Road to Reality

Cover Page......Page 1
Title Page......Page 4
4 Magical complex numbers......Page 6
9 Fourier decomposition and hyperfunctions......Page 7
14 Calculus on manifolds......Page 8
18 Minkowskian geometry......Page 9
22 Quantum algebra, geometry, and spin......Page 10
26 Quantum Weld theory......Page 11
30 Gravity’s role in quantum state reduction......Page 12
33 More radical perspectives; twistor theory......Page 13
Index......Page 14
Preface......Page 16
Acknowledgements......Page 24
Notation......Page 27
ISBN 0224044478......Page 5
Prologue......Page 30
1.1 The quest for the forces that shape the world......Page 36
1.2 Mathematical truth......Page 38
1.3 Is Plato’s mathematical world ‘real’?......Page 41
1.4 Three worlds and three deep mysteries......Page 46
1.5 The Good, the True, and the Beautiful......Page 51
2.1 The Pythagorean theorem......Page 54
2.2 Euclid’s postulates......Page 57
2.3 Similar-areas proof of the Pythagorean theorem......Page 60
2.4 Hyperbolic geometry: conformal picture......Page 62
2.5 Other representations of hyperbolic geometry......Page 66
2.6 Historical aspects of hyperbolic geometry......Page 71
2.7 Relation to physical space......Page 75
3.1 A Pythagorean catastrophe?......Page 80
3.2 The real-number system......Page 83
3.3 Real numbers in the physical world......Page 88
3.4 Do natural numbers need the physical world?......Page 92
3.5 Discrete numbers in the physical world......Page 94
4.1 The magic number ‘i’......Page 100
4.2 Solving equations with complex numbers......Page 103
4.3 Convergence of power series......Page 105
4.4 Caspar Wessel’s complex plane......Page 110
4.5 How to construct the Mandelbrot set......Page 112
5.1 Geometry of complex algebra......Page 115
5.2 The idea of the complex logarithm......Page 119
5.3 Multiple valuedness, natural logarithms......Page 121
5.4 Complex powers......Page 125
5.5 Some relations to modern particle physics......Page 129
6.1 What makes an honest function?......Page 132
6.2 Slopes of functions......Page 134
smooth functions......Page 136
6.4 The ‘Eulerian’ notion of a function?......Page 141
6.5 The rules of differentiation......Page 143
6.6 Integration......Page 145
7.1 Complex smoothness; holomorphic functions......Page 151
7.2 Contour integration......Page 152
7.3 Power series from complex smoothness......Page 156
7.4 Analytic continuation......Page 158
8.1 The idea of a Riemann surface......Page 164
8.2 Conformal mappings......Page 167
8.3 The Riemann sphere......Page 171
8.4 The genus of a compact Riemann surface......Page 174
8.5 The Riemann mapping theorem......Page 177
9.1 Fourier series......Page 182
9.2 Functions on a circle......Page 186
9.3 Frequency splitting on the Riemann sphere......Page 190
9.4 The Fourier transform......Page 193
9.5 Frequency splitting from the Fourier transform......Page 195
9.6 What kind of function is appropriate?......Page 197
9.7 Hyperfunctions......Page 201
10.1 Complex dimensions and real dimensions......Page 208
10.2 Smoothness, partial derivatives......Page 210
10.3 Vector fields and 1-forms......Page 214
10.4 Components, scalar products......Page 219
10.5 The Cauchy–Riemann equations......Page 222
11.1 The algebra of quaternions......Page 227
11.2 The physical role of quaternions?......Page 229
11.3 Geometry of quaternions......Page 232
11.4 How to compose rotations......Page 235
11.5 Clifford algebras......Page 237
11.6 Grassmann algebras......Page 240
12.1 Why study higher-dimensional manifolds?......Page 246
12.2 Manifolds and coordinate patches......Page 250
12.3 Scalars, vectors, and covectors......Page 252
12.4 Grassmann products......Page 256
12.5 Integrals of forms......Page 258
12.6 Exterior derivative......Page 260
12.7 Volume element; summation convention......Page 266
12.8 Tensors: abstract-index and diagrammatic notation......Page 268
12.9 Complex manifolds......Page 272
13.1 Groups of transformations......Page 276
