Topics in mathematical physics

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TopicsinMathematicalPhysics:AJourneyThroughDifferentialEquationsYisongYangPolytechnicInstituteofNewYorkUniversityNonlinearequationsplayimportantrolesinfundamentalscienceandpresentgreatchallengesandopportunitiestoresearchmathematicians.Intheoreticalphysics,manyprofoundconcepts,predictions,andadvanceswerepioneeredthroughmathematicalinsightsgainedfromthestudyoftheequationsgoverningbasicphysicallaws.Whydosomematerialsdemonstratezeroelectricresistancewhencooled(superconductivity)?HowdothetwostrandsinanentangledDNAdoublehelixbecomeseparatedwhenheated(DNAdenaturation)?Whydoestheuniversehaveafinitepast(bigbangcosmology)?Whyarealltheelectricchargesintegermultiplesofaminimalunit(chargequantization)?Whyisthatthebasicconstituentsofmatterknownasquarkscanneverbefoundinisolation(quarkconfinement)?Thesearesomeoftheexemplarysituationswheremathematicalinvestigationisessential.Thepurposeofthisseriesoflectures,tobegivenattheAppliedMathematicsGraduateSummerSchoolofBeijingUniversityinJuly2011,istoprovideavista-typeoverviewofabroadrangeofnonlinearequationsarisinginclassicalfieldtheory.Emphasiswillbegiventothemathematicalstructureandphysicaldescriptionsofvariousbasicproblemsandtotheappreciationofthepoweroffunctionalanalysis,althoughcloseattentionwillalsobegiventothelinkswithotherareasofmathematics,includinggeometry,algebra,andtopology.Hereisatableofcontentsofthenotes.Contents1Hamiltoniansystems31.1Motionofamassiveparticle..............................31.2ThevortexmodelofKirchhoff............................51.3TheN-bodyproblem..................................71.4Hamiltonianfunctionandthermodynamics.....................81.5HamiltonianmodelingofDNAdenaturation.....................102TheSchr¨odingerEquationandQuantumMechanics122.1Pathtoquantummechanics..............................122.2TheSchr¨odingerequation...............................142.3QuantumN-bodyproblem..............................202.4TheHartree–Fockmethod...............................222.5TheThomas–Fermiapproach.............................241 3TheMaxwellequations,Diracmonopole,etc263.1TheMaxwellequationsandelectromagneticduality................263.2TheDiracmonopoleandDiracstrings........................273.3Chargedparticleinanelectromagneticfield.....................283.4RemovalofDiracstrings,existenceofmonopole,andchargequantization....304Abeliangaugefieldequations314.1Spacetime,Lorentzinvarianceandcovariance,etc..................314.2Relativisticfieldequations...............................344.3Couplednonlinearhyperbolicandellipticequations.................374.4AbelianHiggsmodel..................................385TheGinzburg–Landauequationsforsuperconductivity415.1HeuristicproofoftheMeissnereffect.........................415.2Energypartition,fluxquantization,andtopologicalproperties...........425.3Vortex-lines,solitons,andparticles..........................445.4Frommonopoleconfinementtoquarkconfinement.................486Non-Abeliangaugefieldequations496.1TheYang–Millstheory.................................506.2TheSchwingerdyons..................................516.3The’tHooft–PolyakovmonopoleandJulia–Zeedyon................526.4TheGlashow–Weinberg–Salamelectroweakmodel.................567TheEinsteinequationsandrelatedtopics577.1Einsteinfieldequations................................577.2Cosmologicalconsequences..............................617.3StaticsolutionofSchwarzschild............................657.4ADMmassandrelatedtopics.............................708ChargedvorticesandtheChern–Simonsequations718.1TheJulia–Zeetheorem.................................728.2TheChern–Simonstermandduallychargedvortices................738.3TheRubakov–Tavkhelidzeproblem..........................769TheSkyrmemodelandrelatedtopics789.1TheDerricktheoremandconsequences........................789.2TheSkyrmemodel...................................809.3KnotsintheFaddeevmodel..............................819.4Commentsonfractional-exponentgrowthlawsandknotenergies.........8410Stringsandbranes8510.1Relativisticmotionofafreeparticle.........................8610.2TheNambu–Gotostrings...............................8710.3p-branes.........................................8910.4ThePolyakovstring,conformalanomaly,andcriticaldimension..........912 11TheBorn–Infeldgeometrictheoryofelectromagnetism9311.1Formalism........................................9311.2TheBorn–InfeldtheoryandageneralizedBernsteinproblem...........9611.3Chargeconfinementandnonlinearelectrostatics...................981HamiltoniansystemsInthissection,webeginourstudywiththeHamiltonianorLagrangianformalismofclassicalmechanics,whichistheconceptualfoundationofalllaterdevelopments.1.1MotionofamassiveparticleConsiderthemotionofapointparticleofmassmandcoordinates(qi)=qinapotentialfieldV(q,t)describedbyNewtonianmechanics.Theequationsofmotionare∂Vmq¨i=−,i=1,2,···,n,(1.1)∂qiwhere·denotestimederivative.Since∂V−∇V=−(1.2)∂qidefinesthedirectionalongwhichthepotentialenergyVdecreasesmostrapidly,theequation(1.1)saysthattheparticleisacceleratedalongthedirectionoftheflowofthesteepestdescentofV.WiththeLagrangianfunctionXn1˙i2L(q,q,t˙)=m(q)−V(q,t),(1.3)2i=1whichissimplythedifferenceofthekineticandpotentialenergies,(1.1)aresimplytheEuler–LagrangeequationsoftheactionZt2L(q(t),q˙(t),t)dt(1.4)t1overtheadmissiblespaceoftrajectories{q(t)|t10isaparameter.Itisclearthattheflow-linesareconcentriccirclesaroundtheorigin.LetCrbeanyoneofsuchcirclesofradiusr>0.Then,wehaveIv·ds=γ,(1.29)Crwhichsaysthecirculationalonganyflowlineorthestrengthofanyvortextubecontainingthecenterofthevortextakestheconstantvalueγ.Inotherwords,thequantityγgivesthecirculationorstrengthofthevortexcenteredattheorigin.Furthermore,wecanalsocomputethevorticityfielddirectly,w=−∆U=−(∂2+∂2)U=γδ(x),(1.30)12whichclearlyrevealsapointvortexattheorigingivenbytheDiracfunctionandjustifiesagainthequantityγasthestrengthofthepointvortex.FollowingthemodelofKirchhoff,thedynamicalinteractionofNpointvorticeslocatedatx=x(t)∈R2ofrespectivestrengthsγ’s(i=1,···,N)attimetisgovernedbytheinteractioniiipotential1XU(x1,···,xN)=−γiγi0ln|xi−xi0|,(1.31)2π1≤i0istheuniversalgravitationalconstantandXNmimi0U(x1,···,xn)=−G(1.36)0|xi−xi0|1≤i.(1.60)Γ(2d−1)1/2a2Basedontheaboveformalism,PeyrardandBishop[107]succeededinfindingathermo-dynamicaldescriptionoftheDNAdenaturationphenomenon.Using(1.60)andnumericalevaluation,itisshown[107]thatthebasemeanstretchinghy`iincreasessignificantlyasthetemperatureclimbstoaparticularlevelwhichisanunambiguousindicationofDNAdenatura-tion.Anotherinterestingby-productofsuchacalculationisthat,sincethedependenceofthegroundstateontheabsolutetemperatureT=(kBβ)−1isthroughtheparameterdgivenearlier,agreatervalueoftheelasticconstantκleadstoahigherDNAdenaturationtemperature,whichiswhatobserved[46,107]inlaboratory.Inparticular,wehaveseenthatthedynamicsoftheDNAmoleculeiseffectivelydescribedbythereducedHamiltonianthatcontainsthe‘out-of-phase’motionofthebasesonlygivenintermsofthey-variablesasnX11√H=my˙2+κ(y−y)2+D(e−a2yi−1)2.(1.61)iii−122i=1See[106]forareviewofrelatedtopicsanddirections.Thisexampleshowshowasimplesystemofordinarydifferentialequationsmaybeusedtoinvestigateafundamentalprobleminbiophysics.21niTosavespace,weusedx(say)todenotedx···dxandusextodenotethevectorcoordinates(x)orasinglevariableinterchangeablyifthereisnoriskofconfusioninthecontext.11 2TheSchr¨odingerEquationandQuantumMechanicsQuantummechanics(QM)wasdevelopedatthebeginningofthelastcenturyaimedatexplainingphysicalphenomenaatmicroscopicscales(smallmassandsmalldistance)andbaseduponseveralcelebratedexperimentswhichcouldnotbeexplainedwithintheconceptualframeworkofclassicalphysics.Eventoday,QMremainsachallengetohumanintuitionandcontinuestostunawaystudentswhostudyit.See[84,96]forsomeextrareadingofthemathematicalformulationandhistoryofQMand[35,55,90,115]fortextbookintroductionstoQMorientedtowardspeopleinterestedinitsphysicalorigins,mathematicalstructure,andcomputationaldetails.ReferringtoQM,inPrefaceofhisbook[55],Griffithsstates“thereisnogeneralconsensusastowhatitsfundamentalprinciplesare,howitshouldbetaught,orwhatitreallymeans”andquotesRichardFeynman’swords“IthinkIcansaysafelythatnobodyunderstandsquantummechanics.”AllthesesoundratherpessimisticaboutQM.Nevertheless,QMisoneofthegreatestsuccessesofmodernphysics.Inthissection,wewillfocusontheSchr¨odingerequationwhichisthecoreofQMandattempttoachieveareasonablelevelofunderstandingofsomebasicsofQM.2.1PathtoquantummechanicsWestartwithabriefdiscussionofseveralmilestoneearly-daydiscoveriesthatledtotheformu-lationofQM.Wewillmainlyfollowthepresentationin[55]inthefirsttwosubsections.ThephotoelectriceffectPlaceapieceofmetalinavacuumtubeandshootabeamoflightontoit.Theelectronsinthemetalmaybecomesufficientlyenergizedtobeemittedfromthemetal.Thisistheso-calledphotoelectriceffectandfindswiderangeofapplicationsintoday’selectronics.NowmeasuretheenergycarriedbyanemittedelectronanddenoteitbyEe.ItisknownthatEemaybewrittenasthedifferenceoftwoquantities,oneisproportionaltothefrequency,ν,ofthelightbeamsothattheproportioalityconstant,h,isuniversalandindependentofthemetal,theother,φ,dependsonthemetalbutisindependentofthelightfrequency.Therefore,wehaveEe=hν−φ.(2.1)Einstein’spostulateLight,aspecialformofelectromagneticwaves,iscomposedofparticlescalledphotons.Eachphotoncarriesanamountofenergyequaltohν.Thatis,E=hν.(2.2)Whenthephotonhitsanelectroninametal,theelectronreceivesthisamountofenergy,consumestheamountofthebindingenergyofthemetaltotheelectrontoescapefromthemetal,andbecomesanemittedelectronoftheenergygivenby(2.1).MeasurementsInphysics,frequencyνismeasuredinhertzwithunitsecond−1(timespersecond),andangularfrequencyωisrelatedtoνbyω=2πν(radianspersecond).Hence,intermsofω,theEinsteinformulabecomeshE=~ω,~=.(2.3)2π12 RecallthatenergyismeasuredinunitofJoulesandoneJouleisequaltooneNewton×meter.Theconstant~in(2.3),calledthePlanckconstant,isatinynumberoftheunitofJoules×secondandacceptedtobe~=1.05457×10−34.(2.4)(Historically,hiscalledthePlanckconstant,and~theDiracconstantorextendedPlanckconstant.)TheComptoneffectAfterthe1905postulateofEinsteinthatlightiscomposedofphotons,physicistsbegantowonderwhetheraphotonmightexhibitits(kinetic)momentumininteraction(i.e.,incollisionwithanotherparticle).In1922,ComptonandDebyecamewithaverysimple(andbold)mathematicaldescriptionofthis,whichwasthenexperimentallyobservedbyComptonhimselfin1923andfurtherprovedbyY.H.Woo,3thenagraduatestudentofCompton.Insimpleterms,whenaphotonhitsanelectron,itbehaveslikeindeedlikeaparticlecollideswithanotherparticlesothatoneobservesenergyaswellasmomentumconservationrelations,whichisevidencedbyawavelengthshiftafterthecollision.Mathematically,wewritetheenergyofaphotonbytheEinsteinformula,E=mc2,wherecisthespeedoflightinvacuumandmisthe‘virtualrestmass’ofphoton(notethataphotoninfacthasnorestmass).Inviewof(2.3),wehaveE=mc2=~ω.(2.5)Recallthatthewavenumber(alsocalledtheangularwavenumber)k,wavelengthλ,frequencyν,angularfrequencyω,andspeedcofaphotonarerelatedby2πωk=,c=λν=λ.(2.6)λ2πConsequently,themomentumofthephotonisgivenbyEωp=mc==~cc=~k.(2.7)ThedeBrogliewave-particledualityhypothesisIn1924,deBroglieformulatedhiscelebratedwave-particledualityhypothesisinhisPh.D.thesiswhichequalizeswavesandparticles,takestheEinsteinformula(2.3)andtheCompton–Debyeformula(2.7)asthetwoaxioms,andreiteratesthewaveandparticlecharacteristicsofallinteractionsinnature:E=~ω,(2.8)p=~k.(2.9)3HereiswhatfoundinWikipediaaboutY.H.Woo:WugraduatedfromtheDepartmentofPhysicsofNanjingHigherNormalSchool(laterrenamedNationalCentralUniversityandNanjingUniversity),andwaslaterassociatedwiththeDepartmentofPhysicsatTsinghuaUniversity.HewasoncethepresidentofNationalCentralUniversity(laterrenamedNanjingUniversityandreinstatedinTaiwan)andJiaotongUniversityinShanghai.WhenhewasagraduatestudentattheUniversityofChicagohestudiedx-rayandelectronscattering,andverifiedtheComptoneffectwhichgaveArthurComptontheNobelPrizeinPhysics.13 Inotherwords,aparticleofenergyEandmomentumpbehaveslikeawaveofwavenumberkandawaveofwavenumberkbehaveslikeaparticleofenergyEandmomentumpsuchthatE,p,andkarerelatedthrough(2.8)and(2.9).2.2TheSchr¨odingerequationWiththeabovepreparation,wearenowreadytoderivetheSchr¨odingerequationwhichwasfirstpublishedbySchr¨odingerin1926.Considerastationarywavedistributedoverthex-axisofwavenumberk(thewavehaskrepeatedcyclesoverthestandardangular(cell)interval[0,2π])whosesimplestformisgivenbyikxCe.(2.10)Switchontime-dependencesothatthewavemovestoright(say)atvelocityv>0.Weseefrom(2.10)thatthewaveisrepresentedbyik(x−vt)φ(x,t)=Ce.(2.11)Noticethatwecanextend(2.6)as2πωk=,v=λν=λ.(2.12)λ2πCombining(2.11)and(2.12),wehavei(kx−ωt)φ(x,t)=Ce.(2.13)Formally,inviewof(2.13),thedeBrogliemomentum(2.9)canbereadoffasaneigenvalueoftheoperator−i~∂.Thatis,∂x∂−i~φ=(~k)φ=pφ,(2.14)∂xsothat2∂22−i~φ=(~k)φ=pφ.(2.15)∂xSimilarly,thedeBroglieenergy(2.8)canbereadoffasaneigenvalueoftheoperatori~∂.That∂tis,∂i~φ=(~ω)φ=Eφ.(2.16)∂tForafreeparticleofmassm>0,weknowthatthereholdstheclassicalrelationp2E=.(2.17)2mInviewof(2.15)–(2.17),wearriveatthefreeSchr¨odingerequation∂φ~2∂2φi~=−.(2.18)∂t2m∂x214 ForaparticlemovinginapotentialfieldV,theenergy-momentumrelation(2.17)becomesp2E=+V.(2.19)2mThereforetheSchr¨odingerequation(2.18)forafreeparticleismodifiedintoform∂φ~2∂2φi~=−+Vφ.(2.20)∂t2m∂x2ThisiscalledtheSchr¨odingerwaveequationwhosesolution,φ,iscalledawavefunction.StatisticalinterpretationofthewavefunctionbyBornConsiderthe1DSchr¨odingerequation(2.20)describingaparticleofmassmandassumethatφisa‘normalized’solutionof(2.20)whichsatisfiesnormalizationconditionZ|φ(x,t)|2dx=1(2.21)andcharacterizesthe‘state’oftheparticle.AccordingtoBorn,themathematicalmeaningofsuchawavefunctionisthatρ(x,t)=|φ(x,t)|2givestheprobabilitydensityofthelocationoftheparticleattimet.Inotherwords,theprobabilityoffindingtheparticleinaninterval(a,b)attimetisZbP({a0suchthatV2(x)≤−Γ,∀x.(2.43)Then(2.42)and(2.43)leadusto0ΓP(t)≤−P(t).(2.44)~Ifaparticleispresentinitially,P(0)=1,then(2.44)impliesthat−ΓtP(t)≤e~,t>0.(2.45)Inotherwords,inabulksituation,wewillobservelossofparticles,whichsuggeststhatweencounterunstableparticles.HigherdimensionalextensionsItisimmediatethatourdiscussionaboutthe1DSchr¨odingerequationscanbeextendedtoarbitrarilyhighdimensions.Forthispurpose,weconsidertheMinkowskispaceofdimension(n+1)withcoordinatest=x0,x=(x1,···,xn),fortimeandspace,respectively.WeusetheGreekletterµ,ν,etc,todenotethespacetimeindices,µ,ν=0,1,···,n,theLatinletteri,j,k,etc,todenotethespaceindices,i,j,k=1,···,n,and∇todenotethegradientoperatoronfunctionsdependingonx1,···,xn.TheSchr¨odingerequationthatquantum-mechanicallygovernsaparticleofmassminRnisgivenby∂φ~2i~=−∆φ+Vφ.(2.46)∂t2mTheenergyandmomentumoperatorsare,respectively,givenby∂Eˆ=i~,pˆ=−i~∇.(2.47)∂tThetotalenergyoperator,ortheHamiltonian,isHˆ=1pˆ2+V.(2.48)2mTheprobabilitycurrentj=(jµ)=(j0,j)=(ρ,ji)isdefinedbyi~j=i(φ∂iφ−φ∂iφ),i=1,···,n.(2.49)2mTheconservationlawrelatingprobabilitydensityρ(or‘charge’)andprobabilitycurrentj(or‘current’)readsµ∂∂µj=0,orρ+∇·j=0.(2.50)∂tMoreover,theEhrenfesttheoremsaysdhpˆi=−h∇Vi.(2.51)dt18 Notealsothat,inapplications,thepotentialfunctionVmaybeself-inducedbythewavefunctionφ.Forexample,V=|φ|2.SteadystateandenergyspectrumWereturntotheone-dimensionalsituationagainandlookforsolutionof(2.20)intheseparableformφ(x,t)=T(t)u(x).(2.52)Followingstandardpath,wehaveT0(t)~2u00(x)i~=−+V(x)=E,(2.53)T(t)2mu(x)whereEisaconstant.IntermsofE,wehave−iEtT(t)=e~(2.54)anduandEsatisfytherelation~2d2−+Vu=Huˆ=Eu.(2.55)2mdx2Supposethatthequantumstateoftheparticleisdescribedbytheseparablesolution(2.52).Thenthenormalizationcondition(2.21)becomesZ|u|2dx=1.