Covariant EBK quantization of the electromagnetic

arXiv:physics/0405065v1[physics.class-ph]14May2004CovariantEBKquantizationoftheelectromagnetictwo-bodyproblemJaymeDeLuca∗UniversidadeFederaldeS˜aoCarlos,DepartamentodeF´ısicaRodoviaWashingtonLuis,km235CaixaPostal676,S˜aoCarlos,S˜aoPaulo13565-905(Dated:February2,2008)AbstractWediscussamethodtotransformthecovariantFokkeractionintoanimplicittwo-degree-of-freedomHamiltonianfortheelectromagnetictwo-bodyproblemwitharbitrarymasses.Thisdynamicalsystemappeared100yearsagoanditwaspopularizedinthe1940’sbythestillin-completeWheelerandFeynmanprogramtoquantizeitasameanstoovercomethedivergenciesofperturbativeQED.Ourfinite-dimensionalimplicitHamiltonianisclosedandinvolvesnoseriesexpansions.TheHamiltonianformalismisthenusedtomotivateanEBKquantizationbasedontheclassicaltrajectorieswithanon-perturbativeformulathatpredictsenergiesfreeofinfinities.∗correspondingauthor;emailaddress:[e-mail]ltoniandescriptionoftwo-bodydynamicsissurprisinglyrestrictivewithinrelativityphysics:IfLorentztransformationsaretoberepresentedbycanonicaltransfor-mations,onlynon-interactingtwo-bodymotioncanbedescribed.Thisisthecontentoftheno-interactiontheoremof1964,whichlaterin1984wasprovedfortheusualLagrangiande-scriptionaswell[1].AcovariantversionofHamiltoniandynamics,constraintdynamics,wasinventedtoovercomethisgroup-theoreticalobstacle,butthistoohaslimitedapplicability[2].Inthisworkwestartfromarelativisticphysicaltheory:thetime-symmetricrelativisticaction-at-a-distanceelectrodynamics,forwhichnoconstraintdescriptionexistsatpresent.WediscussthepassagefromthecovariantFokkerLagrangiantoanimplicitHamiltonianformalismdefinedinclosedformwithoutpowerexpansions.Wederiveatwo-degreeoffree-domHamiltonianfortheelectromagnetictwo-bodyproblem[3]witharbitrarymassesandwitheitherrepulsiveorattractiveinteraction.InourdescriptionaLorentztransformationisrepresentedbyacanonicaltransformationfollowedbyarescalingoftheevolutionparam-eter.WeusethiscovariantHamiltonianformalismtomotivateanEBKquantizationoftheelectromagnetictwo-bodyproblemthatusestheclassicaltrajectoriestoapproximatethequantumenergieswithaformulathatisnon-perturbativeandfreeofinfinitequantities.In1903,Schwarzchildproposedarelativisticinteractionbetweenchargesthatwastimereversiblepreciselybecauseitinvolvedretardedandadvancedinteractionssymmetrically[4].Thesamemodelreappearedinthe1920sintheworkofTetrodeandFokker[5]anditfinallybecameelectromagnetictheoryafterWheelerandFeynmanshowedthatthisdirect-interactiontheorycandescribealltheclassicalelectromagneticphenomena(i.e.theclassicallawsofCoulomb,Faraday,Amp`ere,andBiot-Savart)[6,7].Inparticular,WheelerandFeynmanshowedin1945thatinthelimitwheretheelectroninteractswithacompletelyabsorbinguniverse,theresponseofthisuniversetotheelectron’sfieldisequivalenttothelocalLorentz-Diracself-interactiontheory[8]withouttheneedofmassrenormalization[6].Completeabsorptionisaddedtothetheoryasanapproximationtouncoupleawayfromthedetailedneutral-delaydynamicsoftheotherchargesoftheuniverse.ForotherapproximationsseeRef.[9].TheWheelerandFeynmanprogram[10]toquantizetheaction-at-distanceelectrodynamicsandovercometheinfinitiesofQEDisstillnotimplementedbecauseofthelackofaHamiltoniandescription,whichwasamotivationforthepresent2work.Therelativisticaction-at-a-distanceelectrodynamicsisbetterregardedasamany-bodyelectromagnetictheory,onceitisbasedonaparametrization-independentactioninvolvingtwo-bodyinteractionsonly,withoutthemediationbyfields[6,7].Theisolatedelectromag-netictwo-bodyproblem,awayfromtheotherchargesoftheuniverse,isatime-reversibledynamicalsystemdefinedbytheactionSF=−Zm1ds1−Zm2ds2−e1e2ZZδ(||x1−x2||2)˙x1·˙x2ds1ds2,(1)wherexi(si)representsthefour-positionofparticlei=1,2parametrizedbyitsarc-lengthsi,doublebarsstandforthefour-vectormodulus||x1−x2||2≡(x1−x2)·(x1−x2),andthedotindicatestheMinkowskiscalarproductoffour-vectorswiththemetrictensorgμν(g00=1,g11=g22=g33=−1)(thespeedoflightisc=1).TheintegrationinEq.(1)cannotberestrictedtoasegmentoftrajectorygoingfromaninitialtoafinaltime,becauseLagrangian(1)attheend-pointsinvolvesthefutureandpastlight-cones,whichareoutofthesegment.ThisdifferencefromourPoincar´e-invariantLagrangiantotheusualHamilton’sprincipleofGalilei-invariantmechanicsdemandsthataction(1)bedefinedbythewholeorbit.AsensiblemathematicalvariationofEq.(1)isalongtrajectoryvariationsδxi(t)thatvanishatbotht=±∞,suchthatδSFisasensiblefinitequantitytobeminimized.Thisdifferenceisdiscussedin[11],andweshallhenceforthignoreitandderiveourequationsofmotionformallybyextremizingaction(1).Theattractivetwo-bodyproblemisdefinedbyEq.(1)withe1=−e2≡e(Hydrogenatom),whiletherepulsiveproblemisdefinedbyEq.(1)withe1=e2≡e.Fortheelectromagnetictwo-bodyproblemwitharbitrarymasses(Eq.(1)),theonlyknownanalyticalsolutionisthecircularorbitfortheattractiveproblem[12,13].AcomprehensivediscussionofliteratureontheFokkerLagrangianisgiveninRef.[3].Someexistenceanduniquenessresultsforthistwo-bodysystemarereviewedinAppendixI.Thispaperisdividedasfollows:InSectionIIweconstructaHamiltonianthatisgeneralenoughtodescrib