13.2 Subgroups and simple groups......Page 279
13.3 Linear transformations and matrices......Page 283
13.4 Determinants and traces......Page 289
13.5 Eigenvalues and eigenvectors......Page 292
13.6 Representation theory and Lie algebras......Page 295
13.7 Tensor representation spaces; reducibility......Page 299
13.8 Orthogonal groups......Page 304
13.9 Unitary groups......Page 310
13.10 Symplectic groups......Page 315
14.1 DiVerentiation on a manifold?......Page 321
14.2 Parallel transport......Page 323
14.3 Covariant derivative......Page 327
14.4 Curvature and torsion......Page 330
14.5 Geodesics, parallelograms, and curvature......Page 332
14.6 Lie derivative......Page 338
14.7 What a metric can do for you......Page 346
14.8 Symplectic manifolds......Page 350
15.1 Some physical motivations for fibre bundles......Page 354
15.2 The mathematical idea of a bundle......Page 357
15.3 Cross-sections of bundles......Page 360
15.4 The CliVord bundle......Page 363
15.5 Complex vector bundles, (co)tangent bundles......Page 367
15.6 Projective spaces......Page 370
15.7 Non-triviality in a bundle connection......Page 374
15.8 Bundle curvature......Page 378
16.1 Finite fields......Page 386
16.2 A finite or infinite geometry for physics?......Page 388
16.3 Different sizes of inWnity......Page 393
16.4 Cantor’s diagonal slash......Page 396
16.5 Puzzles in the foundations of mathematics......Page 400
16.6 Turing machines and Go¨del’s theorem......Page 403
16.7 Sizes of infinity in physics......Page 407
17.1 The spacetime of Aristotelian physics......Page 412
17.2 Spacetime for Galilean relativity......Page 414
17.3 Newtonian dynamics in spacetime terms......Page 417
17.4 The principle of equivalence......Page 419
17.5 Cartan’s ‘Newtonian spacetime’......Page 423
17.6 The fixed finite speed of light......Page 428
17.7 Light cones......Page 430
17.8 The abandonment of absolute time......Page 433
17.9 The spacetime of Einstein’s general relativity......Page 437
18.1 Euclidean and Minkowskian 4-space......Page 441
18.2 The symmetry groups of Minkowski space......Page 444
18.3 Lorentzian orthogonality; the ‘clock paradox’......Page 446
18.4 Hyperbolic geometry in Minkowski space......Page 451
18.5 The celestial sphere as a Riemann sphere......Page 457
18.6 Newtonian energy and (angular) momentum......Page 460
18.7 Relativistic energy and (angular) momentum......Page 463
19.1 Evolution away from Newtonian dynamics......Page 469
19.2 Maxwell’s electromagnetic theory......Page 471
19.3 Conservation and flux laws in Maxwell theory......Page 475
19.4 The Maxwell field as gauge curvature......Page 478
19.5 The energy–momentum tensor......Page 484
19.6 Einstein’s field equation......Page 487
19.7 Further issues: cosmological constant; Weyl tensor......Page 491
19.8 Gravitational field energy......Page 493
20.1 The magical Lagrangian formalism......Page 500
20.2 The more symmetrical Hamiltonian picture......Page 504
20.3 Small oscillations......Page 507
20.4 Hamiltonian dynamics as symplectic geometry......Page 512
20.5 Lagrangian treatment of fields......Page 515
20.6 How Lagrangians drive modern theory......Page 518
21.1 Non-commuting variables......Page 522
21.2 Quantum Hamiltonians......Page 525
21.3 Schro¨dinger’s equation......Page 527
21.4 Quantum theory’s experimental background......Page 529
21.5 Understanding wave–particle duality......Page 534
21.6 What is quantum ‘reality’?......Page 536
21.7 The ‘holistic’ nature of a wavefunction......Page 540
21.8 The mysterious ‘quantum jumps’......Page 545
21.9 Probability distribution in a wavefunction......Page 546
21.10 Position states......Page 549
21.11 Momentum-space description......Page 550
22.1 The quantum procedures U and R......Page 556
22.2 The linearity of U and its problems for R......Page 559
22.3 Unitary structure, Hilbert space, Dirac notation......Page 562
22.4 Unitary evolution: Schro¨dinger and Heisenberg......Page 564