(2.56)Using(2.55),(2.56),andthefactthatEmustberealotherwiseφisnotnormalizable(why?),wegetZZhHˆi=φHφˆdx=E|u|2dx=E,(2.57)whichshouldnotcomeasasurprise.Furthermore,sincethesecondmomentofHˆissimplyZZhHˆ2i=φHˆ2φdx=E2|u|2dx=E2,(2.58)weseethatthevarianceσ2ofthemeasurementsofHˆiszerobecauseHˆσ2=hHˆ2i−hHˆi2=E2−E2=0.(2.59)HˆSucharesultimpliesthat,whenaparticleoccupiesaseparablestate,itsenergyisadefinitevalue.Nowconsiderthegeneralsituation.Forsimplicity,weassumethatHˆhas{En}asitscom-pleteenergyspectrumsothatthecorrespondingenergyeigenstates{un}formanorthonormalbasiswithrespecttothenaturalL2-innerproduct.Therefore,anarbitrarysolutionφ(x,t)of(2.20)canberepresentedasX∞En−itφ(x,t)=cne~un(x).(2.60)n=119 Therefore,inviewofthenormalizationcondition(2.21),wearriveatX∞|c|2=1.(2.61)nn=1Thisresultisamazingsinceitremindsusofthetotalprobabilityofadiscreterandomvariablewhoseprobabilitymassdensityfunctionisgivenbythesequence{|cn|2}.InQM,indeed,sucharandomvariableispostulatedasthemeasuredvalueofenergyfortheparticlethatoccupiesthestategivenby(2.60).Inotherwords,ifweuseEtodenotetherandomreadingoftheenergyoftheparticleoccupyingthestate(2.60),thenEmayonlytakeE1,E2,···,En,···,aspossiblevalues.Furthermore,ifthesevaluesaredistinct,thenP({E=E})=|c|2,n=1,2,···.(2.62)nnInQM,theabovestatementappearsasamajorpostulatewhichisalsoreferredtoasthe‘generalizedstatisticalinterpretation’ofeigenstaterepresentation.Inparticular,whenφ(x,t)itselfisseparableasgivenin(2.52),sinceEisaneigenvalueitself,weseethatEtakesthesinglevalueEwithprobabilityone.Thus,werecovertheearlierobservationmadeonseparablestate.Finally,using(2.60),wecancomputetheexpectedvalueoftheenergyoperatorHˆimmedi-ately:ZhHˆi=φHφˆdxX∞=E|c|2.(2.63)nnn=1Itisinterestingtonotethat(2.63)isconsistentwiththepostulate(2.62).Infact,wemayalsounderstandthattheinterpretation(2.62)ismotivated(orsupported)by(2.63).Notealsothattheright-handsideof(2.63)isindependentoftimet.Thus,(2.63)maybeviewedasaQMversionoftheenergyconservationlaw.2.3QuantumN-bodyproblemConsidertheQMdescriptionofNparticlesofrespectivemassesmiandelectricchargesQi(i=1,···,N)interactingsolelythroughtheCoulombforce.ThelocationsoftheseNparticlesareatxi∈R3.Thuswecanexpresstheirrespectivemomentaaspˆi=−i~∇xi,i=1,···,N.(2.64)ThepotentialfunctionisgivenbyXNQiQi0V(x1,···,xN)=.(2.65)0|xi−xi0|1≤i0(whichhappenstobecontinuous)andE<0(whichhappenstobediscrete).SincetheCoulombpotentialvanishesatinfinity,thestatewithE<0indicatesthattheelectronisinastatewhichliesinsidethe‘potentialwell’ofthenucleus,theproton,whichiscalledaboundstate,anddescribesasituationwhentheelectronandproton‘bind’toformacompositeparticle,thehydrogen.Likewise,thestatewithE>0indicatesthattheelectronisinastatewhichliesoutsidethepotentialwelloftheproton,whichiscalledascatteringstate,anddescribesasituationwhentheelectronandprotoninteractastwocharged‘free’particleswhichdonotappeartohavethecharacteristicsofacompositeparticle,namely,ahydrogenatom.Hencewewillbeinterestedintheboundstatesituationonly.Restrictingtosphericallysymmetricconfigurations,itcanbeshown[55]thatthebound-stateenergyspectrumof(2.68)isgivenbyme4E1En=−=,n=1,2,···,(2.69)2~2n2n2knownastheBohrformula.Theground-stateenergy,E1,isabout−13.6eVwhichiswhatisneededtoionizeahydrogenatom.Supposethatthehydrogenatomabsorbsoremitsanamountofenergy,Eδ,sothattheinitialandfinalenergiesareEiandEf,respectively.Then!11Eδ=Ei−Ef=E12−2.(2.70)nnifItwillbeinstructivetoexamineinsomedetailthatthehydrogenatomismadetoemitenergythroughtheformoflight.ThePlankformulastatesthatthefrequencyνofthelightobeyscEδ=hν=h,(2.71)λwherecisthespeedandλisthewavelengthoflight.Substituting(2.71)into(2.70),weobtainthecelebratedRydbergformula!1me411λ=4πc~3n2−n2,ni>nf.(2.72)fi21 Specifically,transitionstothegroundstatenf=1giverisetoultraviolet(higher-frequency)lightswith1me41=1−,n=2,3,···,(2.73)λ4πc~3n2calledtheLymanseries;transitionstothefirstexcitedstatenf=2leadtovisible(medium-frequency)lightswithn=3,4,···,calledtheBalmerseries;transitionstothesecondexcitedstatenf=3correspondtoinfrared(lower-frequency)lightswithn=4,5,···,calledthePaschenseries.Theserieswithnf=4,5,6arenamedunderBrackett,Pfund,andHumphreys,respec-tively.TheRydbergformulawaspresentedbyJohannesRobertRydberg(aSwedishexperimen-talphysicistatLundUniversity)in1888,4manyyearsbeforetheformulationoftheSchr¨odingerequationandQM.Themodelforhelium,withZ=2,immediatelybecomesmoredifficultbecausetheHamil-toniantakestheform~222e2~222e2e2Hˆ=−∇x+−∇x++,(2.74)2m1|x|2m2|x||x−x|1212inwhichthelasttermrenderstheproblemnon-separable.5Themodelforlithium,withZ=3,sharesthesamedifficulty.Thus,weseethatthequantumN-bodyproblemisimportantforparticlephysicsandquantumchemistrybutdifficulttodealwithwhenN≥2.Awayoutofthisistodevelopapproximationmethods.Alongthisdirection,twowell-knownapproachesaretheHartree–FockmethodandtheThomas–Fermimodel,bothbasedonvariationaltechniques.Wewillbrieflydiscusstheseapproachesbelow.Wenotethat,whiletheclassicalN-bodyproblemisnonlinear,itsQMversion,whichasksaboutthespectrumoftheN-bodyHamiltonian,becomeslinear.2.4TheHartree–FockmethodTomotivateourdiscussion,weuse{En}todenotethecompletesequenceofeigenvaluesoftheHamiltonianHˆsothatE1≤E2≤···≤En≤···,(2.75)and{un}thecorrespondingeigenstateswhichformanorthonormalbasis.Letψbeanynor-malizedfunction,satisfyingZhψ|ψi=|ψ|2dx=1.(2.76)4TheoriginalRydbergformulareads!111=R−,ni>n,λn2n2ffi7−1whereRistheRydbergconstantwiththeobservedvalueR=1.097×10m,whichamazinglyagreeswiththatgivenin(2.72)intermsofthefundamentalconstantsc,e,m,~.5Onemightbetemptedtoignoretheinter-electroninteractionspelledoutbythelastterm.Inthissituation,theHamiltonianisthesumofthetwohydrogenHamiltonians,whichisseparableandrenderstheground-stateenergyE1=8(−13.6)=−108.8eV.Thisvalueismuchlowerthantheexperimentallydeterminedvalue−79eVforhelium.Hence,weseethattheinter-electronpotential,whichindeedaddsapositivecontribution,shouldnotbeneglected.22 Thus,theexpansionX∞ψ=cnun,(2.77)n=1givesusX∞|c|2=1.(2.78)nn=1Consequently,wehaveZX∞X∞hψ|Hˆ|ψi=ψHψˆdx=E|c|2≥E|c|2=E.(2.79)nn1n1n=1n=1Inotherwords,thelowesteigen-pair(E1,u1)maybeobtainedfromsolvingtheminimizationproblemnominhψ|Hˆ|ψi|hψ|ψi=1.(2.80)Inpractice,itisoftenhardtoapproach(2.80)directlyduetolackofcompactness.Instead,onemaycomeupwithareasonablygoodwave-functionconfiguration,atrialapproximation,dependingonfinitelymanyparameters,sayα1,···,αm,oftheformψ(x)=ψ(α1,···,αm)(x).(2.81)Thenonesolvestheminimizationproblemnominhψ(α1,···,αm|Hˆ|ψ(α1,···,αm)i|hψ(α1,···,αm)|ψ(α1,···,αm)i=1,(2.82)involvingmultivariablefunctionsofα1,···,αmonly.WenowrewritetheHamiltonian(2.67)asXZ1XHˆ=Hˆi+Vij,(2.83)2i=1i6=jwhere~2Ze2e2Hˆ=−∇2−,V=,i6=j,i,j=1,···,Z,(2.84)i2mxi|x|ij|x−x|iijarethei-thelectronHamiltonian,withoutinter-electroninteraction,andtheinter-electronCoulombpotentialbetweenthei-andj-thelectrons,respectively.PSincethenon-interactingHamiltonianZHˆallowsseparationofvariables,wearepromptedi=1itousethetrialconfigurationψ(x,···,x)=φ(x)···φ(x),x,···,x∈R3,(2.85)1Z11ZZ1ZknownastheHartreeproduct,whereφ1,···,φZareunknowns.Inordertoimplementthenormalizationconditionhψ|ψi=1,weimposeZZhφ|φi=|φ|2(x)dx=|φ|2(x)dx=1,i=1,···,Z.(2.86)iiiiii23 Inserting(2.85)andusing(2.86),wearriveatZI(φ1,···,φZ)=ψHψˆdx1···dxZXZZ1XZ=φiHˆiφidxi+φiφjVijφiφjdxidxj2i=1i6=jXZZ22XZ22~2Ze2e|φi(x)||φj(y)|=|∇φi|−|φi|dx+dxdy,2m|x|2|x−y|i=1i6=j(2.87)wherewehaverenamedthedummyvariableswithx,y∈R3.Consequently,weareledtoconsideringthereducedconstrainedminimizationproblemmin{I(φ1,···,φZ)|hφi|φii=1,i=1,···,Z},(2.88)whosesolutionsmaybeobtainedbysolvingthefollowingsystemofnonlinearintegro-differentialequations~Ze2e2XZZ|φ(y)|2∆φ+φ+λφ=jdyφ,i=1,···,Z,(2.89)iiiii2m|x|2|x−y|j6=iwiththeLagrangemultipliersλ1,···,λZappearingaseigenvalues.Thus,inparticular,weseethat,inordertosolvealinearproblemwithinteractingpotential,weareofferedahighlynontrivialnonlinearproblemtotackleinstead.NotethattheabovediscussionoftheHartree–Fockmethodisover-simplified.SinceelectronsarefermionswhichobeythePauliexclusionprinciple,thewavefunctiontobeconsideredshouldhavebeentakentobeskew-symmetricinsteadofbeingsymmetric,withrespecttopermutationsofx1,···,xZandφ1,···,φZ,whichgiverisetotheirjointwavefunctionψ(x1,···,xZ).Thus,practically,weneedtoconsidertheproblemwiththeredesignedskew-symmetricwavefunctionφ1(x1)φ1(x2)···φ1(xZ)1φ2(x1)φ2(x2)···φ2(xZ)ψ(x1,···,xZ)=√············,(2.90)Z!φZ(x1)φZ(x2)···φZ(xZ)knownastheSlaterdeterminant[131],whichmakestheproblemmorecomplicated.2.5TheThomas–FermiapproachTheHartree–FockmethodiseffectivewhentheatomnumberZissmall.WhenZislarge,theproblemquicklybecomesdifficultandalternativemethodsaretobedeveloped.TheThomas[140]andFermi[45]approachissuchamethodwhichtreatselectronsasastaticelectrongascloudsurroundinganucleusandsubjecttoacontinuouslydistributedelectrostaticpotential.Theelectronatxassumesthemaximumenergy,say−eA,whereAisaconstantotherwisetheelectronswillnotbestayinginthestaticstate.Lettheelectrostaticpotentialbeφ(x).Then24 −eφ(x)willbethepotentialenergycarriedbytheelectron.Thus,ifweusep(x)todenotethemaximummomentumoftheelectron,wehavetherelationp2(x)−eA=−eφ(x).(2.91)2mOntheotherhand,letn(x)bethenumberofelectronsperunitvolumeofthespace,whichistakentobeatinydomain,sayδΩ,centeredaroundx.Thenp(x)isapproximatelyaconstantoverδΩ.Weassumethatallstatesinthemomentumspaceareoccupiedbytheelectronswhichtakeupavolume4π3p(x)(2.92)3inthemomentumspace.Sinceeachstatecanbeoccupiedbyexactlyoneelectron,duetoPauli’sexclusionprinciple,wearriveattheelectronnumbercount(inδΩ)4πp3(x)n(x)=23,(2.93)h3wherehisthePlanckconstantandthefactor2takesaccountofthetwopossiblespinsoftheelectrons.Inserting(2.91)into(2.93),wehave8π3n(x)=(2me[φ(x)−A])2.(2.94)3h3Ontheotherhand,weknowthattheelectrostaticpotentialfunctionφandtheelectronnumberdensitynarerelatedthroughthePoissonequation∆φ=4πen,(2.95)where−ne=ρisthechargedensity(cf.§3).Inviewof(2.94)and(2.95),weobtaintheself-consistencyequation2332πe3∆φ=α(φ−A)2,α=(2me)2,(2.96)3h3whichservesasthegoverningequationoftheThomas–Fermimethod,alsocalledtheThomas–Fermiequation.Ofcourse,ameaningfulsolutionmustsatisfyφ≥A.SincetheelectroncloudsurroundsanucleusofchargeZe,weseethatφshouldbehavelikeacentralCoulombpotential,Ze/|x|,neartheorigin.Hence,wehavethesingularboundaryconditionlim|x|φ(x)=Ze.(2.97)|x|→0Besides,ifweassumetheelectroncloudisconcentratedinaboundeddomain,sayΩ,thenn=0on∂Ω.Thus,(2.94)leadstotheboundaryconditionφ(x)=A,x∈∂Ω.(2.98)ItisinterestingtonotethattheThomas–Fermisemi-classicaltreatmentofthequantumN-bodywhichislinearturnstheproblembackintoanonlinearproblem.See[74,75,76]forthemathematicalworkontheThomas–Fermimodelandmanyimportantextensions.25 3TheMaxwellequations,Diracmonopole,etcInthissection,westartwiththeelectromagneticdualityintheMaxwellequations.WethenpresenttheDiracmonopoleandDiracstrings.Next,weconsiderthemotionofachargedparticleinanelectromagneticfieldandintroducethenotionofgaugefields.Finally,wederiveDirac’schargequantizationformula.3.1TheMaxwellequationsandelectromagneticdualityLetthevectorfieldsEandBdenotetheelectricandmagneticfields,respectively,whichareinducedfromthepresenceofanelectricchargedensitydistribution,ρ,andacurrentdensity,j.ThenthesefieldsaregovernedintheHeaviside–LorentzrationalizedunitsbytheMaxwellequations∇·E=ρ,(3.1)∂E∇×B−=j,(3.2)∂t∇·B=0,(3.3)∂B∇×E+=0.(3.4)∂tInvacuumwhereρ=0,j=0,theseequationsareinvariantunderthedualcorrespondenceE7→B,B7→−E.(3.5)Inotherwords,inanotherworld‘dual’totheoriginalone,electricityandmagnetismareseenasmagnetismandelectricity,andviceversa.Thispropertyiscalledelectromagneticdualityordualsymmetryofelectricityandmagnetism.However,suchasymmetryisbrokeninthepresenceofanexternalcharge-currentdensity,(ρ,j).Diracproposedaprocedurethatmaybeusedtosymmetrize(3.1)–(3.4),whichstartsfromtheextendedequations∇·E=ρe,(3.6)∂E∇×B−=je,(3.7)∂t∇·B=ρm,(3.8)∂B∇×E+=jm,(3.9)∂twhereρe,jeandρm,jmdenotetheelectricandmagneticsourcetermsrespectively.Theseequa-tionsareinvariantunderthetransformation(3.5)ifthesourcetermsaretransformedaccordingly,ρe7→ρm,je7→jm,ρm7→−ρe,jm7→−je.(3.10)Hence,withthenewmagneticsourcetermsin(3.8)and(3.9),dualitybetweenelectricityandmagnetismisagainachievedasinthevacuumcase.In(3.6)–(3.9),ρeandjearetheusualelectricchargeandcurrentdensityfunctions,butρmandjm,whicharethemagneticchargeandcurrentdensityfunctions,introducecompletelynewideas,bothtechnicallyandphysically.26 3.2TheDiracmonopoleandDiracstringsInordertomotivateourstudy,considertheclassicalsituationthatelectromagnetismisgeneratedfromanidealpointelectricchargeqlyingattheorigin,ρe=4πqδ(x),je=0,ρm=0,jm=0.(3.11)Itisclearthat,inserting(3.11),thesystem(3.6)–(3.9)canbesolvedtoyieldB=0andqE=x,(3.12)|x|3whichisthewell-knownCoulomblawinstaticelectricity.Wenowconsiderthecaseofapointmagneticchargeg,oramonopole,attheorigin,ρe=0,je=0,ρm=4πgδ(x),jm=0.(3.13)HenceE=0andg1B=x=−g∇.(3.14)|x|3|x|Consequently,themagneticfluxthroughaspherecenteredattheoriginandofradiusr>0isZΦ=B·dS=4πg,(3.15)|x|=rwhichisindependentofrandisidenticaltotheGausslawforstaticelectricity.Nevertheless,weshowbelowthroughquantummechanicsthattheintroductionofamagneticchargeyieldsdrasticallynewphysicsbecauseelectricandmagneticfieldsareinduceddifferentlyfromagaugevectorpotential.Wenowevaluatetheenergyofamonopole.RecallthatthetotalenergyofanelectromagneticfieldwithelectriccomponentEandmagneticcomponentBisgivenbyZ122E=(E+B)dx.(3.16)2R3Inserting(3.14)into(3.16)andusingr=|x|,wehaveZ∞1E=2πg2dr=∞.(3.17)r20Thisenergyblow-upseemstosuggestthattheideaofamagneticmonopoleencountersanunacceptableobstacle.However,sincetheCoulomblawexpressedin(3.12)forapointelectricchargealsoleadstoadivergentenergyofthesameform,(3.17),theinfiniteenergyproblemforamonopoleisnotamoreseriousonethanthatforapointelectricchargewhichhasbeenusedeffectivelyasgoodapproximationforvariousparticlemodels.Wenowstudythemagneticfieldgeneratedfromamonopolemoreclosely.Recallthatfortheelectricfieldgeneratedfromapointelectricchargeq,theCoulomblaw(3.12)givesusascalarpotentialfunctionφ=−q/|x|suchthatE=∇φholdseverywhereawayfromthepointelectriccharge.27 Similarly,weconsiderthemagneticfieldBgeneratedfromamonopoleofchargeg,givenin(3.14).Basedontheclassicalknowledgeonmagneticfield,weknowthatBshouldbesolenoidal.Thatis,thereshouldexistavectorfieldAsuchthatB=∇×A,(3.18)exceptattheoriginwherethepointmonopoleisplaced.Unfortunately,using(3.15)andtheStokestheorem,itiseasytoseethat(3.18)cannotholdeverywhereonanyclosedspherecenteredattheorigin.Inotherwords,anysuchspherewouldcontainasingularpointatwhich(3.18)fails.Shrinkingaspheretotheoriginwouldgiveusacontinuouslocusofsingularpointswhichisastringthatlinkstheorigintoinfinity.SuchastringiscalledaDiracstring.Itmaybecheckeddirectlythat−x2x1A+=(A+,A+,A+),A+=g,A+=g,A+=0,(3.19)1231|x|(|x|+x3)2|x|(|x|+x3)3satisfies(3.18)everywhereexceptonthenegativex3-axis,x1=0,x2=0,x3≤0.Thatis,withA+,theDiracstringS−isthenegativex3-axis.Similarly,x2−x1A−=(A−,A−,A−),A−=g,A−=g,A−=0,(3.20)1231|x|(|x|−x3)2|x|(|x|−x3)3satisfies(3.18)everywhereexceptonthepositivex3-axis,x1=0,x2=0,x3≥0.Thatis,withA−,theDiracstringS+isthepositivex3-axis.Asaconsequence,A+andA−donotagreeawayfromS+∪S−becausethereholdsA+−A−=a=(a,a,a),x∈R3(S+∪S−),(3.21)123where−2gx22gx1a1=(x1)2+(x2)2,a2=(x1)2+(x2)2,a3=0.(3.22)InordertounderstandthephysicalmeaningassociatedwiththeappearanceoftheDiracstringswhichgiverisetothepuzzlingambiguity(3.21),weneedtoconsiderthemotionofanelectricchargeinanelectromagneticfield.3.3ChargedparticleinanelectromagneticfieldConsiderapointparticleofmassmandelectricchargeQmovinginanelectricfieldEandamagneticfieldB,inadditiontoapotentialfieldV,intheEuclideanspaceR3sothatx=(x1,x2,x3)givesthelocationoftheparticle.Theequationofmotionismx¨=Q(E+x˙×B)−∇V,(3.23)whereQEistheelectricforceandQx˙×BistheLorentzforceofthemagneticfieldBexertedontheparticleofvelocityx˙.LetBandEberepresentedbyavectorpotentialAandascalarpotentialΨasfollowsB=∇×A,∂AE=∇Ψ−.