The Bootstrap Program for Integrable Quantum Field

arXiv:hep-th/0110261v230Oct2001The“BootstrapProgram”forIntegrableQuantumFieldTheoriesin1+1Dim∗H.Babujian†‡andM.Karowski§Institutf¨urTheoretischePhysikFreieUniversit¨atBerlin,Arnimallee14,14195Berlin,GermanyFebruary1,2008AbstractThepurposeofthe“bootstrapprogram”istoconstructintegrablequantumfieldtheoriesin1+1dimensionsintermsoftheirWightmanfunctionsexplicitly.Asaninputtheintegrabilityandgeneralassumptionsoflocalquantumfieldtheoriesareused.Theobjectistobeachievedintreesteps:1)TheS-matrixisobtainedusingaqualitativeknowledgeoftheparticlespectrumandtheYang-Baxterequations.2)Matrixelementsoflocaloperatorsarecalculatedbymeansofthe“formfactorprogram”usingtheS-matrixasaninput.3)TheWightmanfunctionsarecalculatedbytakingsumsoverintermediatestates.Thefirststephasbeenperformedforalargenumberofmodelsandalsothesecondoneforseveralmodels.Thethirdstepisunsolveduptonow.Heretheprogramisillustratedintermsofthesine-GordonmodelaliasthemassiveThirringmodel.Exploitingthe“off-shell”BetheAnsatzweproposegeneralformulaeforformfactors.Forexamplethen-particlematrixelementforallhighercurrentsaregivenandinparticularalleigenvaluesofthehigherconservedchargesarecalculated.Furthermorequantumoperatorequationsareobtainedintermsoftheirmatrixelements,inparticularthequantumsine-Gordonfieldequation.Exactexpressionsforthefinitewavefunctionandmassrenormalizationconstantsarecalculated.∗Toappearin’FROMINTEGRABLEMODELSTOGAUGETHEORIES’,inHonorofSergeiMatinyan.Eds.V.G.GurzadyanandA.G.Sedrakian(WorldScientificPublications).BasedontalkspresentendattheConferenceatNorAmberd,Yerevan(Armenia),Septemberthe19thto25th,2001.†Permanentaddress:YerevanPhysicsInstitute,AlikhanianBrothers2,Yerevan,375036Armenia.‡e-mail:[e-mail],[e-mail]§e-mail:[e-mail]11IntroductionMorethanfiftyyearsago,Heisenberg[1]pointedouttheimportanceofstudyinganalyticcontinuationsofscatteringamplitudesintothecomplexmomentumplane.ThefirstconcreteinvestigationsinthisdirectionwerecarriedoutbyJost[2]andBargmann[3],initiallyfornon-relativisticscatteringprocesses.Theoriginalideasturnedouttobeveryfruitfulandleadtointerestingresultson-shell,i.e.fortheS-matrix[4],aswellasoff-shell,thatisforthetwo-particleformfactors[5].Onceonerestrictsonesattentionto1+1dimensionalintegrabletheories,then-particlescatteringmatrixfactoriesintotwoparticleS-matricesandtheapproach,nowusuallyreferredtoasthe’bootstrapprogram’,revealsitsfullstrength.The‘bootstrapprogram[6].forintegrablequantumfieldtheoriesin1+1-dimensions’doesnotstartwithanyclassicalLagrangian,butthisprogramclassifiesintegrablequan-tumfieldtheoreticmodelsandinadditionitprovidestheirexplicitexactsolutionsintermofallWightmanfunctions.Wehavecontactwiththeclassicalmodelsonly,whenattheendwecompareourexactresultswithFeynmangraphexpansionswhicharebasedontheLagrangians.Oneoftheauthors(M.K.)etal.[7].formulatedtheon-shellprogrami.e.theex-actdeterminationofthescatteringmatrixusingtheYang-Baxterequations.Off-shellquantities,namelyformfactorswerefirstinvestigatedbyVergelesandGryanik[8]inthesinh-GordonmodelandbyWeisz[9]inthesine-Gordonmodel.Theconceptofgeneral-izedformfactorswasintroducedbyoneoftheauthors(M.K.)etal.[10].Inthisarticleconsistencyequationswereformulatedwhichareexpectedtobesatisfiedbytheseobjects.Thereafterthisapproachwasdevelopedfurtherandstudiedinthecontextofseveralex-plicitmodelsbySmirnov[11]whoproposedtheformfactorequations(i)−(v)(seebelow)asextensionsofsimilarformulaeintheoriginalarticle[10].Theformulaewereprovenbytheauthorsetal.[12].Thereisalargenumberofmorerecentpapers[13]-[49]onformfactors.Alsothereisaniceapplication[50,51]offormfactorsincondensedmatterphysics.TheonedimensionalMottinsulatorscanbedescribedintermsofthequantumsine-Gordonmodel.FinallytheWightmanfunctionsareobtainedbytakingsumsandintegralsoverinter-mediatestates.Theexplicitevaluation1ofalltheseintegralsandsumsremainsanopenchallengeforalmostalltheories,excepttheIsingmodel[52,53].AprogresstowardsasolutionofthisproblemhasrecentlybeenachievedbyKorepinetal.[54].Anentirelydifferentmethod,theBetheAnsatz[55],wasinitiallyformulatedinordertosolvetheeigenvalueproblemforcertainintegrableHamiltonians.TheapproachhasfoundapplicationsinthecontextofnumerousmodelsandhasledtoadetailedstudyofvariousmassspectraandS-matrices.Theoriginaltechniqueshavebeenrefinedintoseveraldirections,ofwhichinparticulartheso-called“off-shell”BetheAnsatz,which1Ofcourseonemayalsoadoptaverypracticalpointofviewandresorttothewell-knownfactthatthefirsttermsoftheseriesexpansionofcorrelationfunctionsintermsofformfactorsdecreaseveryrapidly.Consequentlycorrelationsfunctionsmaybeapproximatedveryoftenquitewellbysimplyincludingthelowparticlenumberformfactorintotheexpansion.2wasoriginallyformulatedbyoneoftheauthors(H.B.)[56,57],willbeexploitedforourpurposes.ThisversionoftheBetheAnsatzpavesthewaytoextendtheapproachtotheoff-shellphysicsandopensuptheintriguingpossibilitytomergethetwomethods,thatistheformfactorapproachandtheBetheAnsatz.Thebasisforthisopportunityliesintheobservation,thatthe“off-shell”BetheAnsatzcapturestheve