22.5 Quantum ‘observables’......Page 567
22.6 YES/NO measurements; projectors......Page 571
22.7 Null measurements; helicity......Page 573
22.8 Spin and spinors......Page 578
22.9 The Riemann sphere of two-state systems......Page 582
22.10 Higher spin: Majorana picture......Page 588
22.11 Spherical harmonics......Page 591
22.12 Relativistic quantum angular momentum......Page 595
22.13 The general isolated quantum object......Page 599
23.1 Quantum mechanics of many-particle systems......Page 607
23.2 Hugeness of many-particle state space......Page 609
23.3 Quantum entanglement; Bell inequalities......Page 611
23.4 Bohm-type EPR experiments......Page 614
23.5 Hardy’s EPR example: almost probability-free......Page 618
23.6 Two mysteries of quantum entanglement......Page 620
23.7 Bosons and fermions......Page 623
23.8 The quantum states of bosons and fermions......Page 625
23.9 Quantum teleportation......Page 627
23.10 Quanglement......Page 632
24.1 Tension between quantum theory and relativity......Page 638
24.2 Why do antiparticles imply quantum fields?......Page 639
24.3 Energy positivity in quantum mechanics......Page 641
24.4 Difficulties with the relativistic energy formula......Page 643
24.5 The non-invariance of ›=›t......Page 645
24.6 Clifford–Dirac square root of wave operator......Page 647
24.7 The Dirac equation......Page 649
24.8 Dirac’s route to the positron......Page 651
25.1 The origins of modern particle physics......Page 656
25.2 The zigzag picture of the electron......Page 657
25.3 Electroweak interactions; reflection asymmetry......Page 661
25.4 Charge conjugation, parity, and time reversal......Page 667
25.5 The electroweak symmetry group......Page 669
25.6 Strongly interacting particles......Page 674
25.7 ‘Coloured quarks’......Page 677
25.8 Beyond the standard model?......Page 680
26.1 Fundamental status of QFT in modern theory......Page 684
26.2 Creation and annihilation operators......Page 686
26.3 Infinite-dimensional algebras......Page 689
26.4 Antiparticles in QFT......Page 691
26.5 Alternative vacua......Page 693
26.6 Interactions: Lagrangians and path integrals......Page 694
26.7 Divergent path integrals: Feynman’s response......Page 699
26.8 Constructing Feynman graphs; the S-matrix......Page 701
26.9 Renormalization......Page 704
26.10 Feynman graphs from Lagrangians......Page 709
26.11 Feynman graphs and the choice of vacuum......Page 710
27.1 Time symmetry in dynamical evolution......Page 715
27.2 Submicroscopic ingredients......Page 717
27.3 Entropy......Page 719
27.4 The robustness of the entropy concept......Page 721
27.5 Derivation of the second law—or not?......Page 725
27.6 Is the whole universe an ‘isolated system’?......Page 728
27.7 The role of the Big Bang......Page 731
27.8 Black holes......Page 736
27.9 Event horizons and spacetime singularities......Page 741
27.10 Black-hole entropy......Page 743
27.11 Cosmology......Page 746
27.12 Conformal diagrams......Page 752
27.13 Our extraordinarily special Big Bang......Page 755
28.1 Early-universe spontaneous symmetry breaking......Page 764
28.2 Cosmic topological defects......Page 768
28.3 Problems for early-universe symmetry breaking......Page 771
28.4 Inflationary cosmology......Page 775
28.5 Are the motivations for inflation valid?......Page 782
28.6 The anthropic principle......Page 786
28.7 The Big Bang’s special nature: an anthropic key?......Page 791
28.8 The Weyl curvature hypothesis......Page 794
28.9 The Hartle–Hawking ‘no-boundary’ proposal......Page 798
28.10 Cosmological parameters: observational status?......Page 801
29.1 The conventional ontologies of quantum theory......Page 811
29.2 Unconventional ontologies for quantum theory......Page 814
29.3 The density matrix......Page 820
29.4 Density matrices for spin ½: the Bloch sphere......Page 822
29.5 The density matrix in EPR situations......Page 826
29.6 FAPP philosophy of environmental decoherence......Page 831
