Classical Field Theory: On Electrodynamics, Non-Abelian Gauge Theories and Gravitation

443 Pages·2012·2.82 MB·English

by  Florian Scheck (auth.)

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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
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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