Quantum Field Theory on the Noncommutative Plane w

arXiv:hep-th/9904132v120Apr1999HIP-1999-19/THApril20,1999QuantumFieldTheoryontheNoncommutativePlanewithEq(2)SymmetryM.Chaichian,A.DemichevaandP.PreˇsnajderbHighEnergyPhysicsDivision,DepartmentofPhysics,UniversityofHelsinkiandHelsinkiInstituteofPhysics,P.O.Box9,FIN-00014Helsinki,FinlandAbstractWestudypropertiesofascalarquantumfieldtheoryonthetwo-dimensionalnoncommutativeplanewithEq(2)quantumsymmetry.Westartfromtheconsiderationofafirstlyquantizedquantumparticleonthenoncommutativeplane.Thenwedefinequantumfieldsdependingonnoncommutativecoordi-natesandconstructafieldtheoreticalactionusingtheEq(2)-invariantmeasureonthenoncommutativeplane.Withthehelpofthepartialwavedecompo-sitionweshowthatthisquantumfieldtheorycanbeconsideredasasecondquantizationoftheparticletheoryonthenoncommutativeplaneandthatthisfieldtheoryhas(contrarytothecommonbelief)evenmoresevereultravioletdivergencesthanitscounterpartontheusualcommutativeplane.Finally,weintroducethesymmetrytransformationsofphysicalstatesonnoncommu-tativespacesanddiscussthemindetailforthecaseoftheEq(2)quantumgroup.PACS:03.70aPermanentaddress:NuclearPhysicsInstitute,MoscowStateUniversity,119899,Moscow,RussiabPermanentaddress:DepartmentofTheoreticalPhysics,ComeniusUniversity,Mlynsk´adolina,SK-84215Bratislava,Slovakia1IntroductionItisgenerallybelievedthatthepictureofspace-timeasamanifoldMshouldbreakdownatveryshortdistancesoftheorderofthePlancklength.Onepossibleapproachtothedescriptionofphysicalphenomenaatsmalldistancesisbasedonnoncommutativegeometryofspace-time.TherehavebeeninvestigationsinthecontextofConnes’approach[1]togravityandtheStandardModelofelectroweakandstronginteractions[2,3]andintheframeworkofthestringtheory[4].Anotherapproachstartingfromstudyofarelationbetweenmeasurementsatverysmalldistancesandblackholeformationshasbeendevelopedinthepioneeringworks[5].OnemorepossibilityisbasedonQuantumGrouptheory(see,e.g.,[6]).Theessenceofthenoncommutativegeometryconsistsinreformulatingfirstthege-ometryintermsofcommutativealgebrasandmodulesofsmoothfunctions,andthengeneralizingthemtotheirnoncommutativeanalogs.Ifthenotionsofthenoncommuta-tivegeometryareuseddirectlyforthedescriptionofthespace-time,thenotionofpointsaselementarygeometricalentityislostandonemayexpectthatanultravioletcutoffappears.Asiswellknownfromthestandardquantummechanics,aquantizationofanycom-pactspace,inparticularasphere,leadstofinite-dimensionalrepresentationsofthecor-respondingoperators,sothatinthiscaseanycalculationisreducedtomanipulationswithfinite-dimensionalmatricesandthusthereissimplynoplaceforUV-divergences(see[7,8,9]andrefs.therein).Thingsarenotsoeasyinthecaseofnoncompactman-ifolds.Thequantizationleadstoinfinite-dimensionalrepresentationsandwehavenoguaranteethatnoncommutativityofthespace-timecoordinatesremovesUV-divergences.Inourprecedingpaper[10]wehaveshownthatultravioletbehaviourofafieldtheoryonanoncommutativespace-timeissensitivetothetopologyofthespace-time,namelytoitscompactness.Weconsideredtheoriesonatwo-dimensionalplanewithHeisenberg-likecommutationrelationsamongcoordinates(seealso[5,11])andonanoncommutativecylinder.Whiletheformerretainsthedivergenttadpoles(asanordinaryQFT),thelatterprovestobeUV-finite.WearguedthattheunderlyingreasonforsuchaUV-behaviourofthemodelsisrelatedtothepropertiesofthecompletecoordinate-momentumquantummechanicalalgebraandtothefactthatthemomentadegreesoffreedomareassociatedtothefullynoncompactHeisenberg-Weylgroupmanifoldinthefirstcaseandtothecylinderinthesecondcase(thecylinderhasonecompactdimension).Usingthesequalitativearguments,wesupposedthatthequantumfieldtheorycon-structedontheq-deformedplane[12,13,14]withEq(2)-symmetryalsohasUV-divergences.WehaveprovedthatindeedtherearenokinematicalreasonsforthismodeltobeUV-finite:theGreenfunctionofthefreetheoryontheq-planeissingular.Moreover,wehaveshownthattheinteractionwithanexternalfielddoesproducedivergenttadpole.How-ever,inthepaper[10]weuseddecompositionofthefieldsontheq-planeintheso-calleddistortedplanewaves(q-deformedexponentialfunctions).Thismakesdifficultmatchingtheq-deformedfieldtheorywiththecorrespondingfirstlyquantizedquantummechanicsofparticlesontheq-deformedplaneandduetotheabsenceoftheadditivitypropertyfortheq-exponentials,makesanexplicitcalculationofnontrivial(e.g.,ϕ4-)verticesimpos-sible.Thustheresultsof[10]haveleftopenthepossibilitythatthecompleteinteractingtheoryontheq-planeisUV-finitebecauseof(dynamical)propertiesofthecorrespondingϕ4-vertices.1Inthispaperweuseanotherdecompositionofthefields,namely,thedecompositioninpartialwaves,similartotherecentlyproposed“sphericalfieldtheory”[15]oncommutativespaces.ThisdecompositiontogetherwiththeHaar(Eq(2)-invariant)measureandq-deformedintegralusedforthedefinitionofthefieldtheoreticalaction,allowstopresentthefieldtheoryontheq-deformedplaneasalatticetheoryofinfinitenumberofinteractingone-dimensionalfields(partialwaves).Theresultingfieldtheoreticaldegreesoffreedomareintransparentcorrespondencewiththespectrumofoperatorsinthefirstlyquantizedversionofthemodel.Thecalculationofthetadpolewiththeaccountoftheϕ4-vertexshowsthatUV