∂t28 Atclassicallevel,ΨandAdonotcontributetotheunderlyingphysicsbecausetheydonotmakeappearanceinthegoverningequations(3.1)–(3.4)and(3.23).However,atquantum-mechanicallevel,theydomakeobservablecontributions.SuchaphenomenonwaspredictedbyAharonovandBohm[5,6]andisknownastheAharonov–Bohmeffect.Thus,inordertoexplorethemeaningofthe‘ambiguity’ofthevectorpotentialassociatedwith(3.21),weneedtoconsidertheQMdescriptionofthemotionofthechargedparticle.Usingy=(yi)=mx˙=m(x˙i)todenotethemechanicalmomentumvector,theequation(3.23)becomes∂Ψ∂Aij∂Aj∂Ai∂Vy˙i=Q−+Qx˙−−∂xi∂t∂xi∂xj∂xidAi∂Ψ˙j∂Aj∂V=−Q+Q+Qx−,dt∂xi∂xi∂xiwhichmayberecastintotheformd∂˙j(yi+QAi)=i(QΨ+QxAj−V),dt∂xord∂L∂L=,i=1,2,3,(3.24)dt∂x˙i∂xiifwedefinethefunctionLtobeL(x,x˙,t)=1m(x˙i)2+QΨ+Qx˙iA−Vi212=mx˙+QΨ(x,t)+Qx˙·A(x,t)−V(x,t).(3.25)2Inotherwords,theformula(3.25)givesustheLagrangianfunctionoftheproblem.ItisinterestingtonotethatthemomentumvectorhasacorrectionduetothepresenceoftheelectromagneticfieldthroughthevectorpotentialA,∂Lpi==˙yi+QAi,i=1,2,3.(3.26)∂x˙iHencetheHamiltonianfunctionbecomes˙i12H=pix−L=y˙i−QΨ+V2m12=(pi−QAi)−QΨ+V.(3.27)2mFinally,ifweuseA=(Aµ)(µ=0,1,2,3)todenoteavectorwithfourcomponents,A=(Ψ,A),theHamiltonianfunction(3.27)takestheform12H=(pi−QAi)−QA0+V.(3.28)2mFromtheHamiltonian(3.28)andthecorrespondence(2.47),wehave∂ψ12i~=−(~∂i−iQAi)ψ−QA0ψ+Vψ.(3.29)∂t2m29 Thus,ifweintroducethegauge-covariantderivativesQDµψ=∂µψ−iAµψ,µ=0,1,2,3,(3.30)~thenthegaugedSchr¨odingerequation(3.29)assumesanelegantform,~2i~Dψ=−D2ψ+Vψ.(3.31)0i2mNotethat(3.29)or(3.31)issemi-quantummechanicalinthesensethatthepointparticleofmassmistreatedquantummechanicallybytheSchr¨odingerequationbuttheelectromagneticfieldisaclassicalfield,throughthecouplingofthevectorpotentialAµ,alsocalledthegaugefield,whichwillbemademorespecificinthenextsection.Itcanbeexaminedthat(3.31)ininvariantunderthetransformationiω~ψ7→eψ,Aµ7→Aµ+∂µω,(3.32)Qwhichisalsocalledthegaugetransformation,gaugeequivalence,orgaugesymmetry.Twogaugeequivalentfieldconfigurations,(ψ,A)and(ψ0,A0),describeidenticalphysics.µµInviewofdifferentialgeometry,theabovestructuredefinesacomplexlinebundleξovertheMinkowskispacetimeR3,1wherethesymmetrygroupisU(1)={eiω|ω∈R}sothatψisacross-sectionandAµaconnectionwhichobeythetransformationproperty00~−123,1ψ=Ωψ,Aµ=Aµ−iΩ∂µΩ,Ω∈C(R,U(1)).(3.33)QItwillbeconvenienttoconsidertheprobleminthecontextofsuchaglobaltransformationproperty.3.4RemovalofDiracstrings,existenceofmonopole,andchargequantizationWearenowpreparedtostudytherelationbetweenthevectorpotentialsA+andA−inducedfromapointmagneticchargegplacedattheorigin.From(3.22),weseethatifweuse(r,θ,ϕ)todenotethesphericalcoordinateswhereθistheazimuthangleandϕtheinclinationangle,thenx2a=2g∂θ,a=2g∂θ,θ=tan−1.(3.34)1122x1Thus,insertingΩ=eiωinto+−~−1Aµ−Aµ=−iΩ∂µΩ,(3.35)QandassumingωdependsonθonlyonR3(S+∪S−),wehave,inviewof(3.34),therelation~∂ω=2g.(3.36)Q∂θIntegratingtheaboveequationandusingtherequirementofthesingle-valuednessofΩ,wearriveat~(2πn)=2g(2π),n∈N,(3.37)Q30 whichleadstotheDiracchargequantizationformula~gQ=n,n∈N.(3.38)2Consequently,whenthecondition(3.38)holds,themagneticfieldawayfromapointmagneticchargegiswelldefinedeverywhereandisgeneratedpiecewisefromsuitablegaugepotentialsdefinedontheircorrespondingdomains.Inparticular,theDiracstringsareseentobeartifactsandareremoved.Consequently,likepointelectriccharges,magneticmonopolesarealsotrulypointmagneticcharges,whichwillsimplybereferredtoasmonopolesfromnowon.Animmediatepopular-scienceimplicationoftheformula(3.38)isthattheexistenceofasinglemonopoleintheuniversewouldpredictthatallelectricchargesareintegermultiplesofabasicunitcharge.Indeed,thisiswhatobservedinnaturesinceallelectricchargesaremeasuredtobethemultiplesofthechargeoftheelectron.Althoughamonopolehasneverbeenfoundinnature,therearesomerecentheatedactivitiesamongexperimentalphysicistsleadingtothediscoveryofmonopole-likestructuresincondensed-mattersystems[15,27,49,93].Thechargequantizationformula(3.38)wasoriginallyderivedbyDirac[36]throughacom-putationoftheQMeigen-spectrumofthemomentumoperatorassociatedwiththemotion(3.23).Thederivationherealongtheformalismofacomplexlinebundle(oraU(1)-principalbundle)wasworkedoutbyT.T.WuandC.N.Yang[149,150]inthemid1970’swhichlaysafirmmathematicalfoundationformonopolesinparticularandtopologicalsolitons[89,152]ingeneral.4AbeliangaugefieldequationsInthissection,wepresentaconcisediscussionoftheAbelianHiggsfieldtheory.Westartwithlayingthenecessaryfoundationsfordevelopingrelativisticfieldtheory.WethenderivetheAbelianHiggsmodel,alongwithadiscussionofvariousnotionsofsymmetrybreakings.4.1Spacetime,Lorentzinvarianceandcovariance,etcLetVbeann-dimensionalspaceoverR,withitsdualspace(thevectorspaceofalllinearfunctionalsoverV)denotedbyV0.Foragivenbasis,B={u1,···,un},overV,weuseB0={u0,···,u0}⊂V0todenotethedualbasisoverV0sothat1nu0(u)=hu,u0i=δ,i,j=1,···,n.(4.1)jiijijForaninvertiblelineartransformationA:V→V,C={v1,···,vn}={Au1,···,Aun}isanotherbasisforV.SupposethedualbasiscorrespondingtoCisC0={v0,···,v0}andtheadjointofA1nisA0:V0→V0.Thenwehaveδ=hv,v0i=hAu,v0i=hu,A0v0i,i,j=1,···,n,(4.2)ijijijijwhichleadsustotheconclusionthatA0v0=u0or(A0)−1u0=v0,i=1,···,n.(4.3)iiii31 Thus,thedualbasistransformationfollowsareverseddirection.Forthisreason,wesaythatthebasesinVaretransformedinacovariantway,whilethebasesinV0aretransformedinacontravariantway.IfweexpressAexplicitlyintermsofamatrix(Aij)byXnAuj=Aijui,i=1,···,n,(4.4)i=1PnthenforanyvectorexpandedwithrespecttothebasesBi.e.,u=i=1xiui,withcoordinatevector(x1,···,xn),wehaveXnXnXnXnv=Au=xiAjiuj=Aijxjui.(4.5)i=1j=1i=1j=1Inotherwords,ifwewritethecoordinatesofvunderBas(y1,···,yn),wehavethesame‘forward’relationshipXnyi=Aijxj,i=1,···,n.(4.6)j=1Thus,wemaycallthecoordinatevectorsforvectorsinVthecovariantvectors.Likewise,foraPPvectoru0=nx0u0,correspondingly,weseethatv0=(A0)−1u0=ny0u0satisfiesi=1iii=1iiXnXnXny0=A−1x0orx0=A0y0=Ay0.(4.7)ijijiijjjijj=1i=1i=1Hence,weseethatthecorrespondingcoordinatevectorsaretransformedina‘backward’mannerwiththetransposedmatrix.Forthisreason,thecoordinatevectorsforvectorsinV0arecalledthecontravariantvectors.Simplyspeaking,covariantandcontravariantvectorstransformthemselveswithrespecttothecolumnandrowindicesofatransformationmatrix,respectively.Analogously,wehavecovariant,contravariant,andmixedtensors,aswell.Notethat,sinceV00=V,theterms‘covariant’and‘contravariant’arerelativeandinter-changeable,sincethecolumnandrowindicesofamatrixareinterchangeable.Nowwecomebacktofieldtheory.TheMinkowskispacetimeR3,1ischosentohavethe4×4metricmatrixgivenasg=(gµν)=diag{1,−1,−1,−1},µ,ν=0,1,2,3.(4.8)µTheLorentztransformationare4×4invertiblematrices,oftheform(Lν)(µ,νaretherowandcolumnindices,respectively),whichleaveginvariant,i.e.,g=LtgL,org=LµgLν(α,β,µ,ν=0,1,2,3),(4.9)αβαµνβwheresummationconventionisassumedoverrepeatedindices.Thusgµνinparticularisacontravariant2-tensor.From(4.9),wehaveg−1=(L−1)g−1(L−1)t.(4.10)32 Sincematrixtranspositionswitchestheindicesofcolumnsandrowsofamatrix,weseethatg−1isacovariant2-tensor,writtenasgµν,whichhappenstobeidenticaltog,g−1=(gµν)=(g)=g.(4.11)µνNowthecoordinatesoftheMinkowskispaceR3,1iswrittenasxµ,whichisacovariantvector,ofcourse,whichistransformedaccordingtoyµ=Lµxν.(4.12)νWemayusegµνandgtoraiseandlowerindices,whichtransformcontravariantandcovariantµνquantitiestotheircounterparts(i.e.,covariantandcontravariantquantities).Forexample,x=gxν,(x)=(x,x,x,x)=(x0,−x1,−x2,−x3).(4.13)µµνµ0123Asaconsequenceof(4.9),(4.12),and(4.13),wehavetheinvarianceyyµ=gLνxαLµxβ=xαgxβ=xxµ,(4.14)µµναβαβµforthecontractionofanycoordinatevectorundertheLorentztransformations,whichisessentialforrelativity.ThesetofallLorentztransformations,ortheLorentzgroup,isgeneratedbytheLorentzboosts6andspatialrotations.Forexample,theLorentzboostalongthex1-axisconnectingtwoinertialcoordinateframes,(xµ)andyµ,movingapartwithrelativespeedv(|v|0isacouplingconstantresemblinganelectricchargeasbefore,andrequirethatthevectorfieldAµobeythetransformationrule1Aµ7→Aµ+∂µω(4.28)eandbehavelikeacontravariantvectorfieldundertheLorentztransformations.Itcanbedirectlycheckedthat,underalocal(x-dependent)phaseshiftiω(x)φ(x)7→eφ(x),(4.29)Dµφchangesitselfcovariantlyaccordingtoiω(x)Dµφ7→eDµφ(4.30)sothatthemodifiedLagrangianindeedbecomesinvariantunderthegaugetransformationcon-sistingof(4.28)and(4.29).Unfortunately,theLagrangian(4.27)isincompletebecauseitcannotgiverisetoanequationofmotionforthenewlyintroducedgaugefieldAµwhichisanadditionaldynamicalvariable.InordertoderiveasuitabledynamiclawforthemotionofAµ,wecompareAµwithφanddemandthatthequalifiedLagrangiancontainquadratictermsofthefirst-orderderivativesofAµ.Sincethesetermsshouldbeinvariantunder(4.28),weseethataminimalwaytodosoistointroducethecontravariant2-tensorFµν=∂µAν−∂νAµ,µ,ν=0,1,2,3.(4.31)UsingcontractiontoobserveLorentzinvariance,wearriveattheminimallymodifiedcompleteLagrangeactiondensity1µν1µ2L=−FµνF+DµφDφ−V(|φ|),(4.32)42wherefactor1isforconvenience,whichisinvariantunderthegaugeandLorentztransformations4andgovernstheevolutionofthefieldsφandAµ.TheEuler–LagrangeequationsoftheupdatedLagrangian(4.32)areDDµφ=−2V0(|φ|2)φ,(4.33)µ∂Fµν=−Jµ,(4.34)νwhereµiµµJ=e(φDφ−φDφ).(4.35)2Theequation(4.33)isagaugedwaveequationwhichextends(4.22)or(4.23).Itwillbeinstruc-tivetorecognizewhatiscontainedintheequation(4.34).Todoso,wecanfirstuse(4.33)tocheckthatJµisaconserved‘4-current’satisfying∂Jµ=0.(4.36)µThus,ifweintroducethechargedensityρandcurrentdensityjbysettingρ=J0=ie(φD0φ−φD0φ),j=(Ji),Ji=ie(φDiφ−φDiφ),i=1,2,3,(4.37)2235 wehavetheconservationlaw∂ρ+∇·j=0,(4.38)∂twhichindicatesthat(4.34)istobeidentifiedwiththeMaxwellequations.Forthispurpose,weintroducetheelectricandmagneticfieldsE=(Ei)andB=(Bi)bysetting0−E1−E2−E3µνE10−B3B2(F)=231.(4.39)EB0−BE3−B2B10Wecancheckthattheµ=0componentof(4.34)issimply∇·E=ρ,whichistheGausslaw(3.1).Thespatialcomponentsof(4.34),withµ=i=1,2,3,rendertheequation∂E+j=∇×B,(4.40)∂twhichis(3.2).HencewehavepartiallyrecovertheMaxwellequations(thepartwithchargeandcurrentsources).Inordertorecoverthesource-freepart,i.e.,theequations(3.3)and(3.4),wenotethat(4.31)impliestheBianchiidentity∂γFµν+∂µFνγ+∂νFγµ=0,(4.41)whichleadto(3.3)and(3.4).Inotherwords,(3.3)and(3.4)automaticallyholdasaconsequenceofthedefinition(4.31).Insummary,(4.34)isindeedtheMaxwellequations.Therefore,wehaveseenthattheMaxwellequationscanbederivedasaconsequenceofimposinggaugesymmetry,whichisagreatlessontohave.Atthisspot,itisworthwhiletopauseandmakesomecommentsaboutwhatwehavejustlearned.Thespacetimewherethefieldsaredefinedoniscalledtheexternalspace.Thescalarfieldmaybeviewedasacross-sectionofaprincipal(complexline)bundleoverthespacetime.Thebundleintowhichthescalarfieldtakesvaluesiniscalltheinternalspace.TheexternalspacehastheLorentzgroupasthesymmetrygroupwhichleadstorelativity.TheinternalspacehastheU(1)groupasthesymmetrygroup,whoselocalinvarianceorgaugeinvarianceleadstotheintroductionofgaugefieldwhichgenerateselectromagnetism.Theexternalspacesymmetryiscalledexternalsymmetryandtheinternalspacesymmetryiscalledinternalsymmetry.LocalU(1)internalsymmetryrequiresthepresenceofelectromagnetism.TheLorentzsymmetrydiscussedisaglobalexternalsymmetrywhichgivesrisetospecialrelativity.Whensuchaglobalexternalsymmetryiselevatedintoalocalone,newfieldsarise.Indeed,inthissituation,weareledtogeneralrelativityandpresenceofgravitation,asshownbyEinstein.Furthermore,itisforeseeablethat,whentheinternalsymmetryismodifiedtobegivenbylargergaugegroupssuchasSU(N)(N≥2),otherphysicalforcesmaybegenerated.Infact,thisisthecaseandtheforcesgeneratedcanbeweakandstrongforcesfornuclearinteractions.Thisprocedureexhaustsallfourknownforcesinnature:gravitational,electromagnetic,weak,andstrongforces.Inotherwords,wecandrawtheconclusionthatexternalsymmetryleadstothepresenceofgravityandinternalsymmetryleadstothepresenceofelectromagnetic,weak,andstronginteractions.Wealsoremarkthat,ifthecomplexscalarfieldφbecomesreal-valued,thechargedensityρgivenin(4.37)vanishesidentically.Inotherwords,areal-valuedscalarfieldleadstoanelectricallyneutralsituationandacomplex-valuedscalarfieldmaybeusedtogenerateelectricity.36 Moreover,weremarkthatthecharge(4.37)isgeneratedbyascalarfieldbutthechargeeinthedefinitionofthegauge-covariantderivativeisswitchedon‘byhand’.Notethat11X1X3X1−FFµν=−FFµν=F2−F2=(E2−B2).(4.42)µνµν0iij42220≤µ<ν≤3i=11≤i0aresuitableparameters,andTc>0isacriticaltemperature.Vacuumsolutions,orgroundstates,arethelowestenergystaticsolutions.Inhightemper-ature,T>Tc,wehavem2(T)>0and,inviewoftheKlein–Gordonequation,thequantitym(T)>0isthemassoftworealscalarparticlesrepresentedbyφ1andφ2whereφ=φ1+iφ2andhigherordertermsofφ1andφ2describeself-interactions.TheonlyminimumoftheHamiltoniandensity1212214H=|∇φ|+m(T)|φ|+λ|φ|(4.58)228isφv=0,(4.59)whichistheuniquevacuumstateoftheproblem.ThisvacuumstateisofcourseinvariantundertheU(1)-symmetrygroupφ7→eiωφ.Symmetryandsymmetrybreaking38 Ingeneral,givenasymmetrygroup,theLagrangiandensityshouldbeinvariantifthevacuumstateisalreadyinvariant,basedonsomeconsiderationfromquantumfieldtheory.SuchastatementisknownastheColemantheorem(theinvarianceofthevacuumstateimpliestheinvarianceoftheuniverse).IfboththevacuumstateandtheLagrangiandensityareinvariant,wesaythatthereisexactsymmetry.Ifthevacuumstateisnon-invariant,theLagrangiandensitymaybenon-invariantorinvariant.Inbothcases,wesaythatthesymmetryasawholeisbroken.Theformercaseisreferredtoasexplicitsymmetry-breakingandthelattercaseisreferredtoasspontaneoussymmetry-breaking,whichisoneofthefundamentalphenomenainlow-temperaturephysics.Toexplainsomeoftheaboveconcepts,weassumethatT0,θ∈R,(4.60)Tcλwhichgiveusacontinuousfamilyofdistinctvacuumstates(acircle)labeledbyθ.Sinceforω6=2kπφ7→eiωφ(4.61)transformsanygivenvacuumstate,φv,θ,toadifferentone,φv,θ+ω,weobservethenon-invarianceofvacuumstatesalthoughtheLagrangiandensityisstillinvariant.Consequently,thesymmetryisspontaneouslybroken.Thequantityφ0measuresthescaleofthebrokensymmetry.Illustration.Considerwhathappenswhenexertingpressureonanuprightstick.GoldstoneparticlesandHiggsmechanismWecontinuetoconsiderthesystematlowtemperature,TTc,thevacuumstateisthezerostateandm(T)clearlydefinesmass.ForT0.(4.66)However,sincetheφ2termisabsent(thehigher=ordertermsdescribeinteractions),theseφ-22particlesaremasslessandarecalledtheGoldstoneparticles.Hence,weseethatspontaneoussymmetry-breakingleadstothepresenceoftheGoldstoneparticles,namely,particlesofzeromassinsteadofparticlesofimaginarymass.ThisstatementisknownastheGoldstonetheorem.TheGoldstoneparticlesaremassless,andhence,arecurious.Weseeinthefollowingthattheseparticlesmayberemovedfromthesystemwhengaugefieldsareswitchedon.Forthispurpose,wereturntothelocallyinvariantLagrangiandensitytogetL=−1FFµν+1DφDµφ−λ(|φ|2−φ2)2,(4.67)µνµ0428andweconsiderfluctuationsaroundthevacuumstateφv=φ0,(Aµ)v=0.(4.68)Usingtheearlierdecompositionofφintoφ0,φ1,φ2,weobtaintheLagrangiandensityfortheinteractionofthefieldsφ1,φ2,andAµasfollows,1µν1µ1µ122µλ22L=−FµνF+∂µφ1∂φ1+∂µφ2∂φ2+eφ0AµA−φ0φ1+Linter,(4.69)42222whereLintercontainsalloff-diagonalinteractiontermsinvolvingthemixedproductsofthefieldsφ1,φ2,Aµ,andtheirderivatives.Recallthatthedefinitionofthegaugetransformation.Hence,φ1,φ2,Aµtransformthemselvesaccordingtotheruleφ+φ+iφ7→φ+φ0+iφ0,A7→A0,012012µµφ0=φcosω−φsinω+φ(cosω−1),1120φ0=φsinω+φcosω+φsinω,212001Aµ=Aµ+∂µω.eFromtheLagrangiandensity,weseethatφ2remainsmassless.Besides,thegaugefieldAµbecomesmassive(amassofeφ0).However,usingtheaboveexpression,wecanfindasuitablegaugetransformationsothatφ0=0.Forexample,wemaychoose2φ2ω=−arctan.