Quantum field theory as eigenvalue problem

arXiv:gr-qc/0303037v110Mar2003QuantumfieldtheoryaseigenvalueproblemArnoldNeumaierInstitutf¨urMathematik,Universit¨atWienStrudlhofgasse4,A-1090Wien,Austriaemail:[e-mail]://∼neum/March10,2003Abstract.Amathematicallywell-defined,manifestlycovarianttheoryofclassicalandquan-tumfieldisgiven,basedonEuclideanPoissonalgebrasandageneralizationoftheEhrenfestequation,whichimpliesthestationaryactionprinciple.Thetheoryopensaconstructivespec-tralapproachtofindingphysicalstatesbothinrelativisticquantumfieldtheoriesandforflexiblephenomenologicalfew-particleapproximations.Inparticular,weobtainaLorentz-covariantphenomenologicalmultiparticlequantumdynam-icsforelectromagneticandgravitationalinteractionwhichprovidesarepresentationofthePoincar´egroupwithoutnegativeenergystates.Thedynamicsreducesinthenonrelativis-ticlimittothetraditionalHamiltonianmultiparticledescriptionwithstandardNewtonandCoulombforces.ThekeythatallowsustoovercomethetraditionalproblemsincanonicalquantizationisthefactthatweusethealgebraoflinearoperatorsonaspaceofwavefunctionsslightlybiggerthantraditionalFockspaces.Keywords:action,axiomaticphysics,classicalfield,classicallimit,conservativequantumstate,constrainedSchr¨odingerequation,covariantinteraction,deformationquantization,density,multiparticleDiracequation,Ehrenfestequation,eigenvalueproblem,electrodynamics,EuclideanPoissonalgebra,expectation,formfactors,gravitation,Hamiltoniandynamics,integral,Liouvilleequation,manifestlycovarianttheory,momentumstate,noncommutativealgebra,nonrelativisticlimit,phasespacequantization,phenomenologicalrelativisticdynamics,physicalsystem,quantumfieldtheory,quantumgravity,relativisticCoulombpotential,relativisticNewtonpotential,runningcouplingconstants,self-energy,stationaryactionprinciple,Wightmanaxioms,WignertransformE-printArchiveNo.:gr-qc/03030372003PACSClassification:03.70,04.60.-m,11.10.Cd,11.10.Ef2000MSCClassification:primary81T05,secondary70S05,81S30,81V10,81V1711Introduction...theancients(aswearetoldbyPappus)esteemedthescienceofmechanicsofgreatestimportanceintheinvestigationofnaturalthings,andthemoderns,rejectingsubstantialformsandoccultqualities,haveendeavoredtosubjectthephenomenaofnaturetothelawsofmathematics...IsaacNewton,1686[30]Renormalizedquantumelectrodynamicsisbyfarthemostsuccessfultheorywehavetoday.Thisveryimpressivefact,however,doesnotmakethewholesituationlessstrange.Westartoutfromequationswhichdonotmakesense.Weapplycertainprescriptionstotheirsolutionsandendupwithapowerseriesofwhichwedonotknowthatitmakessense.Thefirstfewtermsofthisseries,however,givethebestpredictionsweknow.ResJost,1965[16]Inthemorethan300yearsthatpassedsinceNewtonwrotethisinhisPrincipiaMathe-matica,themodernshavebeenverysuccessfulattheendeavortosubjectthephenomenaofnaturetothelawsofmathematics–withexceptionofquantumfieldtheory.Asthesecondquote(whichcouldhaveaswellbeenwrittenin2002)shows,quantumfieldtheorysofarresistedaquantitative,mathematicallyrigorousfoundation.Inthepresentpaper,anaxiomaticapproachisoutlinedthat,Ibelieve,providesfoun-dationsonwhichquantumfieldtheorycanbegivenarigorousmathematicaltreatment.Thepresentpapergivestheelementarypartandexhibitstheconnectionstothetradi-tionalsettings.Adeeperstudyoftheconsequencesanduseoftheconceptspresentedherewillbegivenelsewhere.Inthenewapproach,each(classicalorquantum)conservativephysicalsystemischar-acterizedbytwoHermitianquantities:adensityandanaction.AgeneralizedLiouvilleequationdefinesthedynamicsandimpliesEhrenfestequationsforexpectations.Foraclassical(butnotaquantum)fieldtheory,theEhrenfestequationsinasymplecticPoissonalgebraimplythetraditionalfieldequationsbythestationaryactionprinciple.Inparticular,alltraditionalsystemsderivablefromthestationaryactionprinciplecanbemodelledinoursetting.Inasimilarway,onecangetfromsuitableLie-PoissonalgebrasrelativisticandnonrelativisticEulerequations,Vlasov-Maxwell,Vlasov-Einstein,andEuler-Poincar´eequations.Anewphasespacequantizationprinciple(generalizingtheWignertransform)allowsthesimplequantizationofarbitraryPoissonalgebras,withagoodclassicallimit.AlargeclassofPoincar´einvariantactionsonspaceswithareduciblerepresentationofthePoincar´egroupisexhibited.SinceitismanifestlycovariantbutpossessesaHamil-2toniannonrelativisticlimit,itappearstobewell-suitedforphenomenologicalmodelingofrelativisticfew-particledynamics.Inparticular,weobtainaLorentz-covariantphenomenologicalmultiparticlequantumdynamicsforelectromagneticandgravitationalinteractionwhichreducesinthenon-relativisticlimittothetraditionalHamiltonianmultiparticledescriptionwithstandardNewtonandCoulombforces.Thekeythatallowsustoovercomethetraditionalprob-lemsincanonicalquantizationisthefactthatweusethealgebraoflinearoperatorsonaspaceofwavefunctionsslightlybiggerthantraditionalFockspaces.Foraquantumsystem,iftheactionistranslationinvariant,onecanfindpurestatesofgivenmassdescribingisolatedsystemsinarestframebysolvingaconstrainedSchr¨odingerequation.Thisopensaconstructivespectralapproachtofindingphysicalstatesbothinrelativisticquantu