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The Road to Reality by Roger Penrose

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The Road to Reality by Roger Penrose

The Road to Reality by Roger Penrose

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  • 1. THE ROAD TO REALITY
  • 2. BY ROGER PENROSE The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics Shadows of the Mind: A Search for the Missing Science of Consciousness
  • 3. Roger Penrose T H E R O A D TO REALITY A Complete Guide to the Laws of the Universe JONATHAN CAPE LONDON
  • 4. Published by Jonathan Cape 2004 2 4 6 8 10 9 7 5 3 1 Copyright ß Roger Penrose 2004 Roger Penrose has asserted his right under the Copyright, Designs and Patents Act 1988 to be identified as the author of this work This book is sold subject to the condition that it shall not, by way of trade or otherwise, be lent, resold, hired out, or otherwise circulated without the publisher’s prior consent in any form of binding or cover other than that in which it is published and without a similar condition including this condition being imposed on the subsequent purchaser First published in Great Britain in 2004 by Jonathan Cape Random House, 20 Vauxhall Bridge Road, London SW1V 2SA Random House Australia (Pty) Limited 20 Alfred Street, Milsons Point, Sydney, New South Wales 2061, Australia Random House New Zealand Limited 18 Poland Road, Glenfield, Auckland 10, New Zealand Random House South Africa (Pty) Limited Endulini, 5A Jubilee Road, Parktown 2193, South Africa The Random House Group Limited Reg. No. 954009 www.randomhouse.co.uk A CIP catalogue record for this book is available from the British Library ISBN 0–224–04447–8 Papers used by The Random House Group Limited are natural, recyclable products made from wood grown in sustainable forests; the manufacturing processes conform to the environmental regulations of the country of origin Printed and bound in Great Britain by William Clowes, Beccles, Suffolk
  • 5. Contents Preface xv Acknowledgements xxiii Notation xxvi Prologue 1 1 The roots of science 7 1.1 1.2 1.3 1.4 1.5 The quest for the forces that shape the world Mathematical truth Is Plato’s mathematical world ‘real’? Three worlds and three deep mysteries The Good, the True, and the Beautiful 2 An ancient theorem and a modern question 2.1 2.2 2.3 2.4 2.5 2.6 2.7 The Pythagorean theorem Euclid’s postulates Similar-areas proof of the Pythagorean theorem Hyperbolic geometry: conformal picture Other representations of hyperbolic geometry Historical aspects of hyperbolic geometry Relation to physical space 3 Kinds of number in the physical world 3.1 3.2 3.3 3.4 3.5 A Pythagorean catastrophe? The real-number system Real numbers in the physical world Do natural numbers need the physical world? Discrete numbers in the physical world 4 Magical complex numbers 4.1 4.2 7 9 12 17 22 25 25 28 31 33 37 42 46 51 51 54 59 63 65 71 The magic number ‘i’ Solving equations with complex numbers v 71 74
  • 6. Contents 4.3 4.4 4.5 Convergence of power series Caspar Wessel’s complex plane How to construct the Mandelbrot set 5 Geometry of logarithms, powers, and roots 5.1 5.2 5.3 5.4 5.5 Geometry of complex algebra The idea of the complex logarithm Multiple valuedness, natural logarithms Complex powers Some relations to modern particle physics 6 Real-number calculus 6.1 6.2 6.3 6.4 6.5 6.6 What makes an honest function? Slopes of functions Higher derivatives; C1 -smooth functions The ‘Eulerian’ notion of a function? The rules of diVerentiation Integration 7 Complex-number calculus 7.1 7.2 7.3 7.4 Complex smoothness; holomorphic functions Contour integration Power series from complex smoothness Analytic continuation 8 Riemann surfaces and complex mappings 8.1 8.2 8.3 8.4 8.5 76 81 83 86 86 90 92 96 100 103 103 105 107 112 114 116 122 122 123 127 129 135 The idea of a Riemann surface Conformal mappings The Riemann sphere The genus of a compact Riemann surface The Riemann mapping theorem 135 138 142 145 148 9 Fourier decomposition and hyperfunctions 153 9.1 9.2 9.3 9.4 9.5 9.6 9.7 vi Fourier series Functions on a circle Frequency splitting on the Riemann sphere The Fourier transform Frequency splitting from the Fourier transform What kind of function is appropriate? Hyperfunctions 153 157 161 164 166 168 172
