics of macroscopic objects, Il Nuovo Cimento A42, pp. 390–402, 1966. Y. J. Ng & H. va n Da m, Limit to space-time measurement,
Modern Physics Letters A 9, pp. 335–340, 1994. Y. J. Ng & H. va n Da m, Modern Physics
Letters A Remarks on gravitational sources, 10, pp. 2801–2808, 1995. The discussion is neatly
summarised in Y. J. Ng & H. va n Da m,Comment on ‘Uncertainty in measurements of distance’, preprint at arxiv.org/abs/gr-qc/0209021. See also Y. J. Ng, Spacetime foam, preprint
at arxiv.org/abs/gr-qc/0201022. Cited on pages 87 and 93.
95
L. J. G a ray, Spacetime foam as a quantum thermal bath, Physics Review Letters 80,
pp. 2508–2511, 1998, preprint at arxiv.org/abs/gr-qc/9801024. Cited on page 88.
96
G . A m elino-C amelia & T. P ira n, Planck-scale deformation of Lorentz symmetry as
a solution to the UHECR and the TeV-? paradoxes, preprint at arxiv.org/astro-ph/0008107,
2000. Cited on page 88.
97
R . P. Wo odard, How far are we from the quantum theory of gravity?, preprint at arxiv.org/
abs/0907.4238. For a different point of view, see L. Sm olin, Generic predictions of quantum
theories of gravity, preprint at arxiv.org/abs/hep-th/0605052. Cited on pages 88 and 304.
98
A similar point of view, often called monism, was proposed by Ba ruch Spinoz a, Ethics Demonstrated in Geometrical Order, 1677, originally in Latin; an affordable French edition is Ba ruch Spinoza, L’Ethique, Folio-Gallimard, 1954. For a discussion of his ideas,
especially his monism, see D on G a rret editor, The Cambridge Companion to Spinoza,
Cambridge University Press, 1996, or any general text on the history of philosophy. Cited
on page 88.
99
See the lucid discussion by G . F. R. E llis & T. Rothm an, Lost horizons, American
Journal of Physics 61, pp. 883–893, 1993. Cited on pages 93, 97, and 98.
100 See, for example, the Hollywood film Contact by Robert Zemeckis, based on the book by
C a rl Sag a n, Contact, Simon & Schuster, 1985. Cited on page 98.
101 See, for example, the international bestseller by Stephen Haw king, A Brief History of
Time – From the Big Bang to Black Holes, 1988. Cited on page 101.
102 L. Rosenfeld, Quantentheorie und Gravitation, in H. - J. Tred er, editor, Entstehung,
Entwicklung und Perspektiven der Einsteinschen Gravitationstheorie, Springer Verlag, 1966.
Cited on page 103.
103 Holography in high-energy physics is connected with the work of ’t Hooft and Susskind.
See for example G . ’ t Ho oft, Dimensional reduction in quantum gravity, pp. 284–296,
in A . A li, J. E llis & S . R a ndjbar-Daemi, Salaamfeest, 1993, or the much-cited paper
by L. Susskind, The world as a hologram, Journal of Mathematical Physics 36, pp. 6377–
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
G . A m elino-C amelia, J. E llis, N . E . Mav romatos, D. V. Na nop oulos &
S. Sa ka r, Potential sensitivity of gamma-ray-burster observations to wave dispersion in
vacuo, Nature 393, pp. 763–765, 1998, preprint at arxiv.org/abs/astro-ph/9712103. Cited on
page 87.
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437
6396, 1995, preprint at arxiv.org/abs/hep-th/9409089. A good modern overview is Ref. 30.
Cited on pages 105 and 113.
104 D. B ohm & B. J. Hiley, On the intuitive understanding of nonlocality as implied by
quantum theory, Foundations of Physics 5, pp. 93–109, 1975. Cited on page 106.
105 S. L loy d, Computational capacity of the universe, Physical Review Letters 88, p. 237901,
2002. Cited on page 107.
106 G ot tfried Wilhelm L eib niz, La Monadologie, 1714. Written in French, it is avail-
able freely at www.uqac.uquebec.ca/zone30/Classiques_des_sciences_sociales and in various other languages on other websites. Cited on page 108.
107 See, for example, H. Wussing & P. S. A lexa ndrov editors, Die Hilbertschen Probleme,
Akademische Verlagsgesellschaft Geest & Portig, 1983, or B en H. Ya ndell, The Honours
Class: Hilbert’s Problems and their Solvers, A.K. Peters, 2002. Cited on page 108.
108 A large part of the study of dualities in string and M theory can be seen as investiga-
109 See L. Susskind & J. Ug lum, Black holes, interactions, and strings, preprint at arxiv.
110 M. P l anck, Über irreversible Strahlungsvorgänge, Sitzungsberichte der Kaiserlichen
Akademie der Wissenschaften zu Berlin pp. 440–480, 1899. Today it is commonplace to
use Dirac’s ℏ = ℎ/2π instead of Planck’s ℎ, which Planck originally called ?. Cited on page
118.
111 P. Facchi & S. Pasca zio, Quantum Zeno and inverse quantum Zeno effects, pp. 147–217,
in E . Wolf editor, Progress in Optics, 42, 2001. Cited on page 121.
112 A ristotle, Of Generation and Corruption, book I, part 2. See Jea n-Paul D umont,
Les écoles présocratiques, Folio Essais, Gallimard, p. 427, 1991. Cited on page 121.
113 See for example the speculative model of vacuum as composed of Planck-size spheres pro-
posed by F. Winterberg, Zeitschrift für Naturforschung 52a, p. 183, 1997. Cited on page
122.
114 The Greek salt-and-water argument and the fish argument are given by Lucrece, in full Titus
Lucretius Carus, De natura rerum, c. 60 b ce. Cited on pages 123 and 138.
115 J. H. S chwarz, The second superstring revolution, Colloquium-level lecture presented at
the Sakharov Conference in Moscow, May 1996, preprint at arxiv.org/abs/hep-th/9607067.
Cited on pages 124 and 125.
116 Sim plicius, Commentary on the Physics of Aristotle, 140, 34. This text is cited in Jea n-
Paul D umont, Les écoles présocratiques, Folio Essais, Gallimard, p. 379, 1991. Cited on
page 124.
117 D. Oliv e & C . Montonen, Magnetic monopoles as gauge particles, Physics Letters 72B,
pp. 117–120, 1977. Cited on page 125.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
org/abs/hep-th/9410074, or L. Susskind, String theory and the principle of black hole
complementarity, Physical Review Letters 71, pp. 2367–2368, 1993, and M. Ka rliner,
I. Klebanov & L. Susskind, Size and shape of strings, International Journal of Modern Physics A 3, pp. 1981–1996, 1988, as well as L. Susskind, Structure of hadrons implied
by duality, Physical Review D 1, pp. 1182–1186, 1970. Cited on pages 117 and 132.
Motion Mountain – The Adventure of Physics
tions into the detailed consequences of extremal identity. For a review of dualities, see
P. C . A rg yres, Dualities in supersymmetric field theories, Nuclear Physics Proceedings
Supplement 61, pp. 149–157, 1998, preprint at arxiv.org/abs/hep-th/9705076. A classical
version of duality is discussed by M. C . B. A b dall a, A . L. G a delka & I. V. Va ncea,
Duality between coordinates and the Dirac field, preprint at arxiv.org/abs/hep-th/0002217.
Cited on page 113.
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118 A famous fragment from D io g enes L a ërtius (IX 72) quotes Democritus as follows:
‘By convention hot, by convention cold, but in reality, atoms and void; and also in reality
we know nothing, since truth is at the bottom.’ Cited on page 127.
119 This famous statement is found at the beginning of chapter XI, ‘The Physical Universe’,
in A rthur E d dington, The Philosophy of Physical Science, Cambridge, 1939. Cited on
page 128.
120 P l ato, Parmenides, c. 370 b ce. It has been translated into most languages. Reading it
aloud, like a song, is a beautiful experience. A pale reflection of these ideas is Bohm’s
concept of ‘unbroken wholeness’. Cited on page 128.
121 P. G ib bs, Event-symmetric physics, preprint at arxiv.org/abs/hep-th/9505089; see also his
website www.weburbia.com/pg/contents.htm. Cited on page 128.
122 J. B. Hartle, & S. W. Hawking, Path integral derivation of black hole radiance, Phys-
123 J. Ma d d ox, When entropy does not seem extensive, Nature 365, p. 103, 1993. The issue is
now explored in all textbooks discussing black holes. John Maddox (b. 1925 Penllergaer,
d. 1999 Abergavenny) was famous for being one of the few people who was knowledgeable
in most natural sciences. Cited on page 131.
124 L. B om b elli, R . K. Koul, J. L ee & R . D. S orkin, Quantum source of entropy of black
holes, Physical Review D 34, pp. 373–383, 1986. Cited on page 131.
boundaries, Nature 365, p. 792, 1993. Cited on page 131.
126 See the classic text by P ierre-Gilles d e G ennes, Scaling Concepts in Polymer Physics,
Cornell University Press, 1979. Cited on page 132.
127 See for example S. Maj id, Introduction to braided geometry and ?-Minkowski space, pre-
print at arxiv.org/abs/hep-th/9410241, or S. Maj id, Duality principle and braided geometry, preprint at arxiv.org/abs/hep-th/9409057. Cited on pages 133 and 134.
128 The relation between spin and statistics has been studied recently by M. V. B erry &
J. M. Rob b ins, Quantum indistinguishability: spin–statistics without relativity or field theory?, in R . C . Hilb orn & G . M. Tino editors, Spin–Statistics Connection and Commutation Relations, American Institute of Physics, 2000. Cited on page 135.
129 A . G reg ori, Entropy, string theory, and our world, preprint at arxiv.org/abs/hep-th/
0207195. Cited on pages 136 and 137.
130 String cosmology is a pastime for many. Examples include N . E . Mav romatos, String
cosmology, preprint at arxiv.org/abs/hep-th/0111275, and N . G . Sa nchez, New developments in string gravity and string cosmology – a summary report, preprint at arxiv.org/abs/
hep-th/0209016. Cited on page 137.
131 On the present record, see en.wkipedia.org/wiki/Ultra-high-energy_cosmic_ray and fr.
wkipedia.org/wiki/Zetta-particule. Cited on page 138.
132 P. F. Mende, String theory at short distance and the principle of equivalence, preprint at
arxiv.org/abs/hep-th/9210001. Cited on page 138.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
125 The analogy between polymers and black holes is due to G . Web er, Thermodynamics at
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ical Review D 13, pp. 2188–2203, 1976. See also A . Strom inger & C . Va fa, Microscopic origin of Bekenstein–Hawking entropy, Physics Letters B 379, pp. 99–104, 1996,
preprint at arxiv.org/abs/hep-th/9601029. For another derivation of black hole entropy,
see G . T. Horow itz & J. Polchinski, A correspondence principle for black holes and
strings, Physical Review D 55, pp. 6189–6197, 1997, preprint at arxiv.org/abs/hep-th/
9612146. Cited on pages 131 and 140.
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133 An example is given by A . A . Sl avnov, Fermi–Bose duality via extra dimension, preprint
134
135
136
137
139
140
142
143
144
145
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at arxiv.org/abs/hep-th/9512101. See also the standard work by Michael Stone editor,
Bosonization, World Scientific, 1994. Cited on page 138.
A weave model of space-time appears in certain approaches to quantum gravity, such as
Ref. 27. On a slightly different topic, see also S. A . Maj or, A spin network primer, preprint
at arxiv.org/abs/gr-qc/9905020. Cited on page 138.
L. Sm olin & Y. Wa n, Propagation and interaction of chiral states in quantum gravity, preprint at arxiv.org/abs/0710.1548, and references therein. Cited on page 138.
A good introduction into his work is the paper D. Kreimer, New mathematical structures
in renormalisable quantum field theories, Annals of Physics 303, pp. 179–202, 2003, erratum
ibid. 305, p. 79, 2003, preprint at arxiv.org/abs/hep-th/0211136. Cited on page 139.
Introductions to holography include E . A lva rez, J. C onde & L. Herna ndez, Rudiments of holography, preprint at arxiv.org/abs/hep-th/0205075, and Ref. 30. The importance of holography in theoretical high-energy physics was underlined by the discovery of
J. Ma ldacena, The large N limit of superconformal field theories and supergravity, preprint at arxiv.org/abs/hep-th/9711200. Cited on page 139.
X. - G . Wen, From new states of matter to a unification of light and electrons, preprint at
arxiv.org/abs/0508020. Cited on page 139.
J. S. Av rin, A visualizable representation of the elementary particles, Journal of Knot Theory and Its Ramifications 14, pp. 131–176, 2005. Cited on pages 139 and 347.
The well-known ribbon model is presented in S. B ilson-Thompson, A topological
model of composite preons, preprint at arxiv.org/hep-ph/0503213; S. B ilson-Thompson,
F. Ma rkop oulou & L. Sm olin, Quantum gravity and the standard model, preprint
at arxiv.org/hep-th/0603022; S. B ilson-Thompson, J. Hackett, L. Kauffman
& L. Sm olin, Particle identifications from symmetries of braided ribbon network invariants, preprint at arxiv.org/abs/0804.0037; S. B ilson-Thompson, J. Hackett &
L. Kauffman, Particle topology, braids, and braided belts, preprint at arxiv.org/abs/0903.
1376. Cited on pages 139, 165, and 347.
R . J. Finkelstein, A field theory of knotted solitons, preprint at arxiv.org/abs/hep-th/
0701124. See also R . J. Finkelstein, Trefoil solitons, elementary fermions, and SUq (2),
preprint at arxiv.org/abs/hep-th/0602098, R. J. Finkelstein & A . C. C a david, Masses
and interactions of q-fermionic knots, preprint at arxiv.org/abs/hep-th/0507022, and
R . J. Finkelstein, A knot model suggested by the standard electroweak theory, preprint
at arxiv.org/abs/hep-th/0408218. Cited on pages 139 and 347.
L ouis H. Kauffman, Knotsand Physics, World Scientific, 1991. A wonderful book. Cited
on pages 139 and 275.
S. K. Ng, On a knot model of the π+ meson, preprint at arxiv.org/abs/hep-th/0210024, and
S. K. Ng, On a classification of mesons, preprint at arxiv.org/abs/hep-ph/0212334. Cited on
pages 139 and 347.
For a good introduction to superstrings, see the lectures by B. Zw iebach, String theory for
pedestrians, agenda.cern.ch/fullAgenda.php?ida=a063319. For an old introduction to superstrings, see the famous text by M. B. G reen, J. H. S chwarz & E . Wit ten, Superstring Theory, Cambridge University Press, volumes 1 and 2, 1987. Like all the other books
on superstrings, they contain no statement that is applicable to or agrees with the strand
model. Cited on pages 139 and 348.
See A . Sen, An introduction to duality symmetries in string theory, in Les Houches Summer
School: Unity from Duality: Gravity, Gauge Theory and Strings (Les Houches, France, 2001),
440
146
147
148
149
150
152
154
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copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
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Springer Verlag, 76, pp. 241–322, 2002. Cited on page 139.
Brian Greene regularly uses the name string conjecture. For example, he did so in a podium
discussion at TED in 2009; the video of the podium discussion can be downloaded at www.
ted.org. Cited on page 140.
L. Susskind, Some speculations about black hole entropy in string theory, preprint at arxiv.
org/abs/hep-th/9309145. G . T. Horow itz & J. Polchinski, A correspondence principle
for black holes and strings, Physical Review D 55, pp. 6189–6197, 1997, preprint at arxiv.org/
abs/hep-th/9612146. Cited on pages 140 and 444.
F. Wilcz ek, Getting its from bits, Nature 397, pp. 303–306, 1999. Cited on page 141.
M. R . D ougl as, Understanding the landscape, preprint at arxiv.org/abs/hep-th/0602266;
his earlier papers also make the point. For the larger estimate, see W. Tay lor & Y. N . Wa ng, The F-theory geometry with most flux vacua, preprint at arxiv.org/abs/1511.
03209. Cited on page 141.
The difficulties of the string conjecture are discussed in the well-known internet blog
by P eter Woit, Not even wrong, at www.math.columbia.edu/~woit/blog. Several Nobel Prize winners for particle physics dismiss the string conjecture: Martin Veltman, Sheldon Glashow, Burton Richter, Richard Feynman and since 2009 also Steven Weinberg are
among those who did so publicly. Cited on pages 142 and 166.
The present volume was originally started with the aim to clarify the basic principles of
string theory and to simplify it as much as possible. In particular, the first six chapters and
the last chapter were conceived, structured and written with that aim. They are older than
the strand model. Later on, the project took an unexpected direction, as explained in Ref. 19.
Cited on page 142.
Searches for background-free approaches are described by E . Wit ten, Quantum background independence in string theory, preprint at arxiv.org/abs/hep-th/9306122 and
E . Wit ten, On background-independent open string theory, preprint at arxiv.org/abs/
hep-th/9208027. Cited on page 143.
In fact, no other candidate model that fulfils all requirements for the final theory is available
in the literature so far. This might change in the future, though. Cited on page 149.
S. C a rlip, The small scale structure of spacetime, preprint at arxiv.org/abs/1009.1136. This
paper deduces the existence of fluctuating lines in vacuum from a number of arguments
that are completely independent of the strand model. Steven Carlip has dedicated much of
his research to the exploration of this topic. One summary is S. C a rlip, Spontaneous dimensional reduction in quantum gravity, preprint at arxiv.org/abs/1605.05694; its is also instructive to read his review S. C a rlip, Dimension and dimensional reduction in quantum
gravity, Classical and Quantum Gravity 34, p. 193001, 2017, preprint at arxiv.org/abs/1705.
05417. With the strand model in the back of one’s mind, these results are even more fascinating. Cited on pages 161 and 300.
