Roger Penrose THE ROAD TO REALITY A Complete Guide to the Laws of the Universe UNIVERSIT4T ST. GALLEN HOCHSCHULE FUR WIRTSCHAFTS-, RECHTS- UND SOZlALWtSSENSCHAFTEN BIBLIOTHEK JONATHAN CAPE LONDON
Preface Acknowledgements No tat ion Prologue 1 The roots of science 1.1 The quest for the forces that shape the world 1.2 Mathematical truth 1.3 Is Plato s mathematical world real? 1.4 Three worlds and three deep mysteries 1.5 The Good, the True, and the Beautiful 2 An ancient theorem and a modern question 2.1 The Pythagorean theorem 2.2 Euclid s postulates 2.3 Similar-areas proof of the Pythagorean theorem 2.4 Hyperbolic geometry: conformal picture 2.5 Other representations of hyperbolic geometry 2.6 Historical aspects of hyperbolic geometry 2.7 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 The magic number i 4.2 Solving equations with complex numbers xv xxiii xxvi 1 7 7 9 12 17 22 25 25 28 31 33 37 42 46 51 51 54 59 63 65 71 71 74 V
4.3 Convergence of power series 4.4 Caspar Wessel s complex plane 4.5 How to construct the Mandelbrot set 5 Geometry of logarithms, powers, and roots 5.1 Geometry of complex algebra 5.2 The idea of the complex logarithm 5.3 Multiple valuedness, natural logarithms 5.4 Complex powers 5.5 Some relations to modern particle physics 6 Real-number calculus 6.1 What makes an honest function? 6.2 Slopes of functions 6.3 Higher derivatives; Coo-smooth functions 6.4 The Eulerian notion of a function? 6.5 The rules of differentiation 6.6 Integration 7 Complex-number calculus 7.1 Complex smoothness; holomorphic functions 7.2 Contour integration 7.3 Power series from complex smoothness 7.4 Analytic continuation 8 Riemann surfaces and complex mappings 8.1 The idea of a Riemann surface 8.2 Conformal mappings 8.3 The Riemann sphere 8.4 The genus of a compact Riemann surface 8.5 The Riemann mapping theorem 9 Fourier decomposition and hyperfunctions 9.1 Fourier series 9.2 Functions on a circle 9.3 Frequency splitting on the Riemann sphere 9.4 The Fourier transform 9.5 Frequency splitting from the Fourier transform 9.6 What kind of function is appropriate? 9.7 Hyperfunctions 76 81 83 86 86 90 92 96 100 103 103 105 107 112 114 116 122 122 123 127 129 135 135 138 142 145 148 153 153 157 161 164 166 168 172 vi
10 Surfaces 179 10.1 10.2 Complex dimensions and real dimensions Smoothness, partial derivatives 10.3 Vector fields and 1-forms 10.4 Components, scalar products 10.5 The Cauchy-Riemann equations 179 181 185 190 193 11 Hypercomplex numbers 11.1 The algebra of quaternions 11.2 11.3 The physical role of quaternions? Geometry of quaternions 11.4 How to compose rotations 11.5 Clifford algebras 11.6 Grassmann algebras 12 Manifolds of n dimensions 12.1 Why study higher-dimensional manifolds? 12.2 Manifolds and coordinate patches 12.3 Scalars, vectors, and covectors 12.4 Grassmann products 12.5 Integrals of forms 12.6 Exterior derivative 12.7 Volume element; summation convention 12.8 12.9 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 Differentiation on a manifold? 14.2 Parallel transport 14.3 Covariant derivative 14.4 Curvature and torsion 198 198 200 203 206 208 21 1 217 217 22 1 223 227 229 23 1 237 239 243 241 247 250 254 260 263 266 270 275 28 1 286 292 292 294 298 301 vii
14.5 Geodesics, parallelograms, and curvature 14.6 Lie derivative 14.7 What a metric can do for you 14.8 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 fibre bundles The mathematical idea of a bundle Cross-sections of bundles The Clifford bundle Complex vector bundles, (co)tangent bundles Projective spaces Non-triviality in a bundle connection Bundle curvature 16 The ladder of infinity 16.1 Finite fields 16.2 A finite or infinite geometry for physics? 16.3 Different sizes of infinity 16.4 Cantor s diagonal slash 16.5 Puzzles in the foundations of mathematics 16.6 Turing machines and Godel s theorem 16.7 Sizes of infinity 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 fixed finite speed of light Light cones The abandonment of absolute time The spacetime for Einstein s general relativity 18 Minkowskian geometry 18.1 Euclidean and Minkowskian 4-space 18.2 The symmetry groups of Minkowski space 18.3 Lorentzian orthogonality; the clock paradox 18.4 Hyperbolic geometry in Minkowski space 18.5 The celestial sphere as a Riemann sphere 18.6 Newtonian energy and (angular) momentum 18.7 Relativistic energy and (angular) momentum 303 309 317 32 1 325 325 328 33 1 334 338 34 1 345 349 357 357 359 364 367 37 1 374 378 383 383 385 388 390 394 399 40 1 404 408 412 412 415 417 422 428 43 1 434 viii
