Tài liệu MOTION MOUNTAIN part VI doc

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 25.34, available as free pdf at

www.motionmountain.net

Editio vicesima quinta.

Proprietas scriptoris © Chrestophori Schiller

primo anno Olympiadis trigesimae.

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.

Twenty-fifth edition.

the first year of the 30th 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 at no charge.

To Britta, Esther and Justus Aaron

τῷ ἐμοὶ δαὶμονι

Die Menschen stärken, die Sachen klären.

PREFACE

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 description 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 unification. 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 thinking 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 produces intense and beautiful emotions. Enjoy them!

Munich, 11 December 2012.

8 preface

Galilean physics, heat and electricity

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).

Special relativity

Adventures: light,

magnetism, length

contraction, time

dilation and

E0 = mc2 (vol. II).

Quantum theory

Adventures: death,

reproduction, biology,

chemistry, evolution,

enjoying colours and

art, all high-tech

business, medicine

(vol. IV and V).

Quantum

theory with gravity

Adventures: bouncing

neutrons, under-

standing tree

growth (vol. V).

Final, unified description of

motion

Adventures: understanding

motion, intense joy with

thinking, calculating

couplings and

masses, catching

a glimpse

of bliss

(vol. VI).

G c h, e, k

PHYSICS:

Describing motion

with the least action principle.

Quantum field theory

Adventures: building

accelerators, understanding quarks, stars,

bombs and the basis of

life, matter, radiation

(vol. V).

How do

everyday,

fast and large

things move?

How do small

things move?

What are things?

Why does motion

occur? What are

space, time and

quantum particles?

General relativity

Adventures: the

night sky, measuring curved space,

exploring black

holes and the

universe, space

and time (vol. II).

Classical gravity

Adventures:

climbing, skiing,

space travel,

the wonders of

astronomy and

geology (vol. I).

FIGURE 1 A complete map of physics: the connections are defined by the speed of light c, the

gravitational constant G, the Planck constant h, the Boltzmann constant k and the elementary charge e.

Using this file

Text in green, as found in many marginal notes, marks a link that can be clicked in a pdf

reader. Such green links are either bibliographic references, footnotes, cross references

to other pages, challenge solutions, or pointers to websites.

Solutions and hints for challenges are given in the appendix. Challenges are classified

as research level (r), difficult (d), standard student level (s) and easy (e). Challenges for

which no solution has yet been included in the book are marked (ny).

This sixth volume of the Motion Mountain series has been typeset in a way that printing the file in black and white gives the smallest possible reduction in reading pleasure.

preface 9

Feedback and support

This text is and will remain free to download from the internet. I would be delighted to

receive an email from you at [email protected], especially on the following issues:

Challenge 1 s — What was missing or hard to follow and should be clarified?

— What should be corrected?

In order to simplify annotations, the pdf file allows adding yellow sticker notes in

Adobe Reader. Alternatively, you can provide feedback on www.motionmountain.net/

wiki. Help on the specific points listed on the www.motionmountain.net/help.html web

page would be particularly welcome. All feedback will be used to improve the next edition. On behalf of all readers, thank you in advance for your input. For a particularly

useful contribution you will be mentioned – if you want – in the acknowledgements,

receive a reward, or both.

Your donation to the 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. 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.

A paper edition of this book, printed on demand and delivered by mail to any address, can be ordered at www.lulu.com/spotlight/motionmountain. But above all, enjoy

the reading!

Contents

16 1 From millennium physics to unification

Against a final theory 19 • What went wrong in the past 20 • How to find the final

theory of motion 21

24 2Physics in limit statements

24 Simplifying physics as much as possible

Everyday, or Galilean, physics in one statement 24 • Special relativity in one statement 25 • Quantum theory in one statement 26 • Thermodynamics in one statement 27 • General relativity in one statement 28 • Deducing general relativity 29

• Deducing universal gravitation 32 • The size of physical systems in general relativity 32 • A mechanical analogy for the maximum force 33