David Deutsch states that any good explanation must be ‘hard to vary’. This must also apply
to a unified model, as it claims to explain everything that is observed. See D. D eu tsch,
A new way to explain explanation, video talk at www.ted.org. Cited on pages 164 and 406.
L. B om b elli, J. L ee, D. Meyer & R . D. S orkin, Space-time as a causal set, Physical
Review Letters 59, pp. 521–524, 1987. See also the review by J. Henson, The causal set
approach to quantum gravity, preprint at arxiv.org/abs/gr-qc/0601121. Cited on pages 165
and 299.
D. Finkelstein, Homotopy approach to quantum gravity, International Journal of Theoretical Physics 47, pp. 534–552, 2008. Cited on pages 165 and 299.
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158 L. H. Kauffman & S. J. L om onaco, Quantum knots, preprint at arxiv.org/abs/
quant-ph/0403228. See also S. J. L om onaco & L. H. Kauffman, Quantum knots
and mosaics, preprint at arxiv.org/abs/quant-ph/0805.0339. Cited on page 165.
159 Im m anuel Ka nt, Critik der reinen Vernunft, 1781, is a famous but long book that every
philospher pretends to have read. In his book, Kant introduced the ‘a priori’ existence of
space and time. Cited on page 168.
160 The literature on circularity is rare. For two interesting exceptions, see L. H. Kauffman,
Knot logic, downloadable from www2.math.uic.edu/~kauffman, and L. H. Kauffman,
Reflexivity and eigenform, Constructivist Foundations 4, pp. 121–137, 2009. Cited on page
169.
161 Information on the belt trick is scattered across many books and few papers. The best source
162 There is an interesting exploration behind this analogy between a non-dissipative system
Challenge 234 e
163 D. B ohm, R . S chiller & J. Tiom no, A causal interpretation of the Pauli equation (A),
Supplementi al Nuovo Cimento 1, pp. 48 – 66, 1955, and D. B ohm & R . S chiller, A
causal interpretation of the Pauli equation (B), Supplementi al Nuovo Cimento 1, pp. 67–91,
1955. The authors explore an unusual way to interpret the wavefunction, which is of little
interest here; but doing so, they give and explore the description of Pauli spinors in terms
of Euler angles. Cited on page 198.
164 R icha rd P. Fey nman, QED – The Strange Theory of Light and Matter, Princeton Uni-
versity Press 1988. This is one of the best summaries of quantum theory ever written. Every
physicist should read it. Cited on pages 199, 204, 216, 222, and 441.
165 S. Ko chen & E . P. Specker, The problem of hidden variables in quantum mechanics, 17,
pp. 59–87, 1967. This is a classic paper. Cited on page 202.
166 A . A spect, J. Da li bard & G . Ro g er, Experimental tests of Bell’s inequalities using
time-varying analyzers, Physical Review Letters 49, pp. 1804–1807, 1982, Cited on page
206.
167 L. Kauffman, New invariants of knot theory, American Mathematical Monthly 95,
pp. 195–242, 1987. See also the image at the start of chapter 6 of L ouis H. Kauffman,
On Knots, Princeton University Press, 1987. Cited on page 207.
168 The details on the speed of photons are explained in any textbook on quantum electrody-
namics. The issue is also explained by Feynman in Ref. 164 on page 89. Cited on page 210.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
– a free quantum particle moving in vacuum – and a dissipative system – a macroscopic
body drawn through a viscous liquid, say honey. The first question is to discover why this
analogy is possible at all. (A careful distinction between the cases with spin 0, spin 1 and
spin 1/2 are necessary.) The second question is the exploration of the motion of bodies of
general shape in viscous fluids at low Reynolds numbers and under constant force. For the
best overview of this question, see the beautiful article by O. G onz alez, A . B. A. G ra f
& J. H. Ma d d o cks, Dynamics of a rigid body in a Stokes fluid, Journal of Fluid Mechanics
519, pp. 133–160, 2004. Cited on pages 195 and 360.
Motion Mountain – The Adventure of Physics
of information on this topic are websites. For belt trick visualizations see www.evl.uic.edu/
hypercomplex/html/dirac.html, www.evl.uic.edu/hypercomplex/html/handshake.html, or
www.gregegan.net/APPLETS/21/21.html. For an excellent literature summary and more
movies, see www.math.utah.edu/~palais/links.html. None of these sites or the cited references seem to mention that there are many ways to perform the belt trick; this seems to
be hidden knowledge. In September 2009, Greg Egan took up my suggestion and changed
his applet to show an additional version of the belt trick. Cited on pages 176 and 178.
442
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169 J. - M. L év y -L eblond, Nonrelativitic particles and wave equations, Communic-
ations in Mathematical Physics 6, pp. 286–311, 1967. See also A . G a lind o &
C . Sá nchez d el R ío, Intrinsic magnetic moment as a nonrelativistic phenomenon,
American Journal of Physics 29, pp. 582–584, 1961, and V. I. Fushchich, A . G. Nikitin
& V. A . Sa lo g ub, On the non-relativistic motion equations in the Hamiltonian form, Reports on Mathematical Physics 13, pp. 175–185, 1978. Cited on page 213.
170 L. L erner, Derivation of the Dirac equation from a relativistic representation of spin,
European Journal of Phyics 17, pp. 172–175, 1996. Cited on pages 213 and 214.
171 E . P. Bat tey-P ratt & T. J. R acey, Geometric model for fundamental particles, Inter-
national Journal of Theoretical Physics 19, pp. 437–475, 1980. Without knowing this work,
C. Schiller had deduced the same results in 2008. Cited on pages 213, 214, and 453.
172 A . A b ra ham, Prinzipien der Dynamik des Elektrons, Annalen der Physik 10, pp. 105–179,
Bewegung in der relativistischen Quantenmechanik, Berliner Berichte pp. 418–428, 1930,
and Zur Quantendynamik des Elektrons, Berliner Berichte pp. 63–72, 1931. Numerous subsequent papers discuss these publications. Cited on page 216.
174 See for example the book by Ma rtin R ivas, Kinematic Theory of Spinning Particles,
Springer, 2001. Cited on page 216.
175 The basic papers in the field of stochastic quantization are W. Weiz el, Ableitung der
Quantentheorie aus einem klassischen, kausal determinierten Modell, Zeitschrift für Physik
A 134, pp. 264–285, 1953, W. Weiz el, Ableitung der Quantentheorie aus einem klassischen
Modell – II, Zeitschrift für Physik A 135, pp. 270–273, 1954, W. Weiz el, Ableitung der
quantenmechanischen Wellengleichung des Mehrteilchensystems aus einem klassischen Modell, Zeitschrift für Physik A 136, pp. 582–604, 1954. This work was taken up by E . Nelson,
Derivation of the Schrödinger equation from Newtonian mechanics, Physical Review 150,
pp. 1079–1085, 1969, and in E dward Nelson, Quantum Fluctuations, Princeton Univer-
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
173 The concept of Zitterbewegung was formulated in E . S chrödinger, Über die kräftefreie
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1903, J. Frenkel, Die Elektrodynamik des rotierenden Elektrons, Zeitschrift für Physik
37, pp. 243–262, 1926, L. H. Thom as, The motion of a spinning electron, Nature April 10,
p. 514, 1926, and L. H. Thom as, The kinematics of an electron with an axis, Philosophical
Magazine 3, pp. 1–22, 1927. See also W. E . Bay lis, Surprising symmetries in relativistic charge dynamics, preprint at arxiv.org/abs/physics/0410197. See also W. E . Bay lis,
Quantum/classical interface: a geometric approach from the classical side, pp. 127–154
and W. E . Bay lis, Geometry of paravector space with applications to relativistic physics,
pp. 363–387 in Computational Noncommutative Algebra and Applications, Proceedings of
the NATO Advanced Study Institute, NATO Science Series II, vol. 136, ed. J. B y rnes,
Kluwer Academic 2004. W. E . Bay lis, R . C a b rera & D. Keselica, Quantum/
classical interface: fermion spin, preprint at arxiv.org/abs/0710.3144. D. Hestenes, Zitterbewegung Modelling, Foundations of Physics 23, pp. 365–386, 1993. D. Hestenes,
Zitterbewegung in quantum mechanics – a research program, preprint at arxiv.org/abs/
0802.2728. See also D. Hestenes, Reading the electron clock, preprint at arxiv.org/abs/
0802.3227 and his webpage modelingnts.la.asu.edu/html/GAinQM.html. A . L oing er &
A. Sparza ni, Dirac equation without Dirac matrices, Il Nuovo Cimento 39, pp. 1140–
1145, 1965. D. B ohm, P. Hillion, T. Takabayasi & J. - P. Vig ier, Relativistic rotators
and bilocal theory, Progress of Theoretical Physics 23, pp. 496–511, 1960. A . C ha llinor,
A. L asenb y, S. Gill & C . D oran, A relativistic, causal account of a spin measurement,
Physics Letters A 218, pp. 128–138, 1996. E . Sa nta mato, The role of Dirac equation in
the classical mechanics of the relativistic top, preprint at arxiv.org/abs/0808.3237. Cited on
pages 215 and 216.
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178
179
180
182
183
184
185
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181
sity Press 1985, also downloadable at www.math.princeton.edu/~nelson/books.html, and
the book E dwa rd Nelson, Stochastic Quantization, Princeton University Press 1985. See
also L. Fritsche & M. Haug k, A new look at the derivation of the Schrödinger equation from Newtonian mechanics, Annalen der Physik 12, pp. 371–402, 2003. A summary
of Nelson’s approach is also given in F. Ma rkop oulou & L. Sm olin, Quantum theory from quantum gravity, Physical Review D 70, p. 124029, 2004, preprint at arxiv.org/
abs/gr-qc/0311059. See also the important criticism by T. C . Wa llstrom, Inequivalence
between the Schrödinger equation and the Madelung hydrodynamic equation, Physical Review A 49, pp. 1613–1617, 1994, and T. C . Wa llstrom, The stochastic mechanics of the
Pauli equation, Transactions of the American Mathematical Society 318, pp. 749–762, 1990.
A proposed answer is L. Sm olin, Could quantum mechanics be an approximation to another theory?, preprint at arxiv.org/quant-ph/abs/0609109. See also S. K. Srinivasan &
E . C . G. Sudarshan, A direct derivation of the Dirac equation via quaternion measures,
Journal of Physics A 29, pp. 5181–5186, 1996. Cited on page 216.
Julia n S chw inger, Quantum Mechanics – Symbolism of Atomic Measurements,
Springer, 2001. Cited on page 219.
H. Nikolić,How (not) to teach Lorentz covariance of the Dirac equation, European Journal
of Physics 35, p. 035003, 2014, preprint at arxiv.org/abs/1309.7070. Cited on page 219.
For such an attempt, see the proposal by M. R a iner, Resolution of simple singularities
yielding particle symmetries in space-time, Journal of Mathematical Physics 35, pp. 646–
655, 1994. Cited on page 222.
C . S chiller, Deducing the three gauge interactions from the three Reidemeister moves, preprint at arxiv.org/abs/0905.3905. Cited on pages 222 and 224.
G . T. Horow itz & J. Polchinski, Gauge/gravity duality, preprint at arxiv.org/
abs/gr-qc/0602037. Note also the statement in the introduction that a graviton might
be a composite of two spin-1 bosons, which is somewhat reproduced by the strand
model of the graviton. A more concrete approach to gauge–gravity duality is made by
M. va n R aamsd onk, Building up spacetime with quantum entanglement, preprint at
arxiv.org/1005.3035. This approach to gauge–gravity duality is close to that of the strand
model. Cited on page 224.
K. R eidemeister, Elementare Begründung der Knotentheorie, Abhandlungen aus dem
Mathematischen Seminar der Universität Hamburg 5, pp. 24–32, 1926. Cited on pages 224
and 275.
For an attempt to reconcile braided particle models and SU(5) GUT, see D. C a rtin, Braids
as a representation space of SU(5), preprint at arxiv.org/pdf/1506.08067. Cited on page 242.
Sheld on G l ashow, confirmed this to the author in an email; R icha rd Fey nman,
makes the point in Ja mes G leick, Genius: The Life and Science of Richard Feynman, Vintage Books, 1991, page 288 and also in Rob ert C rease & C ha rles Ma nn, The Second
Creation: Makers of the Revolution in Twentieth-Century Physics, Macmillan Publishing,
page 418; Ma rtin Veltm an, writes this in his Nobel Prize Lecture, available on www.
nobel.org. Cited on page 255.
For some of the background on this topic, see F. Wilcz ek & A . Z ee Appearance of gauge
structures in simple dynamical systems, Physical Review Letters 52, pp. 2111–2114, 1984,
A . S ha pere & F. Wilcz ek, Self-propulsion at low Reynolds number, Physical Review
Letters 58, pp. 2051–2054, 1987, and A . Sha pere & F. Wilcz ek, Gauge kinematics of
deformable bodies, American Journal of Physics 57, pp. 514–518, 1989. Cited on page 272.
R . B rit to, F. C achazo, B. Feng & E . Wit ten, Direct proof of tree-level recursion relation in Yang–Mills theory, preprint at arxiv.org/abs/hep-th/0501052. Cited on page 273.
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443
444
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186 D. V. A hluwalia-Kha lilova, Operational indistinguishability of double special re-
187
188
189
191
192
193
195
196
197
198
199
200
201
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lativity from special relativity, Classical and Quantum Gravity 22, pp. 1433–1450, 2005, preprint at arxiv.org/abs/gr-qc/0212128; see also N . Ja fa ri & A . Sha riati, Doubly special
relativity: a new relativity or not?, preprint at arxiv.org/abs/gr-qc/0602075. Cited on page
280.
E . Verlinde, On the origin of gravity and the laws of Newton, preprint at arxiv.org/abs/
1001.0785. Cited on page 281.
G . - L. L esag e, Lucrèce Newtonien, Nouveaux mémoires de l’Académie Royale des
Sciences et Belles Lettres pp. 404–431, 1747, or www3.bbaw.de/bibliothek/digital/
struktur/03-nouv/1782/jpg-0600/00000495.htm. See
also
en.wikipedia.org/wiki/
Le_Sage’s_theory_of_gravitation. In fact, the first to propose the idea of gravitation as
a result of small particles pushing masses was Nicolas Fatio de Duillier in 1688. Cited on
page 283.
G . ’ t Ho oft, Dimensional reduction in quantum gravity, preprint at arxiv.org/abs/gr-qc/
9310026. Many of the ideas of this paper become easier to understand and to argue when
the strand model is used. Cited on page 288.
S. C a rlip, Logarithmic corrections to black hole entropy from the Cardy formula, Classical
and Quantum Gravity 17, pp. 4175–4186, 2000, preprint at arxiv.org/abs/gr-qc/0005017.
Cited on page 289.
D. N . Pag e, The Bekenstein Bound, preprint at arxiv.org/abs/1804.10623. Cited on page
291.
On the limit for angular momentum of black holes, see Ref. 34. Cited on page 291.
F. Ta m burini, C. C uofano, M. D ell a Va lle & R. G ilmozzi, No quantum gravity
signature from the farthest quasars, preprint at arxiv.org/abs/1108.6005. Cited on page 296.
B. P. A b b ot t & al., (LIGO Scientific Collaboration and Virgo Collaboration) Observation of gravitational waves from a binary black hole merger, Physical Review Letters
116, p. 061102, 2016, also available for free download at journals.aps.org/prl/pdf/10.1103/
PhysRevLett.116.061102. See also the website www.ligo.caltech.edu. More about this discovery and its implications is told in volume II of the Motion Mountain series. Cited on page
297.
On torsion, see the excellent review by R . T. Ha m mond, New fields in general relativity,
Contemporary Physics 36, pp. 103–114, 1995. Cited on page 299.
H. Kleinert, & J. Z a a nen, World nematic crystal model of gravity explaining the absence of torsion, Physics Letters A 324, pp. 361–365, 2004. Cited on page 299.
The analogy between the situation around line defects and general relativity is explained in E kkeha rt Kröner, Kontinuumstheorie der Versetzungen und Eigenspannungen, Springer, 1958, These ideas have been taken up and pursued by J. D. E shelby,
B. A . B ilb y, and many others after them. Cited on page 299.
Loop quantum gravity is a vast research field. The complete literature is available at arxiv.
org/archive/gr-qc. Cited on page 299.
G . ’ t Ho oft, Crystalline Gravity, International Journal of Modern Physics A 24,
pp. 3243–3255, 2009, and also G . ’ t Ho oft, A locally finite model of gravity, preprint
at arxiv.org/abs/0804.0328. Cited on page 299.
L. Susskind, New concepts for old black holes, preprint at arxiv.org/abs/1311.3335, and also
reference Ref. 147. Cited on page 300.
M. B ot ta C antcheff, Spacetime geometry as statistic ensemble of strings, preprint at
arxiv.org/abs/1105.3658. Cited on page 300.
b ib liography
445
202 N . A rka ni-Hamed, L. Motl, A . Nicolis & C . Va fa, The string landscape, black
holes and gravity as the weakest force, preprint at arxiv.org/abs/hep-th/0601001. The paper
contradicts the strand model in multiple ways. Cited on page 301.
203 M. va n R aamsd onk, Comments on quantum gravity and entanglement, preprint at
arxiv.org/abs/0907.2939. Cited on page 301.
204 W. H. Zurek & K. S. Thorne, Statistical mechanical origin of the entropy of a rotating,
charged black hole, Physical Review Letters 54, pp. 2171–2175, 1985. Cited on page 301.
205 M. Sha p oshnikov & C . Wet terich, Asymptotic safety of gravity and the Higgs boson
mass, preprint at arxiv.org/abs/0912.0208. Cited on page 302.
206 M. M. Anb er & J. F. D ono ghue, On the running of the gravitational constant, preprint
at arxiv.org/abs/1111.2875. Cited on page 302.