  • 7. Contents 10 Surfaces 179 10.1 10.2 10.3 10.4 10.5 179 181 185 190 193 Complex dimensions and real dimensions Smoothness, partial derivatives Vector Welds and 1-forms Components, scalar products The Cauchy–Riemann equations 11 Hypercomplex numbers 11.1 11.2 11.3 11.4 11.5 11.6 The algebra of quaternions The physical role of quaternions? Geometry of quaternions How to compose rotations CliVord algebras Grassmann algebras 12 Manifolds of n dimensions 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9 Why study higher-dimensional manifolds? Manifolds and coordinate patches Scalars, vectors, and covectors Grassmann products Integrals of forms Exterior derivative Volume element; summation convention Tensors; abstract-index and diagrammatic notation Complex manifolds 13 Symmetry groups 13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8 13.9 13.10 Groups of transformations Subgroups and simple groups Linear transformations and matrices Determinants and traces Eigenvalues and eigenvectors Representation theory and Lie algebras Tensor representation spaces; reducibility Orthogonal groups Unitary groups Symplectic groups 14 Calculus on manifolds 14.1 14.2 14.3 14.4 DiVerentiation on a manifold? Parallel transport Covariant derivative Curvature and torsion 198 198 200 203 206 208 211 217 217 221 223 227 229 231 237 239 243 247 247 250 254 260 263 266 270 275 281 286 292 292 294 298 301 vii
  • 8. Contents 14.5 14.6 14.7 14.8 Geodesics, parallelograms, and curvature Lie derivative What a metric can do for you Symplectic manifolds 15 Fibre bundles and gauge connections 15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8 Some physical motivations for Wbre bundles The mathematical idea of a bundle Cross-sections of bundles The CliVord bundle Complex vector bundles, (co)tangent bundles Projective spaces Non-triviality in a bundle connection Bundle curvature 16 The ladder of inWnity 16.1 16.2 16.3 16.4 16.5 16.6 16.7 Finite Welds A Wnite or inWnite geometry for physics? DiVerent sizes of inWnity Cantor’s diagonal slash Puzzles in the foundations of mathematics ¨ Turing machines and Godel’s theorem Sizes of inWnity in physics 17 Spacetime 17.1 17.2 17.3 17.4 17.5 17.6 17.7 17.8 17.9 The spacetime of Aristotelian physics Spacetime for Galilean relativity Newtonian dynamics in spacetime terms The principle of equivalence Cartan’s ‘Newtonian spacetime’ The Wxed Wnite speed of light Light cones The abandonment of absolute time The spacetime for Einstein’s general relativity 18 Minkowskian geometry 18.1 18.2 18.3 18.4 18.5 18.6 18.7 viii Euclidean and Minkowskian 4-space The symmetry groups of Minkowski space Lorentzian orthogonality; the ‘clock paradox’ Hyperbolic geometry in Minkowski space The celestial sphere as a Riemann sphere Newtonian energy and (angular) momentum Relativistic energy and (angular) momentum 303 309 317 321 325 325 328 331 334 338 341 345 349 357 357 359 364 367 371 374 378 383 383 385 388 390 394 399 401 404 408 412 412 415 417 422 428 431 434
  • 9. Contents 19 The classical Welds of Maxwell and Einstein 19.1 19.2 19.3 19.4 19.5 19.6 19.7 19.8 Evolution away from Newtonian dynamics Maxwell’s electromagnetic theory Conservation and Xux laws in Maxwell theory The Maxwell Weld as gauge curvature The energy–momentum tensor Einstein’s Weld equation Further issues: cosmological constant; Weyl tensor Gravitational Weld energy 20 Lagrangians and Hamiltonians 20.1 20.2 20.3 20.4 20.5 20.6 The magical Lagrangian formalism The more symmetrical Hamiltonian picture Small oscillations Hamiltonian dynamics as symplectic geometry Lagrangian treatment of Welds How Lagrangians drive modern theory 21 The quantum particle 21.1 21.2 21.3 21.4 21.5 21.6 21.7 21.8 21.9 21.10 21.11 Non-commuting variables Quantum Hamiltonians ¨ Schrodinger’s equation Quantum theory’s experimental background Understanding wave–particle duality What is quantum ‘reality’? The ‘holistic’ nature of a wavefunction The mysterious ‘quantum jumps’ Probability distribution in a wavefunction Position states