(4.70)φ0+φ1Ifweusethephasefunctionωdeterminedaboveinthetransformationandthenewfieldvariablesφ01,A0µ,andsuppresstheprimesign0,weseethattheLagrangianbecomes1µν1µ122µλ22L=−FµνF+∂µφ1∂φ1+eφ0AµA−φ0φ1+Linter,(4.71)4222whereLintercontainsalloff-diagonalinteractiontermsinvolvingthemixedproductsofthefieldsφ1,Aµ,andtheirderivatives.Thusweseethat,insuchafixedgauge,weareleftwithamassive40 realscalarfieldandamassivegaugefieldandthemasslessGoldstoneparticleiseliminated.Inotherwords,spontaneousbreakingofacontinuoussymmetrydoesnotleadtotheappearanceofamasslessGoldstoneparticlebuttothedisappearanceofascalarfieldandtheappearanceofamassivegaugefield.ThisstatementisknownastheHiggsmechanism7andthemassivescalarparticlesarecalledtheHiggsparticles.Inparticular,theLagrangiandensitywehaveseeniscommonlyreferredtoastheAbelianHiggsmodelandthecomplexscalarfieldφiscalledtheHiggsfield.5TheGinzburg–LandauequationsforsuperconductivityInthissection,weconsidertheGinzburg–LandauequationsforsuperconductivityandAbrikosov’svorticesandtheirtopologicalcharacterizations.Wewillendwithanexcursiontothemonopoleandquarkconfinementproblem.Inthestaticsituation,∂0=0,A0=0,(5.1)theAbelianHiggsmodelisthewell-knownGinzburg–Landautheoryforsuperconductivity.ThecomplexscalarfieldφgivesrisetodensitydistributionofsuperconductingelectronpairsknownastheCooperpairsandthefactthattheelectromagneticfieldbehaveslikeamassivefieldduetotheHiggsmechanismissimplyaconsequenceoftheMeissnereffectwhichisequivalenttosayingthatthemagneticfieldbecomesmassiveandcannotpenetrateasuperconductor.Forsimplicity,wefocushereonthetwo-dimensionalsituation.ThisisthemostinterestingsituationbecauseitallowsustoconsidertheAbrikosov[1]orNielsen–Olesen[99]vortices.5.1HeuristicproofoftheMeissnereffectWestartwitha‘proof’oftheMeissnereffectwhichstatesthatasuperconductorscreensanexternalmagneticfieldwhenthismagneticfieldisnotstrongenoughtodestroyitssupercon-ductingphaseandthatthesuperconductingphaseinthesuperconductormaybeswitchedtothenormal-conductingphasepermittingthefullpenetrationoftheexternalmagneticfieldwhenthismagneticfieldismadestrongenough.Forthispurpose,werecallthattheenergydensityorHamiltonianofthetwo-dimensionalGinzburg–Landautheoryinthepresenceofaconstantexternalmagneticfield,Hext>0(say),isgivenby121212λ222H(φ,Aj)=F12+|D1φ|+|D2φ|+(|φ|−φ0)−F12Hext.(5.2)2228Thenormalandcompletelysuperconductingphasesarerepresentedby(φn,Anj)and(φs,Asj),respectively,sothatnnssφ=0,F12=Hext;φ=φ0,Aj=0,j=1,2.(5.3)Therefore,wehavennλ412ssH(φ,Aj)=φ0−Hext,H(φ,Aj)=0.(5.4)827Thisfundamentalmechanisminfactwasduetotheindependentworkbythreedifferentgroupsofpeople:EnglertandBrout[40],Higgs[58],andGuralnik,Hagen,andKibble[57],publishedinthesamejournalandatthesametime.41 Consequently,whenHextsatisfies1√Hext<φ2λ,(5.5)02wehaveH(φn,Anj)>H(φs,Asj)and(φs,Asj)isenergeticallyfavoredover(φn,Anj).Thusthesssuperconductorisinthesuperconductingphasedescribedby(φ,Aj)andthemagneticfieldiscompletelyexpelledfromthesuperconductor,Fs=0.Ontheotherhand,when121√Hext>φ2λ,(5.6)02nnssnnsswehaveH(φ,Aj)0islargeenough,whereS1denotesthecircleinR2centeredattheRoriginandofradiusR.ThereforeΓmaybeviewedasanelementinthefundamentalgroupπ(S1)=Z(5.14)1andrepresentedbyanintegerN.Infact,thisintegerNisthewindingnumberofφaroundS1RandmaybeexpressedbytheintegralZ1N=dlnφ.(5.15)2πiS1RItisinterestingtonotethatthecontinuousdependenceoftheright-handsideoftheaboverelationwithrespecttoRimpliesthatitisactuallyindependentofR.AnimportantconsequenceofsuchanobservationisthefamousfluxquantizationconditionZΦ=F12dx=2πN(5.16)R2whichfollowsfromZZZZFdx+idlnφ=Adx+iφ−1∂φdx12jjjj|x|≤R|x|=R|x|=R|x|=RZ≤|φ−1||Dφ|dsA|x|=RZ≤Ce−δRds|x|=R=2πRCe−δR→0asR→∞.(5.17)Notethat,whenthetheoryisformulatedinthelanguageofacomplexlinebundlesayξ,sothatφisacrosssection,Aisaconnection1-form,F=dAisthecurvature,andDAisthebundleconnection,thentheintegerNisnothingbutthefirstChernclassc1(ξ),whichcompletelyclassifiesthelinebundleuptoanisomorphism.Thatis,Φ=N=c1(ξ).(5.18)2πAnimportantopenquestioniswhetherforanygivenN∈ZthereisasolutiontotheconstrainedminimizationproblemZEN≡infE(φ,A)F12dx=2πN.(5.19)R2Theproblemisknowntobesolvableonlywhenλ=1duetoTaubes[66,138,139].43 5.3Vortex-lines,solitons,andparticlesInthetheoryofsuperconductivity,thecomplexscalarfieldisan‘order’parameterwhichcharac-terizesthetwophases,superconductingandnormalstates,ofasolid.Mathematically,|φ|2givesrisetothedensityofsuperconductingelectronpairs,alsocalledtheCooperpairs,sothatφ6=0indicatesthepresenceofelectronpairsandonsetofsuperconductivityandφ=0indicatestheabsenceofelectronpairsandthedominanceofnormalstate.Asaconsequence,whentheorderparameterφissuchthatitisnonvanishingsomewherebutvanishingelsewhere,wearethenhavinga‘mixedstate’.Recallthat,accordingtotheMeissnereffect,asuperconductorscreensthemagneticfield.Inotherwords,thepresenceofsuperconductivitypreventsthepenetrationofamagneticfield.Therefore,inamixedstate,themagneticfield,F12,alwayshasitsmaximumpenetrationatthespotswhereφ=0.Orequivalently,|F12|assumesitslocalmaximumvaluesatthezerosofφ.SinceF12maybeinterpretedasavorticityfield,thezerosofφgiverisetocentersofvorticesorlocationsofvortex-linesdistributedoverR2.Infact,althoughitisnotobvious,itmaybecheckedthat|Dφ|2+|Dφ|2alsoattainsitslocalmaximaatthezerosofφ.12Hence,wehaveseenthattheenergydensity121212λ22H(φ,A)(x)=F12+|D1φ|+|D2φ|+(|φ|−1)(x)(5.20)2228attainsitslocalmaximaatthezerosofφaswell.However,sinceenergyandmassareequivalent,wehaveobservedmassconcentrationcenteredatthezerosofφ.Inotherwords,wehaveproducedadistributionofsolitonswhichmaybeidentifiedwith‘particles’inquantumfieldtheory.Wenowshowthatthepresenceofzerosofφisessentialforasolutiontobenontrivial:If(φ,A)isafinite-energysolutionoftheAbelianHiggsequationssothatφnevervanishes,then(φ,A)isgauge-equivalenttothetrivialsolutionA≡0,φ≡1.Hereisaquickproof.Sinceφnevervanishes,wemayrewriteφasiωφ=ϕe(5.21)forgloballydefinedreal-valuedsmoothfunctionsϕandωoverR2.Infact,wemayassumeϕ>0.Usingthegaugetransformationφ7→φe−iω,A7→A−∂ω,(5.22)jjjweseethat(φ,A)becomes‘unitary’inthesensethatφisrealvalued,φ=ϕ,(5.23)andwesaythatwehavechosena‘unitarygauge.’Inunitarygauge,theequationsofmotiondecomposesignificantlytotaketheform2λ2∆ϕ=|A|ϕ+(ϕ−1)ϕ,22Ak∂kϕ+(∂kAk)ϕ=0,∂F=ϕ2A,1122∂F=−ϕ2A.212144 Fromthelasttwoequations,wehaveZZ(A∂F−A∂F)dx=|A|2ϕ2dx.(5.24)21121212R2R2Integratingfurtherbypartsanddroppingtheboundaryterms,weobtainZ(F2+|A|2ϕ2)dx=0.(5.25)12R2Sinceϕnevervanishes,wehaveA≡0.Returningtothegoverningequations,wearriveatthesingleremainingequationλ2∆ϕ=ϕ(ϕ−1),(5.26)2whichmayberewrittenasλ∆(ϕ−1)=ϕ(ϕ+1)(ϕ−1).(5.27)2Usingtheboundaryconditionϕ−1→0as|x|→∞andthemaximumprinciple,wededuceϕ≡1asclaimed.Estimateofenergyfrombelow–topologicallowerboundAusefulidentityinvolvinggauge-covariantderivativesis|Dφ|2=|Dφ±iDφ|2±F|φ|2±i(∂[φDφ]−∂[φDφ]).(5.28)j12121221ThereforethetotalenergysatisfiesZ1212122E(φ,A)≥min{λ,1}F12+|Djφ|+(|φ|−1)dxR2228Z112121=min{λ,1}F12±(|φ|−1)+|D1φ±iD2φ|±F12dxR22222≥min{λ,1}π|N|,(5.29)whereΦ=2πNandthesigns±followN=±|N|.Inthesequel,wefocusonN≥0forconvenience.Whenλ=1,wehaveE(φ,A)≥πN(5.30)andsuchanenergylowerboundissaturatedifandonlyif(φ,A)satisfiestheself-dualsystemofequationsD1φ+iD2φ=0,(5.31)12F12=(1−|φ|),(5.32)2whichisareductionoftheoriginalequationsofmotionandisalsooftencalledaBPSsystemafterBogomol’nyi[10]andPrasadandSommerfield[113]whofirstderivedtheseequations.StructureofBPSsystemItwillbeconvenienttocomplexifyourvariablesanduse1112A=A1+iA2,A1=(A+A),A2=(A−A),z=x+ix,(5.33)22i11∂=(∂1−i∂2),∂=(∂1+i∂2),∂1=∂+∂,∂2=i(∂−∂),(5.34)22∆=∂2+∂2=4∂∂=4∂∂.(5.35)1245 HencethefirstequationintheBPSsystemassumestheformi∂φ=Aφ.(5.36)2Toseewhatthisrelationmeans,werecallthe∂-Poincar´elemmawhichstatesthattheequation∂ω(z)=iα(z)(5.37)overadiskB⊂Calwayshasasolution.Infact,thissolutionmayberepresentedby[66]Z1α(z0)0ω(z)=dz∧dz0.(5.38)2πBz0−zNowletψsolve∂ψ=iAlocally.Thenweseethatthecomplex-valuedfunctionf=φe−ψ2satisfiestheCauchy–Riemannequation∂f=∂(φe−ψ)=e−ψ(∂φ−φ∂ψ)=0.(5.39)Thereforef(z)isanalytic.Inparticular,f(andhenceφ)mayonlyhaveisolatedzeroswithintegermultiplicities.Inotherwords,ifz0isazeroofφ,thenφ(z)=(z−z)nh(z)(5.40)0forznearz0andthefunctionh(z)nevervanishes,wherenisapositiveintegerwhichisalsothelocalwindingnumberofφaroundz0.FromthesecondequationoftheBPSsystem,weseeclearlythatthevorticityfieldF12achievesitmaximumvalueatz0aswell,1max{F12}=F12(z0)=(5.41)2sothatthepointz0definesthecenterofa(magnetic)vortex.Theintegernisalsocalledthelocalvortexcharge.Besides,since|φ|→1as|x|→∞,weseethatφcanonlyhaveafinitenumberofzerosoverC.Assumethatthezerosofφandtheirrespectivemultiplicitiesarez1,n1,z2,n2,···,zk,nk.(5.42)Countingmultiplicitiesofthesezeros(i.e.,azeroofmultiplicitymiscountedasmzeros),thetotalvortexchargeisthetotalnumberofzerosofφ,sayN(φ),XkN(φ)=ns.(5.43)s=1Ontheotherhand,awayfromthezerosofφ,thefirstequationintheBPSsystemisA=−i2∂lnφ.(5.44)Therefore,there,wecanrepresentF12as212F12=∂1A2−∂2A1=−i(∂A−∂A)=−2∂∂ln|φ|=−∆ln|φ|.(5.45)246 InsertingtheaboveintothesecondequationintheBPSsystem,wehave∆ln|φ|2+1−|φ|2=0(awayfromthezerosofφ).(5.46)Nowdefineu=ln|φ|2.(5.47)Then,nearzs(s=1,2,···,k),wehaveu(z)=2nsln|z−zs|+aregularterm.(5.48)Consequently,wearriveattheLiouvilletypeequationXk∆u=eu−1+4πnδ(z−z)inC=R2(5.49)sss=1subjecttotheboundaryconditionu→0as|x|=|z|→∞(5.50)(since|φ|→1as|x|→∞),whichmaybesolvedbyvarioustechniques.Conversely,foranygivendata{(zs,ns)},thesolutionoftheaboveellipticequationgivesrisetoasolutionpair(φ,A)whichrepresentsmultiplydistributedvorticesat{zs}withthecorrespondinglocalvortexcharges{ns}.Formally,itmaybeeasilyconvincedthatZ∆udx=0.(5.51)R2Hence,fromtheequation,weobtainZZXk(1−|φ|2)dx=(1−eu)dx=4πn=4πN(φ).(5.52)sR2R2s=1IntegratingthesecondequationintheBPSsystemandinsertingtheaboverelation,wegetthebeautifulresultZZ112N=c1(ξ)=F12dx=(1−|φ|)dx=N(φ).(5.53)2πR24πR2Inotherwords,thetotalvortexnumberisnothingbutthefirstChernclassofthesolutionwehaveseenbefore,whichalsodeterminesthetotalmagneticflux,Φ=2πN.Letusrecordtheimportantconclusionthatoursolutioncarriestheminimumenergy,EN=πN.(5.54)Notethatsuchanexactresultisonlyknownforλ=1butunknownforλ6=1.Itisworthmentioningthatλclassifiessuperconductivitysothatλ<1correspondstotypeIandλ>1correspondstotypeIIsuperconductivity,respectively.Estimateofenergyfromabove–topologicalupperbound47 Using(φ,A)todenoteasolutionoftheBPSsystemwithN=±|N|asatrialfieldconfigu-rationpair,wehaveZ1212122E(φ,A)≤max{λ,1}F12+|Djφ|+(|φ|−1)dxR2228Z112121=max{λ,1}F12±(|φ|−1)+|D1φ±iD2φ|±F12dxR22222≤max{λ,1}π|N|.(5.55)Therefore,wehaveobtainedthefollowinglowerandupperboundsmin{λ,1}π|N|≤EN≤max{λ,1}π|N|,(5.56)whichimpliesthattheenergyEgrowsinproportiontothetotalvortexnumberNandsuggeststhatthesevorticesmayindeedbeviewedasparticles.ItisalsoseenthatanonvanishingNisessentialfortheexistenceofanontrivialsolution.Recently,itisdemonstratedin[21]thatEisasymptoticallylikeπN2lnλforλlarge.N2EnergygapAninterestingfactcontainedin(5.56)isthattheGinzburg–Landauequationshavenonontrivialenergy-minimizingsolutionwithanenergyintheopenintervalI=(0,min{λ,1}π).(5.57)Sucharesultmaybeviewedasanenergyormassgaptheorematclassicallevel.5.4FrommonopoleconfinementtoquarkconfinementConsidertwomasses,m1andm2,initiallyplacedrdistanceaway.Thegravitationalattractiveforcebetweenthemassesisthenm1m2F(r)=G,(5.58)r2followingNewton’slaw.Thus,theworkorenergyneededtocompletelysplitthemsothattheyeventuallystayawayasnon-interacting‘freemasses’isZ∞m1m2W=F(ρ)dρ=G.(5.59)rrOfcourse,wecandothesamethingforapairofmonopoleandanti-monopolebecausethemagneticforcebetweenthemobeysthesameinverse-squarelaw.However,whenthemonopoleandanti-monopoleareplacedinatype-IIsuperconductor,wewillencounteranentirelydifferentsituation.DuetothepresenceofsuperconductingCooperpairsformedatlowtemperatureasaresultofelectroncondensation,themagneticfieldwillberepelledfromthebulkregionofthesuperconductorduetotheMeissnereffectandsqueezedintothintubesofforcelines,intheformofvortex-lines,whichpenetratethroughthespotswheresuperconductivityisdestroyed,manifestedbythevanishingoftheorderparameter.Sincethestrengthsofthesevortex-linesareconstantwithrespecttotheseparationdistanceofthemonopoleandanti-monopole,thebindingforcebetweenthemremainsconstant.Consequently,theworkneededtoplacethemonopoleandmonopole,initiallyseparatedatadistancer0away,atafurtherdistancerawayisW(r)=K(r−r0),r>r0,(5.60)48 whichisalinearfunctionofr.Inparticular,theworkneededtocompletelysplitthemonopolepairsothatthemonopoleandanti-monopoleareseenasisolatedfreeentitieswillbeinfinite.Thus,itwillbepracticallyimpossibletoseparatethemonopoleandanti-monopoleimmersedinatype-IIsuperconductor.Inotherwords,themonopoleandanti-monopoleareconfined.Moreprecisely,weseethat,duetotheMeissnereffect,themonopoleandanti-monopoleplacedinatype-IIsuperconductorinteractwitheachotherthroughnarrowlyformedvortex-lineswhichgiverisetoaconstantinter-monopolebindingforceandthelinearlawofpotentialexpressedin(5.60).Asaresult,itisimpossibletoseparatesuchapairofmonopoleandanti-monopole,andthemonopoleconfinementphenomenontakesplace.Suchaconfinementpictureisalsocalledlinearconfinement.Afundamentalpuzzleinphysics,knownasthequarkconfinement[54],isthatquarks,whichmakeupelementaryparticlessuchasmesonsandbaryons,cannotbeobservedinisolation.Awellacceptedconfinementmechanism,exactlyknownasthelinearconfinementmodel,interest-inglystatesthat,whenonetriestoseparateapairofquarks,suchasaquarkandananti-quarkconstitutingameson,theenergyconsumedwouldgrowlinearlywithrespecttothethesep-arationdistancebetweenthequarkssothatitwouldrequireaninfiniteamountofenergyinordertosplitthepair.Thequarkandanti-quarkmayberegardedasapairofsourceandsinkofcolor-chargedforcefields.Thesourceandsinkinteractthroughcolor-chargedfluxeswhicharescreenedinthebulkofspacebutformthintubesintheformofcolor-chargedvortex-linessothatthestrengthoftheforceremainsconstantoverarbitrarydistance,resultinginalineardependencerelationforthepotentialenergywithregardtotheseparationdistance.Suchasit-uationissimilartotheabove-describedmagneticmonopoleandanti-monopolepairimmersedinatype-IIsuperconductor.WehaveseenthatthemagneticfluxesmediatingtheinteractingmonopolesarenotgovernedbytheMaxwellequations,whichwouldotherwisegiverisetoaninverse-square-powerlawtypeofdecayoftheforcesandleadtonon-confinement,butratherbytheGinzburg–LandauequationsorthestaticAbelianHiggsmodelinthetemporalgauge,whichproducenarrowlydistributedvortex-lines,knownastheAbrikosovvortices[1]ortheNielsen–Olesenstrings[99],aspresented,leadingtoalinearconfinementresult,(5.60).Inspiredbytheabove-describedmonopoleconfinementinatype-IIsuperconductor,Man-delstam[86,87],Nambu[95],and’tHooft[135,137]proposedinthe1970sthatthegroundstateofquantum-chromodynamics(QCD)isacondensateofchromomagnetic(color-charged)monopoles,causingthechromoelectricfluxesbetweenquarkstobesqueezedintonarrowlyformedtubesorvortex-lines,similartotheelectroncondensationinthebulkofasupercon-ductor,intheformoftheCooperpairs,resultingintheformationofcolor-chargedflux-tubesorvortex-lineswhichmediatetheinteractionbetweenquarks,followinganon-AbelianversionoftheMeissnereffect,calledthe‘dualMeissnereffect’,whichisresponsibleforthescreeningofchromoelectricfluxes[124,125].Thus,wehaveseenthatthenotionofmagneticmonopolesisnotonlyatheoreticalcon-structionasaresultoftheelectromagneticdualitybutalsosuppliesasauseful‘theoreticalphenomenon’thatprovidesacrucialhinttoahopefulsolutionofoneofthegreatestpuzzlesofmodernphysics–quarkconfinement.6Non-AbeliangaugefieldequationsInthelasttwosections,wediscussedagaugefieldtheorywiththeAbeliangroupU(1).TheYang–Millstheoryisnowagenericnameforthegaugefieldtheorywithanarbitrarynon-Abelian49 LiegroupG.6.1TheYang–MillstheoryInthissubsection,wepresentashortintroductiontotheYang–Millstheory.Fornotationalconvenience,weshallconcentrateonthespecificcasewhereGiseithertheorthogonalmatrixgroup,O(n),orunitarymatrixgroup,U(n),whichissufficientforallphysicalapplications.LetφbeascalarfieldovertheMinkowskispacetimeandtakevaluesineitherRnorCn,whichistherepresentationspaceofG(withG=O(n)orU(n)).Weuse†todenotetheHermitiantransposeorHermitianconjugateinRnorCn.Then|φ|2=φφ†.WemaystartfromtheLagrangiandensity1µ†2L=(∂µφ)(∂φ)−V(|φ|).(6.1)2ItisclearthatLisinvariantundertheglobalsymmetrygroupG,φ7→Ωφ,Ω∈G.