A Unified Grand Tour of Theoretical Physics Second Edition A Unified Grand Tour of Theoretical Physics Second Edition Ian D Lawrie Reader in Theoretical Physics The University of Leeds Institute of Physics Publishing Bristol and Philadelphia (cid:1)c IOPPublishingLtd2002 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical,photocopying,recordingorotherwise,withoutthepriorpermission of the publisher. Multiple copying is permitted in accordance with the terms of licences issued by the Copyright Licensing Agency under the terms of its agreementwiththeCommitteeofVice-ChancellorsandPrincipals. BritishLibraryCataloguing-in-PublicationData AcataloguerecordforthisbookisavailablefromtheBritishLibrary. ISBN0750306041 LibraryofCongressCataloging-in-PublicationDataareavailable FirstEditionpublished1990 FirstEditionreprinted1994,1998 CommissioningEditor:JamesRevill ProductionEditor:SimonLaurenson ProductionControl:SarahPlenty CoverDesign:Fre´de´riqueSwist MarketingExecutive:LauraSerratrice Published by Institute of Physics Publishing, wholly owned by The Institute of Physics,London InstituteofPhysicsPublishing,DiracHouse,TempleBack,BristolBS16BE,UK US Office: Institute of Physics Publishing, The Public Ledger Building, Suite 1035,150SouthIndependenceMallWest,Philadelphia,PA19106,USA TypesetinLATEX2ε byText2Text,Torquay,Devon PrintedintheUKbyMPGBooksLtd,Bodmin,Cornwall Contents PrefacetotheSecondEdition xi PrefacetotheFirstEdition xiii GlossaryofMathematicalSymbols xv 1 Introduction:TheWaysofNature 1 2 Geometry 6 2.0 TheSpecialandGeneralTheoriesofRelativity 7 2.0.1 Thespecialtheory 7 2.0.2 Thegeneraltheory 12 2.1 SpacetimeasaDifferentiableManifold 15 2.1.1 Topologyoftherealline andof d 16 2.1.2 DifferentiablespacetimemRanifoldR 19 2.1.3 Summaryandexamples 21 2.2 Tensors 23 2.3 ExtraGeometricalStructures 28 2.3.1 Theaffineconnection 29 2.3.2 Geodesics 33 2.3.3 TheRiemanncurvaturetensor 34 2.3.4 Themetric 36 2.3.5 Themetricconnection 38 2.4 WhatistheStructureofOurSpacetime? 39 3 ClassicalPhysicsinGalileanandMinkowskiSpacetimes 45 3.1 TheActionPrincipleinGalileanSpacetime 46 3.2 SymmetriesandConservationLaws 50 3.3 TheHamiltonian 52 3.4 PoissonBracketsandTranslationOperators 53 3.5 TheActionPrincipleinMinkowskiSpacetime 56 3.6 ClassicalElectrodynamics 61 3.7 GeometryinClassicalPhysics 64 3.7.1 Moreontensors 65 3.7.2 Differentialforms,dualtensorsandMaxwell’sequations 67 vi Contents 3.7.3 Configurationspaceanditsrelatives 73 3.7.4 Thesymplecticgeometryofphasespace 75 4 GeneralRelativityandGravitation 83 4.1 ThePrincipleofEquivalence 83 4.2 GravitationalForces 84 4.3 TheFieldEquationsofGeneralRelativity 87 4.4 TheGravitationalFieldofaSphericalBody 91 4.4.1 TheSchwarzschildsolution 91 4.4.2 Timenearamassivebody 93 4.4.3 Distancesnearamassivebody 95 4.4.4 Particletrajectoriesnearamassivebody 96 4.5 BlackandWhiteHoles 97 5 QuantumTheory 107 5.0 WaveMechanics 108 5.1 TheHilbertSpaceofStateVectors 111 5.2 OperatorsandObservableQuantities 114 5.3 SpacetimeTranslationsandthePropertiesofOperators 116 5.4 QuantizationofaClassicalSystem 121 5.5 AnExample:TheOne-DimensionalHarmonicOscillator 123 6 SecondQuantizationandQuantumFieldTheory 130 6.1 TheOccupation-NumberRepresentation 131 6.2 FieldOperatorsandObservables 134 6.3 EquationofMotionandLagrangianFormalismforFieldOperators135 6.4 SecondQuantizationforFermions 137 7 RelativisticWaveEquationsandFieldTheories 140 7.1 TheKlein–GordonEquation 141 7.2 ScalarFieldTheoryforFreeParticles 144 7.3 TheDiracEquationandSpin-1 Particles 146 2 7.3.1 TheDiracequation 146 7.3.2 Lorentzcovarianceandspin 148 7.3.3 Somepropertiesoftheγ matrices 152 7.3.4 ConjugatewavefunctionandtheDiracaction 153 7.3.5 Probabilitycurrentandbilinearcovariants 153 7.3.6 Plane-wavesolutions 155 7.3.7 Masslessspin-1 particles 156 2 7.4 SpinorFieldTheory 157 7.5 WeylandMajoranaSpinors 159 7.6 ParticlesofSpin1and2 163 7.6.1 Photonsandmassivespin-1particles 163 7.6.2 Gravitons 166 7.7 WaveEquationsinCurvedSpacetime 