Momentum-space description 22 Quantum algebra, geometry, and spin 22.1 22.2 22.3 22.4 22.5 22.6 22.7 22.8 22.9 22.10 22.11 The quantum procedures U and R The linearity of U and its problems for R Unitary structure, Hilbert space, Dirac notation ¨ Unitary evolution: Schrodinger and Heisenberg Quantum ‘observables’ yes/no measurements; projectors Null measurements; helicity Spin and spinors The Riemann sphere of two-state systems Higher spin: Majorana picture Spherical harmonics 440 440 442 446 449 455 458 462 464 471 471 475 478 483 486 489 493 493 496 498 500 505 507 511 516 517 520 521 527 527 530 533 535 538 542 544 549 553 559 562 ix
  • 10. Contents 22.12 22.13 Relativistic quantum angular momentum The general isolated quantum object 23 The entangled quantum world 23.1 23.2 23.3 23.4 23.5 23.6 23.7 23.8 23.9 23.10 Quantum mechanics of many-particle systems Hugeness of many-particle state space Quantum entanglement; Bell inequalities Bohm-type EPR experiments Hardy’s EPR example: almost probability-free Two mysteries of quantum entanglement Bosons and fermions The quantum states of bosons and fermions Quantum teleportation Quanglement 24 Dirac’s electron and antiparticles 24.1 24.2 24.3 24.4 24.5 24.6 24.7 24.8 Tension between quantum theory and relativity Why do antiparticles imply quantum Welds? Energy positivity in quantum mechanics DiYculties with the relativistic energy formula The non-invariance of ]=]t CliVord–Dirac square root of wave operator The Dirac equation Dirac’s route to the positron 25 The standard model of particle physics 25.1 25.2 25.3 25.4 25.5 25.6 25.7 25.8 The origins of modern particle physics The zigzag picture of the electron Electroweak interactions; reXection asymmetry Charge conjugation, parity, and time reversal The electroweak symmetry group Strongly interacting particles ‘Coloured quarks’ Beyond the standard model? 26 Quantum Weld theory 26.1 26.2 26.3 26.4 26.5 26.6 26.7 26.8 26.9 x Fundamental status of QFT in modern theory Creation and annihilation operators InWnite-dimensional algebras Antiparticles in QFT Alternative vacua Interactions: Lagrangians and path integrals Divergent path integrals: Feynman’s response Constructing Feynman graphs; the S-matrix Renormalization 566 570 578 578 580 582 585 589 591 594 596 598 603 609 609 610 612 614 616 618 620 622 627 627 628 632 638 640 645 648 651 655 655 657 660 662 664 665 670 672 675
  • 11. Contents 26.10 26.11 Feynman graphs from Lagrangians Feynman graphs and the choice of vacuum 27 The Big Bang and its thermodynamic legacy 27.1 27.2 27.3 27.4 27.5 27.6 27.7 27.8 27.9 27.10 27.11 27.12 27.13 Time symmetry in dynamical evolution Submicroscopic ingredients Entropy The robustness of the entropy concept Derivation of the second law—or not? Is the whole universe an ‘isolated system’? The role of the Big Bang Black holes Event horizons and spacetime singularities Black-hole entropy Cosmology Conformal diagrams Our extraordinarily special Big Bang 28 Speculative theories of the early universe 28.1 28.2 28.3 28.4 28.5 28.6 28.7 28.8 28.9 28.10 Early-universe spontaneous symmetry breaking Cosmic topological defects Problems for early-universe symmetry breaking InXationary cosmology Are the motivations for inXation valid? The anthropic principle The Big Bang’s special nature: an anthropic key? The Weyl curvature hypothesis The Hartle–Hawking ‘no-boundary’ proposal Cosmological parameters: observational status? 29 The measurement paradox 29.1 29.2 29.3 29.4 29.5 29.6 29.7 29.8 29.9 The conventional ontologies of quantum theory Unconventional ontologies for quantum theory The density matrix Density matrices for spin 1: the Bloch sphere 2 The density matrix in EPR situations FAPP philosophy of environmental decoherence ¨ Schrodinger’s cat with ‘Copenhagen’ ontology Can other conventional ontologies resolve the ‘cat’? Which unconventional ontologies may help? 30 Gravity’s role in quantum state reduction 30.1 30.2 Is today’s quantum theory here to stay? Clues from cosmological time asymmetry 680 681 686 686 688 690 692 696 699 702 707 712 714 717 723 726 735 735 739 742 746 753 757 762 765 769 772 782 782 785 791 793 797 802 804 806 810 816 816 817 xi