29.7 Schro¨dinger’s cat with ‘Copenhagen’ ontology......Page 833
29.8 Can other conventional ontologies resolve the ‘cat’?......Page 835
29.9 Which unconventional ontologies may help?......Page 839
30.1 Is today’s quantum theory here to stay?......Page 845
30.2 Clues from cosmological time asymmetry......Page 846
30.3 Time-asymmetry in quantum state reduction......Page 848
30.4 Hawking’s black-hole temperature......Page 852
30.5 Black-hole temperature from complex periodicity......Page 856
30.6 Killing vectors, energy flow—and time travel!......Page 862
30.7 Energy outflow from negative-energy orbits......Page 865
30.8 Hawking explosions......Page 867
30.9 A more radical perspective......Page 871
30.10 Schro¨dinger’s lump......Page 875
30.11 Fundamental conflict with Einstein’s principles......Page 878
30.12 Preferred Schro¨dinger–Newton states?......Page 882
30.13 FELIX and related proposals......Page 885
30.14 Origin of fluctuations in the early universe......Page 890
31.1 Unexplained parameters......Page 898
31.2 Supersymmetry......Page 902
31.3 The algebra and geometry of supersymmetry......Page 906
31.4 Higher-dimensional spacetime......Page 909
31.5 The original hadronic string theory......Page 913
31.6 Towards a string theory of the world......Page 916
31.7 String motivation for extra spacetime dimensions......Page 919
31.8 String theory as quantum gravity?......Page 921
31.9 String dynamics......Page 924
31.10 Why don’t we see the extra space dimensions?......Page 926
31.11 Should we accept the quantum-stability argument?......Page 931
31.12 Classical instability of extra dimensions......Page 934
31.13 Is string QFT finite?......Page 936
31.14 The magical Calabi–Yau spaces; M-theory......Page 939
31.15 Strings and black-hole entropy......Page 945
31.16 The ‘holographic principle’......Page 949
31.17 The D-brane perspective......Page 952
31.18 The physical status of string theory?......Page 955
32.1 Canonical quantum gravity......Page 963
32.2 The chiral input to Ashtekar’s variables......Page 964
32.3 The form of Ashtekar’s variables......Page 967
32.4 Loop variables......Page 970
32.5 The mathematics of knots and links......Page 972
32.6 Spin networks......Page 975
32.7 Status of loop quantum gravity?......Page 981
33.1 Theories where geometry has discrete elements......Page 987
33.2 Twistors as light rays......Page 991
33.3 Conformal group; compactified Minkowski space......Page 997
33.4 Twistors as higher-dimensional spinors......Page 1001
33.5 Basic twistor geometry and coordinates......Page 1003
33.6 Geometry of twistors as spinning massless particles......Page 1007
33.7 Twistor quantum theory......Page 1011
33.8 Twistor description of massless fields......Page 1014
33.9 Twistor sheaf cohomology......Page 1016
33.10 Twistors and positive/negative frequency splitting......Page 1022
33.11 The non-linear graviton......Page 1024
33.12 Twistors and general relativity......Page 1029
33.13 Towards a twistor theory of particle physics......Page 1030
33.14 The future of twistor theory?......Page 1032
34.1 Great theories of 20th century physics—and beyond?......Page 1039
34.2 Mathematically driven fundamental physics......Page 1043
34.3 The role of fashion in physical theory......Page 1046
34.4 Can a wrong theory be experimentally refuted?......Page 1049
34.5 Whence may we expect our next physical revolution?......Page 1053
34.6 What is reality?......Page 1056
34.7 The roles of mentality in physical theory......Page 1059
34.8 Our long mathematical road to reality......Page 1062
34.9 Beauty and miracles......Page 1067
34.10 Deep questions answered, deeper questions posed......Page 1072
Epilogue......Page 1077
Bibliography......Page 1079
B......Page 1110
C......Page 1111
D......Page 1112
F......Page 1113
G......Page 1114
I......Page 1115
M......Page 1116
N......Page 1117
O,P......Page 1118
Q......Page 1119
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T......Page 1122
Y,Z......Page 1123