The General Correlation Function in the Schwinger

arXiv:hep-th/0106071v225Oct2001TheGeneralCorrelationFunctionintheSchwingerModelonaTorusS.Azakov∗February1,2008InstituteofPhysics,AzerbaijanAcademyofSciences,Baku,AzerbaijanAbstractIntheframeworkoftheEuclideanpathintegralapproachwederivetheex-actformulaforthegeneralN-pointchiraldensitiescorrelatorintheSchwingermodelonatorus.PACSnumber(s):11.10Mn...Fieldtheory,Schwingersourcetheory;11.15Tk...Gaugefieldtheories,othernonperturbativetechniques.Keywords:Schwingermodel,zeromodes,torus,fermioniccorrelatorLosAlamosDatabaseNumber:hep-th/0106071Fullpostaladdress:InstituteofPhysics,HuseinJavidave.33,Baku,370143,Azer-baijanTel.99412391424,fax[phone]IntroductionTheSchwingermodel[1](SM)(two-dimensionalQEDwithmasslessfermions)onaEuclideantorusT2isexactlysoluble[2][3],[4]andinmanycalculationsitwouldbeusefultohaveanexpressionforthegeneralN-pointcorrelationfunctionofchiraldensities.Itiswellknownthatinthismodelthe”photon”acquiresamassduetochiralanomalyandfermionsdisappearfromthephysicalspectrum.∗e-mailaddress:[e-mail]1TherearesomefeaturesoftheSMwhichhavesimiliaritywiththoseofQCD.Fermioncondensate,massgeneration,dynamicalsymmetrybreakingandconfine-mentareamongthem.Inbothmodelsinstantonsaresupposedtoberesponsibleforsomenontrivialvacuumexpectationvalues[5],[6],[7].Workonatorusisdesirablebyseveralreasons.Firstly,bydefiningthemodelonafinitevolumewegetridofinfraredproblems.Compactificationmakesmathe-maticalmanipulationsmorerigorous,topologicalrelationsbecomemorepreciseandtransparent.Oneshouldrememberthatthefermionpathintegralshavenomeaningunlessdefinedusingadiscretebasis.Secondly,inthiscasewehaveamodelwithnontrivialtopologyinwhichwecanfindexplicitlyfermioniczeromodesandGreen’sfunctionsinalltopologicalsectors.Apresenceoftopologicallynon-trivialconfigurationsofthegaugefield(in-stantons)andfermioniczeromodesallowsinpathintegralframeworktoreproducethestructureoftheSMfoundintheoperatorformalism[8].Thirdly,acompactificationonatorusallowstofindfinitetemperatureandfinitesizeeffectsandisappropriatetothesystematicanalysisofthelatticeapproximtion.Itisatoruswhichismostnaturallyapproximatedbythefinitecubicallatticeonwhichthenumericalcalculationsareperformed[9].Finally,torusandcircleareparticularlyappropriateforstudyingtherelationbe-tweentheHamiltonianandpathintegralapproachinthegaugetheorywithmasslessfermions[10],[11].ThankstoitsfullsolubilitytheSMmayalsobeusedtotestvariousideasrelatedtononperturbativestructureofquantumfieldtheory.BardakciandCrescimannowerethefirstwhousingpathintegralapproachex-ploredtheroleofnontrivialtopologicalconfigurationsintwo-dimensionalfermionicmodelrelevantinthecontextofstringcompactification[12].Theywereabletoshowthatcertaincorrelationfunctionswhich,beingzeroforthetrivialtopology,consid-erablychangebynontrivialtopologicaleffect.FollowingtheideasofBardakciandCrescimannoManias,NaonandTrobostudiedthebehaviourofcorrelationfunc-tionsoffermionbilinearoperatorsintheSMwithtopologicallynontrivialgaugeconfigurationsintheinfinitespace-time[13].Two-andfour-pointcorrelationfunc-tionsintheSMonthetorushavebeencalculatedin[2],[14].Theauthorsofthepaper[15]consideredasix-pointcorrelationfunctioninthismodelwhichduetosometechnicaldifficultiestheymanagedtocalculateonlyatfinitetemperaturebutintheinfinitespace.TheyalsomadeaconjectureabouttheexplicitexpressionfortheN-pointcorrelatorofchiraldensitiesagainatthefinitetemperaturesbutintheinfinitespace.Thepaperisorganizedasfollows.InSection2webrieflyreviewtheresultsobtainedbeforefortheSMonthetorusinEuclidean(pathintegral)approach2[2],[3],[4]andrelevantforthepresentconsideration.InadditiontotheseresultswegivesomenewinformationwhichconcernsthepossiblechoiceofthezeromodesandfermionicGreen’sfunctionsinthenontrivialtopologicalsectors.Section3isdevotedtothediscussionofmodulartransformation.Theinvariancewithrespecttothistransformationhelpsusinthesequeltodeterminesomeproportionalityconstants.Insection4whichisthecentralpartofthepresentworkwegetourmainresult.Thelastsectionisreservedforconclusionsandthediscussionofpossibledirectionsofthefutureinvestigation.2AbriefreviewofpathintegralformulationoftheSMontheEuclideantorusTheSMactionontheEuclideantorusT2,(0≤x1≤L1,0≤x2≤L2)readsS=ZT2d2x12F212(x)+¯ψ(x)γμ(∂μ−ieAμ(x))ψ(x),(1)whereF12(x)=∂1A2(x)−∂2A1(x)isafieldstrength.Weuseconventionsandnotationsusedin[3],[2].Theevaluationofquantummechanicalexpectationvalues(QMEV)inthepathintegralformulationhΩ[¯ψ,ψ,Aμ]i=1ZZD[ψ,¯ψ,Aμ]Ω[¯ψ,ψ,Aμ]e−S[¯ψ,ψ,Aμ],(2)wheretheZfactor(thepartitionfunction):Z=ZD[ψ,¯ψ,Aμ]e−S[¯ψ,ψ,Aμ],(3)getsthefollowingformforQMEVoffermionfields[2]hψα1(x1)¯ψβ1(y1)···ψαN(xN)¯ψβN(yN)i=Z−1Xk=0,±1,···,±NL|k|1ZAkDAe−S[A]det′[L1γμ(∂μ−ieAμ)]×XPi(−1)piXPj(−1)pjˆχ(1)αi1(xi1)···ˆχ(|k|)αi|k|(xi|k|)¯ˆχ(1)βj1(yj1)···¯ˆχ(|k|)βj|k|(yj|k|)×S(k)αi|k|+1βj|k|+1(xi|k|+1,yj|k|+1;A)···S(k)αiNβjN(xiN,yjN;A).(4)HerewehavealreadyperformedthefermionintegrationRD[ψ,¯ψ]overthefermionicGrassmannvariables.Thesumistakenoverallpossiblepermutations3Pi=(i1,i2,...,iN)andPj=(j1,j2,...,jN)of(1,2,...,N),(−1)pi((−1)pj)isaparityofthepermutationPi(Pj).Letusdiscusst