(6.2)However,asintheAbeliancase,ifthegroupelementΩisreplacedbyalocalonedependingonspacetimepoints,Ω=Ω(x),(6.3)theinvarianceofLisnolongervalidandamodificationistobedevised.ThusweareagainmotivatedtoconsiderthederivativeDµφ=∂µφ+Aµφ,(6.4)wherewenaturallychooseAµtobeanelementintheLiealgebraGofGwhichhasanobviousrepresentationoverthespaceofφ.ThedynamicalterminLbecomes1µ†(Dµφ)(Dφ).(6.5)2Ofcourse,theinvarianceoftheaboveunderthelocaltransformationφ(x)7→φ0(x)=Ω(x)φ(x)(6.6)isensuredifDµφtransformsitselfcovariantlyaccordingtoDφ7→D0φ0=∂φ0+A0φ0=∂(Ωφ)+A0Ωφ=Ω(Dφ).(6.7)µµµµµµµComparingtheaboveresults,weconcludethatAµshouldobeythefollowingruleoftransfor-mation,A7→A0=ΩAΩ−1−(∂Ω)Ω−1.(6.8)µµµµItiseasilyexaminedthattheU(1)gaugefieldtheorypresentedinthelastsectioniscontainedhereasaspecialcase(theLiealgebraistheimaginaryaxisiR).Thus,ingeneral,theLiealgebravaluedfieldAµisalsoagaugefield.InordertoincludedynamicsforthegaugefieldAµ,weneedtointroduceinvariantquadratictermsinvolvingderivativesofAµ.Forthispurpose,recallthatthereisastandardinnerproductoverthespaceofn×nmatrices,i.e.,(A,B)=Tr(AB†).(6.9)50 ForanyA∈G,sinceA†=−A,weseethat|A|2=(A,A)=−Tr(A2).(6.10)IncompleteanalogywiththeelectromagneticfieldintheAbeliancase,wecanexaminethenon-commutativityofthegauge-covariantderivativestogetDµDνφ−DνDµφ=(∂µAν−∂νAµ+[Aµ,Aν])φ,(6.11)where[·,·]istheLiebracket(orcommutator)ofG.Hencewearemotivatedtodefinetheskew-symmetricYang–Millsfield(curvature)2-tensorFµνasFµν=∂µAν−∂νAµ+[Aµ,Aν].(6.12)ItiscleartoseethatFµνtransformsitselfaccordingtoF7→F0=∂A0−∂A0+[A0,A0]=ΩFΩ−1.(6.13)µνµνµννµµνµνHenceweobtaintheanalogousinvariantterm,1µνTr(FµνF).(6.14)4Hence,wearriveatthefinalformofourlocallygauge-invariantLagrangianactiondensity1µν1µ†2L=Tr(FµνF)+(Dµφ)(Dφ)−V(|φ|).(6.15)42whichdefinesanon-AbeliangaugefieldtheorycalledtheYang–Millstheory.TheEuler–LagrangeequationsoftheaboveactionarecalledtheYang–Millsequations.InthesituationwherethepotentialdensityVintroducesaspontaneouslybrokensymmetry,thetheoryiscalledtheYang–Mills–HiggstheoryandtheequationsaretheYang–Mills–Higgsequations.Inthecasewherethemattercomponent(containingφ)isneglected,theactiondensitybecomes1µνL=Tr(FµνF),(6.16)4whichissimplycalledthe(pure)Yang–Millstheory.TheEuler–LagrangeequationsofsuchaLagrangianarecalledthe(pure)Yang–Millsequations,whicharenon-AbelianextensionoftheMaxwellequationsinvacuum.LiketheMaxwellelectromagneticfield,theYang–Millsfieldsarealsomediating(force)fields.Inthenon-Abeliancasethecommutatorsintroducenonlinearityandnewphysicsappears:thesenon-Abeliangaugefieldsareinfactnuclearforcefieldswhichbecomesignificantonlyinshortdistances.Moreprecisely,liketheU(1)groupgivingrisetoelectromagneticinteractions,theSU(2)groupgivesrisetoweak,SU(3)strong,andSU(5)grandunifiedinteractions.6.2TheSchwingerdyonsInordertomotivateourpresentationofsomewell-knownparticle-likesolutionsoftheYang–Mills–Higgsequations,webrieflydiscusstheSchwingerdyonswhicharehypotheticalpointparticlescarryingbothelectricandmagneticchargesandwereintroducedbySchwinger[123]tomodelquarks.51 Considerthemotionofadyonwithmassmandelectricandmagneticchargesq1andg1intheelectromagneticfield(E,B)ofanotherdyonwithelectricandmagneticchargesq2andg2placedattheorigin.Weassumethattheseconddyonissoheavythatitstaysstationarythroughoutourstudy.Solving(3.6)–(3.9)fortheseconddyon,weobtainthegeneratedelectricandmagneticfieldsxxE=q2|x|3,B=g2|x|3.(6.17)Ontheotherhand,thenon-relativisticmotionofthefirstdyonisgovernedbytheLorentzforcesothatmx¨=q1(E+x˙×B)+g1(B−x˙×E).(6.18)Inserting(6.17)into(6.18),wearriveatxxmx¨=(q1q2+g1g2)|x|3+(q1g2−q2g1)x˙×|x|3.(6.19)Using(6.19),wearriveattheconservedtotalangularmomentumJdefinedbyxJ=x×mx˙−(q1g2−q2g1).(6.20)|x|Itisimportanttonoticethat,inthespecialcaseofthemotionofanelectricallychargedparticleinthemagneticfieldgeneratedbyamonopole,wehaveg1=0,q2=0and(6.20)becomesxJ=x×mx˙−q1g2.(6.21)|x|Werecallthat,inthiscase,theDiracchargequantizationformulareadsng2q1=,n=0,±1,±2,···.(6.22)2Asanimmediateconsequenceofthesimilaritybetween(6.20)and(6.21),weobtainthecelebratedSchwingerchargequantizationformulafordyonsng2q1−g1q2=,n=0,±1,±2,···.(6.23)2LikethatofaDiracmonopole,aSchwingerdyonalsocarriesinfiniteenergy.6.3The’tHooft–PolyakovmonopoleandJulia–ZeedyonIntheabovestudy,wediscussedmonopolesanddyonsintermsoftheMaxwellequationsforelectromagnetismwhichisatheoryofAbeliangaugefields.Infactitismorenaturalformonopolesanddyonstoexistinnon-Abeliangaugefield-theoreticalmodelsbecausenonvanish-ingcommutatorsthemselvesarenowpresentaselectricandmagneticsourceterms.Inotherwords,non-Abelianmonopolesanddyonsareself-inducedandinevitable.Inthissection,weshallpresentthesimplestnon-AbeliandyonsknownastheJulia–Zeedyons[67],whichcontainasspecialsolutionsthe’tHooft–Polyakovmonopoles.Considerthesimplestnon-AbelianLiegroupSO(3),whichhasasetofgenerators{ta}(a=1,2,3)satisfying[ta,tb]=abctc.Consequently,theso(3)-valuedquantitiesA=AataandB=Batgiverisetoacommutator,a[A,B]=AaBbt.(6.24)abcc52 Forconvenience,wemayviewAandBastwo3-vectors,A=(Aa)andB=(Ba).Then,by(6.24),[A,B]correspondstothevectorcross-product,A×B.Withtheseinmind,wemakethefollowingintroductiontotheSO(3)(orSU(2)sinceSO(3)andSU(2)haveidenticalLiealgebras)Yang–Mills–Higgstheory.LetAµ=(Aaµ)(µ=0,1,2,3)andφ=(φa)(a=1,2,3)beagaugeandmatterHiggsfields,respectively,interactingthroughtheactiondensity1µν1µλ22L=−F·Fµν+Dφ·Dµφ−(|φ|−1),(6.25)424wherethefieldstrengthtensorFµνisdefinedbyFµν=∂µAν−∂νAµ−eAµ×Aν,(6.26)andthegauge-covariantderivativeDµisdefinedbyDµφ=∂µφ−eAµ×φ.(6.27)Basedonconsiderationoninteractions[67,134,136],itisrecognizedthattheelectromagneticfieldFµνisdefinedbytheformula11Fµν=φ·Fµν−3φ·(Dµφ×Dνφ).(6.28)|φ|e|φ|Theequationsofmotionof(6.25)canbederivedasDνF=−eφ×Dφ,µνµDµDφ=−λφ(|φ|2−1).(6.29)µWeareinterestedinstaticsolutionsof(6.29).Ingeneral,thisisadifficultproblem.Herewecanonlyconsiderradiallysymmetricsolutions.Setr=|x|.FollowingJuliaandZee[67],themostgeneralradiallysymmetricsolutionsof(6.29)areoftheformxaAa=J(r),0er2xbAa=(K(r)−1),iabier2xaφa=H(r),a,b,c=1,2,3.(6.30)er2Inserting(6.30)into(6.29)andusingprimetodenotedifferentiationwithrespecttotheradialvariabler,wehaver2J00=2JK2,2002212rH=2HK−λrH1−H,e2r2r2K00=K(K2−J2+H2−1).(6.31)53 Weneedtospecifyboundaryconditionsfortheseequations.First,weseefrom(6.30)andregularityrequirementthatH,J,KmustsatisfyH(r)J(r)lim,,K(r)=(0,0,1).(6.32)r→0ererSecondly,sincetheHamiltoniandensityof(6.25)takestheformH=F0i·F0i+D0φ·D0φ−L1021021021221222λ22=(K)+(u)+(v)+(K−1)+K(u+v)+(u−1),e2r2222e2r4r24(6.33)whereu=H/erandv=J/er.Finiteenergycondition,ZE=Hdx<∞,(6.34)R3andtheformula(6.33)implythatu(r)→1andK(r)→0asr→∞.Besides,itisseenfrom(6.33)thatv(r)→someconstantC0asr→∞.However,C0cannotbedeterminedcompletely.Werecordtheseresultsasfollows,H(r)J(r)lim,,K(r)=(1,C0,0).(6.35)r→∞ererSolutionsintheBPSlimitInserting(6.30)into(6.28),wefindtheelectricandmagneticfields,E=(Ei)andB=(Bi),asfollows,xidJEi=−F0i=,(6.36)erdrr1xiBi=−Fjk=.(6.37)2ijker3ItisinterestingtonotethatbothEiandBiobeytheinverse-squarelaw.From(6.36)weseethat,ifJ=0,E=0andthereisnoelectricfield.Themagneticchargegmaybeobtainedthroughintegrating(6.37),I11g=limB·dS=,(6.38)4πr→∞|x|=rewhichissimilartotheDiracquantizationformulaexceptthatelectricchargeisdoubled.More-over,theequationsofmotion(6.31)aresimplifiedintotheform2002212rH=2HK−λrH1−H,e2r2r2K00=K(K2+H2−1).(6.39)Theseequationscannotbesolvedexplicitlyforgeneralλ>0butanexistencetheoremhasbeenestablishedbyusingfunctionalanalysis[117].HerewepresentafamilyofexplicitsolutionsattheBPSlimitλ=0duetoBogomol’nyi[10]andPrasad–Sommerfield[113].54 Whenλ=0,thesystem(6.39)becomesr2H00=2HK2,r2K00=K(K2+H2−1),(6.40)withtheassociatedenergyZZ4π∞111E=Hdx=(K0)2+(rH0−H)2+(K2−1)2+K2H2dr.(6.41)R3e202r22r2r2Itisclearthatthesystem(6.40)istheEuler–Lagrangeequationsof(6.41).Besides,usingtheboundaryconditions(6.32)and(6.35),wehaveZ∞2224π01102dHKHE=K+KH+rH−H+(K−1)+−dre20r2r2drrr4π≥.(6.42)eHence,wehavetheenergylowerbound,E≥4π/e,whichisattainedwhen(H,K)satisfiesrK0=−KH,rH0=H−(K2−1).(6.43)Ofcourse,anysolutionof(6.43)alsosatisfies(6.40).ItwasMaison[85]whofirstshowedthat(6.40)and(6.43)areactuallyequivalent,whichisatopicwewillnotgetintohere.Wenowobtainafamilyofexplicitsolutionsof(6.43)(hence(6.40)).Introduceachangeofvariablesfrom(H,K)to(U,V),−H=1+rU,K=rV.(6.44)Then(6.43)becomesU0=V2,V0=UV,whichimpliesthatU2−V2=constant.Thus,byvirtueof(6.35)and(6.44),wehaveU2−V2=e2,r>0.(6.45)Inserting(6.45)intoU0=V2andusing(6.32),i.e.,U(r)∼−1/rforr>0small,weobtainanexplicitsolutionof(6.43),H(r)=ercoth(er)−1,erK(r)=,(6.46)sinh(er)whichgivesrisetoasmooth,minimumenergy(mass)E=4π/e,monopole(J=0)solutionoftheequationsofmotion(6.29)attheBPSlimitλ=0throughtheradiallysymmetricprescription(6.30).Whenλ>0,onemayusevariationalmethodstoobtainanexistencetheoryforsolutions.Thesesolutionsarecollectivelyknownasthe’tHooft–Polyakovmonopoles[109,133]whicharesmoothandoffiniteenergy.Wenextpresentacontinuousfamilyofexplicitdyonsolutions.AttheBPSlimit,λ=0,thesystem(6.31)isr2J00=2JK2,r2H00=2HK2,r2K00=K(K2−J2+H2−1),(6.47)55 whichbecomes(6.40)whenwecompressH,JthroughH7→(coshα)H,J7→(sinhα)H,whereαisaconstant.Therefore,using(6.46),wehaveH(r)=coshα(ercoth(er)−1),J(r)=sinhα(ercoth(er)−1),erK(r)=.(6.48)sinh(er)Consequently,inviewof(6.30),wehaveobtainedafamilyofexplicitdyonsolutionsof(6.29).Notethatalltheboundaryconditionsstatedin(6.32)and(6.35)arefulfilledexceptone,namely,H(r)lim=coshα6=1.(6.49)r→∞erHowever,sinceλ=0,(6.49)isofnoharmtothefiniteenergycondition(6.34).Tocomputethetotalelectriccharge,weuse(6.36).WehaveI1q=limE·dS4πr→∞|x|=r10sinhα=lim(rJ(r)−J(r))=.(6.50)er→∞eItisacomforttoseethatthesolutionbecomeselectricallyneutral,q=0,whenα=0and(6.48)reducestothemonopolesolution(6.46).Totalelectricchargemaybenon-quantizedItshouldbenotedthat,sinceαin(6.50)isarbitrary,theelectricchargeqgivenin(6.50)isnotquantizedandmayassumevalueinacontinuum.ThemainreasonforthediscrepancywithwhatexpressedbytheDiracchargequantizationformulaisthattheelectricchargeqhereisthetotalchargeinducedfromacontinuouslydistributedelectricfieldbutnotapurepointcharge.TheHiggsfieldmaygeneratechargesThecalculationcarriedoutherealsoshowsthat,likemass,bothelectricandmagneticchargesmaybegeneratedfromtheHiggsfields.6.4TheGlashow–Weinberg–SalamelectroweakmodelThebosonicLagrangianactiondensityoftheGlashow–Weinberg–Salamelectroweakmodelmaybewrittenas1µν1µνµ†λ222L=−Fµν·F−HµνH+(Dˆµφ)·(Dˆφ)−(|φ|−φ0),(6.51)442whereφnowisaHiggscomplexdoubletlyinginthefundamentalrepresentationspaceofSU(2),FµνandHµνarethegaugefieldsofSU(2)andU(1)withthepotentialAµandBµandthecorrespondingcouplingconstantsgandg0,respectively,forthegeneratorsofSU(2)weusetheconventionalPaulispinmatricesτa(a=1,2,3),10120−i310aτ=,τ=,τ=,τ=(τ),(6.52)10i00−156 thecoordinatevectorsAµ=(Aaµ),andthegauge-covariantderivativesaredefinedbytheex-pressionsDˆφ=∂−igτaAa−ig0Bφ=Dφ−ig0Bφ.(6.53)µµµµµµTherefore,withintheaboveframework,theEuler–Lagrangianequationsof(6.51)areDˆDˆµφ=λ(|φ|2−φ2)φ,(6.54)µ0DµF=ig(φ†τ[Dˆφ]−[Dˆφ]†τφ),(6.55)µννν∂µH=ig0(φ†[Dˆφ]−[Dˆφ]†φ).(6.56)µνννBothdyonandvortexsolutionsoftheseequationshavebeenobtained(cf.[152]andrefer-encestherein).87TheEinsteinequationsandrelatedtopicsWestartfromaquickintroductiontoRiemanniangeometryandaderivationofthemetricenergy-momentumtensor.WenextderivetheEinsteinequationsforgravitation.Wethendis-cusssomedirectcosmologicalconsequencesfromtheEinsteinequations,theoriginofthecos-mologicalconstantanditsinterpretationasvacuummass-energydensity,andtheSchwarzschildblackholesolutionanditsextensions.WeendwithanexcursiontotheADMmassandrelatedtopicssuchasthepositivemasstheoremandthePenroseinequality.7.1EinsteinfieldequationsLet(gµν)bethemetrictensorofspacetime.Thespacetimelineelementorthefirstfundamentalformisdefinedbyds2=gdxµdxν,(7.1)µνwhichisalsoameasurementofthepropertime(see(4.18)foritsflat-spacetimeversion).AfreelymovingparticleinspacetimefollowsacurvethatstationarizestheactionZds.(7.2)Wenowderivetheequationsofmotionfromtheaboveactionprinciple.8ThefollowingparagraphpostedatWikipediaabouttheGlashow–Weinberg–Salammodel,underthesubjecttitle‘UnifiedFieldTheory,’isworthreadinginthecontextofourstudyhere:In1963AmericanphysicistSheldonGlashowproposedthattheweaknuclearforceandelectricityandmagnetismcouldarisefromapartiallyunifiedelectroweaktheory.In1967,PakistaniAbdusSalamandAmericanStevenWeinbergindependentlyrevisedGlashow’stheorybyhavingthemassesfortheWparticleandZparticlearisethroughspontaneoussymmetrybreakingwiththeHiggsmechanism.Thisunifiedtheorywasgovernedbytheexchangeoffourparticles:thephotonforelectromagneticinteractions,aneutralZparticleandtwochargedWparticlesforweakinteraction.Asaresultofthespontaneoussymmetrybreaking,theweakforcebecomesshortrangeandtheZandWbosonsacquiremassesof80.4and91.2GeV,respectively.Theirtheorywasfirstgivenexperimentalsupportbythediscoveryofweakneutralcurrentsin1973.In1983,theZandWbosonswerefirstproducedatCERNbyCarloRubbia’steam.Fortheirinsights,Salam,GlashowandWeinbergwereawardedtheNobelPrizeinPhysicsin1979.CarloRubbiaandSimonvanderMeerreceivedthePrizein1984.57 Weusethenotationxµ(s)todenotethedesiredcurve(trajectoryoftheparticle)andδxµ(s)asmallvariation,bothparametrizedbys.Then,tothefirstorderofvariation,wehaveδ(ds2)=2dsδ(ds)=(δg)dxµdxν+2gdxµδ(dxν)µνµν=(gδxα)dxµdxν+2gdxµd(δxν),(7.3)µν,αµνwhereandinthesequelweusethenotationf,A,F,Tµν,(7.4),αµ,αµν,α,αetc.,todenotetheconventionalpartialderivativewithrespecttothevariablexαofvariousquantities.Usingvµtodenotethecomponentsofthe4-velocity,dxµ(s)vµ(s)=,(7.5)dswethenobtain1µναµdνδ(ds)=gµν,αvvδx+gµνv(δx)ds2dsα1µνdµµν=(δx)gµν,αvv−(gµαv)ds+d(gµνvδx).(7.6)2dsInsertingtheaboveintothestationaryconditionZδds=0(7.7)andusingthefactthatδxµvanishesatthetwoendpointsofthecurve,wearriveattheequationsofmotiondµ1µν(gµαv)−gµν,αvv=0.(7.8)ds2Again,sincegµνissymmetric,wehaveddvµ(gvµ)=g+gvµvνµαµαµα,νdsdsdvµ1=g+(g+g)vµvν.(7.9)µααµ,ναν,µds2Consequentlytheequationsofmotionbecomedvµdvαg+Γvµvν=0or+Γαvµvν=0,(7.10)µααµνµνdsdswhereΓµναandΓαµνarecalledtheChristoffelsymbols,whicharedefinedby1ααβΓµνα=(gµν,α+gµα,ν−gνα,µ),Γµν=gΓµνβ.(7.11)2Thecurvesthataresolutionsof(7.10)arecalledgeodesics.WeseeimmediatelythatΓµνα=Γµαν.Besides,itisalsousefultonotethatthedefinitionofΓµναgivesustheidentityΓµνα+Γνµα=gµν,α.(7.12)58 OneofthemostimportantapplicationsoftheChristoffelsymbolsistheirroleinthedefinitionofcovariantderivativesforcovariantandcontravariantquantities,A=A−ΓβA,µ;αµ,αµαβT=T−ΓβT−ΓβT,µν;αµν,αµαβνναµβAµ=Aµ+ΓµAβ,;α,αβαTµν=Tµν+ΓµTβν+ΓνTµβ.(7.13);α,αβαβαWewillsometimesuse∇αtodenotecovariantderivative.Adirectconsequenceoftheabovedefinitionandtheidentity(7.12)isthatg=g−Γβg−Γβgµν;αµν,αµαβνναµβ=gµν,α−Γµαν−Γναµ=0.(7.14)Similarly,gµν;α=0.Thereforewehaveseenthatthecovariantandcontravariantmetrictensors,gandgµν,behavelikeconstantsundercovariantdifferentiation.µνLetAµbeatestcovariantvector.Following(7.13),weobtainthroughaneasycalculationthecommutatorA−A=[∇,∇]A=RνA,(7.15)µ;α;βµ;β;ααβµµαβνwhereRν=Γν−Γν+ΓγΓν−ΓγΓν(7.16)µαβµβ,αµα,βµβγαµαγβisamixed4-tensorcalledtheRiemanncurvaturetensor.ThereholdthesimplepropertiesRν=−Rν,(7.17)µαβµβαRν+Rν+Rν=0.(7.18)µαβαβµβµαFurthermore,similarto(7.15),forcovariant2-tensors,wehaveγγTµν;α;β−Tµν;β;α=RµαβTγν+RναβTµγ.(7.19)Therefore,inparticular,foracovariantvectorfieldAµ,wehaveγγAµ;ν;α;β−Aµ;ν;β;α=RµαβAγ;ν+RναβAµ;γ.(7.20)Wenowmakepermutationsoftheindicesν,α,βandaddthethreeresultingequations.Inviewof(7.15),theleft-handsideis(Aµ;α;β;ν−Aµ;β;α;ν)+permutationsγ=(RµαβAγ);ν+permutationsγγ=(RµαβAγ;ν+Rµαβ;νAγ)+permutations.(7.21)Inviewof(7.18),theright-handsideisγRµαβAγ;ν+permutations.(7.22)Equating(7.21)and(7.22),wearriveatγRµαβ;νAγ+permutations=0.(7.23)59 SinceAµisarbitrary,wefindthatγγγR+R+R=0.(7.24)µνα;βµαβ;νµβν;αThisresultisalsoknownastheBianchiidentity.TheRiccitensorRisdefinedfromRνthroughcontraction,µνµαβR=Rα.(7.25)µνµναItisclearthatRµνissymmetric.ThescalarcurvatureRisthendefinedbyR=gµνR.