  • 12. Contents 30.3 30.4 30.5 30.6 30.7 30.8 30.9 30.10 30.11 30.12 30.13 30.14 Time-asymmetry in quantum state reduction Hawking’s black-hole temperature Black-hole temperature from complex periodicity Killing vectors, energy Xow—and time travel! Energy outXow from negative-energy orbits Hawking explosions A more radical perspective ¨ Schrodinger’s lump Fundamental conXict with Einstein’s principles ¨ Preferred Schrodinger–Newton states? FELIX and related proposals Origin of Xuctuations in the early universe 31 Supersymmetry, supra-dimensionality, and strings 31.1 31.2 31.3 31.4 31.5 31.6 31.7 31.8 31.9 31.10 31.11 31.12 31.13 31.14 31.15 31.16 31.17 31.18 Unexplained parameters Supersymmetry The algebra and geometry of supersymmetry Higher-dimensional spacetime The original hadronic string theory Towards a string theory of the world String motivation for extra spacetime dimensions String theory as quantum gravity? String dynamics Why don’t we see the extra space dimensions? Should we accept the quantum-stability argument? Classical instability of extra dimensions Is string QFT Wnite? The magical Calabi–Yau spaces; M-theory Strings and black-hole entropy The ‘holographic principle’ The D-brane perspective The physical status of string theory? 32 Einstein’s narrower path; loop variables 32.1 32.2 32.3 32.4 32.5 32.6 32.7 Canonical quantum gravity The chiral input to Ashtekar’s variables The form of Ashtekar’s variables Loop variables The mathematics of knots and links Spin networks Status of loop quantum gravity? 33 More radical perspectives; twistor theory 33.1 33.2 xii Theories where geometry has discrete elements Twistors as light rays 819 823 827 833 836 838 842 846 849 853 856 861 869 869 873 877 880 884 887 890 892 895 897 902 905 907 910 916 920 923 926 934 934 935 938 941 943 946 952 958 958 962
  • 13. Contents 33.3 33.4 33.5 33.6 33.7 33.8 33.9 33.10 33.11 33.12 33.13 33.14 Conformal group; compactiWed Minkowski space Twistors as higher-dimensional spinors Basic twistor geometry and coordinates Geometry of twistors as spinning massless particles Twistor quantum theory Twistor description of massless Welds Twistor sheaf cohomology Twistors and positive/negative frequency splitting The non-linear graviton Twistors and general relativity Towards a twistor theory of particle physics The future of twistor theory? 34 Where lies the road to reality? 34.1 34.2 34.3 34.4 34.5 34.6 34.7 34.8 34.9 34.10 Great theories of 20th century physics—and beyond? Mathematically driven fundamental physics The role of fashion in physical theory Can a wrong theory be experimentally refuted? Whence may we expect our next physical revolution? What is reality? The roles of mentality in physical theory Our long mathematical road to reality Beauty and miracles Deep questions answered, deeper questions posed 968 972 974 978 982 985 987 993 995 1000 1001 1003 1010 1010 1014 1017 1020 1024 1027 1030 1033 1038 1043 Epilogue 1048 Bibliography 1050 Index 1081 xiii
  • 14. I dedicate this book to the memory of DENNIS SCIAMA who showed me the excitement of physics
  • 15. Preface The purpose of this book is to convey to the reader some feeling for what is surely one of the most important and exciting voyages of discovery that humanity has embarked upon. This is the search for the underlying principles that govern the behaviour of our universe. It is a voyage that has lasted for more than two-and-a-half millennia, so it should not surprise us that substantial progress has at last been made. But this journey has proved to be a profoundly diYcult one, and real understanding has, for the most part, come but slowly. This inherent diYculty has led us in many false directions; hence we should learn caution. Yet the 20th century has delivered us extraordinary new insights—some so impressive that many scientists of today have voiced the opinion that we may be close to a basic understanding of all the underlying principles of physics. In my descriptions of the current fundamental theories, the 20th century having now drawn to its close, I shall try to take a more sober view. Not all my opinions may be welcomed by these ‘optimists’, but I