19 The classical fields 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 flux laws in Maxwell theory The Maxwell field as gauge curvature The energy-momentum tensor Einstein s field equation Further issues: cosmological constant; Weyl tensor Gravitational field energy 20 Lagrangians and Hamiltonians 20.1 The magical Lagrangian formalism 20.2 20.3 The more symmetrical Hamiltonian picture Small oscillations 20.4 20.5 Hamiltonian dynamics as symplectic geometry Lagrangian treatment of fields 20.6 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-part icle 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 22.1 1 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 47 1 47 1 475 478 483 486 489 493 493 496 498 500 505 507 511 516 517 520 521 527 527 530 533 535 538 542 544 549 553 559 562 IX
22.12 Relativistic quantum angular momentum 22.13 The general isolated quantum object 23 The entangled quantum world 23.1 Quantum mechanics of many-particle systems 23.2 Hugeness of many-particle state space 23.3 Quantum entanglement; Bell inequalities 23.4 Bohm-type EPR experiments 23.5 Hardy s EPR example: almost probability-free 23.6 Two mysteries of quantum entanglement 23.7 Bosons and fermions 23.8 The quantum states of bosons and fermions 23.9 Quantum teleportation 23.10 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 fields? Energy positivity in quantum mechanics Difficulties with the relativistic energy formula The non-invariance of d/dt Clifford-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; reflection asymmetry Charge conjugation, parity, and time reversal The electroweak symmetry group Strongly interacting particles Coloured quarks Beyond the standard model? 26 Quantum field theory 26.1 26.2 26.3 26.4 26.5 26.6 26.7 26.8 26.9 Fundamental status of QFT in modern theory Creation and annihilation operators Infinite-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 59 1 594 596 598 603 609 609 610 612 614 616 618 620 622 627 627 628 632 638 640 645 648 651 655 655 657 660 662 664 665 670 672 675 X
26.10 Feynman graphs from Lagrangians 26.11 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 Inflationary cosmology Are the motivations for inflation 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 i: the Bloch sphere 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 68 1 686 686 688 690 692 696 699 702 707 712 714 717 723 726 735 735 739 742 746 753 757 762 765 769 772 782 782 78 5 79 1 793 797 802 804 806 810 816 816 817 xi
Co nten ts 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 flow-and time travel! Energy outflow from negative-energy orbits Hawking explosions A more radical perspective Schrodinger s lump Fundamental conflict with Einstein s principles Preferred Schrodinger-Newton states? FELIX and related proposals Origin of fluctuations in the early universe 819 823 827 833 836 838 842 846 849 853 856 86 1 3 1 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 31.1 1 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 finite? 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 Canonical quantum gravity 32.2 The chiral input to Ashtekar s variables 32.3 The form of Ashtekar s variables 32.4 Loop variables 32.5 The mathematics of knots and links 32.6 Spin networks 32.7 Status of loop quantum gravity? 33 More radical perspectives; twistor theory 33.1 Theories where geometry has discrete elements 33.2 Twistors as light rays 869 869 873 877 880 884 887 890 892 895 897 902 905 907 910 916 920 923 926 934 934 935 938 94 1 943 946 952 958 958 962 xii
33.3 33.4 33.5 33.6 33.7 33.8 33.9 33.10 33.1 1 33.12 33.13 33.14 Conformal group; compactified 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 fields Twistor sheaf cohomology Twistors and positivehegative frequency splitting The non-linear graviton Twistors and general relativity Towards a twistor theory of particle physics The future of twistor theory? 968 972 974 978 982 985 987 993 995 1000 1001 1003 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 Epilogue Bibliography Index 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 1010 1010 1014 1017 1020 1024 1027 1030 1033 1038 1043 1048 1050 1081 xiii