Multi-particle States from the Effective Action fo

arXiv:hep-th/9512113v219Dec1995Multi-particleStatesfromtheEffectiveActionforLocalCompositeOperators:AnharmonicOscillatorAnnaOkopi´nskaInstituteofPhysics,WarsawUniversity,BialystokBranch,Lipowa41,15-424Bialystok,PolandAbstractTheeffectiveactionforthelocalcompositeoperatorΦ2(x)inthescalarquantumfieldtheorywithλΦ4interactionisobtainedintheexpansionintwo-particle-point-irreducible(2PPI)diagramsuptofive-loops.Theeffectivepotentialand2-pointGreen’sfunctionsforele-mentaryandcompositefieldsarederived.Thegroundstateenergyaswellasone-andtwo-particleexcitationsarecalculatedforspace-timedimensionn=1,whenthetheoryisequivalenttothequantummechanicsofananharmonicoscillator.Theagreementwiththeexactspectrumoftheoscillatorismuchbetterthanthatobtainedwithintheperturbationtheory.1IntroductionTheformalismoftheeffectiveaction(EA)[1]isusuallyusedinrelativisticquantumfieldtheory;hovewertheformulationisuniversalandprovidesaneffectiveapproachtoanyquantumtheory.Hereweconsiderthetheoryofarealscalarfieldinn-dimensionalEuclideanspace-timewithaclassicalactiongivenbyS[Φ]=Z[12Φ(x)(−∂2+m2)Φ(x)+λΦ4(x)]dnx.(1)Thesimplestcaseofn=1dimensionalspace-time,whichisequivalenttothequantummechanicsoftheanharmonicoscillator(AO),isfrequentlyusedasatestinggroundforvariousfield-theoreticalmethods.HereweshalldiscussthemethodoftheEAforlocalcompositeoperatorswhichprovidesasys-tematicapproximationschemeforvacuumenergyandlowestmulti-particleexcitations.Weshallkeepthedimensionofthespace-timenarbitraryaslongaspossible,setingn=1onlyinthelaststage,wheretheenergiesarecalculated.TheconventionalEA,whichisageneratingfunctionalforone-particle-irreducible(1-PI)Green’sfunctions,isobtainedbyintroducingasourcecou-pledtothequantumfieldΦ(x).BycouplingexternalsourcestobilocalΦ(x)Φ(y)[2]andlocalΦ2(x)[3]fields,thegeneratingfunctionalsforcom-positeoperatorsaredefined(forreviewseeRef.[4]).Foraninteractingtheorytheexactformofanyfunctionalisnotknown,sooneresortstoapproxima-tions.Formulatinganapproximationschemeforageneratingfunctional,aconsistentsetofapproximateGreen’sfunctionscanbeobtainedthroughdifferentiation.Thisiscrucialforarelativisticquantumfieldtheory,wheretheprocessofrenormalizationhastobeperformed.Anyfunctional,ifcal-culatedexactly,containsthesameinformationandgivesthesameresultforaphysicalquantity;however,thesameapproximationschemeforvariousfunctionalswouldresultindifferentapproximationsofGreen’sfunctionsandobservables.Therefore,anappropriatechoiceofageneratingfunctionalforcalculatingaquantityofinterestisimportant.TheconventionalEAisusedfordiscussingavacuumstructureandone-particleexcitations.Forasimul-taneousstudyoftwo-particleexcitations,theEAforcompositeoperatorsismoresuitable,sinceitdeterminestheconventionalEA(byeliminatingtheexpectationvaluesofcompositeoperators)andgeneratesGreen’sfunctionsrelateddirectlytoone-andtwo-particleeigenmodes.2Generatingfunctionalscanbecalculatedintheloopexpansion.Thecon-ventionalEA,Γ[ϕ],isgivenbyasumofone-particleirreducibilevacuumdiagrams[1].TheEAforthebilocalcompositeoperator,Γ[ϕ(x),G(x,y)]isgivenbytwo-particle-irreducibile(2PI)diagrams[2].Theone-loopresult,aftereliminatingthefullpropagatorG(x,y),givestheGaussianapproxima-tionfortheconventionalEA;however,beyondone-loopthegapequationforG(x,y)isahighlynon-trivialintegralequation.TheEAforthelocalcom-positeoperator,Γ[ϕ(x),Δ(x)],canbealsoobtaineddiagramaticaly[5,6]andtheone-loopresultgivestheGaussianapproximation.Calculationsofpost-Gaussiancorrectionsareeasierinthisapproach,sincethegapequationforthevacuumexpectationvalueofthelocalcompositefieldisalgebraic.ThevacuumfunctionalforthecompositeoperatorΦ2(x)isrepresentedbyapathintegralZ[J1,J2]=eW[J1,J2]=ZDΦe−S[Φ]+RJ1(x)Φ(x)dnx+12RJ2(x)Φ2(x)dnx(2)andtheEAisobtainedasaLegendretransformΓ[ϕ,Δ]=W[J1,J2]−ZJ1(x)ϕ(x)dnx−12ZJ2(x)(ϕ2(x)+Δ(x))dnx,(3)whereδWδJ1(x)=ϕ(x),andδWδJ2(x)=12(ϕ2(x)+Δ(x))(4)determinetheexpectationvaluesofthefieldsΦandΦ2,inthepresenceofexternalcurrentsJ1andJ2.TheEAfulfilsδΓδϕ(x)=−J1(x)−J2(x)ϕ(x),(5)andδΓδΔ(x)=−12J2(x).(6)SettingJ1=J2=0,whichreproducesthephysicaltheory(1),resultsinvariationalequationsδΓδϕ(x)=0(7)3andδΓδΔ(x)=0.(8)Theseequationsdeterminethevacuumexpectationvaluesϕ0andΔ0,whicharespace-timeindependent,bytranslationalinvariance.TheconventionalEAcanbeobtainedasΓ[ϕ]=Γ[ϕ,Δ0]withΔ0[ϕ]determinedbyinvertingthegapequation(8).Theeffectivepotential(EP),definedbyV(φ)=−Γ[ϕ]|ϕ(x)=φ=constRdnx,(9)givesthevacuumenergydensityV(ϕ0).Green’sfunctions,generatedfromtheEAforlocalcompositeoperators,provideaconvenienttooltostudymulti-particlestates,sincetheirzeromodesgivedirectlytheexcitationenergiesabovethegroundstate.One-particleeigenmodeisdeterminedbythe2-pointGreen’sfunctionfortheelementaryfieldΓ2(x−y)=δ2Γδϕ(x)δϕ(y)ϕ(x)=ϕ0,Δ(x)=Δ0,(10)whichisaninverseofthefullpropagatorW2(x−y)=TΦ(x)Φ(y)connected.(11)Anappropriatefunctiontostudytwo-particleexcitationisthe2-pointGreen’sfunctionforthecompositefieldΓ4(x−y)=δ2ΓδΔ(x)δΔ(y)ϕ(x)=ϕ0,Δ(x)=Δ0(12)whichisaninverseofthefunctionW4(x−y)=TΦ2(x)Φ2(y)connected,(13)calledpolarisation(densityfluctuaction)propagatorinmany-bodyphysi