207 The 2016 data about modified Newtonian dynamics is found in S. McG augh, F. L elli
208 C . H. L ineweaver & T. M. Dav is, Misconceptions about the big bang, Scientific Amer-
ican pp. 36–45, March 2005. Cited on page 305.
209 Supernova Sea rch Tea m C oll aboration, A . G . R iess & al., Observational
evidence from supernovae for an accelerating universe and a cosmological constant, Astronomical Journal 116, pp. 1009–1038, 1998, preprint at arxiv.org/abs/astro-ph/9805201.
Cited on page 305.
210 Stephen Haw king & Roger P enrose, The Nature of Space and Time, Princeton Uni211 C . Ba l á zs & I. Sz a pudi, Naturalness of the vacuum energy in holographic theories, pre-
print at arxiv.org/abs/hep-th/0603133. See also C . Ba m bi & F. R . Urban, Natural extension of the generalised uncertainty principle, preprint at arxiv.org/abs/0709.1965. The same
point is made by D. A . E asson, P. H. Fra mpton & G . F. Sm o ot, Entropic accelerating
universe, preprint at arxiv.org/abs/1002.4278. Cited on page 307.
212 W. Fischler & L. Susskind, Holography and Cosmology, preprint at arxiv.org/abs/
hep-th/9806039. Cited on page 307.
213 For a review of recent cosmological data, see D. N . Spergel, R . B ea n, O. D oré,
M. R . Nolta, C . L. B ennett, G . Hinshaw, N . Ja rosik, E . Kom atsu,
L. Pag e, H. V. P eiris, L. Verde, C . Ba rnes, M. Ha lpern, R . S. Hill,
A. Ko g u t, M. L im on, S. S. Meyer, N. Od egard, G. S. Tucker, J. L. Weil and,
E . Woll ack & E . L. Wrig ht, Wilkinson Microwave Anisotropy Probe (WMAP) three
year results: implications for cosmology, preprint at arxiv.org/abs/astro-ph/0603449. Cited
on pages 308 and 309.
214 There is a large body of literature that has explored a time-varying cosmological constant,
especially in relation to holography. An example with many references is L. Xu, J. Lu &
W. L i, Time variable cosmological constants from the age of the universe, preprint at arxiv.
org/abs/0905.4773. Cited on page 308.
215 D. Wiltshire, Gravitational energy and cosmic acceleration, preprint at arxiv.org/abs/
0712.3982 and D. Wiltshire, Dark energy without dark energy, preprint at arxiv.org/abs/
0712.3984. Cited on page 308.
216 The attribution to Voltaire could not be confirmed. Cited on page 313.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
versity Press, 1996. Cited on page 306.
Motion Mountain – The Adventure of Physics
& J. S chombert, The radial acceleration relation in rotationally supported galaxies, preprint at arxiv.org/abs/1609.05917, and in F. L elli, S. S. McG augh, J. M. S chombert
& M. S. Pawlowski, One law to rule them all: the radial acceleration relation of galaxies,
preprint at arxiv.org/abs/1610.08981. Cited on page 303.
446
b ib lio graphy
217 V. C red e & C . A . Mey er, The experimental status of glueballs, Progress in Particle and
218
219
220
222
224
225
226 J. W. Moffat & V. T. Thot, A finite electroweak model without a Higgs particle, preprint
at arxiv.org/abs/0812.1991. The ideas go back to D. Ev ens, J. W. Moffat, G . Kleppe
& R . P. Wo oda rd, Nonlocal regularizations of gauge theories, Physical Review D 43,
pp. 499–519, 1991. For more details on how to introduce non-locality while maintaining
current conservation and unitarity, see G . Kleppe & R . P. Wo odard, Non-local YangMills, Nuclear Physics B 388, pp. 81–112, 1992, preprint at arxiv.org/abs/hep-th/9203016.
For a different approach that postulates no specific origin for the W and Z masses, see
J. W. Moffat, Ultraviolet complete electroweak model without a Higgs particle, preprint
at arxiv.org/abs/1006.1859. Cited on page 331.
227 H. B. Nielsen & P. Olesen, A vortex line model for dual strings, Nuclear Physics B 61,
pp. 45–61, 1973. Cited on pages 334 and 382.
228 B. A nd ersson, G . G ustaf son, G . Ing elman & T. Sj östra nd, Parton fragmentation and string dynamics, Physics Reports 97, pp. 31–145, 1983. Cited on page 334.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
223
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221
Nuclear Physics 63, pp. 74–116, 2009. Cited on page 324.
E . Klem p t & A . Z a itsev, Glueballs, hybrids, multiquarks. Experimental facts versus
QCD inspired concepts, Physics Reports 454, 2007, preprint at arxiv.org/abs/0708.4016.
Cited on pages 324 and 344.
R . V. B uniy & T. W. Kepha rt, A model of glueballs, preprint at arxiv.org/pdf/
hep-ph/0209339; R . V. B uniy & T. W. Kepha rt, Universal energy spectrum of tight
knots and links in physics, preprint at arxiv.org/pdf/hep-ph/0408025; R . V. B uniy &
T. W. Kepha rt, Glueballs and the universal energy spectrum of tight knots and links,
preprint at arxiv.org/pdf/hep-ph/0408027. See also J. P. R a lston, The Bohr atom of
glueballs, preprint at arxiv.org/pdf/hep-ph/0301089. Cited on page 324.
A . J. Niem i, Are glueballs knotted closed strings?, pp. 127–129, in H. Sug a numa,
N . Ishii, M. Oka, H. E nyo, T. Hatsuda, T. Kunihiro & K. Ya z a ki editors,
Color confinement and hadrons in quantum chromodynamics, World Scientific, 2003, preprint at arxiv.org/pdf/hep-th/0312133. See also Y. M. C ho, B. S. Pa rk & P. M. Z ha ng,
New interpretation of Skyrme theory, preprint at arxiv.org/pdf/hep-th/0404181; K. Kond o,
A. Ono, A. Shibata, T. Shinohara & T. Mura kami, Glueball mass from quantized
knot solitons and gauge-invariant gluon mass, Jornal of Physics A 39, pp. 13767–13782, 2006,
preprint at arxiv.org/abs/hep-th/0604006. Cited on pages 324 and 345.
See the one million dollar prize described at www.claymath.org/millennium/
Yang-Mills_Theory. Cited on page 325.
For a clear review on the topic and the planned experiments, see E . Fiorini, Measurement
of neutrino mass in double beta decay, Europhysics News 38, pp. 30–34, 2007, downloadable
at www.europhysicsnews.org. Cited on page 328.
For example, see the detailed discussion of neutrino properties at pdg.web.cern.ch or, in
print, in Ref. 231. Cited on page 329.
For a possible third approach, see A . F. Nicholson & D. C. Kennedy, Electroweak theory without Higgs bosons, International Journal of Modern Physics A 15, pp. 1497–1519,
2000, preprint at arxiv.org/abs/hep-ph/9706471. Cited on page 330.
M. Veltman, The Higgs system, lecture slides at www.nikhef.nl/pub/theory/
academiclectures/Higgs.pdf . See also his CERN Yellow Report 97-05, Reflections on the
Higgs system, 1997, and the paper H. Veltm an & M. Veltm an, On the possibility of
resonances in longitudinally polarized vector boson scattering, Acta Physics Polonica B 22,
pp. 669–695, 1991. Cited on page 330.
b ib liography
447
229 C . B. Thorn, Subcritical string and large N QCD, preprint at arxiv.org/abs/0809.1085.
230
231
233
234
236
237
238
239
240
241
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
235
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232
Cited on page 334.
A . J. B uchmann & E . M. Henley, Intrinsic quadrupole moment of the nucleon, Physical Review C 63, p. 015202, 2000. Alfons Buchmann also predicts that the quadrupole moment of the other, strange ? = 1/2 octet baryons is positive, and predicts a prolate structure for all of them (private communication). For the decuplet baryons, with ? = 3/2,
the quadrupole moment can often be measured spectroscopically, and is always negative. The four Δ baryons are thus predicted to have a negative intrinsic quadrupole moment and thus an oblate shape. This explained in A . J. B uchmann & E . M. Henley,
Quadrupole moments of baryons, Physical Review D 65, p. 073017, 2002. For recent updates, see A . J. B uchm ann, Charge form factors and nucleon shape, pp. 110–125, in
C . N . Pa pa nicolas & A ron B ernstein editors, Shape of Hadrons Workshop Conference, Athens, Greece, 27-29 April 2006, AIP Conference Proceedings 904. Cited on pages
338 and 343.
C . Patrignani & al., (Particle Data Group), Chinese Physics C 40, p. 100001, 2016, or
pdg.web.cern.ch. Cited on pages 338, 339, 340, 359, 361, 372, 373, 374, 375, 446, and 448.
A review on Regge trajectories and Chew-Frautschi plots is W. D rechsler, Das ReggePol-Modell, Naturwissenschaften 59, pp. 325–336, 1972. See also the short lecture on
courses.washington.edu/phys55x/Physics557_lec11.htm. Cited on page 338.
Kurt G ot tfried & Victor F. Weisskopf, Concepts of Particle Physics, Clarendon
Press, Oxford, 1984. Cited on page 339.
G . ’ t Ho oft, G. Isid ori, L. Ma ia ni, A. D. Polosa & V. R iq uer, A theory of scalar
mesons, Physics Letters B 662, pp. 424–430, 2008, preprint at arxiv.org/abs/0801.2288. However, other researchers, such as arxiv.org/abs/1404.5673, argue against the tetraquark interpretation. The issue is not closed. Cited on page 344.
M. Karliner, Doubly heavy tetraquarks and baryons, preprint at arxiv.org/abs/1401.4058.
Cited on page 347.
J. Viro & O. Viro, Configurations of skew lines, Leningrad Mathematical Journal 1,
pp. 1027–1050, 1990, and updated preprint at arxiv.org/abs/math.GT/0611374. Cited on page
347.
W. Thom son, On vortex motion, Transactions of the Royal Society in Edinburgh pp. 217–
260, 1868. This famous paper stimulated much work on knot theory. Cited on page 347.
H. Jehle, Flux quantization and particle physics, Physical Review D 6, pp. 441–457, 1972,
and H. Jehle, Flux quantization and fractional charge of quarks, Physical Review D 6,
pp. 2147–2177, 1975. Cited on page 347.
T. R . Mong an, A holographic charged preon model, preprint at arxiv.org/abs/0801.3670.
Cited on page 347.
The arguments can be found in A . H. C ha mseddine, A . C onnes & V. Mukha nov,
Geometry and the quantum: basics, preprint at arxiv.org/abs/1411.0977 and in
A . H . C ha m seddine & A . C onnes, Why the standard model, Journal of Geometry
and Physics 58, pp. 38–47, 2008, preprint at arxiv.org/abs/0706.3688. Cited on page 348.
Jacob’s rings are shown, for example, in the animation on www.prestidigitascience.fr/index.
php?page=anneaux-de-jacob. They are already published in the book by Tom Tit, La science amusante, 1870, and the images were reprinted the popular science books by Edi Lammers, and, almost a century later on, even in the mathematics column and in one of the
books by Martin Gardner. See also www.lhup.edu/~dsimanek/scenario/toytrick.htm. Cited
on page 351.
448
b ib lio graphy
242 R . B oug hez al, J. B. Tausk & J. J. va n d er B ij, Three-loop electroweak corrections to
243
244
245
246
248
249
250
252
253
254
255
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
251
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247
the W-boson mass and sin2 ?eff in the large Higgs mass limit, Nuclear Physics B 725, pp. 3–
14, 2005, preprint at arxiv.org/abs/hep-ph/0504092. Cited on page 359.
The topic of the g-factor of the W boson and of charged fermions is covered in the delightful
paper by Ba rry R . Holstein, How large is the ‘‘natural’’ magnetic moment?, American
Journal of Physics 74, pp. 1104–1111, 2006, preprint at arxiv.org/abs/hep-ph/0607187. Cited
on page 361.
The calculations have been performed in August 2016 by Eric Rawdon. Cited on pages 359
and 361.
The calculations have been performed by Eric Rawdon and Maria Fisher. Cited on page
363.
The quark masses at Planck energy are due to a private communication by Xing Zhizhong and Zhou Shun. They are calculated following the method presented in Quark
mass hierarchy and flavor mixing puzzles, preprint at arxiv.org/abs/1411.2713 and Z hiz hong X ing, He Z ha ng & Shun Z hou, Updated values of running quark and lepton
masses, preprint at arxiv.org/abs/0712.1419. Cited on page 363.
See H. Fritzsch, A . D. Öz er, A scaling law for quark masses, preprint at arxiv.org/abs/
hep-ph/0407308. Cited on page 363.
K. A . Meissner & H. Nicol ai, Neutrinos, axions and conformal symmetry, preprint at
arxiv.org/abs/0803.2814. Cited on pages 366 and 367.
M. Sha p oshnikov, Is there a new physics between electroweak and Planck scale?, preprint
at arxiv.org/abs/0708.3550. Cited on page 367.
Y. D iao, C . E rnst, A . Por & U. Z ieg ler, The roplength of knots are almost linear in
terms of their crossing numbers, preprint at arxiv.org/abs/0912.3282. No citations.
H. Fritzsch & Z. - Z. X ing, Lepton mass hierarchy and neutrino mixing, preprint at
arxiv.org/abs/hep-ph/0601104 Cited on page 374.
The effects of neutrino mixing, i.e., neutrino oscillations, were measured in numerous
experiments from the 1960s onwards; most important were the experiments at SuperKamiokande in Japan and at the Sudbury Neutrino Observatory in Canada. See Ref. 231.
Cited on page 375.
M. Fukugita & T. Yanagida, Baryogenesis without grand unification, Physics Letters
B 174, pp. 45–47, 1986. Cited on page 376.
J. M. C line, Baryogenesis, preprint at arxiv.org/abs/hep-ph/0609145 or the review by
L. C a netti, M. D rew es & M. Sha p oshnikov, Matter and Antimatter in the Universe, preprint at arxiv.org/abs/1204.4186. They explain the arguments that the standard
model with its CKM-CP violation is not sufficient to explain baryogenesis. The opposite view, by the same authors, is found in L. C a net ti, M. D rewes, T. Frossard
& M. Sha p oshnikov, Dark matter, baryogenesis and neutrino oscillations from right
handed neutrinos, preprint at arxiv.org/abs/1208.4607; another opposing view is found in
T. B rauner, CP violation and electroweak baryogenesis in the Standard Model, EPJ Web
of Conferences 0̌ 70/ p. 00078, 2014. Cited on page 376.
Several claims that the coupling constants changed with the age of the universe have
appeared in the literature. The first claim was by J. K. Web b, V. V. Fl a m baum,
C . W. C hurchill, M. J. D rinkwater & J. D. Ba rrow, A search for time variation
of the fine structure constant, Physical Review Letters 82, pp. 884–887, 1999, preprint at
arxiv.org/abs/astro-ph/9803165. None of these claims has been confirmed by subsequent
measurements. Cited on page 380.
b ib liography
449
256 P. P iera nski, S. P rz y byl & A . Stasia k, Tight open knots, European Physical Journal
E 6, pp. 123–128, 2001, preprint at arxiv.org/abs/physics/0103016. No citations.
257 That tight tangles correlate with random tangles was first shown by V. Katritch,
J. B ed nar, D. Michoud, R . G . Sha rein, J. D ub o chet & A . Stasia k, Geometry and physics of knots, Nature 384, pp. 142–145, 1996. It was confirmed by
E . J. Ja nse va n R ensburg, E . Orl a ndini, D. W. Sum ners, M. C . Tesi &
S. G. Whit tington, The writhe of knots in the cubic lattice, Journal of Knot Theory
and its Ramifications 6, pp. 31–44, 1997. No citations.
258 The quasi-quantization of writhe was discovered by V. Katritch, J. B ed na r,
259 The probabilities for random orientations have been calculated with the help of Tyler Spaeth
Wr i t h e
2
Crossing number
3
0
4
0.00046
5
0
6
0
7
0
8
0
9
0
10
0
11
0
12
0
13
0
3
4
5
0.485
0
0.045
0
0.00022
0
0.000007
0
0
0
0
0
0.392
0
0.076
0
0.00011
0
0.000002
0
0
0
0
0
0
0
0
0
0
0
0.000004
0
0.000002
These are the probabilities of knot orientations with a given writhe and crossing number for
the tight open trefoil knot 31 . The smaller numbers are expected to be calculation artefacts.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
and Ronan Lamy, using ideal knot shapes provided by Jason Cantarella. The calculations
yield the following relative sizes of regions with different crossing numbers and writhes for
the open trefoil and the open figure-eight knot.
Motion Mountain – The Adventure of Physics
D. Michoud, R . G . Sha rein, J. D ub o chet & A . Stasiak, Geometry and physics of knots, Nature 384, pp. 142–145, 1996. See also P. P iera nski, In search of ideal
knots, pp. 20–41, and A . Stasiak, J. D ub o chet, V. Katritch & P. P iera nski,
Ideal knots and their relation to the physics of knots, pp. 1–19, both found in A . Stasiak,
V. Katritch & L. H. Kauffman, editors, Ideal Knots, World Scientific, 1998. Most
pedagogic is P. P iera nski & S. P rz y byl, Quasi-quantization of writhe in ideal knots,
European Physical Journal E 6, pp. 117–121, 2001, preprint at arxiv.org/abs/physics/0102067.
See also C . C erf & A . Stasiak, Linear behavior of the writhe versus the number of crossings in rational knots and links, pp. 111–126, in M. I. Monast yrsky editor, Topology
in Molecular Biology, 2007. The quasi-quantization, and in particular the lack of precise
quantization, was confirmed in 2010 by T. A shton, J. C a ntarella, M. P iatek &
E . R aw d on, private communication. No citations.
