Covariant Hamiltonian Dynamics with Negative Energ

arXiv:0705.3074v1[nucl-th]22May2007CovariantHamiltonianDynamicswithNegativeEnergyStatesM.DeSanctisa,baDepartamentodeF´ısica,UniversidadNacionaldeColombia,Bogot´aD.C.,Colombia.bINFNsez.diRoma,P.leA.Moro2,00185Roma,Italy.e-mail:[e-mail]@roma1.infn.itAbstractArelativisticquantummechanicsisstudiedforboundhadronicsys-temsintheframeworkofthePointFormRelativisticHamiltonianDynamics.NegativeenergystatesareintroducedtakingintoaccounttherestrictionsimposedbyacorrectdefinitionofthePoincar´egroupgenerators.Weobtainnonpathological,manifestlycovariantwaveequationsthatdynamicallycontainthecontributionsofthenegativeenergystates.Auxiliarynegativeenergystatesarealsointroduced,speciallyforstudyingtheinteractionsofthehadronicsystemswithexternalprobes.PACSnumber(s):11.30.Cp,24.10.Jv,03.65.Pm1.IntroductionThestudyofhadronicfew-bodysystemsintermsofconstituentparticlesrep-resentsaveryimportanttoolforthedeterminationoftheirintrinsicproper-tiesandtheirinteractionswithexternalprobes.WeusethetermconstituentparticletomeanasystemthattransformsasanirreduciblerepresentationofthePoincar´egroupwithdefinitemass,spinandinternalsymmetryquantumnumber.Inthisrespectwerecallthathistoricallytheinvestigationstartedwiththestudyoflightnucleiintermsofconstituentnucleons.Later,manyeffortshavebeendevotedtothestudyofthehadronsintermsofconstituentquarks.1Forbothcasesnonrelativisticmodelswereinitiallyconsidered,lateron,rel-ativisticcorrectionswereadded,improvingthereproductionoftheexperi-mentaldata.Nowadays,theconstructionofhadroniccovariantconstituentmodelsmustbeconsideredstrictlynecessaryforanaccuratedescriptionofthesesystemsandforthestudyoftheirinteractionswithelectroweakprobes.Inpartic-ular,thepresentwork,thatrepresentsageneralizationoftheRelativisticHamiltonianDynamics,isfocussedonthestudyofquark(andquark-di-quark)models,butitsformaldevelopmentscanbealsoappliedtothestudyoffew-bodynuclearsystems.Asfortherelativisticcovariantquarkmodels,theyrepresenteffectivemodelsthatshouldberelatedtoquantumchromodynamics(QCD),incorporatinginanonperturbativewayitssymmetriesanddynamicalfeatures.Ontheotherhand,thenuclearmodelsrelyonaphenomenologicalmesonexchangetheory.1.1GeneralremarksFromatheoreticalpointofwiew,wenotethat,forbuildingfew-bodyrela-tivisticmodels,twoslightlydifferentapproachescanbefollowed.Thefirstone,isdenominatedRelativisticHamiltonianDynamics(RHD).ThesecondoneisrepresentedbytheRelativisticWaveEquations(RWE).Theaimofthefirstapproach,thatistheRHD,istosatisfythePoincar´eGroupcommutationrulesbydefiningthegeneratorsofthatgroupintermsoftheconstituentparticleoperators.WenowsynteticallyrecallsometechnicalaspectsrelatedtoRHD.InthecaseoffreeparticlesthetotalgeneratorsofthePoincar´egrouparegivenbythesumofthesingleparticlegenerators.ThemainproblemofRHD,notfoundinthenonrelativisticcase,consistsinfulfillingthePoincar´egroupcommutationruleswhentheinteractionisintroducedinthegenerators.Inthisrespect,RHDcanbeformulatedindifferentways,threeofwhichhavebeenintroducedinthepionieristicworkbyDirac[1].TheyarecalledtheInstantForm(IF),theFrontForm(FF)andthePointForm(PF).Themostrelevantdifferenceamongthemisrepresentedbythewayinwhichthegeneratorsdependontheinteraction.IntheIFboththeHamiltonian,i.e.thetimetranslationgenerator,and2theboostgeneratorsaremodified(withrespecttothefreecase)bytheinteraction[2-4].IntheFF,linearcombinationsofthecomponentsofthefour-vectorsareconsideredand,inconsequence,notstandardLorentztransformationsmustbeintroduced.InthisformofRHD,theinteractionmodifiespiecesofboththeboostandofthefour-momentum[3].Finally,inthePF[5-7],thatisthescheemeadoptedinthepresentwork,theinteractionmodifiesthetotalfour-momentum,i.e.theHamiltonianandthethree-momentumofthesystem,thataregivennotonlybythesumofthefour-momentaofthecostituents,butalsoreceiveacontributionthatdependsontheinteractionoperator.Ontheotherhand,theboostisleftfreefrominteraction.Duetothislastproperty,PFRHDhasbeendefinedasmanifestlycovariant[5].Thispointwillbeanalyzedinthenextsubsect.1.2,consideringthedefinitionofmanifestcovarianceadoptedinthepresentwork.Theoretically,therelevantaspectofRHDisthatitsquantummechanicalpropertiesarewelldefined,inanalogytononrelativisticcase.AllthegeneratorsofthePoincar´ealgebraarerepresentedbyhermiticop-eratorssothatthecorrespondingtransformationsareperformedbyunitaryoperators,satisfyingPoincar´ecovarianceandallowing,atthesametime,tousethestandardquantummechanicalproceduresforthecalculationofphys-icalobservables.Theinteractionoperatorisgivenbyaquasipotentialthatis,ingeneral,momentumdependent.NotethatsomeaspectsthataretypicalofquantumfieldtheorieshavebeencompletelyexcludedinRHD:inparticular,thepossibilityofcreatingorde-stroyingparticlesandthepresenceofnegativeenergystatesintheinteractionamplitudes.Inthisworkweshallanalyzeandovercomethissecondproblem.ThequarkmodelsbasedonRHDreproducesomegeneralhadronicfeaturesrelatedtoQCD,likecolourglobalsymmetry,confinementandisospininvari-ancefortheudquarksector.Furthermore,inallthethreeformsofRHDveryencouragingquantitativecalculationshave