expect further changes of direction greater even than those of the last century. The reader will Wnd that in this book I have not shied away from presenting mathematical formulae, despite dire warnings of the severe reduction in readership that this will entail. I have thought seriously about this question, and have come to the conclusion that what I have to say cannot reasonably be conveyed without a certain amount of mathematical notation and the exploration of genuine mathematical concepts. The understanding that we have of the principles that actually underlie the behaviour of our physical world indeed depends upon some appreciation of its mathematics. Some people might take this as a cause for despair, as they will have formed the belief that they have no capacity for mathematics, no matter at how elementary a level. How could it be possible, they might well argue, for them to comprehend the research going on at the cutting edge of physical theory if they cannot even master the manipulation of fractions? Well, I certainly see the diYculty. xv
  • 16. Preface Yet I am an optimist in matters of conveying understanding. Perhaps I am an incurable optimist. I wonder whether those readers who cannot manipulate fractions—or those who claim that they cannot manipulate fractions—are not deluding themselves at least a little, and that a good proportion of them actually have a potential in this direction that they are not aware of. No doubt there are some who, when confronted with a line of mathematical symbols, however simply presented, can see only the stern face of a parent or teacher who tried to force into them a non-comprehending parrot-like apparent competence—a duty, and a duty alone—and no hint of the magic or beauty of the subject might be allowed to come through. Perhaps for some it is too late; but, as I say, I am an optimist and I believe that there are many out there, even among those who could never master the manipulation of fractions, who have the capacity to catch some glimpse of a wonderful world that I believe must be, to a signiWcant degree, genuinely accessible to them. One of my mother’s closest friends, when she was a young girl, was among those who could not grasp fractions. This lady once told me so herself after she had retired from a successful career as a ballet dancer. I was still young, not yet fully launched in my activities as a mathematician, but was recognized as someone who enjoyed working in that subject. ‘It’s all that cancelling’, she said to me, ‘I could just never get the hang of cancelling.’ She was an elegant and highly intelligent woman, and there is no doubt in my mind that the mental qualities that are required in comprehending the sophisticated choreography that is central to ballet are in no way inferior to those which must be brought to bear on a mathematical problem. So, grossly overestimating my expositional abilities, I attempted, as others had done before, to explain to her the simplicity and logical nature of the procedure of ‘cancelling’. I believe that my eVorts were as unsuccessful as were those of others. (Incidentally, her father had been a prominent scientist, and a Fellow of the Royal Society, so she must have had a background adequate for the comprehension of scientiWc matters. Perhaps the ‘stern face’ could have been a factor here, I do not know.) But on reXection, I now wonder whether she, and many others like her, did not have a more rational hang-up—one that with all my mathematical glibness I had not noticed. There is, indeed, a profound issue that one comes up against again and again in mathematics and in mathematical physics, which one Wrst encounters in the seemingly innocent operation of cancelling a common factor from the numerator and denominator of an ordinary numerical fraction. Those for whom the action of cancelling has become second nature, because of repeated familiarity with such operations, may Wnd themselves insensitive to a diYculty that actually lurks behind this seemingly simple xvi
  • 17. Preface procedure. Perhaps many of those who Wnd cancelling mysterious are seeing a certain profound issue more deeply than those of us who press on