Graduate Texts in Physics Graduate Texts in Physics Graduate Texts in Physics publishes core learning/teaching material for graduate- and advanced-levelundergraduatecoursesontopicsofcurrentandemergingfieldswithinphysics, bothpureandapplied.ThesetextbooksservestudentsattheMS-orPhD-levelandtheirin- structorsascomprehensive sourcesof principles, definitions, derivations, experimentsand applications(asrelevant)fortheirmasteryandteaching,respectively.Internationalinscope and relevance,the textbooks correspond to course syllabi sufficiently to serve as required reading.Theirdidacticstyle,comprehensivenessandcoverageoffundamentalmaterialalso makethemsuitableasintroductionsorreferencesforscientistsentering,orrequiringtimely knowledgeof,aresearchfield SeriesEditors ProfessorRichardNeeds CavendishLaboratory JJThomsonAvenue CambridgeCB30HE,UK E-mail:[email protected] ProfessorWilliamT.Rhodes FloridaAtlanticUniversity ImagingTechnologyCenter DepartmentofElectricalEngineering 777GladesRoadSE,Room456 BocaRaton,FL33431,USA E-mail:[email protected] ProfessorH.EugeneStanley BostonUniversity CenterforPolymerStudies DepartmentofPhysics 590CommonwealthAvenue,Room204B Boston,MA02215,USA E-mail:[email protected] Forfurthervolumes: http://www.springer.com/series/8431 Florian Scheck Classical Field Theory On Electrodynamics, Non-Abelian Gauge Theories and Gravitation 123 FlorianScheck JohannesGutenberg-UniversityMainz Germany ISSN1868-4513 ISSN1868-4521(electronic) ISBN978-3-642-27984-3 ISBN978-3-642-27985-0(eBook) DOI10.1007/978-3-642-27985-0 SpringerHeidelbergDordrechtLondonNewYork LibraryofCongressControlNumber:2012937327 (cid:2)c Springer-VerlagBerlinHeidelberg2012 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart ofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations, recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordis- similarmethodologynowknownorhereafterdeveloped.Exemptedfromthislegalreservationare briefexcerptsinconnectionwithreviewsorscholarlyanalysisormaterialsuppliedspecificallyfor thepurposeofbeingenteredandexecutedonacomputersystem,forexclusiveusebythepurchaser ofthework.Duplicationofthispublicationorpartsthereofispermittedonlyundertheprovisions oftheCopyrightLawofthePublisherslocation,initscurrentversion,andpermissionforusemust alwaysbeobtainedfromSpringer.PermissionsforusemaybeobtainedthroughRightsLinkatthe CopyrightClearanceCenter.ViolationsareliabletoprosecutionundertherespectiveCopyright Law. 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Preface Traditionally one begins a course or a textbook on electrodynamics with an ex- tensive discussion of electrostatics, of magnetostatics, and of stationary currents, beforeturningtothefulltime-dependentMaxwelltheoryinlocalform.Inthisbook Ichooseasomewhatdifferentapproach:StartingfromMaxwell’sequationsinin- tegralform,thatis to say,fromthephenomenologicalandexperimentallyverified basisofelectrodynamics,thelocalequationsareformulatedanddiscussedwiththeir generaltime and space dependenceright from the start. Static or stationary situa- tionsappearasspecialcasesforwhichMaxwell’sequationssplitintotwomoreor lessindependentgroupsandthusaredecoupledtoacertainextent. GreatimportanceisattachedtothesymmetriesoftheMaxwellequationsand,in particular,theircovariancewithrespecttoLorentztransformations.Anothercentral issueisthetreatmentofelectrodynamicsintheframeworkofclassicalfieldtheory by means of a Lagrangedensity and Hamilton’sprinciple.Generalprinciplesthat weredevelopedformechanics,appearinadeeperandmoregeneralapplicationthat canserveasamodelandprototypeforanyclassicalfieldtheory.Thefactthatthe fieldsofMaxwelltheory,ingeneral,dependonspaceandtimemakesitnecessaryto enlargetheframeworkoftraditionaltensoranalysisinR3toexteriorcalculusonR4. The venerable vector and tensor analysis that was