450
b ib lio graphy
Wr i t h e
-1
1
0.561
0
0.296
0
0.083
0
0.043
0
0.013
0
0.00008
0
0.000005
0
0.00083
0
0.00065
0
0.00038
0
0.0001
0
0.00004
0
0
0
260 The poster on www.physicsoverflow.org referred to J. P. L estone, Physics based calcula-
tion of the fine structure constant, preprint at arxiv.org/abs/physics/0703151. The preprint
has never been published. Cited on page 385.
261 For a highly questionable, but still intriguing argument based on black hole thermodynam-
ics that claims to deduce the limit ? > ln 3/48π ≈ 1/137.26, see S. Hod, Gravitation, thermodynamics, and the fine-structure constant, International Journal of Modern Physics D
19, pp. 2319–2323, 2010. It might well be that similar or other arguments based on textbook
physics will yield more convincing or even better limits in the future. Cited on page 385.
262 V. A rnold, Topological Invariants of Plane Curves and Caustics, American Mathematical
Society, 1994. Cited on page 391.
263 See M. Pospelov & A . R itz, Electric dipole moments as probes of new physics, preprint
at arxiv.org/abs/hep-ph/0504231. Cited on page 392.
264 D. Hilbert, Über das Unendliche, Mathematische Annalen 95, pp. 161–190, 1925. Cited
on page 401.
265 The Book of Twenty-four Philosophers, c. 1200, is attributed to the god Hermes Trismegistos,
Page 423
but was actually written in the middle ages. The text can be found in F. Hud ry, ed., Liber
viginti quattuor philosophorum, Turnholt, 1997, in the series Corpus Christianorum, Continuatio Mediaevalis, CXLIII a, tome III, part 1, of the Hermes Latinus edition project headed
by P. Lucentini. There is a Spinozian cheat in the quote: instead of ‘nature’, the original says
‘god’. The reason why this substitution is applicable is given above. Cited on page 406.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
These are the probabilities of knot orientations with a given writhe and crossing number for
the tight open figure-eight knot 41 . The limits to the precision of the calculation are clearly
noticeable.
The lack of precision is due to the lack of precision of the available tight knot shapes. The
probabilities for random changes in orientation are then deduced from the values in these
tables. Because the knots are tight, it is a good approximation to assume that Reidemeister
I and Reidemeister II moves can be distinguished without ambiguity – in contrast to the
loose knot case, where this is not possible – from the writhe and crossing numbers of the
start and end orientation. No citations.
Motion Mountain – The Adventure of Physics
Crossing number
4
0
5
0.001
6
0
7
0.0004
8
0
9
0.00038
10
0
11
0.000087
12
0
13
0
14
0
15
0
16
0
0
b ib liography
451
266 As a disappointing example, see G illes D eleuze, Le Pli – Leibniz et le baroque, Les Edi-
tions de Minuit, 1988. In this unintelligible, completely crazy book, the author pretends to
investigate the implications of the idea that the fold (in French ‘le pli’) is the basic entity of
matter and ‘soul’. Cited on page 408.
267 Werner Heisenberg, Der Teil und das Ganze, Piper, 1969. The text shows well how
boring the personal philosophy of an important physicist can be. Cited on page 409.
268 John Barrow wrote to the author saying that he might indeed have been the first to have
Motion Mountain – The Adventure of Physics
used the T-shirt image, in his 1988 Gifford Lectures at Glasgow that were a precursor to his
book John D. Ba rrow, Theories of Everything: The Quest for Ultimate Explanation, 1991.
He added that one can never be sure, though. Cited on page 411.
269 R ené D esca rtes, Discours de la méthode, 1637. He used and discussed the sentence
again in his Méditations métaphysiques 1641, and in his Les principes de la philosophie 1644.
These books influenced many thinkers in the subsequent centuries. Cited on page 414.
270 D. D. Kelly, Sleep and dreaming, in Principles of Neural Science, Elsevier, New York, 1991.
The paper summarises experiments made on numerous humans and shows that even during dreams, people’s estimate of time duration corresponds to that measured by clocks.
Cited on page 414.
271 Astrid Lindgren said this in 1977, in her speech at the fiftieth anniversary of Oetinger Verlag,
her German publisher. The German original is: ‘Alles was an Großem in der Welt geschah,
vollzog sich zuerst in der Phantasie eines Menschen, und wie die Welt von morgen aussehen
wird, hängt in großem Maß von der Einbildungskraft jener ab, die gerade jetzt lesen lernen.’
The statement is found in A strid L ind gren, Deshalb brauchen Kinder Bücher, Oetinger
Almanach Nr. 15, p. 14, 1977. Cited on page 417.
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
CR EDIT S
Acknowled gments
Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
This volume was first published in 2009. No other person helped developing or exploring the
strand model, until in 2014, Sergei Fadeev suggested to rethink the strand models for the W
and Z bosons. His suggestion triggered many improvements, including a much clearer relation
between the three Reidemeister moves and the intermediate gauge bosons of the three gauge
interactions. The results were first included in 2015, in edition 28. The calculation of the fine
structure constant became more involved, but is still possible.
A few other people helped to achieve progress on specific issues or provided encouragement.
In the first half of the text, stimulating discussions in the 1990s with Luca Bombelli helped structuring the chapter on the contradictions between general relativity and quantum theory, as well
as the chapter on the difference between vacuum and matter. In the years up to 2005, stimulating
discussions with Saverio Pascazio, Corrado Massa and especially Steven Carlip helped shaping
the chapter on limit values.
The second half of the text, on the strand model, owes much to Louis Kauffman. The ideas
found in his books and in his papers inspired the ideas of this text long before we met and exchanged mails. His papers – available on www2.math.uic.edu/~kauffman – and his books are all
worth reading; among them is the fascinating paper Knot Logic and the wonderful book Knots
and Physics, World Scientific, 1991. His ideas on knots, on quantum theory, on measurement, on
particle physics, on set theory and on foundational issues convinced me that strands are a promising direction in the search for a final theory. The breadth of Louis’s knowledge and interests,
the depth of his passion and his warm humanity are exemplary.
I thank Eric Rawdon and Maria Fisher for their ropelength calculations. I also thank Claus
Ernst, Andrzej Stasiak, Ralf Metzler and Jason Cantarella for their input and the fruitful discussions we had.
Hans Aschauer, Roland Netz, Gerrit Bauer, Stephan Schiller, Richard Hoffmann, Axel Schenzle, Reinhard Winterhoff, Alden Mead, Franca Jones-Clerici, Damoon Saghian, Frank Sweetser,
Franz Aichinger, Marcus Platzer, Miles Mutka, and a few people who want to remain anonymous
provided valuable help. My parents, Isabella and Peter Schiller, strongly supported the project.
I thank my mathematics and physics teachers in secondary school, Helmut Wunderling, for the
fire he has nurtured inside me.
The typesetting and book design is due to the professional consulting of Ulrich Dirr. The
typography was much improved with the help of Johannes Küster and his Minion Math font.
The design of the book and its website also owe much to the suggestions and support of my wife
Britta.
From 2007 to 2011, the electronic edition and distribution of the Motion Mountain text was
generously supported by the Klaus Tschira Foundation. I also thank the lawmakers and the taxpayers in Germany, who, in contrast to most other countries in the world, allow residents to use
the local university libraries.
cred its
453
Film credits
Image credits
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
The photograph of the east side of the Langtang Lirung peak in the Nepalese Himalayas, shown
on the front cover, is courtesy and copyright by Kevin Hite and found on his blog thegettingthere.
com. The photograph of the Ultra Deep Field project on page 16 is courtesy of NASA. The drawing by Maurits Escher on page 62 is copyright by the M.C. Escher Heirs, c/o Cordon Art, Baarn,
The Netherlands, who kindly gave permission for its use. The passport photograph of Andrei Sakharov on page 78 is copyright by himself. The photograph of an apheresis machine on
page 179 is courtesy of Wikimedia. The drawing by Peter Battey-Pratt and Thomas Racey of the
belt trick on page 213, taken from Ref. 171, is courtesy and copyright by Springer Verlag. The
graph of the running coupling constants on page 378 is courtesy and copyright by Wim de Boer
and taken from his home page at www-ekp.physik.uni-karlsruhe.de/~deboer. On page 410, the
photograph of Cerro Torre is copyright and courtesy of Davide Brighenti, and found on Wikimedia; the photograph of the green hill on the same page is copyright and courtesy of Myriam70,
and found on her site www.flickr.com/photos/myriam70. The photograph on the back cover, of
a basilisk running over water, is courtesy and copyright by the Belgian group TERRA vzw and
found on their website www.terravzw.org. All drawings are copyright by Christoph Schiller.
Motion Mountain – The Adventure of Physics
The animations of the belt trick on page 176 are copyright and courtesy of Greg Egan; they can be
found on his website www.gregegan.net/APPLETS/21/21.html. I am also grateful to Greg Egan
for expanding his applet so as to show a second option out of the many possible ones for the belt
trick.
The beautiful animation of the belt trick on page 177 and the wonderful and so-far unique
animation of the fermion exchange on page 182 are copyright and courtesy of Antonio Martos.
He made them for this text. They can be found at vimeo.com/62228139 and vimeo.com/62143283.
The animation of the rotating blue ball attached to a sheet on page 178 was made for this book
and is part of the software found at www.ariwatch.com/VS/Algorithms/DiracStringTrick.htm.
The colourful animations of the belt trick with many tails on page 179 and page 180 are copyright and courtesy of Jason Hise at www.entropygames.net, and were made for this text and for
the Wikimedia Commons website.
The film of the chain ring trick on page 350 is copyright and courtesy of Franz Aichinger.
The animations and the film of the falling chain ring trick were included into the pdf file with
the help of a copy of the iShowU software sponsored by Neil Clayton from www.shinywhitebox.
com.
NAME INDEX
A
Abdall a
on particles as Moebius
bands 347
C
Cabibbo, Nicola 372
Cabrera, R. 442
Cachazo, F. 443
Cadavid, A.C. 439
Canetti, L. 448
Cantarella, Jason 449, 452
Carlip, Steven 440, 444, 452
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
B
Balachandran, A.P. 432
Balázs, C. 445
Bambi, C. 445
Barnes, C. 445
Baron, J. 435
Barrow, J.D. 448
Barrow, John D. 451
Bateman, H. 429
Battey-Pratt, Peter 442, 453
Bauer, Gerrit 452
Baylis, W.E. 442
Bean, R. 445
Bednar, J. 449
Beenakker, C.W.J. 431
Beig, R. 430
Bekenstein, Jacob 431, 433
on the entropy bound 131
Bennett, C.L. 445
Berlin, Isaiah 138
Bernreuther, W. 434
Bernstein, Aron 447
Berry, M.V. 438
Besso, Michele 68
Bianchi, E. 432
Bianco, C.L. 431
Bij, J.J. van der 448
Bilby, B.A. 444
Bilson-Thompson, Sundance
439
on particles as triple
ribbons 347
Bimonte, G. 432
Blair, D.G. 430
Blandford, R. 429
Boer, Wim de 378, 453
Bohm, David 437, 438, 441,
442
on entanglement 205
on wholeness 106
Bohr, Niels 429
on minimum action 28
on thermodynamic
indeterminacy 30
Bombelli, Luca 300, 434, 438,
440, 452
Bonner, Yelena 78
Botta Cantcheff, Marcelo 444
on fluctuating strings 300
Boughezal, R. 448
Bousso, R. 431
Brauner, T. 448
Brighenti, Davide 410, 453
Brightwell, G. 434
Britto, R. 443
Bronshtein, Matvei 8
Brown, Stuart 429
Buchmann, A.J. 447
Buchmann, Alfons 447
Buniy, R.V. 446
Busch, Wilhelm 395
Byrnes, J. 442
Motion Mountain – The Adventure of Physics
A
Abbott, B.P. 444
Abdalla, M.C.B. 437
Abdo, A.A. 435
Abraham, A. 442
Adams, Douglas 106
Adler, R.J. 433
Ahluwalia, D.V. 433, 444
Aichinger, Franz 350, 452, 453
Akama, K. 435
Alexandrov, P.S. 437
Ali, A. 436
Allen, Woody 414
Aloisio, R. 435
Alvarez, E. 439
Amati, D. 433
Amelino-Camelia, G. 431, 435,
436
Anaxagoras of Clazimenes
on unification 407
Anber, M.M. 445
Andersson, B. 446
Antonio Martos 182
Argyres, P.C. 437
Aristotle 166, 168, 437
on learning 418
on points 121
on vacuum 83
Arkani-Hamed, N. 445
Arnold, V. 450
Aschauer, Hans 452
Ashtekar, A. 431
Ashton, T. 449
Aspect, A. 441
Aspinwall, P. 433
Augustine of Hippo 423
Avrin, Jack 439
na m e ind ex
C
Cartin
E
Easson, D.A. 445
Eddington, Arthur 438
on particle number 127
Egan, Greg 176, 441, 453
Ehlers, Jürgen 432
on point particles 58
Ehrenfest, Paul
on spinors 198
Ehrenreich, H. 431
Einstein, Albert
last published words 86
on continuity 86
on dropping the
continuum 68, 69, 86
on gods 428
on his deathbed 38
on mathematics 107
on modifying general
relativity 163
on thinking 56
on ultimate entities 69
on unification 22
Ellis, G.F.R. 436
Ellis, J. 436
Enyo, H. 446
Ernst, Claus 230, 448, 452
Escher, Maurits 453
Heirs 62
illustrating circularity 61
Eshelby, J.D. 444
Euclid
on points 72
Evens, D. 446
F
Facchi, P. 437
Fadeev, Sergei 251, 411, 452
Fatio de Duillier, Nicolas 444
Faust 110
Feng, B. 443
Feynman, Richard 255, 432,
440, 441, 443
on many-particle wave
functions 204
Finkelstein, David 74, 299,
434, 440
Finkelstein, Robert 439
on fermions as knots 347
Fiorini, E. 446
Fischler, W. 445
Fisher, Maria 363, 448, 452
Flambaum, V.V. 448
Flint, H.T. 434
Frampton, P.H. 445
Fredenhagen, K. 433
Fredriksson, S. 434
Frenkel, J. 442
Friedberg, R. 434
Friedman, J.L. 432
Fritsche, L. 443
Fritzsch, H. 434, 448
Frossard, T. 448
Fukugita, M. 448
Fushchich, V.I. 442
G
Gadelka, A.L. 437
Gaessler, W. 435
Galante, A. 435
Galindo, A. 442
Garay, L. 433, 436
Gardner, Martin 447
Garret, Don 436
Gehrels, N. 429
Gell-Mann, Murray
on strings 142
Gennes, Pierre-Gilles de 438
Gibbons, Gary 429, 430
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
D
Dalibard, J. 441
Dam, H. van 93, 431, 435, 436
Dante Alighieri
on the basic knot 404
Das, A. 434
Davis, T.M. 445
Dehmelt, Hans 435
Deleuze, Gilles 451
Della Valle, M. 444
Democritus
on learning 418
on particles and vacuum
127
Descartes, René 414, 451
Deutsch, David 161, 440
on explanations 164
DeWitt, B.S. 432
DeWitt, C. 432
Diao, Y. 448
Diner, S. 432
Dirac, Paul 214
on unification 22
Dirr, Ulrich 452
Dis, J. van Lith-van 429
DiSessa, A. 429
Donoghue, J.F. 445
Doplicher, S. 433
Doran, C. 442
Doré, O. 445
Douglas, M.R. 440
Drechsler, W. 447
Drewes, M. 448
Drinkwater, M.J. 448
Dubochet, J. 449
Dumont, Jean-Paul 437
Dyson, Freeman 352
Dällenbach, Werner 69
Dürrenmatt, Friedrich 428
