On the problem of interactions in quantum theory

arXiv:quant-ph/9805052v118May1998ONTHEPROBLEMOFINTERACTIONSINQUANTUMTHEORYFelixM.LevLaboratoryofNuclearProblems,JointInstituteforNuclearResearch,Dubna,Moscowregion141980Russia(E-mail:[e-mail])AbstractThestructureofrepresentationsdescribingsystemsoffreeparti-clesinthetheorywiththeinvariancegroupSO(1,4)isinvestigated.Thepropertyoftheparticlestobefreemeansasusualthattherep-resentationdescribingamany-particlesystemisthetensorproductofthecorrespondingsingle-particlerepresentations(i.e.nointeractionisintroduced).Itisshownthatthemassoperatorcontainsonlycon-tinuousspectrumintheinterval(−∞,∞)andsuchrepresentationsareunitarilyequivalenttoonesdescribinginteractions(gravitational,electromagneticetc.).ThismeansthattherearenoboundstatesinthetheoryandtheHilbertspaceofthemany-particlesystemcontainsasubspaceofstateswiththefollowingproperty:theactionoffreerepresentationoperatorsonthesestatesismanifestedintheformofdifferentinteractions.Possibleconsequencesoftheresultsaredis-cussed.PACS:03,65Bz,04.62.+v,11.30-j.1GeneralremarksonquantumtheoriesTheexistingquantumtheoriesareusuallybasedonthefollowingprocedure:theLagrangianofthesystemunderconsiderationiswrittenasL=Lm+Lg+LintwhereLmistheLagrangian1of”matter”,LgistheLagrangianofgaugefieldsandLintistheinteractionLagrangian.ThesymmetryconditionsdonotdefineLintuniquelysinceatleasttheinteractionconstantisarbitrary.NeverthelesssuchanapproachhasturnedouttobehighlysuccessfulinQED,electroweaktheoryandQCD.Atthesametimethedifficultiesinconstructingquantumgravityhavenotbeenovercome.Thepopularideaexpressedbymanyphysicistsisthatsuchnotionsasmatterandinteractionsarenotfundamental—theyareonlymanifestationsofsomepropertiesofspace-time(seee.g.Ref.[1]).Ontheotherhandtheproblemariseswhatisthemeaningofspace-timeonthequantumlevel.Indeed,thereisnooperatorcorrespondingtotime(seee.g.thediscussioninRef.[2])andthelatterisconsideredonlyasaclassicalquantity.Moreover,ithasbecomeclearalreadyin30ththatinrelativisticquantumtheorythereisnooperatorhavingallthepropertiesofthecoordinateoperator(seee.g.Ref.[3]).Asaconsequence,thequantityxintheLagrangiandensityL(x)isnotthecoordinateinMinkowskispacebutsomeparameterwhichbecomesthecoordinateonlyintheclassicallimit.OnecanconsidertheLagrangianonlyasanauxiliarytoolforconstructingallthegeneratorsofthesymmetrygroupintheframeworkofthecanonicalformalism(seee.g.Ref.[4]).Never-thelessthepracticalrealizationofsuchanapproachencountersseriousmathematicalproblems.Oneofthereasonisthattheinteractingfieldoperatorscanbetreatedonlyasoperatorval-ueddistributions[5]andthereforetheproductoftwolocalfieldoperatorsatcoincidingpointsisnotwelldefined.Theproblemofthecorrectdefinitionofsuchproductsisknownastheprob-lemofconstructingcompositeoperators(seee.g.Ref.[6]).So2farthisproblemhasbeensolvedonlyintheframeworkofper-turbationtheoryforspecialmodels.Whenperturbationtheorydoesnotapplytheusualprescriptionsaretoseparatetheargu-mentsoftheoperatorsinquestionandtodefinethecompositeoperatorasalimitofnonlocaloperatorswhentheseparationgoestozero(seee.g.Ref.[7]andreferencestherein).HowevertheLagrangiancontainsproductsoflocaloperatorsatthesamepoint.Ifoneseparatestheargumentsoftheseoperators,theLagrangianimmediatelybecomesnonlocalanditisnotclearhowtousetheNoetherprocedureinthiscase.Asaconsequence,thegeneratorsconstructedinastandardwayareusuallynotwell-defined(seee.g.Ref.[8])andthethe-orycontainsanomalies.Itisalsoworthnotingthatthereexiststhewell-knownparadox:ontheonehandtherenormalizableperturbationtheory,whichisformallybasedoninteractionpic-ture,isinbeautifulagreementwithmanyexperimentaldata,whileontheotherhand,accordingtotheHaagtheorem[9,5],theinteractionpicturedoesnotexist.Oneoftheadvantagesofsuperstringtheoriesisthattheyinvolveproductsofoperatorsatdifferentpointsandthereexisttheorieswithoutanomalies[10].AtthesametimethetheoriesinvolvesuchnotionsasLagrangian,interactionandperturbationtheory.Inchiraltheories(seee.g.Ref.[11]andreferencestherein)theinteractionLagrangianisusuallynotintroducedbutthefieldshavetherangeinsomenonlinearmanifold.Thereexistseriousproblemsinquantizingsuchtheories.Theaboveremarksshowthatatpresentphysicistshavenoclearanswertothequestionswhetherinquantumtheorythenotionsofspace-time,Lagrangianandinteractionareprimary3ortheyaremanifestationsofsomedeeperreasons.AccordingtotheHeisenbergprogram(whichwasverypopularin60th)suchnotionsarerudimentswhichwillnotsurviveinthefuturetheory(seee.g.thediscussioninRef.[12]).Accordingtothisprogram,theonlyfundamentaloperatoristheS-operatordefinedontheHilbertspaceofelementaryparticleswhileboundstatesaremanifestedonlyaspolesinthematrixelementsofthisoperator.Itisalsointerestingtoposethefollowingproblem.IsitpossiblethattheultimatephysicaltheoryisfullydefinedbythechoiceofthesymmetrygroupandthereisnoneedtointroducetheLagrangianandinteractions?Inthepresentpaperweconsideramodelwhichinouropinioncanshedlightonthisproblem.WechoosethedeSittergroupSO(1,4)(morepreciselyitscoveringgroupSO(1,4))asthesym-metrygroup.Werequireasusualthattheelementaryparticlesaredescribedbyunitaryirreduc