The Form Factors and Quantum Equation of Motion in

TheFormFactorsandQuantumEquationofMotioninthesine-GordonModelH.BabujianyzandM.KarowskixInstitutfurTheoretischePhysikFreieUniversitatBerlin,Arnimallee14,14195Berlin,GermanyAbstractUsingthemethodsofthe\formfactorprogramexactexpressionsofallmatrixelementsareobtainedforseveraloperatorsofthequantumsine-GordonmodelaliasthemassiveThirringmodel.Ageneralformulaispresentedwhichprovidesformfactorsintermsofanintegralrepresentation.Inparticularcharge-lessoperatorsasforexamplethecurrentofthetopologicalcharge,theenergymomentumtensorandallhighercurrentsareconsidered.Inthebreathersectoritisfoundthequantumsine-Gordoneldequationholdswithanexactrelationbetweenthe\baremassandthenormalizedmass.Alsoarelationforthetraceoftheenergymomentumisobtained.AllresultsarecomparedwithFeynmangraphexpansionandfullagree-mentisfound.PACS:11.10.-z;11.10.Kk;11.55.DsKeywords:Integrablequantumeldtheory,Formfactors1IntroductionThisworkcontinuesapreviousinvestigation[1]onexactformfactorsforthesine-GordonaliasthemassiveThirringmodel.Coleman[2]hadshownthatthesetwomodelsareequivalentonthequantumlevel.ThecorrespondingclassicalmodelsaredenedbytheirLagrangian’sLSG=12@’@’+2(cos’1)LMT=(i@M)12gjj;j=:TalkgivenbyH.BabujianatNATOAdvancedResearchWorkshop\DynamicalSymmetriesinInte-grableTwo-DimensionalQuantumFieldTheoriesandLatticeModels25-30September,KievUkraine.yPermanentaddress:YerevanPhysicsInstitute,AlikhanianBrothers2,Yerevan,375036Armenia.ze-mail:[e-mail],[e-mail]-mail:[e-mail]1WedonotusetheseclassicalLagrangiansandanyquantizationproceduretoconstructthequantummodels.Wehavecontactwiththeclassicalmodelsonly,whenattheendwecompareourexactresultswithFeynmangraphexpansionswhicharebasedontheLagrangians.The‘formfactorprogram’ispartofthe‘Bootstrapprogramforintegrablequantumeldtheoriesin1+1-dimensions’.ThisprogramclassiesintegrablequantumeldtheoreticmodelsandinadditionitprovidestheirexplicitexactsolutionsintermofallWightmanfunctions.Theseresultsareobtainedinthreesteps:1.TheS-matrixiscalculatedbymeansofgeneralpropertiessuchasunitarityandcrossing,theYang-Baxterequations(whichareaconsequenceofintegrability)andtheadditionalassumptionof‘maximalanalyticity’.Thismeansthatthetwo-particleS-matrixisananalyticfunctioninthephysicalplane(oftheMandelstamvariable(p1+p2)2)andpossessesonlythosepolestherewhichareofphysicalorigin.2.Generalizedformfactorswhicharematrixelementsoflocaloperatorsouthp0m;:::;p01jO(x)jp1;:::;pniinarecalculatedbymeansoftheS-matrix.Moreprecisely,theequations(i)(v)givenbelowonpage6areusedasaninput.TheseequationsfollowfromLSZ-assumptionsandagaintheadditionalassumptionof‘maximalanalyticity’(seealso[1]).3.TheWightmanfunctionsareobtainedbyinsertingacompletesetofintermediatestates.InparticularthetwopointfunctionforahermitianoperatorO(x)readshO(x)O(0)i=1Xn=01n!ZZdp1:::dpn(2)n2!1:::2!nh0jO(0)jp1;:::;pniin2eixPpi:Theon-shellprogrami.e.theexactdeterminationofthescatteringmatrixwasformu-latedin[3].O-shellconsiderationswerecarriedoutin[4]andin[5],wheretheconceptofageneralizedformfactorwasintroducedandconsistencyequationswereformulatedwhichareexpectedtobesatisedbytheseobjects.Thereafterthisapproachwasdevel-opedfurtherandstudiedinthecontextofseveralexplicitmodels(seee.g.[6]1).Morerecentpapersonsolitonicmatrixelementsinthesine-Gordonmodelare[7,8].Thereisaniceapplication[9,10]offormfactorsincondensedmatterphysics.TheonedimensionalMottinsulatorscanbedescribedintermsofthequantumsine-Gordonmodel.Inthepreviouspaper[1]anintegralrepresentationforgeneralmatrixelementsofthefundamentalfermi-eldofthemassiveThirringmodelhasbeenproposed.In[11,12,13]wegeneralizethisformulaandinvestigateinparticularcharge-lesslocaloperators.Thestrategyisasfollows:Forastateofnparticlesofkindiwithmomentapi=msinhiandalocaloperatorO(x)thegeneralizedformfactorisdenedbyh0jO(x)j1(p1);:::;n(pn)iin=eix(p1++pn)O()(1)1Formorereferencessee[1].2for1n.Theshortnotation=(1;:::;n)and=(1;:::;n)hasbeenused.WemaketheAnsatzO()=ZCdz1ZCdzmh(;z)pO(;z)(;z)withtheBethestate(;z)denedbyeq.(5)andtheintegrationcontoursCofgure1.Thescalarfunctionh(;z)isuniquelydeterminedbytheS-matrixwhereasthescalar‘p-function’pO(;z)dependsontheoperator.BymeansoftheAnsatzwetransformtheproperties(i)(v)oftheco-vectorvaluedfunctionO()(seepage6)toproperties(i0)(v0)ofthescalarfunctionpO(;z)whichareeasilysolved.Forexampleweobtainthep-functionsforthelocaloperator2N(x)asp(;z)=NnnXi=1eimXi=1ezinXi=1eimXi=1ezi!:Insection4weproposeinadditionthep-functionsforN5(x),thecurrentj(x),theenergymomentumtensorT(x)andtheinnitelymanyhigherconservedcurrentsJL(x).TheidenticationwiththeoperatorsismadebycomparingtheexactresultswithFeynmangraphexpansions.Propertiesascharge,behaviorunderLorentztransformationsetc.willalsobecomeobvious.2RecallofformulaeInthissectionwerecallsomeformulaewhichweshallneedinthefollowingsectionstopresentourresults.Allthismaterialcanbefoundin[1]includingtheoriginalreferences.2.1TheS-matrixThesine-Go