Motion Mountain – The Adventure of Physics
on fluctuating lines 161,
300
Cartin, D. 443
Cerf, C. 449
Challinor, A. 442
Chamseddine, A.H. 447
Chen, B. 431
Cho, Y.M. 446
Christ, N.H. 434
Christiansen, W.A. 435
Churchill, C.W. 448
Ciafaloni, M. 433
Cicero, Marcus Tullius
life 105
Clay Mathematics Institute
325
Clayton, Neil 453
Cline, J.M. 448
CODATA 428
Conde, J. 439
Connes, A. 447
Connes, Alain 348
Cordon Art 453
Coule, D.H. 435
Crease, Robert 443
Crede, V. 446
Cuofano, C. 444
455
456
G
Gibbs
I
Illy, József 434
Ingelman, G. 446
Inverno, Ray d’ 430
Ishii, N. 446
Isidori, G. 447
J
Jacobson, T. 33, 430
Jaekel, M.-T. 430, 433
Jafari, N. 444
Jammer, Max 429
Janssen, Michel 434
Jarlskog, Cecilia 374
Jarosik, N. 445
Jauch, W. 83, 435
Jehle, Herbert 447
on particles as knots 347
Jerven, Walter 416
Johnson, Samuel 428
Jones-Clerici, Franca 452
Ju, L. 430
K
Kaluza, Theodor
on unification 22
Kant, Immanuel 168, 441
Karliner, M. 437, 447
Karliner, Marek 347
Karolyhazy, F. 436
Katritch, V. 449
Katsuura, K. 435
Kauffman, Lou 214
Kauffman, Louis 439, 441,
449, 452
on commutation relations
207
Kelly, D.D. 451
Kempf, Achim 430, 433
Kennard, E.H. 432
Kennedy, D.C. 446
Kephart, T.W. 446
Keselica, D. 442
Klaus Tschira Foundation 452
Klebanov, I. 437
Kleinert, Hagen 299, 444
Klempt, E. 446
Kleppe, G. 446
Knox, A.J. 434
Kochen, S. 441
Kogut, A. 445
Komatsu, E. 445
Kondo, K. 446
Konishi, K. 433
Kostro, L. 430
Kostro, Ludwik 429
Koul, R.K. 438
Kovtun, P. 431
Kramer, M. 430
Kreimer, Dirk 439
on knots in QED 139
Kronecker, Leopold
life 107
Kröner, Ekkehart 299, 444
Kunihiro, T. 446
Küster, Johannes 452
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
H
Hackett, J. 439
Halpern, M. 445
Hammond, R.T. 444
Harari, H. 434
Hartle, J.B. 438
Hatsuda, T. 446
Hattori, T. 435
Haugk, M. 443
Hawking, Stephen 433, 436,
438, 445
Heath, T. 434
Hegel, Friedrich 142
Heisenberg, Werner 432, 451
on symmetry 409
on thermodynamic
indeterminacy 30
on unification 22
Hellund, E.J. 434
Henley, E.M. 447
Henson, J. 440
Hermes Trismegistos 406, 450
Hernandez, L. 439
Hertz, Heinrich
on everything 407
Hestenes, D. 442
Higgs, Peter 331
Hilbert, David 450
famous mathematical
problems 108
his credo 143
on infinity 401
on unification 22
Hilborn, R.C. 438
Hildebrandt, Dieter
life 218
Hiley, B.J. 437
Hill, E.L. 434
Hill, R.S. 445
Hillion, P. 442
Hillman, L. 435
Hinshaw, G. 445
Hise, Jason 179, 180, 453
Hite, Kevin 453
Hod, S. 450
Hoffer, Eric 66
Hoffmann, Richard 452
Hohm, U. 431
Holstein, Barry R. 448
Honner, John 429
’t Hooft, Gerard 74, 299, 434,
436, 444, 447
Hooke, Robert 282
Horowitz, G.T. 438, 440, 443
Hudry, F. 450
Motion Mountain – The Adventure of Physics
Gibbs, Phil 433, 438
on event symmetry 84, 128
Gill, S. 442
Gilmozzi, R. 444
Glashow, Sheldon 255, 440,
443
Gleick, James 443
Goethe, Johann Wolfgang von
on searching 110
Gonzalez, O. 441
Gottfried, Kurt 447
Graf, A.B.A. 441
Green, M.B. 439
Greene, Brian 431, 440
on popularization 52
on superstrings 140
Gregori, Andrea 438
on particle mass 136
Gregory, R. 434
Grillo, A.F. 435
Gross, D.J. 433
Gunzig, E. 432
Gustafson, G. 446
na m e ind ex
na m e ind ex
L
L ammers
N
Nanopoulos, D.V. 436
NASA 17, 453
Nelson, Edward 442, 443
Netz, Roland 452
Newton, Isaac 282
Ng Sze Kui 347, 439
Ng, Y.J. 93, 431, 435, 436
Nicholson, A.F. 446
Nicolai, H. 448
Nicolis, A. 445
Nielsen, H.B. 446
Niemi, A.J. 446
Nietzsche, Friedrich
on walking 149
Nikitin, A.G. 442
Nikolić, H. 443
Nikolić, Hrvoje 219
Nolta, M.R. 445
O
Occam, William of 127
Odegard, N. 445
Ohanian, Hans 431
Oka, M. 446
Olesen, P. 446
Olive, D. 437
Ono, A. 446
Oppenheimer, J. 433
Orlandini, E. 449
Özer, A.D. 448
P
Padmanabhan, T. 432
Paffuti, G. 433
Page, D.N. 444
Page, L. 445
Papanicolas, C.N. 447
Park, B.S. 446
Parmenides 128
Pascazio, Saverio 437, 452
Pati, J.C. 434
Patrignani, C. 447
Pauli, Wolfgang
on gauge theory 248
Pawlowski, M.S. 445
Peiris, H.V. 445
Penrose, Roger 445
Peres, A. 434
Phaedrus 418
Piatek, M. 449
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
M
Maddocks, J.H. 441
Maddox, John 438
life 438
Maggiore, M. 433
Magueijo, J. 435
Maiani, L. 447
Maimonides 64
Majid, S. 438
Major, S.A. 439
Maldacena, J. 439
Mandelbaum, G. 434
Mann, Charles 443
Markopoulou, F. 439, 443
Marmo, G. 432
Marsden, Jerry 272
Martos, Antonio 177, 453
Marx, Groucho
on principles 38
Massa, Corrado 429, 430, 452
Mavromatos, N.E. 436, 438
McGaugh, S. 445
McGaugh, S.S. 445
Mead, Alden 272, 430, 452
Meissner, K.A. 448
Mende, P.F. 433, 438
Mende, Paul
on extension checks 138
Metzler, Ralf 452
Meyer, C.A. 446
Meyer, D. 434, 440
Meyer, S.S. 445
Michoud, D. 449
Mills, Robert 248
Misner, C.W. 430, 432
Moffat, J.W. 446
Monastyrsky, M.I. 449
Mongan, Tom 447
on particles as tangles 347
Montonen, C. 437
Motl, L. 445
Mukhanov, V. 447
Murakami, T. 446
Mutka, Miles 452
Myriam70 410, 453
Méndez, F. 435
Motion Mountain – The Adventure of Physics
L
Lammers, Edi 447
Lamy, Ronan 449
Lange, B. 430
Lao Tse
on motion 408, 416
Lasenby, A. 442
Laughlin, Robert 428
Laërtius, Diogenes 438
Lee Tsung Dao 248, 434
Lee, J. 434, 438, 440
Leibniz, Gottfried Wilhelm
408, 437
on parts 108
on relations in nature 306
Leighton, Robert B. 432
Lelli, F. 445
Lenin (Vladimir Ilyich
Ulyanov) 82
Lerner, L. 442
Lesage, Georges-Louis 444
on universal gravitation
283
Lestone, J.P. 450
Li, W. 445
Li, Y-Q. 431
Lichtenberg, Georg Christoph
on truth 25
Lieu, R. 435
Limon, M. 445
Lindgren, Astrid 417, 451
Lineweaver, C.H. 445
Lloyd, Seth 437
on information 107
Loinger, A. 442
Loll, R. 433
Lomonaco, S.J. 441
Loren, Sophia
on everything 355
Lorentz, Hendrik Antoon 429
on the speed limit 27
Lu, J. 445
Lucentini, P. 450
Lucrece, in full Titus
Lucretius Carus 437
Luzio, E. 435
Lévy-Leblond, J.-M. 441
457
458
P
Pieranski
S
Sabbata, V. de 430
Sabbata, Venzo de 429
Sagan, Carl 436
Saghian, Damoon 452
Sakar, S. 436
Sakharov, Andrei 430, 431
life 78
on matter constituents 121
on maximum particle
mass 39
on minimum length 43
portrait 78
Salam, Abdus 434
on unification 22
Salecker, H. 114, 433
Salogub, V.A. 442
Sanchez, N.G. 438
Sands, Matthew 432
Santamato, E. 442
Santiago, D.I. 433
Schaefer, B.E. 435
Schenzle, Axel 452
Schild, A. 434
Schiller, Britta 452
Schiller, Christoph 429, 430,
432, 443, 453
Schiller, Isabella 452
Schiller, Peter 452
Schiller, R. 441
Schiller, Stephan 452
Schoen, R.M. 430
Schombert, J. 445
Schombert, J.M. 445
Schrödinger, Erwin 442
on thinking 116
Schulmann, Robert 434
Schulz, Charles 135
Schwarz, J.H. 437, 439
Schwinger, Julian 238, 443
Schön, M. 432
Sen, A. 439
Seneca, Lucius Annaeus 412
Shakespeare, William 124, 139,
400
Shalyt-Margolin, A.E. 429
Shapere, Alfred 272, 443
Shaposhnikov, M. 445, 448
Shaposhnikov, Mikhail 302
Sharein, R.G. 449
Shariati, A. 444
Shibata, A. 446
Shinohara, T. 446
Shupe, M.A. 434
Simoni, A. 432
Simplicius 437
Simplicius of Cilicia 124
Sivaram, C. 429, 430
Sjöstrand, T. 446
Slavnov, A.A. 439
Smolin, L. 430, 433, 435, 436,
439, 443
Smoot, G.F. 445
Snyder, H.S. 434
Socrates 418
Son, D.T. 431
Sorabji, R. 434
Sorkin, R.D. 432, 434, 438,
440
Spaeth, Tyler 449
Sparzani, A. 442
Specker, E.P. 441
Spergel, D.N. 445
Spinoza, Baruch 408, 436
Springer Verlag 213, 453
Srinivasan, S.K. 443
Stachel, John J. 434
Stanhope, Philip 403
Starinets, A.O. 431
Stasiak, Andrzej 449, 452
Stewart, Ian 434
Stone, Michael 439
Strominger, A. 438
Sudarshan, E.C.G. 443
Suganuma, H. 446
Sumners, D.W. 449
Supernova Search Team
Collaboration 445
Susskind, L. 444
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
R
Raamsdonk, Mark van 301,
443, 445
Racey, Thomas 442, 453
Ragazzoni, R. 435
Rainer, M. 443
Ralston, J.P. 446
Ramsauer, Carl
life 405
Ramsey, N.F. 434
Randjbar-Daemi, S. 436
Rawdon, Eric 363, 448, 449,
452
Raymer, Michael 433
on the indeterminacy
relation 64
Reidemeister, Kurt 443
on knot deformations 224,
275
Renaud, S. 430, 433
Rensburg, E.J. Janse van 449
Reznik, B. 433
Richter, Burton 440
Riemann, Bernhard
on geometry 38
Riess, A.G. 445
Rindler, Wolfgang 430, 431
Riquer, V. 447
Ritz, A. 450
Rivas, Martin 442
Robbins, J.M. 438
Roberts, J.E. 433
Roger, G. 441
Rosen, N. 434
Rosenfeld, L. 103, 436
Rothman, T. 436
Rovelli, C. 432, 433
Ruffini, Remo 431
Rutherford, Ernest 146
Motion Mountain – The Adventure of Physics
Pieranski, Piotr 449
Piran, T. 436
Pittacus 118
Planck, M. 437
Plato 94, 438
on love 418
on nature’s unity 128
Platzer, Marcus 452
Plotinus 423
Polchinski, J. 438, 440, 443
Polosa, A.D. 447
Pontecorvo, Bruno 375
Por, A. 448
Pospelov, M. 450
Preparata, G. 431
Provero, P. 433
Przybyl, S. 449
na m e ind ex
na m e ind ex
Susskind, Leonard 300, 436,
437, 440, 445
on Planck scale scattering
117
Suzuki, M. 434
Sweetser, Frank 452
Szapudi, I. 445
Szilard, Leo 429
on minimum entropy 30
Sánchez del Río, C. 442
S
Susskind
Wiltshire, D. 308, 445
Winterberg, F. 437
Winterhoff, Reinhard 452
Witten, Edward 439, 440, 443
on duality 137
on infinities 140
on unification 22
Woit, Peter 440
Wolf, C. 435
Wolf, E. 437
Wolff, Barbara 434
Wollack, E. 445
Woodard, R.P. 436, 446
Wright, E.L. 445
Wunderling, Helmut 452
Wussing, H. 437
W
Wald, R.M. 435
Wallstrom, T.C. 443
Wan, Y. 439
Wang, Y.-N. 440
Webb, J.K. 448
Weber, G. 438
Weiland, J.L. 445
Weinberg, Steven 103, 428,
432, 440
on unification 22
Weis, A. 434
Weisskopf, Victor F. 447
Weizel, W. 442
Wen, X.-G. 439
Wetterich, C. 445
Wetterich, Christof 302
Wheeler, John A. 157, 159, 160,
296, 430, 432
life 58
on nature’s principles 405
on topology change 58
on unification 22
really big questions 411
Whittington, S.G. 449
Wigner, Eugene 114, 432, 433
Wikimedia 179, 453
Wilczek, Frank 272, 430, 440,
443
Wilde, Oscar 92, 93
life 91
Wiles, Andrew
on research 410
William of Occam 127
X
Xing Zhi-zhong 448
Xing, Z.-Z. 448
Xing, Zhi-zhong 448
Xu, L. 445
Xue, S.-S. 431
Y
Yanagida, T. 448
Yandell, Ben H. 437
Yang Chen Ning 248
Yazaki, K. 446
Z
Zaanen, J. 444
Zaitsev, A. 446
Zee, A. 443
Zeh, H.D. 435
Zemeckis, Robert 436
Zeno of Elea 70, 119, 121, 128,
218
on motion as an illusion
417
on size 124
Zhang, He 448
Zhang, P.M. 446
Zhao, C. 430
Zhou Shun 448
Zhou, Shun 448
Ziegler, U. 448
Zimmerman, E.J. 114, 433
Zurek, W.H. 445
Zwiebach, B. 439
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
U
Uffink, J. 429
Uglum, J. 437
Unruh, W.G. 433, 435
Urban, F.R. 445
V
Vafa, C. 438, 445
Vancea, I.V. 437
Veltman, H. 446
Veltman, Martin 255, 440,
443, 446
Veneziano, G. 433
Verde, L. 445
Verlinde, Erik 444
on universal gravitation
281
Vigier, J.-P. 442
Viro, Julia 347, 447
Viro, Oleg 347, 447
Voltaire
life 313
Motion Mountain – The Adventure of Physics
T
Takabayasi, T. 442
Tamburini, F. 444
Tanaka, K. 434
Tausk, J.B. 448
Taylor, Gareth 177
Taylor, W. 440
Terence, in full Publius
Terentius Afer
life 115
Tesi, M.C. 449
Thales of Miletus 400
Thomas Aquinas 423
Thomas, L.H. 442
Thomson–Kelvin, William
447
on atoms as knotted
vortices 347
Thorn, C.B. 447
Thorne, K.S. 430, 432, 445
Thot, V.T. 446
Tillich, Paul 90
Tino, G.M. 438
Tiomno, J. 441
Tit, Tom 447
Townsend, P.K. 430
Treder, H.-J. 436
Tregubovich, A.Ya. 429
Tschira, Klaus 452
Tucker, G.S. 445
Turatto, M. 435
Turnbull, D. 431
459
na m e ind ex
460
Z
Zwiebach
Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
SUBJECT INDEX
averaging of strands 187
Avrin, Jack
on particles as Moebius
bands 347
axioms
in a final theory 148
in physics 108, 166–169
axion 273
B
background
continuous 154
dependence 148
differs from physical space
283, 285
illustration of 168
independence 169
space 206, 285
space-time 166
Balinese candle dance 176
ball
tethered 368
Banach–Tarski
paradox/theorem 70, 72
band models 139, 165
Barbero–Immirzi parameter
288
bare quantity 237
baryogenesis 376
baryon
density in universe 309
form factor 343
masses 338
number 315
number conservation 251,
267, 315
number limit 310
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
amoeba and nature 129–130,
170
Anaxagoras of Clazimenes
on unification 407
angle
A
weak mixing 360
acceleration
angular frequency
indeterminacy 28
upper limit, or Planck 36
upper limit, or Planck 36
angular momentum
accelerator, Planck 82
limit for black holes 291
accuracy, maximum 94
lower limit and spin 40
action
upper limit 45–46
as fundamental quantity
anomaly issue 140, 348
401
anti-twister mechanism 180
defined with strands 158,
antimatter
208, 209
belt trick and 179
is change 26
indistinguishability from
lower limit 28–29
matter 53, 77–78, 104
no lower limit for virtual
antiparticle see antimatter
particles 40
apheresis machine 180
principle of least 208
illustration of 179
upper limit 45
aphorism, a physical 114
action, quantum of, ℏ
apple counting 107
as lower limit 36
area
from strands 150
lower limit, or Planck 37
ignored by relativity 57–58 argument
lower limit 28–29
encouraging 22
physics and 8
Aristotle
Planck scales and 60–61
on learning 418
quantum theory implied
on points 121
by 18, 28
on vacuum 83
addition of states 189
arrow
Feynman’s rotating 216
ADM mass 103
Zeno’s flying 121, 218
AdS/CFT correspondence 224
Aspect experiment 205
aether
asymptotic safety 302
useless 70, 157
atoms 116
vortices in 347
Motion Mountain – The Adventure of Physics
Numbers
3
omnipresence in the
standard model 353
462
B
bath
41, 42
microstates 288
no microscopic 301, 367
radiation 290
Schwarzschild 31
shape of 292
size limit 35
sphericity of 292
strand definition 285
universe as inverted 305
universe lifetime and 51
upper power limit 42
blood platelets 180
blurring of tangle 185, 194
Bohm, David
on entanglement 205
on wholeness 106
Bohr, Niels
on minimum action 28
on thermodynamic
indeterminacy 30
Bohr–Einstein discussion 29
Boltzmann constant ? 30, 150
physics and 8
book
perfect, on physics 109
boost see Lorentz boost
border of space 153
Bose–Einstein condensates 49
boson
tangle structure quiz 351
bosonization 138
bosons
as radiation particles 18
definition 174
gauge 317
illustration of 183
illustration of exchange 183
in millennium description
18
masses of W and Z 358
none at Planck scales 77
strand model 183
weak gauge 248
weak intermediate 248
Botta Cantcheff, Marcelo
on fluctuating strings 300
bound see limit
boundary
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
spin 1/2 and 174–180
spin and 134
SU(2) and 245–248
torsion and 299
two options 178
wheels and 219
with 96 tails 180
beta decay, neutrinoless
double 328
big bang
creation and 408
distance in time 93
initial conditions and 136
lack of 148
no creation in 425
not a singularity 101
not an event 92
precision and 53
size of 99
strand illustration of 306
strands and the 306–307
Bilson-Thompson, Sundance
on particles as triple
ribbons 347
biology 169
bit 160
black hole
see also Schwarzschild
radius
as size limit 35
as smallest systems 39
cannot have Planck mass
301, 367
charge limit 291
charged 41
clock limits and 64
definition 31
entropy 131, 287–289
strand illustration of 288
entropy limit 47–48
evaporation 42, 46, 290
falling into a 301
information loss 290
limits 290