33 Planck limits for all physical observables

Physics, mathematics and simplicity 35 • Limits to space, time and size 35 • Mass

and energy limits 36 • Virtual particles – a new definition 37 • Curiosities and fun

challenges about Planck limits 37

41 Cosmological limits for all physical observables

Size and energy dependence 42 • Angular momentum and action 42 • Speed 43

• Force, power and luminosity 43 • The strange charm of the entropy bound 44

• Curiosities and fun challenges about system-dependent limits to observables 45

• Cosmology in one statement 47 • The cosmological limits to observables 47

• Limits to measurement precision and their challenge to thought 48 • No real

numbers 48 • Vacuum and mass: two sides of the same coin 49 • Measurement

precision and the existence of sets 49

50 Summary on limits in nature

52 3 General relativity versus quantum theory

The contradictions 53 • The origin of the contradictions 54 • The domain of contradictions: Planck scales 55 • Resolving the contradictions 57 • The origin of

points 57 • Summary on the clash between the two theories 58

59 4 Does mat ter differ from vacuum?

Farewell to instants of time 59 • Farewell to points in space 61 • The generalized

indeterminacy principle 63 • Farewell to space-time continuity 63 • Farewell

to dimensionality 66 • Farewell to the space-time manifold 66 • Farewell to observables, symmetries and measurements 67 • Can space-time be a lattice? 68 •

A glimpse of quantum geometry 69 • Farewell to point particles 70 • Farewell

to particle properties 71 • A mass limit for elementary particles 72 • Farewell to

massive particles – and to massless vacuum 73 • Matter and vacuum are indistinguishable 75 • Curiosities and fun challenges on Planck scales 76 • Common

constituents 80 • Experimental predictions 81 • Summary on particles and

vacuum 82

84 5 What is the difference between the universe and nothing?

Cosmological scales 84 • Maximum time 85 • Does the universe have a definite

age? 85 • How precise can age measurements be? 86 • Does time exist? 87

• What is the error in the measurement of the age of the universe? 88 • Maximum length 92 • Is the universe really a big place? 92 • The boundary of space

– is the sky a surface? 94 • Does the universe have initial conditions? 94 • Does

the universe contain particles and stars? 95 • Does the universe contain masses

contents 11

and objects? 96 • Do symmetries exist in nature? 97 • Does the universe have a

boundary? 98 • Is the universe a set? – Again 99 • Curiosities and fun challenges

about the universe 100 • Hilbert’s sixth problem settled 102 • The perfect

physics book 102 • Does the universe make sense? 103 • Abandoning sets and

discreteness eliminates contradictions 104 • Extremal scales and open questions

in physics 104 • Is extremal identity a principle of nature? 105 • Summary on the

universe 106 • A physical aphorism 107

108 6 The shape of points – extension in nature

109 The size and shape of elementary particles

Do boxes exist? 109 • Can the Greeks help? – The limitations of knives 109 • Are

cross sections finite? 110 • Can we take a photograph of a point? 111 • What is

the shape of an electron? 112 • Is the shape of an electron fixed? 113 • Summary

of the first argument for extension 114

114 The shape of points in vacuum

Measuring the void 116 • What is the maximum number of particles that fit inside

a piece of vacuum? 116 • Summary of the second argument for extension 116

117 The large, the small and their connection

Is small large? 117 • Unification and total symmetry 118 • Summary of the third

argument for extension 119

120 Does nature have parts?

Does the universe contain anything? 122 • An amoeba 122 • Summary of the

fourth argument for extension 123

123 The entropy of black holes

Summary of the fifth argument for extension 125

125 Exchanging space points or particles at Planck scales

Summary of the sixth argument for extension 126

126 The meaning of spin

Summary of the seventh argument for extension 128

128 Curiosities and fun challenges about extension

Gender preferences in physics 129

130 Checks of extension

Current research based on extended constituents 131 • Superstrings – extension

and a web of dualities 132 • Why superstrings and supermembranes are so appealing 132 • Why the mathematics of strings is so difficult 133 • Testing strings:

couplings and masses 134 • The status of the string conjecture 134 • Summary

on extension in nature 135

138 7 The basis of the strand model

Requirements for a final theory 138 • Introducing strands 140 • From strands to

modern physics 142 • Vacuum 146 • Observables and limits 147 • Particles and

fields 148 • Curiosities and fun challenges about strands 149 • Do strands unify?