(7.26)µνIntheBianchiidentity(7.24),setγ=νandmultiplybygµα.Weobtain(gµαRν)+(gµαRν)+(gµαRν)=0,(7.27)µνα;βµαβ;νµβν;αwhichissimplyβ2Rα;β−R;α=0.(7.28)Multiplyingtheabovebygµα,wehavethefollowingveryimportantresult,Gµν=0,(7.29);νwhereµνµν1µνG=R−gR,(7.30)2oritscovariantpartner,Gµν,iscalledtheEinsteintensor.Wenextconsiderphysicsoverthecurvedspacetimeofmetric(gµν)governedbyamatterfielduwhichiseitherascalarfieldoravectorfieldandgovernedbytheactionZpS=L(x,Du,g)|g|dx,(7.31)wherewehaveemphasizedtheinfluenceofthemetrictensorpg=(gµν)andusedthecanonicalvolumeelement|g|dx.Here|g|istheabsolutevalueofthedeterminantofthemetricg.Sincephysicsisindependentofcoordinates,Lmustbeascalar.Forexample,therealKlein–Gordonactiondensitynowreads1µνL(x,u,Du,g)=g∂µu∂νu−V(u),(7.32)2whichisg-dependent.Inotherwords,physicscannolongerbepurelymaterial.ItiseasilyseenthattheEuler–Lagrangeequations,ortheequationsofmotion,of(7.31)arenow1p∂L∂Lap∂µ|g|=,a=1,2,···,m;u=(u).(7.33)|g|∂(∂µua)∂uaUsingthetranslationinvarianceoftheactionand(7.33),wecanderivetheenergy-momentumtensor,9alsocalledthestresstensor,Tµν,givenasµν∂Lµναµβν∂LµνT=−2−gL=2gg−gL,(7.34)∂gµν∂gαβ9ThefoundationalframeworkthatallowsonetoderiveconservedquantitiesasaresultofsymmetrypropertiesoftheactioniscalledtheNoethertheoremwhichwillnotbeelaboratedhere.60 whichobeystheconservationlawTµν=0.(7.35);νThebasicprinciplewhichledEinsteintowritedownhisfundamentalequationsforgravita-tionstatesthatthegeometryofaspacetimeisdeterminedbythematteritcontains.Mathe-matically,Einstein’sideawastoconsidertheequationQµν=−κTµν,(7.36)whereQµνisa2-tensorgeneratedfromthespacetimemetric(gµν)whichispurelygeometric,Tµνistheenergy-momentumtensorwhichispurelymaterial,κisaconstantcalledtheEinsteingravitationalconstant,andthenegativesigninfrontofκisinsertedforconvenience.Thisequationimposessevererestrictiontothepossibleformofthe2-tensorQµν.Forexample,Qµνshouldalsosatisfythesameconservationlaw(orthedivergence-freecondition),Qµν=0,(7.37);νasTµν(see(7.35)).ThesimplestcandidateforQµνcouldbegµν.However,sincegµνisnon-degenerate,thischoiceisseentobeincorrectbecauseitmakesTµνnon-degenerate,whichisabsurdingeneral.ThenextcandidatecouldbetheRiccicurvatureRµν.SinceRµνdoesnotsatisfytherequiredidentity(7.37),ithastobeabandoned.Consequently,basedonboththecompatibilitycondition(7.37)andsimplicityconsideration,wearenaturallyledtothechoiceoftheEinsteintensor,Gµν,definedin(7.30).ThereforeweobtaintheEinsteinequations,Gµν=−κTµνorG=−κT.(7.38)µνµνItcanbeshownthattheequation(7.38)recoversNewton’slawofgravitation,m1m2F=−G,(7.39)r2whichgivesthemagnitudeofanattractiveforcebetweentwopointparticlesofmassesm1andm2withadistancerapart,inthestaticspacetimeandslowmotionlimit,ifandonlyifκ=8πG.RecallthattheconstantGiscalledtheNewtonuniversalgravitationalconstant,whichisextremelysmallcomparedtootherquantities.Insummary,wehavejustderivedtheEinsteingravitationalfieldequations,Gµν=−8πGTµν.(7.40)7.2CosmologicalconsequencesInmoderncosmology,theuniverseisbelievedtobehomogeneous(thenumberofstarsperunitvolumeisuniformthroughoutlargeregionsofspace)andisotropic(thenumberofstarsperunitsolidangleisthesameinalldirections).ThisbasicpropertyisknownastheCosmologicalPrincipleandhasbeenevidencedbyastronomicalobservations.Adirectimplicationofsuchaprincipleisthatsynchronizedclocksmaybeplacedthroughouttheuniversetogiveauniformmeasurementoftime(Cosmictime).Anotheristhatthespacecurvature,K,isconstantatanyfixedcosmictimet.Hencewehavethefollowingsimplemathematicaldescriptionsforthespace.61 (a)IfK=K(t)>0,thespaceisclosedandmaybedefinedasa3-sphereembeddedintheflatEuclideanspaceoftheform222221x+y+z+w=a,a=a(t)=p.(7.41)K(t)(b)IfK=K(t)<0,thespaceisopenandmaybedefinedsimilarlybytheequation222221−x−y−z+w=a,a=a(t)=p,(7.42)|K(t)|whichisembeddedintheflatMinkowskispacewiththelineelementdx2+dy2+dz2−dw2,(7.43)knownastheanti-deSitterspace(orthe‘adSspace’inshort).10(c)IfK=K(t)=0,thespaceistheEuclideanspaceR3.Inparticularthespaceisopenanditslineelementisgivenbyd`2=dx2+dy2+dz2.(7.44)Usetheconventionalsphericalcoordinates(r,θ,χ)toreplacetheCartesiancoordinates(x,y,z).Wehavex=rcosθsinχ,y=rsinθsinχ,z=rcosχ.(7.45)Thus,inthecases(a)and(b),wehave±r2+w2=a2,±rdr=wdw.(7.46)Substituting(7.45)and(7.46)intothelineelementsoftheEuclideanspaceandoftheMinkowskispacegivenby(7.43),respectively,weobtaintheinducedlineelementd`2ofthespace,a2d`2=dr2+r2dθ2+r2sin2θdχ2.(7.47)(a2∓r2)Finally,inserting(7.47)intothespacetimelineelementandmakingtherescalingr7→ar,wehave2221222222ds=dt−a(t)dr+rdθ+rsinθdχ,(7.48)(1−kr2)wherek=±1ork=0accordingtoK>0,K<0orK=0.ThisisthemostgenerallineelementofahomogeneousandisotropicspacetimeandisknownastheRobertson–Walkermetric.Incosmology,thelarge-scaleviewpointallowsustotreatstarsorgalaxiesasparticlesofaperfect‘gas’thatfillstheuniverseandischaracterizedbyitsmass-energydensityρ,countingbothrestmassandkineticenergyperunitvolume,andpressurep,sothattheassociatedenergy-momentumtensorTµνisgivenbyTµν=(ρ+p)vµvν+pgµν,(7.49)P10IntheMinkowskispacetimeRn,1withthemetricds2=(dx0)2−n(dxi)2,thedeSitteroranti-deSitterPi=1spaceisthehyperbolicsubmanifolddefinedby(x0)2−n(xi)2=∓a2(a>0).i=162 wherevµisthe4-velocityofthegasparticlesandgµνisthespacetimemetric.Thecosmologicalprinciplerequiresthatρandpdependontimetonly.WenowconsidersomepossibleconsequencesofahomogeneousandisotropicuniverseinviewoftheEinsteintheory.From(7.16)and(7.25),wecanrepresenttheRiccitensorintermsoftheChristoffelsymbolsbyR=Γα−Γα+ΓαΓβ−ΓαΓβ.(7.50)µνµα,νµν,αµβανµναβNaturally,welabelourcoordinatesaccordingtox0=t,x1=r,x2=θ,x3=χ.ThenthenonzeroChristoffelsymbolsinducedfromtheRobertson–Walkerlineelementare11a(t)a0(t)Γ0=,Γ0=a(t)a0(t)r2,Γ0=a(t)a0(t)r2sin2θ,11(1−kr2)2233a0(t)krΓ1=,Γ1=,01a(t)11(1−kr2)Γ1=−r(1−kr2),Γ1=−r(1−kr2)sin2θ,2233a0(t)1Γ2=,Γ2=,Γ2=−sinθcosθ,021233a(t)ra0(t)1Γ3=,Γ3=,Γ3=cotθ,(7.51)03a(t)13r23wherea0(t)=da(t)/dt.Inserting(7.51)into(7.50),weseethattheRiccitensorRbecomesµνdiagonalwith3a00aa00+2(a0)2+2kR00=,R11=−2,a1−krR=−(aa00+2(a0)2+2k)r2,22R=−(aa00+2(a0)2+2k)r2sin2θ.(7.52)33Hence,inviewof(7.26),thescalarcurvature(7.26)becomes60002R=(aa+(a)+k).(7.53)a2Ontheotherhand,from(7.10)and(7.51),weseethatthegeodesicsofthemetric(7.48),whicharethetrajectoriesofmovingstarsandgalaxieswhennetlocalinteractionsareneglected,aregivenbyr,θ,χ=constant.Thusin(7.49)wehavev0=1andvi=0,i=1,2,3.ThereforeTµνisalsodiagonalwithpa2T=ρ,T=,T=pa2r2,T=pa2r2sin2θ.(7.54)00111−kr22233Substituting(7.52),(7.53),and(7.54)intotheEinsteinequations(7.40),wearriveatthefollowingtwoequations,3a00=−4πG(ρ+3p),aaa00+2(a0)2+2k=4πG(ρ−p)a2.(7.55)11Itshouldbenotedthat,nowadays,thecomputationofRiemanniantensorshasbeenfacilitatedenormouslybyavailablesymbolicpackages.See[19,88,112]andreferencestherein.63 Eliminatinga00fromtheseequations,weobtainthewell-knownFriedmannequation028π2(a)+k=Gρa.(7.56)3Wecanshowthat,inthecategoryoftime-dependentsolutions,theEinsteincosmologicalequa-tions,(7.55),areinfactequivalenttothesingleFriedmannequation(7.56).Tothisend,recallthatbothsystemsaretobesubjecttotheconservationlawfortheenergy-momentumtensor,namely,Tµν=0or;νa0ρ0+3(ρ+p)=0.(7.57)aDifferentiating(7.56)andusing(7.57),wegetthefirstequationin(7.55).Inserting(7.56)intothefirstequationin(7.55),wegetthesecondequationin(7.55).TherelativerateofchangeoftheradiusoftheuniverseisrecognizedasHubble’s‘constant’,H(t),whichisgivenbya0(t)H(t)=.(7.58)a(t)RecentestimatesforHubble’sconstantputitatabout(18×109years)−1.Inparticular,a0>0atpresent.However,sincethefirstequationin(7.55)indicatesthata00<0everywhere,wecanconcludethata0>0foralltimeinthepast.Inotherwords,theuniversehasundergoneaprocessofexpansioninthepast.Wenowinvestigatewhethertheuniversehasabeginningtime.Forthispurpose,lett0denotethepresenttimeandtdenoteanypasttime,ta0(t0),whichimpliesthata(t)−a(t)>a0(t)(t−t).000Thustheremustbeafinitetimetinthepast,t0isaconstant.Sincefappearsasthecoefficientofdt2in1themetricelement(7.66),wemayrescalethetimecoordinatewitht7→K2ttonormalizeKtounity.Thus,substitutingfh=1into(7.69)andsettingittozero,weobtain[144]01f=(1−f),(7.71)rwhichmaybeintegratedtogiveusCf(r)=1+,(7.72)rwhereCisanintegratingconstant.Wecanexaminethatthepair(f,h)whereh=1/fandfgivenin(7.72)indeedmakes(7.67)–(7.69)vanishidentically.Therefore,themetricelement(7.66)becomes(−1)2C2C22222ds=1+dt−1+dr+r(dθ+sinθdφ).(7.73)rrItisworthnotingthat,inther→∞limit,thismetricassumestheformofthatinthestandardflatMinkowskispacetime,i.e.,ds2=dt2−{dr2+r2(dθ2+sin2θdφ2)}.(7.74)Sinceregularityofthemetricelementrequiresf>0,wemaybetemptedtotakeC>0in(7.72).Unfortunately,orfortunately,thesituationwearefacinghereisnotsosimpleandamoreelaborateconsiderationneedstobecarriedoutsothatthesolutionisphysicallymeaningful.Indeed,onemayargue[144]that,sincethesolutionshouldrecoverthatgivenbytheNewtonlawofgravitationgeneratedfromacentralizedlocalizedmass,sayM,intheregionwhererissufficientlylarge,oneisledtotheinevitableconclusionC=−2GM,(7.75)whereGistheNewtonconstant.Itmaybeinstructivetotakeapauseandfindhowtowork(7.75)out.Forthispurpose,considerthemotionofatestparticlefarawayfromlocalregion.Relativistically,thetrajectoryoftheparticleisparameterizedbythepropertime,ds,intermsofthespacetimecoordinatesxµ(s),andfollowingthegeodesicequations,(7.10).Wemayassumethatthespeedofthemotionisnegligiblecomparedwiththespeedoflight.Thusweareabletotaketheapproximationds≈dt,vµ≈(1,0,0,0).(7.76)Inviewof(7.76),theequations(7.10)becomed2xα+Γα=0,α=0,1,2,3.(7.77)dt200However,byvirtueof(7.66),weseethattheonlynontrivialcomponentof(7.77)isatα=1(x1=r)whichisthesingleequationd2rf0(r)1CC==ff0=−1+,(7.78)dt22h22r2r66 whichagreeswithNewton’slawd2rGM=−(7.79)dt2r2forrsufficientlylargeonlyifCistakentosatisfy(7.75),asstated.Insummary,wehavearrivedatthesolutionrepresentedby(−1)22GM22GM22222ds=1−dt−1−dr+r(dθ+sinθdφ),(7.80)rrwhichisthecelebratedSchwarzschildmetricorSchwarzschildsolutionoftheEinsteinequations.Schwarzschildobtainedthissolutionin1915,thesameyearwhenEinsteinpublishedhisworkofgeneralrelativity.12Therearecoordinatesingularitiesattheradiusr=rs=2GM,(7.81)referredtoastheSchwarzschildradius.Withrs,werewrite(7.80)as−12rs2rs22222ds=1−dt−1−dr+r(dθ+sinθdφ),(7.82)rrThesingularsphere,r=rs,inspace,alsocalledtheSchwarzschildsurface,isaneventhorizon.Thesolution(7.80)representsanempty-spacesolutionorexteriorsolutionwhichisvalidoutsideasphericallydistributedmassivebodyoccupyingtheregiongivenbyr≤R(say).Inotherwords,thesolution(7.80)isvalidforr>R.TogettheinteriorsolutionoftheEinsteinequationsinrR.12HereissomeinterestingreadingprovidedbyWikipedia:Einsteinhimselfwaspleasantlysurprisedtolearnthatthefieldequationsadmittedexactsolutions,becauseoftheirprimafaciecomplexity,andbecausehehimselfhadonlyproducedanapproximatesolution.Einstein’sapproximatesolutionwasgiveninhisfamous1915articleontheadvanceoftheperihelionofMercury.There,Einsteinusedrectangularcoordinatestoapproximatethegravitationalfieldaroundasphericallysymmetric,non-rotating,non-chargedmass.Schwarzschild,incontrast,choseamoreelegant‘polar-like’coordinatesystemandwasabletoproduceanexactsolutionwhichhefirstsetdowninalettertoEinsteinof22December1915,writtenwhileSchwarzschildwasservinginthewarstationedontheRussianfront.Schwarzschildconcludedtheletterbywriting:“Asyousee,thewartreatedmekindlyenough,inspiteoftheheavygunfire,toallowmetogetawayfromitallandtakethiswalkinthelandofyourideas.”In1916,EinsteinwrotetoSchwarzschildonthisresult:“Ihavereadyourpaperwiththeutmostinterest.Ihadnotexpectedthatonecouldformulatetheexactsolutionoftheprobleminsuchasimpleway.Ilikedverymuchyourmathematicaltreatmentofthesubject.NextThursdayIshallpresenttheworktotheAcademywithafewwordsofexplanation.–AlbertEinstein.”SchwarzschildaccomplishedthistriumphwhileservingintheGermanarmyduringWorldWarI.Hediedthefollowingyearfrompemphigus,apainfulautoimmunediseasewhichhedevelopedwhileattheRussianfront.67 However,ifthemassdensityofthegravitatingbodyissohighthatRrs,weseeclearlythatlightissloweddownwhenitisnearagravitatingbody.Thus,agravitatingbodybendslight.Insidetheeventhorizon,r0isemptyspace.Thuswehaveobservedthat,insidetheeventhorizon,itis‘easier’tofalltowardsthecenterthandeviatefromit.Indeed,suchaspaceregion,enclosedbytheeventhorizonandpopularlycalledtheSchwarzschildblackhole,allowsnothing,notevenlight,toescapefromitbutrathertendsto‘collapse’everythingtowardsitscenter.Itmayalsobeofinteresttopresentsomeofthewell-knownextensionsoftheSchwarzschildsolution.TheReissner–NordstromsolutionFirst,considerthesituationwhereanelectrostaticfieldcomesintothepicture.SettingB=0(vanishingmagneticfield)andE=(Q/r3)x(electricfieldgivenbytheCoulomblaw)intheemptyspaceandsolvingtheEinsteinequations(7.40)there,weseethatthemetric(7.80)becomes(−1)22GMGQ222GMGQ222222ds=1−+dt−1−+dr+r(dθ+sinθdφ),rr2rr2(7.88)whichiscalledtheReissner–Nordstrommetric.Aneventhorizonoccurswhen2GMGQ21−+=0,(7.89)rr213Insidetheeventhorizon,rGM2.(7.90)Inthiscase,themetricisregulareverywhereinr>0.Ontheotherhand,whenQ20centeredattheorigin.Underthisrrdiffeomorphism,themetricgjkofMateachendcanberepresentednearinfinitybyg(x)=δ+a(x),x∈R3B,jkjkjkra(x)=O(|x|−1),∂a(x)=O(|x|−2),∂∂a(x)=O(|x|−3),(7.98)jk`jk`mjkandthesecondfundamentalform(hjk)theresatisfiessimilarasymptoticestimatesh(x)=O(|x|−2),∂h(x)=O(|x|−3),x∈R3B.(7.99)jk`jkrWithoutlossofgenerality,weassumeonlyoneendforconvenience.AccordingtoArnowitt,Deser,andMisner[7],thetotalenergyEandthemomentumP`canbedefinedasthelimitsofintegralfluxesZ1kE=lim(∂jgjk−∂kgjj)νdσr,(7.100)16πr→∞∂BrZ1kP`=lim(h`k−δ`khjj)νdσr,(7.101)8πr→∞∂Brwheredσristheareaelementof∂Brandνdenotestheoutnormalvectorto∂Br.ThePositiveEnergyTheorem[103,119,120,121,148]statesthatthetotalenergy(7.100)isboundedfrombelowbythetotalmomentum(7.101)byE≥|P|(7.102)andthatE=0ifandonlyif(M,g)istheEuclideanspace(R3,δ).Inthespecialcasewhenthesecondfundamentalform(hjk)vanishesidentically,P≡0,theenergyEiscalledthetotalmassortheADMmass,MADM,whichisalwaysnonnegative,MADM≥0.ThePositiveMassTheorem[122]statesthatMADM>0,(7.103)unlessthehypersurface(M,g)istheEuclideanspace(R3,δ).Simplyput,theabovetheoremsimplythatnoenergyormassmeansnogeometryornogravitation.70 Notethat,usingtheEinsteinequations(7.40),onemayrelatethescalarcurvatureRgof(M,g)totheenergydensityT00byR+(hk)2−hkhj=16πGT.(7.104)gkjk00Thus,asaconsequenceofthedominantenergycondition(see(7.97)),thevanishingofthesecondfundamentalformnaturallyleadstothepositivityconditionforthescalarcurvature,Rg≥0.(7.105)Naturally,onewouldhopetoboundMADMawayfromzerobysomephysicalinformationinagravitationalsystem.Forexample,onemaystartfromconsideringanisolatedblackholeofmassM>0whosespacetimemetricisknowntobegivenbytheSchwarzschildlineelement(7.80).ItcanbecheckedthatthespatialsliceatanyfixedthasthepropertythatitssecondfundamentalformvanishesandthatitsADMmassisthesameastheblackholemassM.Inthiscase,thesingularsurfaceortheeventhorizon,Σ,oftheblackholeisasphereofradiusrs=2GMwhosesurfaceareahasthevalueArea(Σ)=4πr2=16πG2M2.(7.106)sThePenroseConjecture[105]statesthatthetotalenergyEofthespacetimedefinedin(7.100)isboundedfrombelowbythetotalsurfaceareaofitsapparenthorizonΣ,whichcoincideswiththeeventhorizoninthecaseofaSchwarzschildblackhole,by16πG2E2≥Area(Σ).