lower power limit 46
lower temperature limit 49
magnetic field limit and 44
mass 286
maximum force and 31, 32,
Motion Mountain – The Adventure of Physics
number of leptons 328
number of quarks 320
observed number of 19,
101, 162
quadrupole moment 447
Regge trajectories 339
shape 343
spin 335
strand illustration of
342–344
strand model of 299,
341–344
bath
gluon 269
measurement and 200
perfect 84
photon 228
vacuum as a 155
weak boson 246
beauty
in physics 56
of strands 405
symmetry is not 408
beginning of time 92
Bekenstein’s entropy bound
43, 47–48, 291
Bekenstein, Jacob
on the entropy bound 131
beliefs
about unification 22
in finitude 403
Occam and 127
belt trick 264
and Dirac equation 214
antimatter and 179
Dirac equation and
212–217
illustration of 213
fermions and 331
illustration of 175
parity violation and 179,
247
particle mass and 195,
363–365
phase and 196
quantum theory and 220
saving lives 180
space-time symmetries
and 224
sub j ect ind ex
sub j ect ind ex
B
boxes
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
lower limit 67
mass and vacuum 124
particle rotation and 282
quantum effects and
60–61, 106, 216
table with examples 60
upper limit 51
vacuum and 80
computer
nature is not a 403
conditions
initial, of universe 101, 306
confinement 322
conformal invariance 140
conjecture
no avail 304
conjecture, no avail 301
consciousness
and final theory 21
constant
cosmological see
cosmological constant
coupling see coupling
constant
constants
values of fundamental 150
constituents
common, of particles and
space 86
extended 299
fundamental 147
continuity 70, 158
as time average 170
discreteness and 157
lack of 39, 69, 70, 86, 125
of motion 401
of space and time 68
continuum
see continuity
contradictions between
relativity and quantum
theory 57–63
coordinates
fermionic 133
Grassmann 133
core, tangle
deformation of 224
rotation 222
rotation of 224
Motion Mountain – The Adventure of Physics
electric 228
of objects 117
electric, from strands 152
of point particles 118
elementary ?, physics and
of space 100
8
boxes, limits to 116
fractional 44
braid 133
limit for black holes 291
braid symmetry 134
magnetic, no 230
braiding 351
and mass 352
quantization 386
of tails 373, 374
unit, electric 379
of tails, and mass 366
weak 246–248
brain, and circularity 169
illustration of 247
breaking of SU(2) 250
chirality 229, 385
Broglie, de, wavelength
circularity
lower limit 67
contradictions and 111
Bronshtein cube 8, 301
fundamental 166–169, 187
bucket experiment, resolution
in classical physics 59
218
resolution 218
in modern physics 61
C
in physics 109
Cabibbo angle 372
resolution 128
calculations
classical gravitation 35, 280
non-perturbative 236
classicality
perturbative 236
of measurement apparatus
Calugareanu’s theorem 387,
200
421
climbing
capacitors 44
a green hill 410
capacity
Motion Mountain 410
indeterminacy of 44
symbolism of 427
Carlip, Steven
clocks
on fluctuating lines 161,
limits and Planck time
300
64–66
Casimir effect 57
CODATA 428
catechism, catholic 424
cogito ergo sum 414
categories 106
Coleman–Mandula theorem
centre of group 260
273
Cerro Torre 410
collapse of wave function 201
chain
colour charge 269, 270
film of falling ring 350
strand illustration of 322
illustration of falling ring
three types 270
349
colours in nature, origin of 18,
chain ring trick 352
416
challenge
combination, linear 188
classification 9
complex numbers 272
change
compositeness and strand
is action 26
number 345
nature minimizes 26, 401
Compton wavelength
charge
as displacement limit 29
see also electromagnetism
elementarity and 76, 78
conjugation 242
final theory and 147
463
464
C
corpuscules
D
D-branes 170
dance 176
dangers of a final theory 21
Dante Alighieri
on the basic knot 404
dark energy see cosmological
constant, 19, 162, 304, 307
dark matter 18, 304, 309, 353
challenge 309
is conventional matter 353
death 413
decay
neutrino-less double-beta
353
decoherence 202
defects in vacuum 284, 297
definition
circular 167
deformation
gauge groups and 272
of core 224
of tails 224
degrees of freedom
and boundary 140
and entropy 120
and quantum theory 349
and system surface 49
and volume 84
entropy limit and 41
fundamental 149
in universe 49
of space-time 48
delocalization of W and Z
bosons 331, 332
Democritus
on learning 418
on particles and vacuum
127
denseness 70
density limit for black holes
291
desert
high-energy 350, 354
determinism 84, 85, 410
Deutsch, David
on explanations 164
devils 21
diffeomorphism invariance
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
in neutrinos 376
CP-violating phase 373
CPT invariance 351
CPT symmetry 74, 88, 112
not valid 104
creation 106, 107, 425
is impossible 101
cross sections at Planck scales
117
crossing
as simplest tangle 320
density 188
illustration of 185
illustration of definition
152
in knot theory 153
number, signed 420
of strands 152
orientation average 188
position density 187
switch 150
switch as event 152
switch in space-time 151
switch is observable 346
switch, definition 153, 184
switch, illustration of 151
crystal, nematic world 299
crystals and vacuum 36
cube
Bronshtein 8
physics 8
curiosity 7
current
indeterminacy 44
curvature
see also space-time
around spherical masses
292
of curve 419
space, from strands 283,
292
strand illustration of 283
total 328, 370
upper limit, or Planck 37
curve
rotation, of stars in
galaxies 303
unknotted 316
cutting matter 120
Motion Mountain – The Adventure of Physics
thethered 368
corpuscules ultra-mondains
283
cosmic background radiation
patterns in 107
cosmic string 297, 303
not possible 298
strand illustration of 297
cosmological constant
and strands 307
cosmological constant Λ
307–308
as millennium issue 19, 162
cosmological limits and 51
cosmology and 304
cosmology implied by 18
for flat vacuum 210
from thermodynamics 34
general relativity 51
implies cosmology 50
is dark energy 307
minimum length and 43
problem 143
time variation 311
vacuum density and 51
cosmological limit
see also system-dependent
limit
lowest force 51
to observables 45, 51–52
cosmological scales 90
cosmology 304–312
in one statement 50
Coulomb’s inverse square
relation deduced from
strands 230
counting objects 107
coupling constant
calculation of 381–391
comparison of 384
definition 378
electromagnetic, and
Planck limits 43
illustration of running 378
running and the Higgs
boson 382
covering, topological 71
CP problem, strong 267, 273
CP violation 339, 373, 374, 376
sub j ect ind ex
sub j ect ind ex
D
differences
experiment and gravity
298
with strands 199, 200
doubly special relativity 279,
435
dreams 414–416
duality 167
as an argument for
extension 126
between large and small
124, 125
gravity/gauge 224
space-time see space-time
duality
strings and 437
superstrings and 139
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
E
Eddington, Arthur
on particle number 127
Eddington–Finkelstein
coordinates 71
efficiency
of nature 27
Ehlers, Jürgen
on point particles 58
Ehrenfest, Paul
on spinors 198
Einstein’s field equations
see field equations
Einstein’s hole argument 58
Einstein, Albert
last published words 86
on continuity 86
on dropping the
continuum 68, 69, 86
on gods 428
on his deathbed 38
on mathematics 107
on modifying general
relativity 163
on thinking 56
on ultimate entities 69
on unification 22
Einstein–Bohr discussion 29
Einstein–Brillouin–Keller
quantization 29
electric charge quantum
number 378, 386
electric dipole moment 76,
276, 391–392
electric field
illustration of 231
lower limit 50
upper limit 43, 231
electric potential 196
electrodynamics
from strands 226–243
electromagnetic coupling
constant
see fine structure constant
electromagnetic energy
from strands 230
electromagnetism
cosmological limits 50
from strands 226–243
illustration of 229
Planck limits 43–44, 231
electron
?-factor 199, 236, 242
tangle of 327
electroweak interaction 250
elementary charge see fine
structure constant
elementary particle
see also particle
action limit and 28
cannot be point particle 75
cannot have Planck mass
367
definition 39
properties 313, 355
shape of 116, 119, 120
size limit and 28, 29, 35, 39
size of 117
speed limit and 27
upper energy limit 39
upper mass limit 39
upper momentum limit 39
virtual particles and 40
elements of sets
none in nature 53, 105
elongation 132
emotions
beautiful 7
end
of strand 170
energy
Motion Mountain – The Adventure of Physics
71, 74, 125, 130
differences are approximate
127
dimensions
higher 139, 160
higher, and the final
theory 144
no higher 144, 276, 277,
348
none and superstrings 85
none at Planck scales 53, 71
of space undefined 148
orgin of three 206
origin of spatial 206
dinner parties, physics for 26
dipole moment, electric 76
Dirac equation 73, 212
and belt trick 214
explanation 217, 220
from strands 172
from tangles 212–217
ingredients 216
visualizing the 215
Dirac, Paul
on unification 22
discreteness
continuity and 157
none in nature 111, 403
displacement
indeterminacy 29
limit, quantum 28
distance
defined with strands 158
lower limit, or Planck 37
distinction
none in nature 111
divergence
none in the strand model
276
of QED 237
of vacuum energy 210
dogmas about unification 22
domain walls 297, 303
donate
for this free pdf 9
to this book 9
double beta decay,
neutrinoless 328
double-slit
465
466
E
F
Faust 110
featureless
strands are 153
Fermat’s theorem 410
fermion
as matter particle 18
definition 173, 174, 181
exchange and extension
133–134
from strands 174
illustration of 158
illustration of exchange 181
illustration of wave
function 173
in general relativity 58, 302
in millennium description
18
none at Planck scales 77
spin and extension 134–135
fermionic coordinates 133
Feynman diagram
QED, illustration of 235–237
and braiding 352
high-order QED 139
mechanism for 222–278
strands and 235
strong, strand illustration
of 270
weak 253
weak, illustration of 253
weak, strand illustration of
254
Feynman’s rotating arrow 216
Feynman, Richard
on many-particle wave
functions 204
fiction, no science 354
field
electric 230
magnetic 230
without field 159
field equations
deduced from a drawing
294
from maximum force
32–35
from strands 293–295
films
dreams and 415
Hollywood 98
final theory 405, 428
arguments against 20–21
candidates 22, 139, 165,
299, 347
dangers 21
disinformation 428
extension and 143
higher dimensions and
144, 348
how to find it 144
list of testable predictions
24, 395
modification 147
of motion 20
requirements of 146
steps of the search for a 24
supersymmetry and 144
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
definition 59, 153
from strands 152
fundamental 150
fundamental, illustration
of 147
horizon 305
in dreams 414
symmetry 84, 128
exchange
extension and 133, 134
existence and Planck scales
108
exotic manifold 170
experiment
hard challenge 376
explanation
definition of 164
extension
essential for spin 1/2 177
exchange and 133, 134
final theory and 143
importance of 149
in superstrings 139
of constituents 299
spin and 134
tests of 137
unification and 146
extinction 197
extremal identity 112–113
Motion Mountain – The Adventure of Physics
dark see cosmological
constant
density, critical 308
electromagnetic, from
strands 230
from strands 192, 208, 284
indeterminacy 30, 35, 64
kinetic 208
no region with negative
298
of horizons 286
potential 208
quanton definition 192
speed 27
energy–momentumsystem-dependent limits
and 45
upper limit for elementary
particles 39, 82
energy–momentum tensor 34
ensembles 106
entangled state 202–206
entanglement 202–206, 219
quantum gravity and 301
entropy
at Planck scales 85
Bekenstein’s bound 47–48,
131, 291, 307
Bekenstein–Hawking 289
black hole 47–48, 303
defined with strands 158
lower limit, or Boltzmann
constant ? 30, 36
of black holes 287–289
of gravity 281
of horizons 287–289
of vacuum 47
particle shape and 120
upper limit 291
upper limit to 47–48
equations
non-existence of evolution
148
Escher, Maurits
illustrating circularity 61
essence of universe 110
Euclid
on points 72
Euler angles 198
event
sub j ect ind ex
sub j ect ind ex
F
fine
black hole and 41, 42
electric charge and 41
general relativity implied
by 17, 30–36
principle 34
quantum effects and 36
size limit and 39
why gravitational 41
form factor
of baryons 343
of mesons 338
framing of tangle 419
freedom
asymptotic 272
Frenet frame 419
Frenet ribbon 420
Fulling–Davies–Unruh effect
81, 281, 303
fundamental principle 150
illustration of 398
funnels 169, 406
illustration of 169
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
G
?-factor 199, 236, 242
g-factor
of W boson 361
Galilean physics 26, 59, 401,
415
circular reasoning
resolution 218
gamma-ray bursts 44, 87, 231
gasoline 83
gauge
choice 232–234
covariant derivative 235
freedom, illustration of 232
group, no other 318
interaction 234
interaction antiscreening
383
interaction screening 383
interactions 222–278
interactions, summary 275
symmetry 19, 162, 222,
232–234
symmetry and cores 224
symmetry, not valid 73, 104
theory, non-Abelian 248,
267, 317, 325
transformation 232
U(1) freedom, illustration
of 233
gauge boson
no other types 318
role of 317
strand illustration of 316
weak, illustration of 249
weak, illustration of
incorrect 252
Gedanken experiment see
thought experiment
Gell-Mann matrices 258
Gell-Mann, Murray
on strings 142
gender and physics 137
general relativity
see also field equations
contradicts quantum
theory 57–63
deviations from 303
from maximum force
32–35
from strands 279
horizons and 32
in one statement 30
indeterminacy relation 35
millennium issues and
19–20, 162–163
minimum force 51
non-locality 75
size limit 35
generalizations of the strand
model 163, 405
generalized indeterminacy
principle see
indeterminacy relation,
generalized
generalized uncertainty
principle see
indeterminacy relation,
generalized
generations of quarks 323
generators
of SU(3) and slides 262
Gibbs, Phil
on event symmetry 84, 128
Glashow, Sheldon
Motion Mountain – The Adventure of Physics
table of requirements 147
unmodifiable 163
fine structure constant 18, 152
see also coupling constant,
electromagnetic
charge unit and 50, 378
dead end 385
electrodynamics and 226
estimation of 381
how not to calculate it 385
Planck limits and 43
fine-tuning
none 370–371
finitude
absence of 403
Finkelstein, Robert
on fermions as knots 347
fish in water 123, 138
flavour quantum numbers 315
flavour-changing charged
currents 322
fluctuating lines 300
fluctuations of strands 155, 159
fluid
tangle motion in 195, 360
foam
quantum see foam,
space-time
space-time 311
foam, space-time 157, 296
folds 130, 408, 451
fool
making one of oneself 347
foolishness 21, 347
force
is momentum flow 17, 31
lower limit 46, 51
maximum see force limit,
maximum
no fifth 275
Planck see force limit,
maximum
surface and 31
unlimited for virtual
particles 40
upper limit 42
force limit, maximum ?4 /4?
