The Road to Reality : A Complete Guide to the Laws of the Universe

1123 Pages·2005·13.84 MB·English

by  Roger Penrose

THE ROAD TO REALITY
BY ROGER PENROSE
TheEmperor’sNewMind:
ConcerningComputers,Minds,
andtheLawsofPhysics
ShadowsoftheMind:
ASearchfortheMissingScience
ofConsciousness
Roger Penrose
T R
HE OAD TO
R
EALITY
A Complete Guide to the Laws
of the Universe
JONATHAN CAPE
LONDON
PublishedbyJonathanCape2004
2 4 6 8 10 9 7 5 31
Copyright(cid:1)RogerPenrose2004
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FirstpublishedinGreatBritainin2004by
JonathanCape
RandomHouse,20VauxhallBridgeRoad,
LondonSW1V2SA
RandomHouseAustralia(Pty)Limited
20AlfredStreet,MilsonsPoint,Sydney,
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Contents
Preface xv
Acknowledgements xxiii
Notation xxvi
Prologue 1
1 The roots of science 7
1.1 Thequestfortheforcesthatshapetheworld 7
1.2 Mathematicaltruth 9
1.3 IsPlato’smathematicalworld‘real’? 12
1.4 Threeworldsandthreedeepmysteries 17
1.5 TheGood,theTrue,andtheBeautiful 22
2 An ancient theorem and a modern question 25
2.1 ThePythagoreantheorem 25
2.2 Euclid’spostulates 28
2.3 Similar-areasproofofthePythagoreantheorem 31
2.4 Hyperbolicgeometry:conformalpicture 33
2.5 Otherrepresentationsofhyperbolicgeometry 37
2.6 Historicalaspectsofhyperbolicgeometry 42
2.7 Relationtophysicalspace 46
3 Kinds of number in the physical world 51
3.1 APythagoreancatastrophe? 51
3.2 Thereal-numbersystem 54
3.3 Realnumbersinthephysicalworld 59
3.4 Donaturalnumbersneedthephysicalworld? 63
3.5 Discretenumbersinthephysicalworld 65
4 Magical complex numbers 71
4.1 Themagicnumber‘i’ 71
4.2 Solvingequationswithcomplexnumbers 74
v
Contents
4.3 Convergenceofpowerseries 76
4.4 CasparWessel’scomplexplane 81
4.5 HowtoconstructtheMandelbrotset 83
5 Geometry of logarithms, powers, and roots 86
5.1 Geometryofcomplexalgebra 86
5.2 Theideaofthecomplexlogarithm 90
5.3 Multiplevaluedness,naturallogarithms 92
5.4 Complexpowers 96
5.5 Somerelationstomodernparticlephysics 100
6 Real-number calculus 103
6.1 Whatmakesanhonestfunction? 103
6.2 Slopesoffunctions 105
6.3 Higherderivatives;C1-smoothfunctions 107
6.4 The‘Eulerian’notionofafunction? 112
6.5 TherulesofdiVerentiation 114
6.6 Integration 116
7 Complex-number calculus 122
7.1 Complexsmoothness;holomorphicfunctions 122
7.2 Contourintegration 123
7.3 Powerseriesfromcomplexsmoothness 127
7.4 Analyticcontinuation 129
8 Riemann surfaces and complex mappings 135
8.1 TheideaofaRiemannsurface 135
8.2 Conformalmappings 138
8.3 TheRiemannsphere 142
8.4 ThegenusofacompactRiemannsurface 145
8.5 TheRiemannmappingtheorem 148
9 Fourier decomposition and hyperfunctions 153
9.1 Fourierseries 153
9.2 Functionsonacircle 157
9.3 FrequencysplittingontheRiemannsphere 161
9.4 TheFouriertransform 164
9.5 FrequencysplittingfromtheFouriertransform 166
9.6 Whatkindoffunctionisappropriate? 168
9.7 Hyperfunctions 172
vi
Contents
10 Surfaces 179
10.1 Complexdimensionsandrealdimensions 179
10.2 Smoothness,partialderivatives 181
10.3 VectorWeldsand1-forms 185
10.4 Components,scalarproducts 190
10.5 TheCauchy–Riemannequations 193
11 Hypercomplex numbers 198
11.1 Thealgebraofquaternions 198
11.2 Thephysicalroleofquaternions? 200
11.3 Geometryofquaternions 203
11.4 Howtocomposerotations 206
11.5 CliVordalgebras 208
11.6 Grassmannalgebras 211
12 Manifolds of n dimensions 217
12.1 Whystudyhigher-dimensionalmanifolds? 217
12.2 Manifoldsandcoordinatepatches 221
12.3 Scalars,vectors,andcovectors 223
12.4 Grassmannproducts 227
12.5 Integralsofforms 229
12.6 Exteriorderivative 231
12.7 Volumeelement;summationconvention 237
12.8 Tensors;abstract-indexanddiagrammaticnotation 239
