This page intentionally left blank Thisbookprovidesanintroductiontothemajormathematicalstructuresusedinphysics today.Itcoverstheconceptsandtechniquesneededfortopicssuchasgrouptheory,Lie algebras,topology,Hilbertspacesanddifferentialgeometry.Importanttheoriesofphysics suchasclassicalandquantummechanics,thermodynamics,andspecialandgeneralrela- tivityarealsodevelopedindetail,andpresentedintheappropriatemathematicallanguage. The book is suitable for advanced undergraduate and beginning graduate students in mathematicalandtheoreticalphysics.Itincludesnumerousexercisesandworkedexamples totestthereader’sunderstandingofthevariousconcepts,aswellasextendingthethemes coveredinthemaintext.Theonlyprerequisitesareelementarycalculusandlinearalgebra. Nopriorknowledgeofgrouptheory,abstractvectorspacesortopologyisrequired. PeterSzekeres receivedhisPh.D.fromKing’sCollegeLondonin1964,inthearea of general relativity. He subsequently held research and teaching positions at Cornell University, King’s College and the University of Adelaide, where he stayed from 1971 tillhisrecentretirement.Currentlyheisavisitingresearchfellowatthatinstitution.Heis wellknowninternationallyforhisresearchingeneralrelativityandcosmology,andhasan excellentreputationforhisteachingandlecturing. A Course in Modern Mathematical Physics Groups, Hilbert Space and Differential Geometry Peter Szekeres FormerlyofUniversityofAdelaide cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press TheEdinburghBuilding,Cambridgecb22ru,UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521829601 © P. Szekeres 2004 Thispublicationisincopyright.Subjecttostatutoryexceptionandtotheprovisionof relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. Firstpublishedinprintformat 2004 isbn-13 978-0-511-26167-1eBook (Adobe Reader) isbn-10 0-511-26167-5 eBook (Adobe Reader) isbn-13 978-0-521-82960-1hardback isbn-10 0-521-82960-7 hardback isbn-13 978-0-521-53645-5 isbn-10 0-521-53645-6 CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyofurls forexternalorthird-partyinternetwebsitesreferredtointhispublication,anddoesnot guaranteethatanycontentonsuchwebsitesis,orwillremain,accurateorappropriate. Contents Preface pageix Acknowledgements xiii 1 Setsandstructures 1 1.1 Setsandlogic 2 1.2 Subsets,unionsandintersectionsofsets 5 1.3 Cartesianproductsandrelations 7 1.4 Mappings 10 1.5 Infinitesets 13 1.6 Structures 17 1.7 Categorytheory 23 2 Groups 27 2.1 Elementsofgrouptheory 27 2.2 Transformationandpermutationgroups 30 2.3 Matrixgroups 35 2.4 Homomorphismsandisomorphisms 40 2.5 Normalsubgroupsandfactorgroups 45 2.6 Groupactions 49 2.7 Symmetrygroups 52 3 Vectorspaces 59 3.1 Ringsandfields 59 3.2 Vectorspaces 60 3.3 Vectorspacehomomorphisms 63 3.4 Vectorsubspacesandquotientspaces 66 3.5 Basesofavectorspace 72 3.6 Summationconventionandtransformationofbases 81 3.7 Dualspaces 88 4 Linearoperatorsandmatrices 98 4.1 Eigenspacesandcharacteristicequations 99 4.2 Jordancanonicalform 107 v Contents 4.3 Linearordinarydifferentialequations 116 4.4 Introductiontogrouprepresentationtheory 120 5 Innerproductspaces 126 5.1 Realinnerproductspaces 126 5.2 Complexinnerproductspaces 133 5.3 Representationsoffinitegroups 141 6 Algebras 149 6.1 Algebrasandideals 149 6.2 Complexnumbersandcomplexstructures 152 6.3 QuaternionsandCliffordalgebras 157 6.4 Grassmannalgebras 160 6.5 LiealgebrasandLiegroups 166 7 Tensors 178 7.1 Freevectorspacesandtensorspaces 178 7.2 Multilinearmapsandtensors 186 7.3 Basisrepresentationoftensors 193 7.4 Operationsontensors 198 8 Exterioralgebra 204 8.1 r-Vectorsandr-forms 204 8.2 Basisrepresentationofr-vectors 206 8.3 Exteriorproduct 208 8.4 Interiorproduct 213 8.5 Orientedvectorspaces 215 8.6 TheHodgedual 220 9 Specialrelativity 228 9.1 Minkowskispace-time 228 9.2 Relativistickinematics 235 9.3 Particledynamics 239 9.4 Electrodynamics 244 9.5 Conservationlawsandenergy–stresstensors 251 10 Topology 255 10.1 Euclideantopology 255 10.2 Generaltopologicalspaces 257 10.3 Metricspaces 264 10.4 Inducedtopologies 265 10.5 Hausdorffspaces 269 10.6 Compactspaces 271 vi Contents 10.7 Connectedspaces 273 10.8 Topologicalgroups 276 10.9 Topologicalvectorspaces 279 11 Measuretheoryandintegration 287 11.1 Measurablespacesandfunctions 287 11.2 Measurespaces 292 11.3 Lebesgueintegration 301 12 Distributions 308 12.1 Testfunctionsanddistributions 309 12.2 Operationsondistributions 314 12.3 Fouriertransforms 320 12.4 Green’sfunctions 323 13 Hilbertspaces 330 13.1 Definitionsandexamples 330 13.2 Expansiontheorems 335 13.3 Linearfunctionals 341 13.4 Boundedlinearoperators 344 13.5 Spectraltheory 351 13.6 Unboundedoperators 357 14 Quantummechanics 366 14.1 Basicconcepts 366 14.2 Quantumdynamics 379 14.3 Symmetrytransformations 387 14.4 Quantumstatisticalmechanics 397 15 Differentialgeometry 410 15.1 Differentiablemanifolds 411 15.2 Differentiablemapsandcurves 415 15.3 Tangent,cotangentandtensorspaces 417 15.4 Tangentmapandsubmanifolds 426 15.5 Commutators,flowsandLiederivatives 432 15.6 DistributionsandFrobeniustheorem 440 16 Differentiableforms 447 16.1 Differentialformsandexteriorderivative 447 16.2 Propertiesofexteriorderivative 451 16.3 Frobeniustheorem:dualform 454 16.4 Thermodynamics 457 16.5 Classicalmechanics 464 vii Contents 17 Integrationonmanifolds 481 17.1 Partitionsofunity 482 17.2 Integrationofn-forms 484 17.3 Stokes’theorem 486 17.4 Homologyandcohomology 493 17.5 ThePoincare´lemma 500 18 Connectionsandcurvature 506 18.1 Linearconnectionsandgeodesics 506 18.2 Covariantderivativeoftensorfields 510 18.3 Curvatureandtorsion 512 18.4 Pseudo-Riemannianmanifolds 516 18.5 Equationofgeodesicdeviation 522 18.6 TheRiemanntensoranditssymmetries 524 18.7 Cartanformalism 527 18.8 Generalrelativity 534 18.9 Cosmology 548 18.10 Variationprinciplesinspace-time 553 19 LiegroupsandLiealgebras 559 19.1 Liegroups 559 19.2 Theexponentialmap 564 19.3 Liesubgroups 569 19.4 Liegroupsoftransformations 572 19.5 Groupsofisometries 578 Bibliography 587 Index 589 viii