designed for three-dimensional Euclidean spaces, does not suffice and must be generalized to higher dimensions andtoMinkowskisignature.Whiletheexteriorproductisthegeneralizationofthe vectorproductinR3,Cartan’sexteriorderivativeisthenaturalgeneralizationofthe curlinR3 and,bythesametoken,encompassesthefamiliaroperationsofgradient anddivergence. AmongthemanyapplicationsofMaxwelltheoryIchosesomecharacteristicand, I felt, nowadaysparticularlyrelevantexamplessuch as an extensive discussion of polarizationofelectromagneticwaves,thedescriptionofGaussianbeams(theseare analyticsolutionsoftheHelmholtzequationinparaxialapproximation),andoptics of metamaterials with negative index of refraction. Regarding other, more tradi- vii viii Preface tionalapplicationsIrefertothe well-known,excellenttextbooksbyJ.D. Jackson, byL.D.LandauandE.M.Lifshits,andothers. As a novelfeature I take up in the fifth chaptera furtherdirectionof greatim- portanceforpresent-dayphysics:Theconstructionofnon-Abeliangaugetheories. These Yang–Millstheoriesas they are called1, are essential and indispensable for ourpresentunderstandingofthefundamentalinteractionsofnature.Althoughthese theorieswhichareatthebasisoftheso-calledstandardmodelofelementaryparticle physics,leadusfarintoquantizedfieldtheory,theirconstructionandtheiressential featuresareofaclassicalnature,atleastaslongasoneconsidersonlytheradiation fields,i.e.theanaloguesoftheMaxwellfields,andclassicalscalarfields,butleaves outfermionicmatterparticles.Non-Abeliangaugetheoriesareconstructedfollow- ing the example of Maxwell theory. They bear some similarities to the latter but exhibit also significant differences from it. Even the phenomenonof spontaneous symmetry breaking that preserves us from the appearance of too many massless fields, in essence, is a classical mechanism. In view of the great impact of gauge theoriesonourunderstandingofthefundamentalinteractionsitwouldbealossnot todothisstepwhichbuildsonMaxwelltheoryinamostnaturalmanner. Chapter 6 gives an extensive phenomenologicaland geometric introduction to generalrelativityand,hence,roundsoffthedescriptionofallfundamentalinterac- tionsintheframeworkofclassicalfieldtheory.Heretoo,Iuseconsistentlyamodern geometriclanguagewhich–aftersomeinvestmentindifferentialgeometry–allows fora transparentformulationof Einstein’sequationswhichis betterfocusedto its essentialsthantheoldertensoranalysisformulatedincomponentsonly. Much of the material included in this book was tried out in numerouslectures that I gave at Johannes Gutenberg-Universityover the years. I am grateful to the studentswhohavefollowedthesecourses,fortheirquestionsandcomments,andto theteachingassistantswhotookgoodcareofexerciseclasses,fortheirstimulating questionsandcriticalcomments. IowespecialthankstoImmanuelBlochforthediscussionswehadonGaussian beamsandthefascinatingtopicofmetamaterialswithnegativeindexofrefraction, and for his encouragement to include these modern applications. Special thanks also to Mario Paschke who more than once broughtup originalideasand pointed outsomealmostforgottenbutrelevantreferences. The cooperation with Springer-Verlag in Heidelberg and with the production teamofLE-TeXin Leipzigwasexcellentandveryefficient.Thegreatencourage- mentandeditorialcareofferedbyDr.ThorstenSchneiderfromSpringer-Verlagis gratefullyacknowledged. Mainz,November2011 FlorianScheck 1FirstideaswerepublishedbyOskarKlein,Z.Physik37(1926)895.ItisreportedthatWolfgang Paulidevelopedthemindependentlybutdidnotpublishthem. Contents 1 Maxwell’sEquations . . . . . . . . . . . . . . . . . . . . 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Gradient,CurlandDivergence . . . . . . . . . . . . . . 2 1.3 IntegralTheoremsfortheCaseofR3 . . . . . . . . . . . 7 1.4 Maxwell’sEquationsinIntegralForm . . . . . . . . . . . 