36
4-force and 42
467
468
G
global
H
Haag’s theorem 273
Haag–Kastler axioms 273
hadron
see meson, baryon
heat and horizons 33
Heisenberg picture 207
Heisenberg, Werner
on symmetry 409
on thermodynamic
indeterminacy 30
on unification 22
heresy, religious 424
Hertz, Heinrich
on everything 407
hidden variables 187
hierarchy
of particle masses 195, 357,
366
Higgs boson
2012 update 331
mass 361
mass prediction 331
predictions about 329
strand illustration of 329,
332
Higgs mechanism 366
Hilbert action 295–296
definition 295
Hilbert space 192
Hilbert’s problems 108
Hilbert’s sixth problem 54,
108–109
Hilbert, David
famous mathematical
problems 108
his credo 143
on infinity 401
on unification 22
hill, gentle green 410
hole argument 58
Einstein’s 284
Hollywood films 98
holography 105, 113, 139, 140,
306–308, 431, 436, 439, 445,
447
’t Hooft, Gerard 299
hoop conjecture 31, 35, 286,
303
hopping
from strand to strand 352
horizon 100, 284
see also black hole
and Planck scales 112
behind a 285
cosmic 50, 305
cosmic, diameter of 100,
102–104
cosmic, distance 97–100
cosmic, none 104–105
cosmic, shape of 100–101
electric charge 41
energy 286
entropy 287–289
entropy limit 47–48
heat flow 33
maximum force 31–35,
41–42
maximum power 31–35,
41–42
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
universal 35, 280–283
gravitational constant ? 30
see also force limit,
maximum, see also power,
upper limit
absence from quantum
theory 58
as conversion constant 59
physics and 8
gravitational wave 87
detectors 87
emitted from atoms 302
graviton 296, 320, 324
illustration of 324
strand illustration of 296
gravity
see gravitation
weak, conjecture 301
Greene, Brian
on popularization 52
on superstrings 140
Gregori, Andrea
on particle mass 136
group
centre of 260
slide 262
GUT see grand unification
Motion Mountain – The Adventure of Physics
on fake unification 255
global coordinate systems 68
glueballs 324–325, 340, 345
gluon
as slide 267
Lagrangian 268–269
self-interaction
illustration of 268
waves 318
gluonic waves 318
gluons 257, 266
Gödel’s incompleteness
theorem 20, 106, 167
gods
and Dante 404
and Einstein 70, 428
and integers 107
and Kronecker 107
and Leibniz 408
and Thales 400
and Trismegistos 450
definition of 423
existence proof 424
favorite T-shirt 411
final theory and 21
intervention of 411
things and 400
Goethe, Johann Wolfgang von
on searching 110
grand unification 380, 396
does not exist 242, 255, 273,
276, 319
not final 164
Grassmann coordinates 133
gravitation 281
see also general relativity,
quantum gravity
and double-slit experiment
298
classical 35, 280
entropic 281–283
entropy and 281
from strands 284
maximum force and 30–35
of superposition 298
quantum 297
strand illustration of 280
strands and 280, 287
surface 31, 33, 47
sub j ect ind ex
sub j ect ind ex
H
Hubble
J
Jarlskog invariant 374
Jehle, Herbert
on particles as knots 347
Jerven, Walter 416
K
Kaluza, Theodor
on unification 22
kaons and quantum gravity 88
Kauffman, Louis
on commutation relations
207
key issues
of unification 392
Killing vector field 34
Klein–Gordon equation 212
knife, limitations of 116
knot
closed 318
definition 318
long 316, 318
models of mesons 345
models of nature 165
models of particles 139
open 316, 318
table of properties 419
topological invariant 229
writhe probabilities 449
knots
dimensionality and 71
in strands 170
Kochen–Specker theorem 202
Kovtun-Son-Starinets
conjecture 431
Kreimer, Dirk
on knots in QED 139
Kronecker, Leopold 107
Kruskal–Szekeres coordinates
71
L
Lagrangian
description of physics 56
Dirac 220
electromagnetic field 231
electroweak 252
from strands 209
gluon 268–269
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
I
idea, Platonic
of observables 94
identity, extremal 112–113, 137
illusion
of motion 417
impenetrability of strands 151,
219
incompleteness theorem,
Gödel’s 20, 106, 167
indeterminacy principle
see indeterminacy relation
indeterminacy relation 194
all together 36
equivalence to Planck
limits 36
for capacitors 44
for current 44
for temperature 30
generalized 68
Heisenberg’s 29, 64, 68, 83
in general relativity 35
in quantum theory 29
in special relativity 28
in thermodynamics 30
indivisibility of nature 403
induction: not a problem 412
infinity
absence of 20, 401
as a lie 402
inflation
and strands 307
lack of 307, 310, 311, 397
information
in the universe 106
no loss 290
initial conditions
of the universe 101, 306
integers 107
interaction
definition 223
electroweak 250
from tangles 224
gauge 222
inversion and 113
mixing 250
no fifth 275
interference
from strands 196, 199
illustration of 197, 199, 200
visualized with strands 199
invariant
see also action, quantum
of, see also force limit, see
also Lorentz invariance,
see also Planck units, see
also speed of light
knot 229
maximum force as 17
Planck unit as 24, 26–44
quantum of action as 18
speed of light as 17
topological 229
inversion 113
irreducibility
computational 20
isotropy of the vacuum 68
issue
key, of unification 392
open: ending funnels 170
open: funnel diameter
behaviour under boosts
170
open: lepton tangles 329
open: W and Z tangles 251
Motion Mountain – The Adventure of Physics
nature of 305
no space beyond 71
nothing behind 397
nothing behind a 71
puzzle 286
quantum effects at 91
radius 32
relation 33
singularities and 58
strand illustration of 285
symmetries at the 104
temperature 290
temperature limit 41
thermodynamic properties
293
types 305
Hubble radius 98
Hubble time 95
hydrogen atom 217
illustration of 217
hypercharge, weak 242, 378
469
470
L
L agrangian
360
list
millennium 18, 161
of experimental
predictions 395
of requirements for a final
theory 147
three important kinds 24
Lloyd, Seth
on information 107
locality
lack of 148
need to abandon 75
none at Planck scales 117
quantum theory vs.
general relativity 58
long knots 316
loop quantum gravity 138
loop, twisted
electromagnetism and 226
loops
time-like 297
Loren, Sophia
on everything 355
Lorentz boosts
maximum force and 31
quantum theory and 215
Lorentz invariance
fluctuations and 75
none at Planck scales 71,
83, 87, 88
of strand model 157
of the vacuum 210
quantum gravity and 88
quantum theory and 214
Lorentz symmetry
see Lorentz invariance
Lorentz transformations
from invariance of ? 211
lattices and 74
minimum length and 71
temperature 49
Lorentz, Hendrik Antoon
on the speed limit 27
M
machine, braiding 351
magnetic
monopole, none 230
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
number 315
strand illustration of 327,
365
tangle of 327
Lesage, Georges-Louis
on universal gravitation
283
LHC see Large Hadron
Collider
Lichtenberg, Georg Christoph
on truth 25
lie, infinity as a 402
life
meaning 413
origin of 21
saving with belt trick 180
light
see also speed of light
deflection by the Sun 59
onion 97
propagation and quantum
gravity 87
scattering of 68
Lilliput
no kingdom 349
lily, beauty of 17
limit
cosmological see
cosmological limit
Planck see Planck limits
limits
in nature, summary 54
our human 417
physics in 26
Planck units as 26
size-dependent 45
system dependent, to all
observables 45–52
system-dependent 45
to cutting 117
to measurements 67
to motion 26
to observables, additional
46
to precision see precision
linear combination 188
lines, skew 347
linking number 387, 420
liquid, tangle motion in 195,
Motion Mountain – The Adventure of Physics
none for strands 348
of QED 234
of general relativity 311
of the standard model 277
of the strong interaction
271
of the weak interaction 249
properties 236
strands and 156, 208–209
superstrings and 141, 142
Lagrangian density see
Lagrangian
Lamb shift, gravitational 88
Landau pole 237
Langtang Lirung 410
Lao Tse
on motion 408, 416
Large Hadron Collider
no discoveries 354
no Higgs boson 331
strand model and 411
W and Z scattering 331,
395, 396
large number hypothesis 103
lattice space-time 74
laziness of nature 26
least action principle
from strands 172, 208–209
in nature 17, 26, 56
valid for strands 277, 295
leather trick 321, 323, 363
illustration of 323
Leibniz, Gottfried Wilhelm
on parts 108
on relations in nature 306
length
defined with strands 158
definition 59
indeterminacy 28
intrinsic 122
lower limit, or Planck 37
maximum 98
minimum 158
Lenin (Vladimir Ilyich
Ulyanov) 82
leptogenesis
none 376, 377
lepton
mass ratios 364
sub j ect ind ex
sub j ect ind ex
M
magnetic
form factor 338
from tangles 334
heavy, illustration of 337
illustration of 334
knotted 345
mass sequences 338
pseudoscalar 334
Regge trajectories 339
shape 338
vector 334
metre rule
cosmic horizon and 98
Planck scales and 66
metric
Planck scales and 71
space 72
microstates of a black hole
288
millennium description of
physics 17–20
millennium list
final summary 398, 399
not solved by superstring
conjecture 142
of open issues 18–20,
161–163
millennium problems
from the Clay
Mathematics Institute 325
minimal coupling 196, 199,
234
minimal crossing number 229
minimization of change
see least action
minimum length 158
Minion Math font 452
mixed state 206
mixing
angle, weak 360
angles 372
matrices 372
quark 373
model
non-commutative 348
tangle, of particles 159
model, topological particle
347
modification of final theory
147
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
sequences of mesons 338
upper limit for elementary
particles 39, 78–79
W and Z bosons 358
without mass 159
matchboxes and universe 99
mathematics
of nature, simplicity of 38,
54
matter
density in universe 309
difference from vacuum 64
extension of 85, 115–144
indistinguishable from
vacuum 52–53, 81–82
made of everything 407
made of nothing 407
mixes with vacuum 80, 81
mattress analogy of vacuum
36
maximons 78
maximum force
see force limit, maximum
maximum speed
see speed of light ?
Maxwell’s field equations 226,
232, 240
meaning in life 413
measurement
always electromagnetic 159
averaging in 85
classicality of 200
definition 401
from strands 201
none at Planck scales 73
precision see precision
problem in quantum
theory 21
problem, quantum 21
mechanism for Feynman
diagrams 222–278
membranes 139
Mende, Paul
on extension checks 138
mesons
charmed, illustration of
336
CP violation 339
excited 338
Motion Mountain – The Adventure of Physics
magnetic charge 230
magnetic field
lower limit 50
upper limit 43, 231
magnetic moment
anomalous 238–240
neutrino 329
magnetic vector potential 196
man-years of work in
superstrings 142
manifolds
see also space, space-time
definition 72
exotic 170
lack of 71
none at Planck scales 39
many-particle state 203
Marx, Groucho
on principles 38
mass
absolute value for particles
366
ADM 103
and braiding of tails 366
black hole 286
calculation for neutrinos
370
calculation of 356
crossing switch rate as 367
eigenstate of quark 373
elementary particle 311, 357
flow, upper limit 40
from strands 194, 284, 330,
346
gap 325
generation 250
gravitational 79, 80, 103,
356, 359
hierarchy 195, 357, 366–369
in universe 102
inertial 80, 356, 360
inverse of 125
maximum density 37
measurement 78–82
negative 83
of bosons 358
rate limit 291
ratios of leptons 364
ratios of quarks 362
471
472
M
modified
rare decays 329
tangle of 327
O
object
motion and 59
observables
basic 157
defined with crossing
switches 156
none at Planck scales 73
system-dependent limits
45
unexplained, millennium
issues 18
value definition 170
observer
definition 82
Occam’s razor 127, 166
octonions 272
Olbers’ paradox 42
operator
Hermitean 207
unitary 207
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
N
National Institute for Play 429
natural units see also Planck
limits, see also Planck
units, 37, 150
naturalness
none 370–371
of standard model 152
nature
and description, table of
difference 167
efficiency of 27
event symmetry 84
has no meaning 111
is indivisible 403
is not finite 403
laziness of 26
limits motion 36
made of one strand 407
multiplicity of 128
no infinity in 401
no sets nor elements 53–54
non-locality of 75
not a computer 403
not a set 128
not discrete 403
table of properties 112
unity of 128
vs. people 418
whether deterministic 84,
410
whole in each of its parts
406
nematic world crystal 299
neurobiology 169
neutrino
magnetic moment 329
mass calculation 370
mixing 375
illustration of 375
tangle of 327
neutrinoless double beta
decay 328
neutron
decay and strand model
255
form factor 343
neutron–antineutron
oscillations 276
Newton’s bucket 218
Newtonian physics
see Galilean physics
Nietzsche, Friedrich
on walking 149
night sky 17
meaning of 408
nightmare scenario 354
no avail conjecture 301, 304
no-hair theorem 132, 286
non-commutative model 348
non-locality 148
natural 169
solves contradictions 75
non-perturbative calculations
236
non-zero action 28
norm of quantum state 191
nothing
difference from universe
90
NSA dream 306
number of stars 102
numbers, no real 52, 73
Motion Mountain – The Adventure of Physics
modified Newtonian
dynamics 303
momentum 194
flow is force 17, 31
indeterminacy 29
quanton definition 194
upper limit for elementary
particles 39
momentum indeterminacy 67
monad 108
Mongan, Tom
on particles as tangles 347
monism 408, 436
monopoles, magnetic, none
230
motion 173
as an illusion 417
continuity of 401
essence of 416
fast 8, 22
helical 360
limited in nature 26, 36
limits to 26
none at Planck scales 86
of particles through
vacuum 352
powerful 8, 22
predictability of 401
quantum 174
strand illustration of 350
tiny 8, 22
translational 352
ultimate questions and
400
uniform 8, 22
Motion Mountain 21
climbing 410
nature of 410
supporting the project 9
top of 400
move see Reidemeister move
multi-particle state
see many-particle state
multiplicity
approximate 130
multiverse nonsense 108, 111,
348, 407, 423
muon
?-factor 199, 236, 242
sub j ect ind ex
sub j ect ind ex
order out of chaos 405
origin
human 413
oscillator, harmonic 273
overcrossing 250, 251
O
order
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
disappearance of 228
entangled
illustration of 205
model of 227
speed of 210
stability of 228
physical space differs from
background space 283
physical system 27, 106
see also system-dependent
limit
illustration of 154
physicists
conservative 55, 412
physics 167
approximations and the
sky 408
beauty in 56
book, perfect 109
definition 17
Galilean 26, 59, 401, 415
gender and 137
golden age 22
in four steps 404
in limit statements 26
in the year 2000 18
map of 8
motion limits in 26, 36
progress of 404
simplicity in 409
simplicity of 26–36, 38
table of progress 404
the science of motion 17
unification in one
statement 403
physics cube 8
Planck acceleration 36
Planck accelerator 82
Planck action ℏ
see action, quantum of
Planck angular frequency 36
Planck area 37
Planck constant ℏ
see action, quantum of
Planck curvature 37
Planck density 37, 79
Planck distance 37
Planck energy 41, 57, 82
see also Planck scales
Motion Mountain – The Adventure of Physics
vacuum 123
of four or more strands 341
pair creation 58, 60, 100
properties 313, 355
quantum 159, 175
spectrum, explanation 352,
P
353
pantheism 424
spectrum, prediction 353
parity 242
spin 1 314–319
none at Planck scales 78
stable 174
violation 246–248
table of correspondence
belt trick and 179
with tangles 346
part of nature see parts
tangle as 160
particle
translational motion 352
see also elementary
virtual 40, 153, 157, 172
particle, see also matter, see parts
also virtual particle
are approximate 127
circular definition of
in nature 127, 401
space-time and 59
in nature, lack of 127
common constituents with
none in nature 403
vacuum 88
pastime, unification as 21
definition 59, 173, 313
path
electrically charged 228
helical 360
exchange 74, 77, 133
integral 199
in the millennium
integral formulation 186
description of physics 18
Pauli equation 199, 212
internal structure 346
from tangles 198
intrinsic property list 315
Pauli matrices 198
lower speed limit 51
Pauli, Wolfgang
made of one strand
on gauge theory 248
314–319
Penrose conjecture 286, 303
made of three strands 326
pentaquarks 347
spin of 340
permutation symmetry
made of two strands 319
not valid 74, 104, 133
mass 356
origin of 184
mass, absolute value 366
perturbation theory
motion 59
convergence of 240
motion through vacuum
failure of 331
352
validity of 237
no exchange at horizon
phase 173
scales 104
average 188
no point 32, 39, 41, 58,
CP-violating 373
75–78
quantum 188
none at Planck scales
tangle 226
77–82, 88, 160
Philippine wine dance 176
photography, limits of 118
number in the universe
photon 227
101–104, 107, 110, 112,
affected by quantum
127–129
gravity 87
number that fits in
473
474
P
Pl anck
incompatible with
unification 24, 63
shape of 115
size of 121, 123
poke see also Reidemeister
move
basic
illustration of 247
gauge group
illustration of 245
transfer
illustration of 244
posets 106
position
from strands 173
indeterminacy 66
positron charge 341
potential
and strands 196
electric 230
indeterminacy 44
electromagnetic 232
magnetic 230
power
lower limit to 46
misuse of 21
surface and 31
upper limit 42
upper limit ?5 /4? 30–36
precession 178
precision
does not increase with
energy 126
fun and 114
lack at Planck scales 71
lack of at Planck scales 68
limited by quantum theory
58
limits 77
limits to 52–54
maximum 94
of age measurements 93
of clocks 66, 93–98
of final theory 147
of length measurements 66
predictability
of motion 401
predictions
about axions 273
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
surprising behaviour at
64–89
vacuum and matter at
52–53, 81–82
Planck speed ? see speed of
light ?
Planck time 150
age measurement and 93
as measurement limit
64–66, 93, 97, 114
shutters and 118
value of 60
Zeno effect and 121
Planck units 36, 148, 150, 158
as invariants 26, 37
as key to unification 143
as limits 26, 37, 67
as natural units 67
corrected, definition 37
definition 24
key to unification 54
Planck value 37
see natural units, see
Planck units
Planck volume 37, 44, 67
number in the universe 49
plate trick 176
platelets 180
Plato
on love 418
on nature’s unity 128
Platonic idea
of time 94
play 21
plural 401
and motion 417
Poincaré symmetry
of vacuum 212
point particles
do not exist 32, 39, 41, 58,
75–78
points
as clouds 119
as tubes 123
cross section of 123
do not exist 24, 32, 40, 53,
66–73
exchange 133
in vacuum 121
Motion Mountain – The Adventure of Physics
definition 39
Planck entropy 150
Planck force ?4 /4?