(7.107)InthespecialcasewhenthesecondfundamentalformofthehypersurfaceMvanishes,(7.107)becomes16πG2M2≥Area(Σ),(7.108)ADMwhichisreferredtoastheRiemannianPenroseInequality,forwhichthelowerboundmaybesaturatedonlyintheSchwarzschildlimit[16,17,18,62,63].8ChargedvorticesandtheChern–SimonsequationsWehaveseenthatgaugetheoryinthreedimensionsallowstheexistenceofmagneticallyandelectricallychargedparticle-likesolutionscalleddyons.Naturally,itwillbeinterestingtoknowwhethertherearedyonsintwodimensions.Thatis,whethertherearemagneticallyandelectri-callychargedstaticvorticesingaugetheory.Toanswerthisquestion,JuliaandZeestudiedtheAbelianHiggsgaugefieldtheorymodelintheirnowclassic1975paper[67].Usingaradiallysymmetricfieldconfigurationansatzandassumingasufficientlyfastdecayrateatspatialinfin-ity,theywereabletoconcludethatafinite-energystaticsolutionoftheequationsofmotionoverthe(2+1)-dimensionalMinkowskispacetimemustsatisfythetemporalgaugecondition,andthus,isnecessarilyelectricallyneutral.Thisresult,referredhereastheJulia–Zeetheorem,leadstomanyinterestingconsequences.Forexample,itmakesittransparentthatthestaticAbelianHiggsmodelisexactlytheGinzburg–Landautheory[51]whichispurelymagnetic[66,99].SincetheworkofJuliaandZee[67],ithasbeenaccepted[38,60,65,70,104,142,143]that,inordertoobtainbothelectricallyandmagneticallychargedstaticvortices,oneneedstointroduceintotheLagrangianactiondensitytheChern–Simonstopologicalterms[30,31],whichisanessentialconstructinanyonphysics[146,147].Seealso[47].Thepurposeofthissectionistopresentadiscussionoftheseproblems.71 8.1TheJulia–ZeetheoremRecallthattheclassicalAbelianHiggstheoryoverthe(2+1)-dimensionalspacetimeisgovernedbytheLagrangianactiondensity(4.32)andtheassociatedequationsofmotionare(4.33)–(4.35).Inthestaticsituation,theoperator∂0=0nullifieseverything.Hencetheelectricchargedensityρbecomes0i002ρ=J=(φDφ−φDφ)=−A0|φ|,(8.1)2whereDµ=∂µ+iAµistherenormalizedgauge-covariantderivative,andanontrivialtemporalcomponentofthegaugefield,A0,isnecessaryforthepresenceofelectriccharge.Ontheotherhand,theµ=0componentoftheleft-handsideoftheMaxwellequation(4.34)is∂F0ν=∂(F)=∂2A=∆A.(8.2)νii0i00Consequently,thestaticversionoftheequationsofmotion(4.33)–(4.35)maybewrittenasD2φ=2V0(|φ|2)φ−A2φ,(8.3)i0i∂jFij=(φDiφ−φDiφ),(8.4)2∆A=|φ|2A,(8.5)00inwhich(8.5)istheGausslaw.Moreover,sincetheenergy-momentum(stress)tensorhastheformµ0ν01Tµν=−ηFµµ0Fνν0+(DµφDνφ+DµφDνφ)−ηµνL,(8.6)2theHamiltoniandensityisgivenby1212212122H=T00=|∂iA0|+A0|φ|+Fij+|Diφ|+V(|φ|),(8.7)2242sothatthefinite-energyconditionreadsZHdx<∞.(8.8)R2Withtheaboveformulation,theJulia–Zeetheorem[67]maybestatedas:Supposethat(A,A,φ)isafinite-energysolutionofthestaticAbelianHiggsequations(8.3)–(8.5)overR2.0iTheneitherA0=0everywhereifφisnotidenticallyzeroorA0≡constantandthesolutionisnecessarilyelectricallyneutral.Inotherwords,thestaticAbelianHiggsmodelisexactlytheGinzburg–Landautheory.Hereisaproof[132]ofthetheoremwhichreliesonacrucialchoiceofatestfunctionsothattheargumentisvalidexactlyintwodimensions.Let0≤η≤1beofcompactsupportanddefineforM>0fixedthetruncatedfunctionMifA0>M,AM=Aif|A|≤M,(8.9)000−MifA0<−M.Then,multiplying(8.5)byηAMandintegrating,wehave0Z[η∇A·∇AM+AM∇A·∇η+η|φ|2AMA]dx=0.(8.10)000000R272 Using(8.9)in(8.10),wefindZZη|φ|2A2dx+M2η|φ|2dx0{|A0|M}∩supp(η)Z+η|∇A|2dx0{|A0|0,wenowchooseηtobealogarithmiccutofffunctiongivenas1if|x|R2.ThenZ22π|∇η|dx=.(8.13)R2lnRUsing(8.13)in(8.11)givesZZ|φ|2A2dx+|∇A|2dx00{|A0|M}∩BR{|A0|0arecouplingconstants.TheEuler–Lagrangeequationsoftheproblemare∆u−g2|A|2u−2λ(u2−v2)u=0,(8.43)∇×(∇×A)−ξ(∇×A)+2g2u2A=0,(8.44)whereξistheLagrangianmultiplierassociatedwithvaryingthefunctional8π2I(A,u)=E(A,u)−ξNCS(A,u).(8.45)g2See[118]forarecentstudyand[114]forsomediscussioninthecontextofasurvey.Sofar,arigorousmathematicalstudyofsuchanEuclideanthree-dimensionalChern–Simonsproblemhasnotbeencarriedoutyet.Itshouldbenotedthatthemodel(8.41)lacksgaugeinvariance.Inordertorecoveritsgaugeinvariance,wemayreplacetherealscalarfieldubyacomplexscalarfieldφandusethegauge-covariantderivativeDAφ=∇φ−giAφ(8.46)asbefore.Since|Dφ|2=|∇φ|2+g2|A|2|φ|2+2gA·Im(φ∇φ),(8.47)AweseethattheRubakov–Tavkhelidzeenergy(8.41)isthereal-scalar-fieldversionoftheGinzburg–Landauenergy[51]Z122222E(A,φ)=|∇×A|+|DAφ|+λ(|φ|−v)dx.(8.48)2Inotherwords,wemaysaythatthemodel(8.41)istheGinzburg–Landaumodel(8.48)statedintheunitarygauge.Hencetheequationsofmotion(8.43)–(8.44)aremodifiedintoD2φ=2λ(|φ|2−v2)φ,(8.49)A∇×∇×A=ig(φDAφ−φDAφ)+ξ(∇×A).(8.50)Inthepurelysuperconductinglimitwhen|φ|attainsitsmaximumvalueveverywhere,theequa-tions(8.49)–(8.50)arereducedintothesingleonegoverningthegaugepotentialA:∇×∇×A=−2g2v2A+ξ(∇×A).(8.51)InthespecialcasewhenthereisnoChern–Simonsinvariantpresent,ξ=0,andweareleftwith∇×∇×A=−2g2v2A.(8.52)Finally,recallthattheinducedmagneticfieldBmaybeexpressedasB=∇×A.Thus,inviewof(8.52),wearriveat∆B=2g2v2B.(8.53)ThisequationisknownastheLondonequation[83],discoveredbytheLondonbrothers,14whichclearlyindicatesthattheinducedmagneticfieldinasuperconductorbecomes‘massive’,andthus,fadesoutexponentiallyfastinsidethesuperconductor.SuchastatementgavetheearliestmathematicalproofoftheMeissnereffect.14FritzandHeinzLondon,German-bornphysicists.ThefullsetoftheLondonequations,althoughlinear,actuallyareslightlymorecomplicatedthan(8.53),andgiveriseto(8.53).77 9TheSkyrmemodelandrelatedtopicsTheideathatelementaryparticlesmaybedescribedbycontinuouslydistributedfieldswithlocalizedenergyconcentrations,alsocalledsolitons,hasalonghistory.Asaresult,itwillbeinterestingtoknowwhetherthereexiststaticsolitonsdescribingparticlesatrestorinequilib-rium.9.1TheDerricktheoremandconsequencesWestartfromthestandardKlein–GordonfieldtheorywhoseLagrangianhasbeendefinedearlierwithanarbitrarypotentialdensityV≥0.Inthestaticlimit,theequationsofmotionbecomeasemilinearellipticequation∆u=2V0(|u|2)u,(9.1)whichistheEuler–LagrangeequationoftheHamiltonenergyZ122E(u)=|∇u|+V(|u|)dx.(9.2)Rn2Therefore,thesolutionsaresimplythecriticalpointsoftheenergyfunctional.Supposethatuisacriticalpoint.Thenuλ(x)=u(λx)isacriticalpointaswellwhenλ=1,whichleadsustotheassertiondE(uλ)=0.(9.3)dλλ=1Ontheotherhand,ifweusexλtodenoteλxand∇λtodenotethegradientoperatorwithderivativesintermsofdifferentiationinxλ,thenZZ12122E(uλ)=|∇uλ|+V(uλ)dx=λ|∇λuλ|+V(uλ)dxRn2Rn2ZZ122−n12−n2−n=λ|∇λu(xλ)|+V(u(xλ))λdxλ=λ|∇u|+λV(u)dx.(9.4)Rn2Rn2Combining(9.3)and(9.4),weobtaintheidentityZZ(2−n)|∇u|2dx=2nV(u)dx.(9.5)RnRnConsequently,weseethatthereisnonontrivialsolutionifn≥3whichrulesoutthemostphysicaldimension.ThisstatementisknownastheDerricktheorem.(MathematiciansalsocalledtheaboveintegralidentitythePohozaevidentity–seebelow.)Besides,thecasen=2isinterestingonlyintheabsenceofpotentialenergy,V=0.Onlywhenn=1,thepotentialdensityfunctionVisnotsubjecttoanyrestrictionandlocallyconcentratedstaticsolutionscanindeedbeconstructed(whichareoftencalledkinksordomainwalls).ThePohozaevidentity–anexcursionThisisacleverandsometimesveryusefultoolinanalyzingsemilinearpartialdifferentialequations.Toseewhatitis,weconsidertheellipticboundaryvalueproblem−∆u=λ|u|p−1uinΩ,u=0on∂Ω,(9.6)whereΩisastar-shapedboundeddomaininRnwithsmoothboundaryandλ>0.78 Multiplying∆ubyx·∇u,integratingoverΩ,andusingu=0on∂Ω,wehaveZZZxj∂u∂∂udx=−nu∆udx−xju∂∂∂udxjkkkkjΩZΩZΩZjj=−nu∆udx+δku∂k∂judx+x∂ku∂k∂judxΩZΩZΩZ1j2n2=(1−n)u∆udx+∂j(x|∇u|)dx−|∇u|dxΩ2Ω2ΩZZZ12n2=(1−n)u∆udx+|∇u|(ν(x)·x)dS−|∇u|dxΩ2∂Ω2ΩZZ(n−2)212=|∇u|dx+|∇u|(ν(x)·x)dS,(9.7)2Ω2∂Ωwhereν(x)denotestheunitoutwardnormalattheboundarypointx∈∂Ω.Besides,multiplying|u|p−1ubyx·∇uandintegrating,wehaveZZZZp−11p+1np+1np+1(x·∇u)|u|udx=∇·(x|u|)dx−|u|dx=−|u|dx.p+1Ωp+1Ωp+1Ω(9.8)Multiplyingthedifferentialequationin(9.6)byx·∇uandusing(9.7)and(9.8),weobtainZZZ(n−2)212λnp+1|∇u|dx+|∇u|(ν(x)·x)dS=|u|dx,(9.9)2Ω2∂Ω(p+1)ΩwhichiscalledthePohozaevidentity.WhenΩisstar-shapedabouttheorigin,wehavex·ν(x)≥0.Therefore,wegetZZ(n−2)2λnp+1|∇u|dx≤|u|dx.(9.10)2Ω(p+1)ΩOntheotherhand,multiplyingthedifferentialequationin(9.6)simplybyuandintegrating,wehaveZZ|∇u|2dx=λ|u|p+1dx.(9.11)ΩΩCombiningtheabovetworesults,wearriveatZ(n−2)np+1−|u|dx≤0,(9.12)2(p+1)Ωwhichestablishesthattheonlysolutionisthetrivialsolutionu=0when(n−2)n−>0(9.13)2(p+1)orn+2n≥3,p>.(9.14)n−2Inotherwords,nontrivialsolutionsarepossibleonlywhenn=1,2orn≥3andpsatisfiesn+2thesubcriticalcondition:10isauniversalconstant.HenceEN>0foranyN6=0.DefineS={N∈Z{0}|theFaddeevProblem(9.32)hasasolutionatN}.(9.36)TheFaddeevKnotProblemaskswhetherornotthereholdsS=Z.Asafirststeptowardtheabovequestion,wehave:ThesetSisnotempty.Theproofofthisstatement[77]amountstoestablishingtheSubstantialInequalityfortheFaddeevenergy(9.28)andnotingthatifSisempty,thenthesplittingexpressedin(9.33)and(9.34)willcontinueforever,whichcontradictsthefinitenessandpositivenessofENforanyN.With(9.33),wecanlearnmoreaboutthesolublesetS.Forexample,chooseN0∈Z{0}sothatEN0=min{EN|N∈Z{0}}.(9.37)ThenwemusthaveN0∈SbecauseanontrivialsplittinggivenintheSubstantialInequalitywillbeimpossiblebythedefinitionofN0.Thuswecanstate[77]:TheleastenergypointintheFaddeevenergyspectrum{EN|N∈Z{0}}isattainable.Basedonsomecarefulestimates[78],wealsohave±1∈S.MoreknowledgeaboutthesetScanbededucedfromtheSubstantialInequalityafterwerealizethatthefractionalexponent3inthelowerbound(9.35)isinfactsharpbyestablishing4[77]thesublinearenergyupperbound3EN≤C1|N|4,(9.38)whereC1isauniversalpositiveconstant(cf.[59]forsomeestimatesforthevalueofC1),whichenablesus[77]toobtain:83 ThesetSisaninfinitesubsetofZ.Hereisaquickproofofthisresult.OtherwiseassumethatSisfinite.SetN0=max{N∈S}(9.39)andletN0∈SbesuchthatEN0=min{EN|N∈S},asdefinedearlier.Takingrepeateddecompositionsifnecessary,wemayassumethatalltheintegersN1,N2,···,Nkin(9.33)and(9.34)areinSalready.Hence|N1|,|N2|,···,|Nk|≤N0.Thus,inviewof(9.33),wehaveN≤kN0;inviewof(9.34),wehaveE≥kE.Consequently,NN0EN0EN≥N,(9.40)N0whichcontradicts(9.38)whenNissufficientlylarge.Hence,theassumptionthatSisfiniteisfalse.9.4Commentsonfractional-exponentgrowthlawsandknotenergiesWenoticedthatoneofthecrucialfactsthatguaranteestheexistenceofenergyminimizingFaddeevknotsininfinitelymanyHopfclassesisthesublinearenergygrowthproperty(9.38).Thatis,theminimumvalueoftheFaddeevenergyformapsfromR3intoS2withHopfinvariantNgrowsublinearlyinNwhenNbecomeslarge.Thispropertyimpliesthatcertain“particles”withlargetopologicalchargesmaybeenergeticallypreferredandthattheselargeparticlesarepreventedfromsplittingintoparticleswithsmallertopologicalcharges.Inordertoachieveadeeperunderstandingofthisratherpeculiarproperty,weshallmaketwomoreobservationsintwoextremecases,utilizingtheropelengthenergyandextendingtheFaddeevmodeltohigherdimensions,respectively.RopelengthenergyandgrowthlawItisknownthattheappearanceofknottedstructuresmayalsoberelevanttotheexistenceandstabilityoflargemolecularconformationinpolymersandgelelectrophoresisofDNA.Intheselatterproblems,acrucialgeometricquantitythatmeasuresthe“energy”ofaphysicalknotofknot(orlink)typeK(orsimplyknot)isthe“ropelength”L(K),oftheknotK.Todefineit,weconsiderauniformtubecenteredalongaspacecurveΓ.The“ropelength”L(Γ)ofΓistheratioofthearclengthofΓovertheradiusofthelargestuniformtubecenteredalongΓ.ThenL(K)=inf{L(Γ)|Γ∈K}.(9.41)AcurveΓachievingtheinfimumcarriestheminimumenergyinKandgivesrisetoan“ideal”or“physicallypreferred”knot[69,72],alsocalledatightknot[26].Clearly,thisidealconfigurationdeterminestheshortestpieceoftubethatcanbeclosedtoformtheknot.Similarly,anothercrucialquantitythatmeasuresthegeometriccomplexityofΓistheaveragenumberofcrossingsinplanarprojectionsofthespacecurveΓdenotedbyN(Γ)(say).ThecrossingnumberN(K)oftheknotKisdefinedtobeN(K)=inf{N(Γ)|Γ∈K},(9.42)84 whichisaknotinvariant.NaturallyoneexpectstheenergyandthegeometriccomplexityoftheknotKtobecloselyrelated.Indeed,thecombinedresultsin[20,26]leadtotherelationCN(K)p≤L(K)≤CN(K)p,(9.43)12whereC1,C2>0aretwouniversalconstantsandtheexponentpsatisfies3/4≤p<1sothatintrulythree-dimensionalsituationsthepreferredvalueofpissharplyatp=3/4.Thisrelationstrikinglyresemblesthefractional-exponentgrowthlawfortheFaddeevknotsjustdiscussedandremindsusoncemorethatasublinearenergygrowthlawwithregardtothetopologicalcontentinvolvedisessentialforknottedstructurestooccur.TheFaddeevmodelingeneralHopfdimensionsIntheFaddeevKnotProblem,itistheunderlyingpropertyandstructureofthehomotopygroupπ(S2)andtheFaddeevenergyfunctionalformulathatguaranteethevalidityofthe3associatedsublineargrowthlaw.Generally,itseemsthatsuchapropertymayberelatedtothenotionofquantitativehomotopyintroducedbyGromov[56].Forexample,wemayconsidertheWhiteheadintegralrepresentationoftheHopfinvariantandthe“associated”knotenergyalaFaddeev.Moreprecisely,letu:R4n−1→S2n(n≥1)beadifferentiablemapwhichapproachesaRconstantsufficientlyfastatinfinite.DenotebyΩthevolumeelementofS2nand|S2n|=2nΩ.SThentheintegralrepresentationofuinthehomotopygroupπ(S2n),sayQ(u),whichisthe4n−1Hopfinvariantofu,isgivenbyZ1∗∗Q(u)=v∧u(Ω),dv=u(Ω).(9.44)|S2n|R4n−1WecanintroduceageneralizedFaddeevknotenergyforsuchamapuasfollows,Z21∗2E(u)=|du|+|u(Ω)|dx.(9.45)R4n−12Forthisenergyfunctional,weareabletoestablishthefollowinggeneralizedsublinearenergygrowthestimate[79,80]C|N|(4n−1)/4n≤E≤C|N|(4n−1)/4n,(9.46)1N2whereEN=inf{E(u)|E(u)<∞,Q(u)=N},(9.47)andC1,C2>0areuniversalconstants.Inparticular,weareabletoseethatthefractionalexponentinthegeneralizedgrowthlawistheratioofthedimensionofthedomainspaceandtwiceofthedimensionofthetargetspace.10StringsandbranesInthissection,wepresentabriefintroductiontorelativisticstringsandbranes.Wefirstdiscusstherelativisticmotionofapointparticle.WethengeneralizethisdiscussiontoconsidertheNambu–Gotostringsandbranes.WenextstudythePolyakovstringsandshowthattheirquantizationleadstothecriticaldimensionalitycountsofspacetime.85 10.1RelativisticmotionofafreeparticleLetmbethemassofafreeparticlewithcoordinates(xi)inthespaceRn,dependingontimet.RecallthattheNewtonianactionandLagrangianofthemovingparticleareZXn1i2dfS=Ldt,L=m(˙x),f˙=.(10.1)2dti=1Ontheotherhand,relativistically,themotionoftheparticlefollowsthetrajectorythatextremizestheactionZS=κds,(10.2)whereκisanundeterminedconstantandds2isthemetricelementgivenbyXnds2=c2dt2−(dxi)2=ηdxµdxν,(10.3)µνi=1withc>0thespeedoflight,x0=ct,andηµν=diag(1,−1,···,−1)thestandardMinkowskianmetrictensor.Inviewof(10.2)and(10.3),ifweconsiderthemotionoftheparticleintermsoftimet,wehavevZunuX(˙xi)2S=κct1−dt.(10.4)c2i=1Itisclearthat(10.2)givesthedynamicsof(10.1)inlowspeedwhenκ=−mc.Thus,wearrivedattherelativisticactionandLagrangianvZunuX(˙xi)2S=Ldt,L=−mc2t1−.(10.5)c2i=1Asaconsequence,wecancomputetheassociatedmomentumvector∂Lmx˙ipi==q,i=1,···,n,(10.6)∂x˙iPn(˙xi)21−i=1c2whichleadstotheHamiltonianXnmc2H=px˙i−L=q,(10.7)iPn(˙xi)2i=11−i=1c2whichgivesustherelationXnH2=m2c4+c2p2.(10.8)ii=1Inparticular,whentheparticleisatrest,weobtainthepopular-scienceformulaE=H=mc2.86 10.2TheNambu–Gotostrings2PnForsimplicity,wesetthespeedoflighttounity,c=1,usethenotationx=,andi=1v(t)=x˙.Thentheactionofafreeparticleofmassmoveratimespan[t1,t2]readsZt2pS=−m1−v2dt.(10.9)t1Wenowconsiderthemotionofafreestringofauniformmassdensity,ρ0,parametrizedbyarealparameter,s,withthespatialcoordinatesgivenasaparametrizedcurve,x=x(s,t),s1≤s≤s2,(10.10)atanyfixedtimet.Following(10.9),theactionforthemotionoftheinfinitesimalportion∂xd`=ds≡|x0|ds,s≤s≤s,(10.11)∂s12isgivenbyZt2p−ρ0d`1−v2dt.(10.12)t1IntheNambu–Gototheory[52],theinternalforcesbetweenneighboringpointsalongastringdonotcontributetotheactionsothatthevelocityvectorvisperpendiculartothetangentofthestringcurve.Thus,wehavedx∂x∂x∂xv==+a(s,t),v·=0.(10.13)dt∂t∂s∂sFrom(10.13),wecandeterminethescalarfactora(s,t)andobtainvasfollows,∂x·∂x∂x∂t∂s∂xv=−2.(10.14)∂t∂s∂x∂sIntegrating(10.12),weobtainthetotalactionforaNambu–Gotostring,sZt2Zs22∂xS=−ρ0(1−v2)dsdt.(10.15)t1s1∂sToappreciatethegeometricmeaningoftheaction(10.15),werecallthat,fortwovectors,x=(xµ)=(x0,x)andy=(yµ)=(y0,y),xystandsfortheinnerproductXnxy=xµy=xµηyν=x0y0−xiyi=x0y0−x·y,(10.16)µµνi=1intheMinkowskispacetimeRn,1.Thus,withthenotationf˙=∂f,f0=∂f,(10.17)∂t∂s87 wehave0∂x∂x∂x∂xxx˙=1,0,=−·,(10.18)∂t∂s∂t∂s222∂x02∂xx˙=1−,x=−.(10.19)∂t∂sTherefore,22222∂x2∂x∂x∂x∂x∂x(1−v)=−+·∂s∂s∂t∂s∂t∂s=(˙xx0)2−x˙2x02.(10.20)HencetheNambu–GotoactionbecomesZpA=−ρ0(˙xx0)2−x˙2x02dsdt.(10.21)Besides,wemaycalculatethelineelementoftheembedded2-surfacexµ=xµ(s,t),intheflatMinkowskispacetimeRn,1,parametrizedbytheparameterssandtbyds2=dxµηdxνµν=(˙xµdt+x0µds)η(˙xνdt+x0νds)µν=x˙2dt2+2˙xx0dtds+x02ds2=hduadub,a,b=0,1,(10.22)abwhereu0=t,u1=s,andx˙2xx˙0(hab)=002.