– The millennium list of open issues 150 • Are strands final? – On generalizations

and modifications 152 • Why strands? – Simplicity 154 • Why strands? – The

fundamental circularity of physics 155 • Funnels – an equivalent alternative to

strands 158 • Summary on the fundamental principle of the strand model – and

on continuity 158

160 8 Quantum theory of matter deduced from strands

Strands, vacuum and particles 160 • The belt trick, rotation and spin 1/2 162 • An

aside: the belt trick saves lives 165 • Fermions, spin and statistics 166 • Bosons,

12 contents

spin and statistics 167 • Tangle functions: blurred tangles 168 • Details on fluctuations and averages 170 • Tangle functions are wave functions 170 • Deducing the

Schrödinger equation from tangles 175 • Mass from tangles 178 • Potentials 178 •

Quantum interference from tangles 179 • Deducing the Pauli equation from tangles 180 • Rotating arrows and path integrals 182 • Measurements and wave function collapse 182 • Many-particle states and entanglement 184 • Mixed states 187 •

The dimensionality of space-time 187 • Operators and the Heisenberg picture 188

• Hidden variables and the Kochen–Specker theorem 189 • Lagrangians and the

principle of least action 189 • Special relativity: the vacuum 191 • Special relativity: the invariant limit speed 191 • Dirac’s equation deduced from tangles 193 •

Visualizing spinors and Dirac’s equation using tangles 196 • Quantum mechanics

vs. quantum field theory 198 • A flashback: settling three paradoxes of Galilean

physics 199 • Fun challenges about quantum theory 199 • Summary on quantum theory of matter: millennium issues and experimental predictions 201

203 9 Gauge interactions deduced from strands

Interactions and phase change 203 • Tail deformations versus core deformations 204

207 Electrodynamics and the first Reidemeister move

Strands and the twist, the first Reidemeister move 207 • Can photons decay or

disappear? 208 • Electric charge 209 • Challenge: What knot invariant is electric

charge? 210 • Electric and magnetic fields and potentials 210 • The Lagrangian of

the electromagnetic field 212 • U(1) gauge invariance induced by twists 212 • U(1)

gauge interactions induced by twists 214 • The Lagrangian of QED 215 • Feynman

diagrams and renormalization 215 • Maxwell’s equations 218 • Curiosities and fun

challenges about QED 220 • Summary on QED and experimental predictions 221

223 The weak nuclear interaction and the second Reidemeister move

Strands, pokes and SU(2) 224 • Weak charge and parity violation 225 • Weak

bosons 227 • The Lagrangian of the unbroken SU(2) gauge interaction 228 •

SU(2) breaking 228 • The electroweak Lagrangian 230 • The weak Feynman diagrams 231 • Fun challenges and curiosities about the weak interaction 231 • Summary on the weak interaction and experimental predictions 233

235 The strong nuclear interaction and the third Reidemeister move

Strands and the slide, the third Reidemeister move 235 • From slides to SU(3) 236

• Open challenge: Find a better argument for the gluon tangle 240 • The gluon

Lagrangian 240 • Colour charge 241 • Properties of the strong interaction 243 •

The Lagrangian of QCD 243 • Renormalization of the strong interaction 243 • Curiosities and fun challenges about SU(3) 244 • Summary on the strong interaction

and experimental predictions 244

246 Summary on millennium issues: gauge interactions

Prediction about the number of interactions 246 • Unification of interactions 246

• Predictions about grand unification and supersymmetry 247 • No new observable gravity effects in particle physics 247 • The status of our quest 248

249 10 General relativity deduced from strands

Flat space, special relativity and its limitations 249 • Classical gravitation 250

