Variational Master Field for Large-N Interacting M

arXiv:hep-th/9609216v127Sep1996VariationalMasterFieldforLarge-NInteractingMatrixModels–FreeRandomVariablesonTrialM.Engelhardt1,2,∗,†andS.Levit11DepartmentofCondensedMatterPhysicsWeizmannInstituteofScienceRehovot76100,Israel2Institutf¨urtheoretischePhysikIIIUniversit¨atErlangenStaudtstr.7,91058Erlangen,Germany()Matricesaresaidtobehaveasfreenon-commutingrandomvariablesiftheactionwhichgovernstheirdynamicsconstrainsonlytheireigenvalues,i.e.dependsontracesofpowersofindividualma-trices.Theauthorsuserecentlydevelopedmathematicaltechniquesincombinationwithastandardvariationalprincipletoformulateanewvariationalapproachformatrixmodels.Approximatevaria-tionalsolutionsofinteractinglarge-Nmatrixmodelsarefoundusingthefreerandommatricesasthevariationalspace.SeveralclassesofclassicalandquantummechanicalmatrixmodelswithdifferenttypesofinteractionsareconsideredandthevariationalsolutionscomparedwithexactMonteCarloandanalyticalresults.Impressiveagreementisfoundinamajorityofcases.PACS:02.10.Sp,11.15.Pg,11.80.Fv,12.38.-tKeywords:Matrixmodels,large-Nlimit,variationalmethodI.INTRODUCTIONThemainmotivationbehindthisworklayindevelopingapproximateanalyticaltoolsfordealingwiththelongrangephysicsofQuantumChromodynamics.Thelarge-Ncapproximationisanattractivecandidateforthisproject.Initiatedby’tHooft,Ref.[1],itwasimmediatelyappliedtosolvelarge-Nc1+1dimensionalQCD,Refs.[2],[3],[4],[5]andledtoextremelyfruitfulphenomenologicalinsightsandmodels,Refs.[3],[6],[7].Thefactorizationpropertyofcorrelatorsofgaugeinvariantoperatorssuggestedtheideaofthemasterfield–thegaugefieldconfigurationwhichdominatesthepathintegralinthelarge-Nclimit,cf.Refs.[3],[6],[8].Knowledgeofthemasterfieldshouldallowtocalculateanygaugeinvariantobservableasifitwereaclassicalnon-fluctuatingobject.Aconcreteexampleofamasterfieldwasprovidedbytheexactsolutionofamodelwhichdescribedthedynamics(classicalaswellasquantummechanical)ofasinglehermitianN×Nmatrix,Ref.[9].Itwasfoundthatinthelarge-NlimittheBoltzmannintegral(intheclassicalcase)andthegroundstateproperties(inthequantumcase)ofthismodelwerecompletelydeterminedbyanensembleofmatriceswithafrozendistributionofeigenvaluesgivenbyasolutionofacertainintegralequationofthemeanfieldtype.The“angular”orientationsofthematricesinthisensemble,i.e.theunitarymatricesWinthedecompositionM=WmW+withm=diag(m1,m2,...,mN)werecompletelyfree,i.e.distributedwithequalprobabilityaccordingtotheunbiasedU(N)grouptheoreticalHaarmeasure.Thisworkand,initssequel,Ref.[10]gaverisetoanintenseinterestinsolutionsoflarge-Nmatrixmodelsandmadecontactwiththestudyofsuchmodelsinotherfieldsofphysics,mostnotablynuclearphysics,cf.Ref.[11]and,morerecently,two-dimensionalquantumgravity,cf.Ref.[12]andquantumchaos,cf.Ref.[13].Thegeneralinterestinlarge-Nmatrixmodelshasbeenbehindasteadyprogressinunderstandingsolutionsofvariousversionsandgeneralizationsofsuchmodelsasreportede.g.inRefs.[14],[15],[16],[17],[18].Anewdirectioninthisactivityopenedupwhenthephysicscommunitybecameacquaintedwithrecentadvancesinthemathematicalliterature,associatedwiththeworkofVoiculescu,Ref.[19],onnon-commutativeprobabilitytheory,Ref.[20].Thestatisticalmechanics(classicalorquantummechanical)ofmatrix(non-commuting)variablesor(gauge)fieldsisanexampleofprobabilitytheoryofnon-commutativevariables.Aconceptwhichturnedouttobeespeciallyusefulforthestudyoflarge-Nmatrixandgaugetheories,cf.Refs.[21]and[22],wastheintroductioninRef.[19]oftheso-calledfreerandomvariables.Thesimplestexamplesofsuchvariablesoccurinindependent∗Addressfrom1.10.96:Institutf¨urtheoretischePhysik,Universit¨atT¨ubingen,AufderMorgenstelle14,72076T¨ubingen,Germany.†SupportedinpartbyMINERVAandbytheBundesministeriumf¨urBildungundForschung,Germany.1large-Nmatrixmodels,i.e.intheclassicalstatisticalmechanicsofseveralN×N,N→∞hermitianmatricesMi,i=1,...,D,withtheprobabilitydistribution(theBoltzmannfactor)givenasconst·exp(−N2PiVi(Mi))withVi(Mi)=(1/N)TrPi(Mi)andPipolynomialfunctions.Typical“observables”insuchamodelarecorrelators1NhTr(Mi1...Mik)i≡Z1NTr(Mi1...Mik)exp{DXi=1[Fi−N2Vi(Mi)]}DYk=1Dμ[Mk],(1)withtheintegrationmeasureDμ[Mk]≡NYγ=1d(Mk)γγNYγν=1dRe(Mk)γνdIm(Mk)γν,(2)andFi=−lnRDμ[Mi]exp[−N2Vi(Mi)].WeretheMiregularcommutingvariables,onewouldcallthemindependentsincehMi1...Miki=hMi1i...hMikiforanyselectioni1,...,ik.Forthematrixmodelthisisobviouslynotthecase.TheprobabilitydistributiondependsonlyupontheeigenvaluesmioftheMi.TheexpectationofproductsoftheMiontheotherhandinvolvesalsothe“angular”variablesWiinthedecompositionMi=WimiW+i.Thesevariableshoweverare“free”,meaningthattheyareweightedwiththeunbiasedHaarU(N)measure.TheintegralsovertheWicanthereforebeevaluatedinauniversalmannerindependentofwhatthepotentialsViare.TheremainingintegralsoverthemicanthenbeevaluatedinthelimitN→∞byasaddlepointapproximationasinRef.[9].Theremarkablefactisthatthisprocedurecanbeformulatedintermsofverygeneralrulesrelatinganycorrelatorofindependentmatricestoalinearcombinationofproductsofvariousindividualmomentsofthesematrices,cf.[19].TheseruleswillbereviewedinSectionIIBbelow.Anynon-commutativevariablesdistrib