(10.23)xx˙xFrom(10.21)and(10.23),weseethattheNambu–GotostringactionissimplyasurfaceintegralZZpA=−ρ0|h|dtds=−ρ0dS,(10.24)ΩSwhere|h|istheabsolutevalueofthedeterminantofthematrix(10.23)anddSisthecanonicalareaelementoftheembedded2-surface,(S,{hab}),intheMinkowskispacetime.Asacomparison,theaction(10.21)forapointparticleissimplyapathintegral,ZτZ2√A=−mx˙2dτ=−mdC,(10.25)τ1CwheredCisthelineelementofthepathCparametrizedbyxµ=xµ(τ),τ≤τ≤τ,inthe12Minkowskispacetime.Thereforethemotionofapointparticlefollowsanextremizedpath,theworldline,andthemotionofaNambu–Gotostringfollowsanextremizedsurface,theworldsheet.Ofcourse,both(10.24)and(10.25)areparametrizationinvariant.Returningto(10.21),usingthegeneralizedtimeandstringcoordinates,τandσ,with∂f∂ft=t(τ,σ),s=s(τ,σ),fτ=∂τf=,fσ=∂σf=,(10.26)∂τ∂σ88 andsettingT0tounity,weseethattheNambu–GotostringactionbecomesZpS=−(xτxσ)2−(xτ)2(xσ)2dτdσ.(10.27)UsingPµτandPµσtodenotethegeneralized‘momenta’where∂L∂LpPτ=,Pσ=,L=−(xx)2−(x)2(x)2.(10.28)µµµµτστσ∂xτ∂xσThentheequationsofmotionoftheNambu–Gotostringobtainedfromvaryingtheaction(10.27)maybewrittenintheformoftheconservationlaws∂Pτ∂Pσµµ+=0,µ=0,1,···,n,(10.29)∂τ∂σor,moreexplicitly,!!(xx)∂x−(x)2∂x(xx)∂x−(x)2∂xτσσµστµτστµσσµ∂τp+∂σp=0.(10.30)(xτxσ)2−(xτ)2(xσ)2(xτxσ)2−(xτ)2(xσ)210.3p-branesMoregenerally,consideranembedded(p+1)-dimensionalhypersurface,oramembrane,parametrizedbythecoordinates(ua)(a=0,1,···,p),sothattheinducedmetricelementisgivenby∂xµ∂xνds2=hduadub,h=η.(10.31)ababµν∂ua∂ubTheactionofap-braneisgivenbythevolumeintegralofthehypersurfaceasfollows,ZZpS=−TdV=−T|h|du0du1···dup,(10.32)pp+1pwhereTpisapositiveconstantreferredtoasthetensionofthep-brane.Inparticular,apointparticleisa0-brane,andastringisa1-brane.Wenowconsidera‘time-independentn-brane’,M,intheMinkowskispacetimeRn,1,whichmayberealizedasagraphofafunctiondependingonthespatialcoordinatesonlygivenbyx0=f(x1,···,xn).(10.33)Using(10.31),weseethatthemetrictensor(hij)ofMish=η∂xµ∂xν=∂f∂f−δ,i,j=1,···,n.(10.34)ijµνijijijItcanbecheckedthat|h|=|det(∂f∂f−δ)|=1−|∇f|2,(10.35)ijijwherewehaveassumedthatMisspacelike,|∇f|<1.Hence,ignoringthecouplingconstant,theactionofann-branewhichhappenstobeagraphisZpS=−1−|∇f|2dx,(10.36)89 whoseEuler–Lagrangeequationreads!∇f∇·p=0.(10.37)1−|∇f|2Awell-knowntheoremofChengandYau[29]statesthatallsolutionsof(10.37)overthefulln1nPnispaceRsatisfying|∇f|<1mustbeaffinelinear,f(x,···,x)=i=1aix+b,whereai’sandbareconstants.MinimalhypersurfacesItmaybeinstructivetocomparetheabovestudyofann-branewithitsEuclideanspacecounterpartwherewereplacetheMinkowskispacetimeRn,1withtheEuclideanspaceRn+1sothattheinheritedmetricoftheembeddedn-hypersurfaceMdefinedbythegraphofthefunction(10.33)isgivenbyh=δ∂xµ∂xν=∂f∂f+δ,i,j=1,···,n.(10.38)ijµνijijijConsequently,|h|=det(h)=1+|∇f|2,(10.39)ijsuchthatthecanonicalvolumeofthehypersurfaceMreadsZZppVM=|h|dx=1+|∇f|2dx.(10.40)Minimizing(10.40)givesustheclassicalequation!∇fn∇·p=0,x∈R,(10.41)1+|∇f|2knownastheminimalhypersurfaceequationfornon-parametricminimalhypersurfacesdefinedasthegraphofafunction.TheBernsteintheoremforthisequationstatesthatallentiresolutionsareaffinelinearforn≤7(cf.[101]andreferencestherein).ItwasCalabiwhofirstobservedthattheequations(10.37)and(10.41)areequivalent[24]whenn=2.Aproofofthisfactisasfollows.Letubeasolutionof(10.41)andp=∂1f,q=∂2f.(10.42)pSetw=1+p2+q2.Then(10.41)readspq∂1+∂2=0.(10.43)wwHence,thereisareal-valuedfunctionUsuchthatqp∂1U=P≡−,∂2U=Q≡.(10.44)wwTherefore,wehave2211−P−Q=>0(10.45)w290 andUisspace-like(i.e.,p|∇U|2<1).Insertingtherelationsp=Qw,q=−Pw,andw=1/WwhereW=1−P2−Q2intotheidentity∂2p=∂1q,wearriveatPQ∂1+∂2=0.(10.46)WWThus,Usolves(10.37).Theinversecorrespondencefrom(10.37)to(10.41)maybeestablishedsimilarly.TheaboveequivalencetheoremofCalabicanbeextendedintoarbitraryn-dimensionalset-tings[152]whichgiverisetoarichrangeofopenproblems.10.4ThePolyakovstring,conformalanomaly,andcriticaldimensionConsideramapφ:(M,{h})→(N,{g}),φ(u0,u1,···,um)=(x0,x1,···,xn),(10.47)abµνwhere(M,{hab})and(N,{gµν})are(m+1)-and(n+1)-dimensionalMinkowskimanifoldsparametrizedwiththecoordinates(ua)and(xµ),respectively.ThePolyakovactionissimplythe‘harmonicmap’functionaldefinedasZS=−(Dφ)2dV,(Dφ)2=ghab∂xµ∂xν,a,b=0,1,···,m,(10.48)hµνabpwheredVhisthecanonicalvolumeelementof(M,{hab})givenbydVh=|h|du0du1···dum,orcustomarily,forap-brane,ZZpS=LdV=−τ|det(h)|ghab∂xµ∂xνdu0du1···dup.(10.49)hpabµνabwherewehaveattachedtheconstantτp>0toaccountforthePolyakovp-branetension.Whenp=1,theaction(10.49)definesthePolyakovstring[110]actionwhichisconformallyinvariant(i.e.,theactionisinvariantundertheconformaltransformationofthemetric,hab7→Λhab).SuchaninvariancepropertyisalsocalledtheWeylinvariance.Theobviousadvantageoftheaction(10.49)over(10.32)isthattheformerisquadraticinxµ’sandgivesrisetolinearequationsofmotionforthebranes.Thus,thetheoryismucheasiertoquantize.Inthespecialcasewhen(M,{hab})isregardedasasubmanifoldofthespacetime(Rn,1,{ηµν})sothatthemetrichabisinducedfromthemap(10.47),wehaveh=η∂xµ∂xν,(10.50)abµνabwhichleadsustohabη∂xµ∂xν=p+1.(10.51)µνabConsequently,weseethatthePolyakovp-braneaction(10.49)reducesintotheNambu–Gotop-braneaction(10.32)whenTp=(p+1)τp.ThereisanotherpointofviewregardingtherelationshipbetweentheNambu–GotostringsandthePolyakovstrings:ExtremizingthemetrictensorhabinthePolyakovp-braneaction(10.49),weobtainafterneglectingtheconstantfactorτpthevanishingstresstensorconditionT=2g∂xµ∂xν−hL=0,∀a,b,(10.52)abµνabab91 whichgivesusthesolutionforhabasfollows,2g∂xµ∂xν2µνabhab=≡(∂ax·∂bx).(10.53)LLInserting(10.31)into(10.49),wehaveZ(p−1)p22S=−(p+1)τp|det(∂ax·∂bx)|du,(10.54)Lwhichisclearlyseentobecomeapurevolumeintegralexactlywhenp=1.Inotherwords,insuchacontext,theNambu–GotoandPolyakovstringactionsareequivalent.QuantizationofthePolyakovstringleadstothestringpartitionfunction[110]ZZ1122ϕ2Z=Dϕexp−(26−D)(∂aϕ)+κedu,(10.55)48π2whereD=n+1=26isthespacetimedimension,themetrichabisEuclideanizedintoeϕδabRthroughaWickrotationu07→iu0,κ>0isconstant,andDϕdenotesthepathintegraloverthespaceofallpossibleconformalexponents.Notethat,asalreadyobserved,thePolyakovstringactionisconformalinvariant.However,thepartitionfunction(10.55)clearlyspellsoutthefactthatsuchaconformalinvarianceisnolongervalidwhenD6=26.Suchaphenomenoniscalled‘conformalanomaly’andthevanishingofconformalanomalygivesustheuniqueconditionD=26,(10.56)knownasthecriticaldimensionofbosonicstringtheory.Furthermore,whenfermionsarepresent,Polyakov’scomputation[111]ofthepartitionfunctionofquantizedsupersymmetricstrings,orsuperstrings,givesustheuniqueconditionD=10,(10.57)toavoidconformalanomaly,again.Theseresultsaboutthecriticaldimensionsofspacetimearenowstandardfactsinstringtheory[155].Physicists[25,68,155]furtherconjecturedthatour10-dimensionaluniverse,M10,isaproductofa4-dimensionalspacetime,M4,anda6-dimensionalcompactmanifold,K6,curledupinatinybuthighlysophisticatedway,followingaformalismcalledthe‘stringcompactification’[37,53],sothatthespacetimeM4ismaximallysymmetric(whichimpliesthatMcaneitherbeMinkowski,deSitter,oranti-deSitter)15and4K6isaCalabi–Yaumanifold[22,23,153].16Notealsothat,theactionstemmingoutfrom(10.55),givenbyZ1a2ϕ2L=∂aϕ∂ϕ−κedu,(10.58)2andtheassociatedwaveequationϕ−ϕ=−κ2eϕ,u0=τ,u1=σ,(10.59)ττσσastheequationofmotion,arejointlyknowntodefinetheLiouvillefieldtheory[34,48],whichisintegrable[152]andofindependentinterestasatoymodel.15AmanifoldismaximallysymmetricifithasthesamenumberofsymmetriesasordinaryEuclideanspace.1Moreprecisely,aRiemannianmanifoldismaximallysymmetricifithasn(n+1)(n=dimensionofthemanifold)2linearlyindependentKillingvectorfieldswhichgenerateisometricflowsonthemanifold.16ACalabi–YaumanifoldisacompactK¨ahlermanifoldwithvanishingfirstChernclass.92 11TheBorn–InfeldgeometrictheoryofelectromagnetismAsanaturaldevelopmentofthetopicscoveredintheprevioussection,itwillbeinterestingtopresentanintroductiontotheBorn–Infeldtheory[11,12,13,14]formulatedinthe1933–1934andrevivedoverthelast20yearsduetoitsrelevanceinstringtheory.11.1FormalismRecallthatoneofthemajormotivationsfortheintroductionoftheBorn–Infeldelectromagneticfieldtheory[11,12,13,14]istoovercometheinfinityproblemassociatedwithapointchargesourceintheoriginalMaxwelltheory.Itisobservedthat,sincetheEinsteinmechanicsofspecialrelativitymaybeobtainedfromtheNewtonmechanicsbyreplacingtheclassicalactionfunctionL=1mv2bytherelativisticexpression2rrv21L=mc21−1−=b21−1−mv2,b2=mc2,(11.1)c2b2sothatnophysicalparticleofapositiverestmassmcanmoveataspeedvgreaterthanthespeedoflightc,itwillbeacceptabletoreplacetheactionfunctionoftheMaxwelltheory,122L=(E−B),(11.2)2whereEandBareelectricandmagneticfields,respectively,byacorrespondingexpressionoftheformrL=b21−1−1(E2−B2),(11.3)b2whereb>0isasuitablescalingparameter,oftencalledtheBorn–Infeldparameter.Itisclearthat(11.3)definesanonlineartheoryofelectromagnetismandtheMaxwelltheory,(11.2),mayberecoveredintheweakfieldlimitE,B→0.NotethatthechoiceofsigninfrontoftheLagrangiandensity(11.2)istheoppositeofthatofBornandInfeld[14]andiswidelyadoptedincontemporaryliterature.Thisconventionwillbeobservedthroughoutthenotes.Intrinsically,if(11.2)isreplacedbyL=−1FFµν,then(11.3)takestheform4µνr21µνL=b1−1+FµνF,(11.4)2b2whereFµν=∂µAν−∂νAµ(11.5)istheelectromagneticfieldstrengthcurvatureinducedfromagaugevectorpotentialAµ.Moreprecisely,ifweuseE=(E1,E2,E3),B=(B1,B2,B3)(11.6)todenotetheelectricandmagneticfields,respectively,asearlier,thenthereholdsthestandardidentificationF0i=−Ei,Fij=−ijkBk,i,j,k=1,2,3,(11.7)93 whichhasthefollowingmatrixform,0−E1−E2−E3µνE10−B3B2(F)=231.(11.8)EB0−BE3−B2B10ThedualofFµνreadsµνµν1µναβ∗F=F˜=Fαβ.(11.9)2From(11.5),thereholdsagaintheBianchiidentity∂F˜µν=0.(11.10)µOntheotherhand,itiseasytofindthattheEuler–Lagrangeequationsof(11.4)are∂Pµν=0,(11.11)µFµνPµν=q.(11.12)1+1FFαβ2b2αβCorrespondingtotheelectricfieldEandmagneticfieldB,weintroducetheelectricdisplacementDandmagneticintensityH,D=(D1,D2,D3),H=(H1,H2,H3),(11.13)andmaketheidentificationP0i=−Di,Pij=−ijkHk,i,j,k=1,2,3,(11.14)whichhasthefollowingmatrixform,0−D1−D2−D3µνD10−H3H2(P)=231.(11.15)DH0−HD3−H2H10Inserting(11.8)into(11.10)and(11.15)into(11.11),weobtainthefundamentalgoverningequationsoftheBorn–Infeldelectromagnetictheory,∂B+∇×E=0,∇·B=0,(11.16)∂t∂D−+∇×H=0,∇·D=0,(11.17)∂twhichlookexactlylikethevacuumMaxwellequations,exceptthat,inviewoftherelations(11.8),(11.12),and(11.15),thefieldsE,BandD,Harerelatednonlinearly,ED=q,(11.18)1+1(B2−E2)b2BH=q.(11.19)1+1(B2−E2)b294 Hence,theBorn–InfeldelectromagnetismintroducesE-andB-dependentdielectricsandper-meability‘coefficients’,D=ε(E,B)E,B=µ(E,B)H.(11.20)Ifthereisanexternalcurrentsource,(jµ)=(ρ,j),theequation(11.11)willbereplacedby∂Pµν=jν(11.21)µandequivalently,theequationsin(11.17)become∂D−+∇×H=j,∇·D=ρ,(11.22)∂tWenowexaminethepointchargeproblem.Considertheelectrostaticfieldgeneratedfromapointparticleofelectricchargeqplacedattheorigin.ThenB=0,H=0,andtheBorn–Infeldequationsbecomeasingleone,∇·D=4πqδ(x),(11.23)whichcanbesolvedtogiveusqxD=,(11.24)|x|3whichissingularattheorigin.However,from(11.18),wehaveED=q,(11.25)1−1E2b2whichimpliesthatDE=q1+1D2b2qx=s.(11.26)2|x||x|4+qbInparticular,theelectricfieldEisgloballybounded.Itisinterestingtoseethat,when|x|issufficientlylarge,Egivenin(11.26)approximatesthatgivenbytheCoulomblaw,aconsequenceoftheMaxwellequations.Asfortheenergy,weobtainfromtheLagrangedensity(11.4)theenergy-momentumtensorFγνFνqγµνTµ=−−δµL,(11.27)1+1FFαβ2b2αβwhichgivesusintheelectrostaticcasetheHamiltonianenergydensity021H=T0=bq−11−1E2b2r212=b1+D−1b2s22q1=b1+−1.(11.28)b|x|495 From(11.28),itisseenthatthetotalenergyofapointelectricchargeisnowfinite,ZE=Hdx<∞.(11.29)R3Similarly,wecanconsiderthemagnetostaticfieldgeneratedfromapointmagneticchargegplacedattheoriginofR3.Inthiscase,D=0,E=0,andtheBorn–Infeldequationsbecome∇·B=4πgδ(x).(11.30)From(11.30),wehaveasbefore,gxB=,(11.31)|x|3gxH=s.(11.32)2|x||x|4+gbThusHisaboundedvectorfield.Inviewof(11.27),theHamiltoniandensityofamagnetostaticfieldtakestheform,r212H=b1+B−1.(11.33)b2Inserting(11.31)into(11.33),weseethatthetotalenergyofapointmagneticchargeisalsofiniteintheBorn–Infeldtheory.Wenotethatitisnothardtoextendtheabovediscussiontocoverthesituationofmultiplydistributedpointelectricchargesormagneticmonopoles.11.2TheBorn–InfeldtheoryandageneralizedBernsteinproblemWenowstudysourcelessstaticsolutions.WithE=∇φandB=∇×A,theequationsofmotionoftheBorn–Infeldtheory,(11.16)and(11.17),become∇φ∇·q=0,(11.34)1+1(|∇×A|2−|∇φ|2)b2∇×A∇×q=0.(11.35)1+1(|∇×A|2−|∇φ|2)b2From(11.35),weseethatthereisarealscalarfunctionψsuchthat∇×Aq=∇ψ,(11.36)1+1(|∇×A|2−|∇φ|2)b2whichleadsustotherelation1|∇ψ|2|∇×A|2=1−|∇φ|2.(11.37)b21−1|∇ψ|2b296 Inserting(11.37)into(11.36),weobtainq1−1|∇φ|2b2∇×A=∇ψq.(11.38)1−1|∇ψ|2b2Inviewof(11.37)and(11.38),weseethatthestaticBorn–Infeldequations(11.34)and(11.35)areequivalenttothefollowingcoupledsystemoftwoscalarequations,q1−12|∇ψ|2b∇·∇φq=0,(11.39)1−1|∇φ|2b2q1−12|∇φ|2b∇·∇ψq=0.(11.40)1−1|∇ψ|2b2ThissystemgeneralizestheminimalormaximalhypersurfaceequationsandisthesystemoftheEuler–LagrangeequationsoftheactionfunctionalZrr11A(φ,ψ)=1−1−|∇φ|21−|∇ψ|2dx.(11.41)R3b2b2ItisinterestingtoaskwhethertheBernsteinpropertyholdsfor(11.39)and(11.40)orwhetheranyentiresolution(φ,ψ)oftheseequationsmustbesuchthatφandψareaffinelinearfunctions.However,itiseasilyseenthatanentiresolutionof(11.39)and(11.40)isnotnecessarilyaffinelinear.Infact,acounterexampleisobtainedbysettingφ=ψ=hintheseequationswherehisanarbitraryharmonicfunctionoverthefullspace.Thuswehaveyettoidentifythemostgeneraltrivialsolutionsofthesecoupledequations.Herewewillbesatisfiedwithaproofthatallsolutionsof(11.39)and(11.40)areconstantifthemagneticandelectricfieldsEandBdecaytotheirvacuumstatesE=0andB=0atinfinityatleastatarateoftheformO(r−1),whichisanaturalphysical(finiteenergy)requirement.Indeed,suchaconditionimpliesthatboth∇φand∇ψdecayatinfinityatthesamerate(see(11.36)).Therefore,φandψareboundedfunctions.ApplyingtheHarnackinequalityforstrictlyellipticequationsto(11.39)and(11.40)separately,weconcludethatφandψmustbeconstant.Therefore,wehaveprovedthefactthatE=0andB=0everywhereasexpected.Thisresultagainexcludesthepossibilityofaself-inducedstaticsolutioncarryingeitheranelectricormagneticcharge.ExtendingtheequivalencetheoremofCalabi[24]abouttheminimalandmaximalsurfaceequations,wecanshowthatthecoupledsystemofequations,(11.39)and(11.40),isequivalenttothesystemofequationsp1+|∇g|2∇·∇fp=0,(11.42)1+|∇f|2p1+|∇f|2∇·∇gp=0,(11.43)1+|∇g|2overR2,whichareseentobetheEuler–LagrangeequationsoftheactionfunctionalZppA(f,g)=1+|∇f|21+|∇g|2−1dx.(11.44)R297 Asa‘warm-upquestion’,weaskwhetherthesolutionsof(11.42)–(11.43)offiniteaction,A(f,g)<∞,areconstant.See[126]forsomefurtherdiscussion.11.3ChargeconfinementandnonlinearelectrostaticsInaseriesofworksofPagels–Tomboulis[102],Adler–Piran[2,3,4],andLehmann–Wu[73],theMaxwellequationsweremodifiedtogiverisetoelectricchargeconfinement,17whichmaybeviewedasprovidingatoymodeloflinearconfinement,whichsharesmanycommonfeatureswiththeBorn–Infeldtheorydiscussedabove.Thus,itwillbeinterestingheretodescribebrieflythedifferentialequationproblemsinvolved,inthecontextofthissection.Theelectricconfinementmodelstudiedin[2,3,4,73,102]isbasedonanelectrostaticframeworkwherethemagneticfieldisabsentandelectricfieldisstaticwhichisgeneratedfromachargedensity,ρ.ThustheMaxwellequationsare∇·D=ρ,(11.45)∇×E=0,(11.46)whereDandEaretheelectricdisplacementfieldandelectricfield,respectively.Thesetwoequationscontainnothingnewatthisstage.NowwesupplementtotheseequationswiththeconstitutiveequationD=ε(E)E,(11.47)whichrelatesDwithE,whereandinthesequel,E=|E|andD=|D|.Thus,(11.47)givesustherelationD=ε(E)E.(11.48)In[73],ageneralrelationbetweenEandDisspelledouttobeE=f(D).(11.49)Asaconsequence,wehaveD=f−1(E)=ε(E)E≡g(E).(11.50)Inviewof(11.46),wemayexpressEbyascalarpotential,Ψ,E=∇Ψ.(11.51)Inserting(11.51)into(11.47),wehaveD=ε(|∇Ψ|)∇Ψ.(11.52)Henceweseethatthelinearequation(11.45)isconvertedintothenonlinearPoissonequation∇·(ε(|∇Ψ|)∇Ψ)=ρ.(11.53)Ontheotherhand,from(11.50),wehaveDDg(E)g(|∇Ψ|)ε(E)====.(11.54)Ef(D)E|∇Ψ|17IamgratefultoElliottLiebforintroducingmetothesestudies.98 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