12.9 Complexmanifolds 243
13 Symmetry groups 247
13.1 Groupsoftransformations 247
13.2 Subgroupsandsimplegroups 250
13.3 Lineartransformationsandmatrices 254
13.4 Determinantsandtraces 260
13.5 Eigenvaluesandeigenvectors 263
13.6 RepresentationtheoryandLiealgebras 266
13.7 Tensorrepresentationspaces;reducibility 270
13.8 Orthogonalgroups 275
13.9 Unitarygroups 281
13.10 Symplecticgroups 286
14 Calculus on manifolds 292
14.1 DiVerentiationonamanifold? 292
14.2 Paralleltransport 294
14.3 Covariantderivative 298
14.4 Curvatureandtorsion 301
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Contents
14.5 Geodesics,parallelograms,andcurvature 303
14.6 Liederivative 309
14.7 Whatametriccandoforyou 317
14.8 Symplecticmanifolds 321
15 Fibre bundles and gauge connections 325
15.1 SomephysicalmotivationsforWbrebundles 325
15.2 Themathematicalideaofabundle 328
15.3 Cross-sectionsofbundles 331
15.4 TheCliVordbundle 334
15.5 Complexvectorbundles,(co)tangentbundles 338
15.6 Projectivespaces 341
15.7 Non-trivialityinabundleconnection 345
15.8 Bundlecurvature 349
16 The ladder of inWnity 357
16.1 FiniteWelds 357
16.2 AWniteorinWnitegeometryforphysics? 359
16.3 DiVerentsizesofinWnity 364
16.4 Cantor’sdiagonalslash 367
16.5 Puzzlesinthefoundationsofmathematics 371
16.6 TuringmachinesandGo¨del’stheorem 374
16.7 SizesofinWnityinphysics 378
17 Spacetime 383
17.1 ThespacetimeofAristotelianphysics 383
17.2 SpacetimeforGalileanrelativity 385
17.3 Newtoniandynamicsinspacetimeterms 388
17.4 Theprincipleofequivalence 390
17.5 Cartan’s‘Newtonianspacetime’ 394
17.6 TheWxedWnitespeedoflight 399
17.7 Lightcones 401
17.8 Theabandonmentofabsolutetime 404
17.9 ThespacetimeforEinstein’sgeneralrelativity 408
18 Minkowskian geometry 412
18.1 EuclideanandMinkowskian4-space 412
18.2 ThesymmetrygroupsofMinkowskispace 415
18.3 Lorentzianorthogonality;the‘clockparadox’ 417
18.4 HyperbolicgeometryinMinkowskispace 422
18.5 ThecelestialsphereasaRiemannsphere 428
18.6 Newtonianenergyand(angular)momentum 431
18.7 Relativisticenergyand(angular)momentum 434
viii
Contents
19 The classical Welds of Maxwell and Einstein 440
19.1 EvolutionawayfromNewtoniandynamics 440
19.2 Maxwell’selectromagnetictheory 442
19.3 ConservationandXuxlawsinMaxwelltheory 446
19.4 TheMaxwellWeldasgaugecurvature 449
19.5 Theenergy–momentumtensor 455
19.6 Einstein’sWeldequation 458
19.7 Furtherissues:cosmologicalconstant;Weyltensor 462
19.8 GravitationalWeldenergy 464
20 Lagrangians and Hamiltonians 471
20.1 ThemagicalLagrangianformalism 471
20.2 ThemoresymmetricalHamiltonianpicture 475
20.3 Smalloscillations 478
20.4 Hamiltoniandynamicsassymplecticgeometry 483
20.5 LagrangiantreatmentofWelds 486
20.6 HowLagrangiansdrivemoderntheory 489
21 The quantum particle 493
21.1 Non-commutingvariables 493
21.2 QuantumHamiltonians 496
21.3 Schro¨dinger’sequation 498
21.4 Quantumtheory’sexperimentalbackground 500
21.5 Understandingwave–particleduality 505
21.6 Whatisquantum‘reality’? 507
21.7 The‘holistic’natureofawavefunction 511
21.8 Themysterious‘quantumjumps’ 516
21.9 Probabilitydistributioninawavefunction 517
21.10 Positionstates 520
21.11 Momentum-spacedescription 521
22 Quantum algebra, geometry, and spin 527
22.1 ThequantumproceduresUandR 527
22.2 ThelinearityofUanditsproblemsforR 530
22.3 Unitarystructure,Hilbertspace,Diracnotation 533
22.4 Unitaryevolution:Schro¨dingerandHeisenberg 535
22.5 Quantum‘observables’ 538
22.6 yes/nomeasurements;projectors 542
22.7 Nullmeasurements;helicity 544
22.8 Spinandspinors 549
22.9 TheRiemannsphereoftwo-statesystems 553
22.10 Higherspin:Majoranapicture 559
22.11 Sphericalharmonics 562
ix