Matrix $phi^4$ Models on the Fuzzy Sphere and thei

arXiv:hep-th/0109084v111Sep2001Matrixϕ4ModelsontheFuzzySphereandtheirContinuumLimitsBrianP.Dolan∗†,DenjoeO’Connor‡andP.Preˇsnajder§¶DeptodeF´ısica,Cinvestav,ApartadoPostal70-543,M´exicoD.F.0730,M´exicoFebruary1,2008AbstractWedemonstratethattheUV/IRmixingproblemsfoundrecentlyforascalarϕ4theoryonthefuzzyspherearelocalizedtotadpolediagramsandcanbeovercomebyasuitablemodificationoftheaction.Thismodificationisequivalenttonormalorderingtheϕ4vertex.Inthelimitofthecommutativesphere,theperturbationtheoryofthismodifiedactionmatchesthatofthecommutativetheory.1IntroductionFuzzymodelshavebeenproposed,[1]-[4],asapotentialmethodofdoing“lattice”fieldtheory.Thebasicideaistotakeaclassicalphasespaceoffinitevolume,quantizeitandthusobtainaspacewithafinitenumberofdegreesoffreedom.Thingsareofcoursealittlemorecomplicatedbutessentially∗Permanentaddress:Dept.ofMathematicalPhysics,NUI,Maynooth,Ireland.†Email:[e-mail].‡Email:denjoe@fis.cinvestav.mx§Permanentaddress:Dept.ofTheoreticalPhysics,ComeniusUniversity,Mlynsk´aDolina,Bratislava,Slovakia.¶Email:[e-mail].1thisideaworkswhenthephasespaceisaco-adjointorbit,thesimplestsuchexamplebeingthetwosphereS2,withtheresultingquantizedspaceknownasthefuzzysphere[5].Fieldtheorymodelsonthefuzzyspherethenpossessonlyafinitenumberofmodes.Thesimplestfieldtheorymodelofascalarfieldwithϕ4interactionwasproposedin[1]andhasprovedtobeanexcellenttestinggroundfortheideaofusingfuzzyspacesfordoing‘lattice’studies.Apotentialproblemwithusingthesefuzzyspacestogetafiniteapprox-imationtofieldtheoriesandthusdo‘lattice’physicshasemergedduetothephenomenonofUV/IRmixing[6].TheproblemwasdiscussedintheMoyalplanein[7]wherethefieldspossessaninfinitenumberofmodes,andconse-quentlyaregularizationprocedureisneeded.Thisphenomenonappearstobegenericforfieldtheoriesinnon-commutativespaces.InarecentarticleVaidya[8]pointedoutthatthisUV/IRmixingphe-nomenonispresentintheϕ4modelof[1]onthefuzzysphere.HisworkwasfollowedbythatofChu,MadoreandSteinacker,[9],whocalculatedexplicitlytheone-loopcontributiontothetwopointvertexfunctionandfoundthatinthecommutativelimit,thenon-planardiagramretainsaresidualfinitecon-tributionoverandabovetheexpectedcommutativeterm.Theadditionalterm,canbeseenasanonlocalrotationallyinvariantcontributiontotheeffectiveaction.Thoughthistermisnon-singularforthefuzzyspheretheyshowedthatintheplanarlimititincorporatestheUV/IRmixingsingularityoftheMoyalplane.Theimplicationsofthisresultareveryseriousfortheprogramofusingthematrixmodelapproximationstocontinuumfieldtheoriestostudythenon-perturbativecontinuumbehaviour.Oneimplicationisthatthescalaractionconsideredbytheseauthorscannotbethecorrectfuzzyactionforthe‘lattice’programoutlinedabove.Wethereforereturntotheproblemandstudyhowseriousitinfactisandwhetherithasanaturalsolution.Wefindthatindeedithasaquitenaturalsolutionandthattheproblemdisappearswhentheinteractionterminthematrixactionis“normalordered”,i.e.whentheappropriatesubtractionsassociatedwithWickcontractionsoftheϕ4termareincludedintheaction.Theaction(55)withnormalorderedvertexisthereforethecorrectstartingpointforthefuzzylatticephysicsprogram.Thepaperisorganizedasfollows:Webeginbybrieflyreviewingthe∗-product,thatrealizesmatrixmultiplicationattheleveloffunctionsonthefuzzysphere.Wethenreviewtheone-loopcalculationofthetwo-pointfunction,repeatingthecalculationusingthe∗-product.Nextwediscuss2the4-pointvertexfunctionsanddemonstratethattheproblemanomalouscontributionsinthatcommutativelimitdoesnotarisehere.Wefurthershow,infact,thatthisproblemislocalizedtothecaseofapropagatorreturningtothesamevertexthatitleaves,i.e.thatitisassociatedwithtadpolediagrams.Wefinallyproposeoursolutionfortheeliminationoftheseunwantedcontributions—normalorderingtheϕ4vertex.Thearticleconcludeswithadiscussionandconclusionswherewedefinethematrixmodelwhichhasaslimitingtheorythestandardcontinuummodel.2StarproductsonthefuzzysphereAtthelevelofthefundamentalrepresentationthefuzzysphere[5]canbedefinedastheorbittheadjointactionofSU(2)onarankoneprojectionoperator,ρ,ina2-dimensionalHilbertspace,[3][10],withρ2=ρ,ρ†=ρandTrρ=1.AtapointonthesphereparameterizedbytheunitvectorninR3,obtainedbyrotatingthenorthpolebyg∈SU(2),wehaveρ(n)=gρ0g†(1)Hereρ0=0001istheprojectoratthenorthpole,n0=(0,0,1).Atthelevelofthe(L+1)-dimensionalrepresentationweconstructarankoneprojectorPL(n)astheL-foldsymmetrictensorproductofρ(n).Associatedwithevery(L+1)×(L+1)matrixisafunctiononthefuzzysphere,definedbyFL(n)=TrPL(n)ˆF,(2)andanassociative∗-productbetweentwosuchfunctionsisgivenby(FL∗GL)(n)=TrPL(n)ˆFˆG.(3)Intermsofderivatives(FL∗GL)(n)=LXk=02k(L−k)!L!k!∂A1···∂AkFL(n)KA1B1···KAkBk∂B1···∂BkGL(n),(4)3withKAB=12(PAB+iJAB),wherePAB=δAB−nAnBandJAB=ǫABCnCsoP2=PandJ2=−P(indicesA,B,...areraisedandloweredwiththeEuclideanmetric,δAB,inR3).TheactionoftheSU(2)generatorsonafunctiononthefuzzyspherecanbewrittenasLAFL(n)=TrPL(n)[ˆF,LA],(5)whereLAarethegeneratorsinthe(L+1)-dimensionalmatrixrepresentationandLA=iǫABCnB∂Carethegeneratorsindifferentialform.TheL+1dimensionalmatricescanbeexpandedintermsofanortho-normal