11 1.4.1 TheLawofInduction . . . . . . . . . . . . . . . 11 1.4.2 Gauss’Law . . . . . . . . . . . . . . . . . . . 13 1.4.3 TheLawofBiotandSavart . . . . . . . . . . . . 15 1.4.4 TheLorentzForce . . . . . . . . . . . . . . . . 17 1.4.5 TheContinuityEquation . . . . . . . . . . . . . 18 1.5 Maxwell’sEquationsinLocalForm . . . . . . . . . . . . 21 1.5.1 InductionLawandGauss’Law . . . . . . . . . . . 22 1.5.2 LocalFormoftheLawofBiotandSavart . . . . . . 23 1.5.3 LocalEquationsinAllSystemsofUnits . . . . . . . 24 1.5.4 TheQuestionofPhysicalUnits . . . . . . . . . . . 25 1.5.5 EquationsofElectromagnetisminSISystem . . . . . 28 1.5.6 TheGaussianSystemofUnits . . . . . . . . . . . 29 1.6 ScalarPotentialsandVectorPotentials . . . . . . . . . . . 35 1.6.1 AFewFormulaefromVectorAnalysis . . . . . . . . 35 1.6.2 ConstructionofaVectorField fromItsSourceandItsCurl . . . . . . . . . . . . 40 1.6.3 ScalarPotentialsandVectorPotentials . . . . . . . . 42 1.7 PhenomenologyoftheMaxwellEquations . . . . . . . . . 46 1.7.1 TheFundamentalEquationsandTheirInterpretation . . 47 1.7.2 RelationBetweenDisplacementFieldandElectricField 50 1.7.3 RelationBetweenInductionandMagneticFields . . . 52 1.8 StaticElectricStates . . . . . . . . . . . . . . . . . . 55 1.8.1 PoissonandLaplaceEquations . . . . . . . . . . . 56 ix x Contents 1.8.2 SurfaceCharges,DipolesandDipoleLayers . . . . . 62 1.8.3 TypicalBoundaryValueProblems . . . . . . . . . . 66 1.8.4 MultipoleExpansionofPotentials . . . . . . . . . . 69 1.9 StationaryCurrentsandStaticMagneticStates . . . . . . . . 83 1.9.1 PoissonEquationandVectorPotential . . . . . . . . 84 1.9.2 MagneticDipoleDensityandMagneticMoment . . . . 84 1.9.3 FieldsofMagneticandElectricDipoles . . . . . . . 88 1.9.4 EnergyandEnergyDensity . . . . . . . . . . . . 92 1.9.5 CurrentsandConductivity . . . . . . . . . . . . . 95 2 SymmetriesandCovarianceoftheMaxwellEquations . . . . . 97 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 97 2.2 TheMaxwellEquationsinaFixedFrameofReference . . . . 97 2.2.1 RotationsandDiscreteSpacetimeTransformations . . . 98 2.2.2 Maxwell’sEquationsandExteriorForms . . . . . . . 102 2.3 LorentzCovarianceofMaxwell’sEquations . . . . . . . . . 119 2.3.1 PoincaréandLorentzGroups . . . . . . . . . . . . 120 2.3.2 RelativisticKinematicsandDynamics . . . . . . . . 123 2.3.3 LorentzForceandFieldStrength . . . . . . . . . . 126 2.3.4 CovarianceofMaxwell’sEquations . . . . . . . . . 128 2.3.5 GaugeInvarianceandPotentials . . . . . . . . . . 132 2.4 FieldsofaUniformlyMovingPointCharge . . . . . . . . . 136 2.5 LorentzInvariantExteriorFormsandtheMaxwellEquations . . 141 2.5.1 FieldStrengthTensorandLorentzForce . . . . . . . 142 2.5.2 DifferentialEquationsfortheTwo-Forms!F and!F . . 145 2.5.3 PotentialsandGaugeTransformations . . . . . . . . 148 2.5.4 BehaviourUndertheDiscreteTransformations . . . . 149 2.5.5 *CovariantDerivativeandStructureEquation . . . . . 150 3 MaxwellTheoryasaClassicalFieldTheory . . . . . . . . . . 153 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 153 3.2 LagrangianFunctionandSymmetriesinFiniteSystems . . . . 153 3.2.1 Noether’sTheoremwithStrictInvariance . . . . . . . 155 3.2.2 GeneralizedTheoremofNoether . . . . . . . . . . 156 3.3 LagrangianDensityandEquationsofMotionforaFieldTheory 157 3.4 LagrangianDensityforMaxwellFieldswithSources . . . . . 163 3.5 SymmetriesandNoetherInvariants . . . . . . . . . . . . 168 3.5.1 InvarianceUnderOne-ParameterGroups . . . . . . . 169 3.5.2 GaugeTransformationsandLagrangianDensity . . . . 171 3.5.3 InvarianceUnderTranslations . . . . . . . . . . . 175 3.5.4 InterpretationoftheConservationLaws . . . . . . . 179 3.6 WaveEquationandGreenFunctions . . . . . . . . . . . . 183 3.6.1 SolutionsinNoncovariantForm . . . . . . . . . . 183 3.6.2 SolutionsoftheWaveEquationinCovariantForm . . . 188