168 Contents vii 8 Forces,ConnectionsandGaugeFields 179 8.1 Electromagnetism 179 8.2 Non-AbelianGaugeTheories 185 8.3 Non-AbelianTheoriesandElectromagnetism 192 8.4 RelevanceofNon-AbelianTheoriestoPhysics 193 8.5 TheTheoryofKaluzaandKlein 194 9 InteractingRelativisticFieldTheories 199 9.1 AsymptoticStatesandtheScatteringOperator 200 9.2 ReductionFormulae 202 9.3 PathIntegrals 205 9.3.1 Pathintegralsinnon-relativisticquantummechanics 205 9.3.2 Functionalintegralsinquantumfieldtheory 208 9.4 PerturbationTheory 211 9.5 QuantizationofGaugeFields 214 9.6 Renormalization 218 9.7 QuantumElectrodynamics 224 9.7.1 TheCoulombpotential 224 9.7.2 Vacuumpolarization 227 9.7.3 TheLambshift 229 9.7.4 Therunningcouplingconstant 229 9.7.5 Anomalousmagneticmoments 231 10 EquilibriumStatisticalMechanics 235 10.1 ErgodicTheoryandtheMicrocanonicalEnsemble 236 10.2 TheCanonicalEnsemble 241 10.3 TheGrandCanonicalEnsemble 243 10.4 RelationBetweenStatisticalMechanicsandThermodynamics 245 10.5 QuantumStatisticalMechanics 251 10.6 FieldTheoriesatFiniteTemperature 254 10.7 BlackBodyRadiation 257 10.8 TheClassicalLatticeGas 259 10.9 AnalogiesBetweenFieldTheoryandStatisticalMechanics 261 11 PhaseTransitions 266 11.1 Bose–EinsteinCondensation 266 11.2 CriticalPointsinFluidsandMagnets 269 11.3 TheIsingModelanditsApproximationbyaFieldTheory 274 11.4 Order,DisorderandSpontaneousSymmetryBreaking 276 11.5 TheGinzburg–LandauTheory 279 11.6 TheRenormalizationGroup 281 11.7 TheGinzburg–LandauTheoryofSuperconductors 287 11.7.1 Spontaneousbreakingofcontinuoussymmetries 288 11.7.2 Magneticeffectsinsuperconductors 290 11.7.3 TheHiggsmechanism 291 viii Contents 12 UnifiedGaugeTheoriesoftheFundamentalInteractions 295 12.1 TheWeakInteraction 296 12.2 TheGlashow–Weinberg–SalamModelforLeptons 301 12.3 PhysicalImplicationsoftheModelforLeptons 306 12.4 HadronicParticlesintheElectroweakTheory 308 12.4.1 Quarks 308 12.4.2 Quarksintheelectroweaktheory 312 12.5 ColourandQuantumChromodynamics 314 12.6 GrandUnifiedTheories 319 12.7 Supersymmetry 328 12.7.1 TheWess–Zuminomodel 329 12.7.2 Superfields 330 12.7.3 Spontaneoussupersymmetrybreaking 332 12.7.4 Thesupersymmetryalgebra 335 12.7.5 Supersymmetricgaugetheoriesandsupergravity 340 12.7.6 Somealgebraicdetails 343 13 SolitonsandSoOn 346 13.1 DomainWallsandKinks 347 13.2 TheSine–GordonSolitons 355 13.3 VorticesandStrings 359 13.4 MagneticMonopoles 369 14 TheEarlyUniverse 379 14.1 TheRobertson–WalkerMetric 380 14.2 TheFriedmann–LemaˆıtreModels 385 14.3 Matter,RadiationandtheAgeoftheUniverse 390 14.4 TheFairlyEarlyUniverse 393 14.5 Nucleosynthesis 401 14.6 RecombinationandtheHorizonProblem 404 14.7 TheFlatnessProblem 405 14.8 TheVeryEarlyUniverse 406 15 AnIntroductiontoStringTheory 425 15.1 TheRelativisticPointParticle 427 15.2 TheFreeClassicalString 431 15.2.1 Thestringaction 431 15.2.2 Weylinvarianceandgaugefixing 434 15.2.3 TheEuclideanworldsheetandconformalinvariance 437 15.2.4 Modeexpansions 440 15.2.5 Ausefultransformation 445 15.3 QuantizationoftheFreeBosonicString 447 15.3.1 ThequantumVirasoroalgebra 449 15.3.2 Quantumgaugefixing 454 15.3.3 Thecriticalspacetimedimension 458 Contents ix 15.3.4 TheghostHilbertspace 462 15.3.5 TheBRSTcohomology 464 15.4 PhysicsoftheFreeBosonicString 470 15.4.1 Themassspectrum 470 15.4.2 Vertexoperators 475 15.4.3 Stringsandquantumgravity 478 15.5 FurtherDevelopments 481 15.5.1 Stringinteractions 481 15.5.2 Superstrings 485 15.5.3 Theramificationsofcompactification 489 15.6 TheLastWord? 495 SomeSnapshotsoftheTour 501 AppendixA SomeMathematicalNotes 518 A.1 DeltaFunctionsandFunctionalDifferentiation 518 A.2 TheLevi-CivitaTensorDensity 520 A.3 VectorSpacesandHilbertSpaces 521 A.4 Gauss’Theorem 523 A.5 SurfaceAreaandVolumeofad-DimensionalSphere 524 A.6 GaussianIntegrals 524 A.7 GrassmannVariables 525 AppendixB SomeElementsofGroupTheory 528 AppendixC NaturalUnits 540 AppendixD ScatteringCross-SectionsandParticleDecayRates 544 Bibliography 548 References 552 Index 555