• Deducing universal gravitation from black hole properties 251 • Summary on

universal gravitation from strands 252 • Curved space 253 • Horizons and black

holes 254 • Is there something behind a horizon? 255 • Energy of black hole horizons 255 • The nature of black holes 256 • Entropy of horizons 256 • Temperature,

contents 13

radiation and evaporation of black holes 258 • Black hole limits 259 • Curvature

around black holes 260 • The field equations of general relativity 261 • Equations

from no equation 262 • The Hilbert action of general relativity 263 • Space-time

foam 263 • Gravitons and gravitational waves 263 • Open challenge: Improve the

argument for the graviton tangle 264 • Other defects in vacuum 264 • Torsion,

curiosities and challenges about general relativity 265 • Predictions of the strand

model about general relativity 268

269 Cosmology

The finiteness of the universe 269 • The big bang 271 • The cosmological

constant 272 • The value of the matter density 273 • Open challenge: Are the

conventional energy and matter densities correct? 274 • The topology of the universe 274 • Predictions of the strand model about cosmology 274 • Summary on

millennium issues: relativity and cosmology 275

277 11 The particle spectrum deduced from strands

Particles and quantum numbers from tangles 277

279 Particles made of one strand

Unknotted curves 279 • Gauge bosons 280 • Complicated knots 281 • Closed

tangles: knots 282 • Summary on tangles made of one strand 282

282 Particles made of two strands

Quarks 284 • Quark generations 286 • The graviton 287 • Glueballs 287 • The

mass gap problem and the Clay Mathematics Institute 288 • A puzzle 289 • Summary on two-stranded tangles 289

289 Particles made of three strands

Leptons 290 • Open challenge: Find better arguments for the lepton tangles 292 •

The Higgs boson – in 2009 292 • The Higgs boson – summer 2012 update 294 •

Quark-antiquark mesons 295 • Meson form factors 297 • Meson masses, excited

mesons and quark confinement 299 • CP violation in mesons 300 • Other threestranded tangles and glueballs 302 • Summary on three-stranded tangles 302

302 Tangles of four and more strands

Baryons 303 • Tetraquarks and exotic mesons 303 • Other tangles made of four

or more strands 305 • Summary on tangles made of four or more strands 307

308 Fun challenges and curiosities about particle tangles

Motion through the vacuum – and the speed of light 309

313 Summary on millennium issues and predictions about particles

Predictions about dark matter and the LHC 314

315 12Part icle propert ies deduced from strands

315 The masses of the elementary particles

General properties of particle mass values 316 • Boson mass ratios and the weak

mixing angle 316 • Quark mass ratios 318 • Lepton mass ratios 320 • Mass ratios across particle families 321 • Predictions about absolute mass values and the

mass hierarchy 322 • Open issue: calculate masses ab initio 324 • Summary on

elementary particle masses and millennium issues 325

326 Mixing angles

Quark mixing 326 • A challenge 328 • CP-violation in quarks 328 • Neutrino

mixing 328 • CP-violation in neutrinos 329 • Open challenge: Calculate mixing

angles and phases ab initio 330 • Summary on mixing angles and the millennium

list 330

331 Coupling constants and unification

14 contents

Strands imply unification 333 • General expectations about coupling constants 333

• First hint: charge quantization and topological writhe 334 • Second hint: the

energy dependence of physical quantities 335 • Third hint: the running of the coupling constants at low energy 335 • Fourth hint: predictions at low energy, independent of particle content 336 • The running of the coupling constants near Planck

energy 337 • On estimating the fine structure constant from knot shapes 337 •

Fifth hint: 3d-writhe 338 • Sixth hint: torsion 339 • Seventh hint: linking number 339 • Eighth hint: estimating the fine structure constant from phase effects 339

• Ninth hint: a calculation approach for two coupling constants 340 • Open challenge: Calculate coupling constants ab initio 340 • Summary on coupling constants and millennium issues 341

342 The final summary on the millennium issues

343 Experimental predictions of the strand model

346 13 The top of Motion Mountain

346 Our path to the top

Everyday life: the rule of infinity 346 • Relativity and quantum theory: the absence

of infinity 347 • Unification: the absence of finitude 349

350 New sights

The beauty of strands 350 • Can the strand model be generalized? 351 • What

is nature? 352 • Quantum theory and the nature of matter 353 • Cosmology 353

• Musings about unification and strands 354 • The elimination of induction 358

• What is still hidden? 359

359 A return path: je rêve, donc je suis

361 What is motion?

363 Postface

364 a Knot geometry

367 Challenge hints and solutions

371 Bibliography

392 Credits

Acknowledgments 392 • Film credits 392 • Image credits 393

394 Name index

400 Subject index

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 interactions appear naturally,

and that a complete description of motion is possible.

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