see force limit, maximum
Planck length 37, 150
see also Planck scales,
Planck energy
antimatter and 53
as measurement limit
66–73
duality and 113
extremal identity and 113
mass limit and 78
shutters and 118
space-time lattices and 74
value of 60
Planck limits
see also Planck units,
natural units
curiosities and challenges
40–44
definition 36
electromagnetic 43–44
Planck mass 78, 367
definition 39
does not exist as black hole
301, 367
does not exist as
elementary particles 367
stand model 367
strand model of 367
Planck mass density 37
Planck momentum
definition 39
Planck scales
as domain of
contradictions 59
definition 37
general relativity and
quantum theory at 59–63
large symmetry at 125
nature at 64–89
no dimensions at 71
no events at 69
no measurements at 73–74
no observables at 73–74
no space-time at 72
no supersymmetry at 78
no symmetries at 73–74
sub j ect ind ex
sub j ect ind ex
P
preon
QED
271–274
226–243
flavour 315
lepton number 315
parity 315
spin 315
quantum particle
properties 313
quantum state 185
quantum theory
and space-time curvature
81
contradicts general
relativity 57–63
displacement limit 28, 29
implied by quantum of
action 18
in one statement 28
measurement problem 21
millennium issues and 19,
162
no infinity in 401
non-zero action 28
of matter 172
space-time curvature and
57
vacuum and 79
quark
flavour change
illustration of 323
mixing
illustration of 372
model acceptance 335
Planck energy table 364
ropelength table 364
quarks
are elementary 75
fractional charge 44
generations of 323
mass ratios 362
mesons and 334
mixing 372, 373
strand illustration of 321,
362
tangles 320–324
quasars 231
quaternions 272
qubit 161
R
race, in quantum gravity 87
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
Q
QCD
convergence of 240
quantities
bare 237
quantum action principle
209, 219
quantum effects
are due to extension 407
quantum field theory 217, 383
as approximation of the
strand model 275
strand hopping and 352
quantum fluctuations 106
quantum foam see foam,
space-time
quantum geometry 52, 75, 113
quantum gravity 297
QED and 243
definition of 296
does not exist 304
effects on photons 87
entanglement and 301
entropy and 288
experiments in 87–88
extension and 138, 288
finite entropy 131
from strands 311
gravity waves and 87
is unobservable 304
loop 299
Lorentz symmetry and 88
minimum distance and 37
minimum power and 47
no such theory 23
non-locality 75
Planck scales and 73
predictions 303, 397
predictions about 277
strands and 293, 298
topology and 310
quantum groups 396
quantum lattices 106
quantum measurement
from tangles 199
quantum mechanics see also
quantum theory, 217
quantum numbers
all 315
baryon number 315
charge(s) 315
Motion Mountain – The Adventure of Physics
about cosmology 310
about dark matter 353
about general relativity 302
about grand unification
276
about mesons 338
about supersymmetry 276
about the number of
interactions 275
about the strong
interaction 273
of the strand model 395
on charge quantization 378
on coupling constants 381
on the weak interaction
255
preon models 434
pride 21
principle
fundamental 150
fundamental, illustration
of 147
of least action 17, 26, 56,
208, 277
of least change 17, 26, 56
of maximum force see
force limit, maximum
of non-zero action 28
quantum action 219
probability density 191
process
fundamental 152
projection, minimal 386
propagator 216
properties
intrinsic 85, 313
unexplained, as
millennium issues 18
proton
charge 341
decay 276
form factor 343
mass 343
tangle of 341
puzzle about strands 325
475
476
R
resolution in measurements
118
Reynolds number 360
ribbon
framing 419
models 139, 165, 347
ribbon, mathematical 387
Ricci scalar 295
Ricci tensor 34
Riemann, Bernhard
on geometry 38
ring chain trick 352
rope braiding 351
ropelength 359
measured in diameters 359
rotation
of tangle cores 222
tethered 180
rotation curve
of stars in galaxies 303
rule, superselection 190
running
of coupling 238, 253, 272
of coupling constants 383
of coupling, data 382
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
S
S-duality 124
safety, asymptotic 302
Sakharov, Andrei 78
on matter constituents 121
on maximum particle
mass 39
on minimum length 43
portrait 78
Salam, Abdus
on unification 22
scalar multiplication 189
scalar product 191
scale
extremal, nature at 111
scales, cosmological 90
scattering
by vacuum 82
of longitudinal W and Z
bosons 330, 332
to determine mass 80
to determine size 76, 117
Schrödinger equation 73, 192,
212
Schrödinger picture 186, 207
Schrödinger, Erwin
on thinking 116
Schwarzschild black hole 285
see black hole
Schwarzschild radius
see also black hole
as limit of general
relativity 59
as measurement limit 67,
79, 80
definition 31
entropy and 131
extension and 115
lack of sets and 106
mass and vacuum 124
table with examples 60
science fiction, no 354
scissor trick 176
see-saw mechanism 366, 376
self-linking number 420
sets
not useful to describe
nature 128
not useful to describe
universe 53–54, 105–106
shape
of points 115
touching 119
sheet
and belt trick 178
shivering 151
and divergences 276
short-time average 185
shutter
limits of a 118
time table 118
SI units 150
simplicity
of physics 38
simplification
as guiding idea 409
single atom 83
singularities
horizon and 58
none in nature 84, 148
none in the strand model
291
Motion Mountain – The Adventure of Physics
Raychaudhuri equation 34,
293
Raymer, Michael
on the indeterminacy
relation 64
real numbers, no 52, 73
reductionism
and the final theory 21
and the strand model 160
Regge slope
illustration of 339
Regge trajectories 338
region
of negative energy 303
Raychaudhuri region of negative energy
not possible 298
regions of negative energy 297
regularization
non-local 331
weak bosons and 331
Reidemeister move
and gauge bosons 317
first or type I or twist
225–243
illustration of types 225
second or type II or poke
225, 244–256
third or type III or slide
225, 257–274, 317
Reidemeister’s theorem 275
Reidemeister, Kurt
on knot deformations 224,
275
relativity
as approximation of the
strand model 311
doubly special 435
general see general
relativity
no infinity in 401
special see special relativity
summary on 311
renormalization
at Planck scales 74
of QCD 272
of QED 236
of gravity 303
requirements
for a final theory 146
sub j ect ind ex
sub j ect ind ex
S
size
speed
lower limit 46, 51
of energy 27
of light ?
from strands 209–212
physics and 8
strands and 210
tangles and 351
special relativity implied
by maximum 17, 27
unlimited for virtual
particles 40
upper limit 27, 36
spin
and strand number 340
at Planck scales 77
belt trick and 134, 299
entangled
illustration of 204
extension and 134
foam 138, 299
from strands 174
from tangles 174
general relativity and 58
importance of 134
many-particle
illustration of 203
minimal action and 40
operator 198
orientation 173
superposition
illustration of 201
three-dimensionality and
207
without topology change
432
spin–statistics theorem 174
spinor 198, 215, 218
visualization 215
sponsor
this book 9
this free pdf 9
standard model
and the final theory 23
standard model of particle
physics
millennium description of
physics 18, 146
strands and 152
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
background space 285
physical, from strands 166
shivering 151, 154, 170, 351
topological 72
topology change, not
needed 182
space-time see also curvature
as statistical average 75
as thermodynamic limit 75
circular definition of
particles and 59
continuity 39, 69, 70, 86,
125
curvature 34
curvature and quantum
theory 57, 81
discrete and continuous
157
duality 113, 124, 224
elasticity 36
entropy of 47
foam 157, 296, 311
motion of 48
must be fluctuating 75
no dimensionality at
Planck scales 71
non-commutative 276
not a lattice 74, 106
not a manifold 72
results from upper energy
speed 27
shivering 151, 154, 170, 351
symmetries 224
topology change, not
needed 182
spatial order 68
special relativity
double or deformed 302
doubly special 279
falsified by minimum
length 71
implied by maximum
speed 17
in one statement 27–28
massive tangles and 352
strands and 279
spectrum
of particles, explanation
353
Motion Mountain – The Adventure of Physics
none inside black holes
286
none predicted in nature
303
size
indeterminacy 35
system-dependent limits
and 45
size limit
due to cosmology 50
due to general relativity 35
due to quantum theory 29
due to special relativity 28
skew lines 347
sky, at night 17
slide see Reidemeister move
group 262
illustration of observable
263
illustration of
unobservable 262
slide transfer
illustration of 257
slit
double, and gravity 298
double, and strands 199,
200
Sokolov–Ternov effect 303,
304
space see also background, see
also vacuum
airless, breathing in 86
background 154, 166
border of 153
constituents of 121
curved 284
definition 401
in dreams 415
isotropy and strands 156,
157, 187, 210, 212
mathematical 72
metric 72
no points in 121
none at Planck scales 88
not a lattice 74
not a manifold 72
physical 154, 285
physical, definition of 157
physical, differs from
477
478
S
star
principle 73
superselection rules 190
superstrings
basic principles of 142
black hole entropy and 289
conjecture 140
dimensionality and 85
joke 141
Lagrangian 141
not an explanation 165
overview 139–144
status 142
summary 142
vs. strands 348
supersymmetry
final theory and 144
not correct 74, 78, 133, 164,
242, 255, 276, 319, 353, 396
required 22
strings and 139
support
this book 9
this free pdf 9
surface
force and 31
gravity 31, 33, 47, 290
physical 31, 43
surprises in nature 85, 410
Susskind, Leonard
on Planck scale scattering
117
switch of crossing 150
symmetry
at the horizon 104
beauty and 408
between large and small
124
breaking 250
event 84
no higher 277
none at Planck scales 74,
148
space-time 224
total 125
system, physical
table of correspondence
with strands 155
table of correspondence
with tangles 299
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
extension and 146
foundations 150
fundamental principle 150
generalizations of 163, 405
history of 411
illustration of fundamental
event 147
illustration of fundamental
principle 147
is natural 152
list of predictions 395
of Planck mass 367
of the cosmological
constant 307
other research and the 412
simplicity of 165
slow acceptance 412
table of predictions 395
vs. superstrings 348
stress-energy tensor 42
string nets 139
string trick see also belt trick,
176
quantum theory and 217,
220
strings see superstrings
strong interaction
conservation properties
271
strong nuclear interaction
257–274
SU(2) 245, 378
SU(2) breaking 250
SU(2) field
classical waves 317
SU(3)
and slides 261–268
multiplication table 260
properties 258–261
SU(3) field
classical waves 318
supergravity
not correct 242, 255, 276
supermembranes 139
superparticles 276
superposition
from strands 188
gravitational field of 298
of clocks 58
Motion Mountain – The Adventure of Physics
star
rotation curve in galaxies
303
Stark effect, gravitational 88
stars
in the universe 101
number of 102
state
in dreams 415
Stern–Gerlach experiment
199
stones 174
strand
as cloud 153
averaging 187
braiding and mass 352
definition 153
density 157, 170
diameter 150
diameter behaviour under
boosts 154
diameter, issues 153
ends 170
evolution equation 187
fluctuations, conditions on
186, 294
hopping 352
illustration of fluctuations
186
impenetrability 151, 300
impenetrability of 219
knotted 170
linear combination
illustration of 191
puzzle 325
scalar multiplication
illustration of 190
spacing 157
substance of a 159
translation in other
languages 149
vs. superstring 348
strand model
anomalous magnetic
moment and 238–240
basis of 146
beauty of 405
checking the 171
confirmation of 355
sub j ect ind ex
sub j ect ind ex
Szilard, Leo
on minimum entropy 30
S
Szil ard
of everything 129
of everything does not
exist 405
physical, definition 21
thermodynamics in one
statement 30
thinking
extreme 21
mechanism of 18
Thomson–Kelvin, William
on atoms as knotted
vortices 347
thought experiment
constituents and 121
on extension 137
on force and power 32
on shape 120
on vacuum 80, 123
time 166
beginning of 92
coordinate 66
defined with strands 158
definition 59, 401
does not exist 94
in dreams 414
indeterminacy 44, 64
issue of 151
lower limit, or Planck 36
maximum 91
measurement 58
Platonic idea 94
proper, end of 66
time-like loops 297
tombstone 411
topological models 347
topological space 72
topological writhe 229
definition 386
topology change
of space-time, not needed
182
topology of the universe 310
toroidal black holes 297
torsion
in general relativity 299,
303
of curve 419
total, of a curve 420
translation invariance 68
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
rational, of high
complexity 324
table of properties 419
tails 173, 175
three-stranded 341
tight 358
tight, illustration of 358
topological invariant 229
trivial 319
two-stranded 325
tangle classes
one strand, illustration of
314
three strands, illustration
of 326
two strands, illustration of
319
tangle model
of particles 159
tau
tangle of 327
technicolour 396
temperature
indeterminacy 30
lower limit 49
upper limit 41
vacuum 281
tetraquark
strand illustration of 345
tetraquarks 344
theorem
Banach–Tarski see
Banach–Tarski
paradox/theorem
Calugareanu 421
Coleman–Mandula 273
Fermat’s last 410
Gödel’s see
incompleteness theorem,
Gödel’s
Kochen–Specker 202
no-hair 132, 286
Reidemeister’s 275
spin–statistics 174
Weinberg–Witten 273
theory
final 428
final, not of everything 405
freedom of 408
Motion Mountain – The Adventure of Physics
T
T-duality 124
T-shirt 411
first use 451
tachyon 27
tail
braiding 250
deformations 224
essential for spin 1/2 177
model for particle 134
shifting 374
Tait number 420
tangle 156
alternating 386
as particle 159
blurred 185, 194
braided 341
chirality 385
coloured, illustration of
269
core 173
core deformation
illustration of 223
core rotation 222
illustration of 223
family 335
family of leptons 364
four-stranded 345
framing of 419
function 184, 185
functions are wave
functions 187–194
ideal 358, 419
lepton 327
locally knotted 320
moving
illustration of 193, 211
of four or more strands 341
one-stranded 318–319
phase 226
prime 320
quark 320
rational 320, 341
rational, as coloured
fermion 270
rational, definition 270
479
480
T
trick
unmodifiability of final
theory 163
Unruh effect see
Fulling–Davies–Unruh
effect
Unruh radiation see
Fulling–Davies–Unruh
effect
V
vacuum see also space
as a bath 155
breathing in 86
common constituents with
particles 55, 61, 63, 81, 86,
88, 132
defects in 284, 297
difference from matter 64
elasticity of 36
energy density 19, 57, 162,
210, 307
entropy bound 47–48
entropy of 47
illustration of schematic
156
indistinguishable from
matter 52–53, 81–82
Lorentz invariance 210
made of everything 407
mass error 79–81
mixes with matter 80
none at Planck scales 88
Poincaré symmetry 212
relativity vs. quantum
theory 57–63
strand model 157, 172
tangle function 187
temperature 281
uniqueness of 157, 172, 210,
277, 284, 354
variables, hidden 202
vector
binormal 419
normal 419
Verlinde, Erik
on universal gravitation
281
violence and infinity 402
virtual particles 40, 153, 157,
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
steps of the search for 24
three key issues 392
unified models
assumptions of 166
complexity of 165
requirements of 146
uniqueness 406
uniqueness
of the unified model 406
unitarity
violation in W and Z
boson scattering 330
units
Planck’s natural see Planck
units, natural units
universal gravitation 35, 280
universe
age error of 93, 94
age of 91
U
as inverted black hole 305
U(1) 226–235, 242, 378
definition of 111, 113
U(1) invariance 196
difference from nothing 90
U-duality 124
essence of 110
uncertainty see indeterminacy
fate of 98
uncertainty principle see
finiteness of 304
indeterminacy relation
has no meaning 111
uncertainty relation see
horizon of 305
indeterminacy relation
information and 107
unification see also final
initial conditions 101, 306
theory
luminosity limit 42
arguments against 20–21
mass of 102–104
as lack of finitude 403
matter density 309
as pastime 21
no boundary 104–105
as riddle 21, 24
no information in 107
beliefs and dogmas 22
not a container 129
difficulty of 300
not a physical system 106
disinformation 428
not a set 53–54, 105–106
grand see grand
oscillating 91
unification
power limit 47
is possible 125
radius 100
key to 54
sense of 110
millennium issues and 18
size of 98
of interactions 275, 276, 381
strand illustration of 305
quantum theory and
strand model 304
relativity 56
system dependent limits
reason for failure 255
and 45–52
table of properties 110
requiring extreme
topology of 310
thinking 21
volume of 100
simplicity of 22, 38
Motion Mountain – The Adventure of Physics
trick
belt 176
dirty Higgs 331
leather 363
plate 176
scissor 176
tubes in space 123
tuning, fine
none 370–371
Turing machines 106
twist see Reidemeister move,
317, 387, 420
and gauge
illustration of 233
generalized 234
twist transfer
illustration of 226
twisted loop 226
sub j ect ind ex
sub j ect ind ex
172
at Planck scales 77
viscous fluids
tangles and 195, 360
volume 72
lower limit, or Planck 37
of the universe 100
vortices in the aether 347
V
viscous
crystal, nematic 299
of dreams 414
origin of 413
wormholes 170, 296, 297, 310
do not exist 298, 303
writhe 387
2d 420
3d 421
topological 229, 420
topological, definition 386
writhing number 421
Y
Yang–Mills theory see gauge
theory, non-Abelian
Yukawa coupling 366
Yukawa mechanism 366
Z
Z boson 251
mass 361
mass of 358
strand model 248
two mass values 331
Zeno effect, quantum 28, 121
Zeno of Elea
on motion as an illusion
417
on size 124
Zeno’s argument against
motion 70, 119, 121
resolution 218
zero-point energy 57, 72
Zitterbewegung 216
copyright © Christoph Schiller June 1990–September 2018 free pdf file available at www.motionmountain.net
weak bosons 248
weak charge 246–248
weak current
absence of
flavour-changing 347
weak gravity conjecture 301
weak hypercharge 242, 378
weak interaction 244–256
weak mixing angle 360
Weinberg, Steven
on unification 22
Weinberg–Witten theorem
273
Wheeler, John A.
on ‘it from bit’ 160
on mass without mass 159
on nature’s principles 405
on space-time foam 157,
296
on topology change 58
on unification 22
really big questions 411
Wheeler–DeWitt equation 73
wheels
none in nature 219
wholeness, Bohm’s unbroken
106, 438
Wightman axioms 273
Wiles, Andrew
on research 410
Witten, Edward
on duality 137
on infinities 140
on unification 22
words and physics 409
world
Motion Mountain – The Adventure of Physics
W
W boson
g-factor of 361
mass 361
mass of 358
strand model 248
two mass values 331
W polarization
illustration of 330
walls
limitations to 117
water flow
upper limit 40
wave function 173, 185
and crossing size 188
as blurred tangles 217
as rotating cloud 187
collapse 201
collapse from tangles 199
definition 187
is a tangle function
187–194
visualizing the 215
wave, gravitational
emitted from atoms 302
waves
gluonic 318
481
MOTION MOUNTAIN
The Adventure of Physics – Vol. VI
The Strand Model –
A Speculation on Unification
What is the origin of colours?
Which problems in physics are unsolved since the year 2000
and what might be their solution?
At what distance between two points does it become
impossible to find room for a third one in between?
Why do change and motion exist?
What is the most fantastic voyage possible?
Answering these and other questions, this book
gives an entertaining and mind-twisting introduction
to the search for the final theory of physics. The
search leads to the strand model: Based on a
simple principle, strands reproduce quantum theory,
the standard model of particle physics and general
relativity. Strands leave no room for alternative
theories, agree with all experimental data and
allow estimating the fine structure constant.
Christoph Schiller, PhD Université Libre de Bruxelles,
is a physicist and physics popularizer. This entertaining
book is for students, teachers and anybody interested
in modern research about fundamental physics.
Pdf file available free of charge at
www.motionmountain.net
