A non-perturbative approach to non-commutative sca

arXiv:hep-th/0501174v31Apr2005LMU-TPS05/01Anon-perturbativeapproachtonon-commutativescalarfieldtheoryHaroldSteinacker∗Departmentf¨urPhysikLudwig–Maximilians–Universit¨atM¨unchenTheresienstr.37,D-80333M¨unchen,GermanyAbstractNon-commutativeEuclideanscalarfieldtheoryisshowntohaveaneigenvaluesectorwhichisdominatedbyawell-definedeigenvaluedensity,andcanbedescribedbyamatrixmodel.ThisisestablishedusingregularizationsofR2nθviafuzzyspacesforthefreeandweaklycoupledcase,andextendsnaturallytothenon-perturbativedomain.Itallowstostudytherenormalizationoftheeffectivepotentialusingmatrixmodeltechniques,andiscloselyrelatedtoUV/IRmixing.Inparticularwefindaphasetransitionfortheφ4modelatstrongcoupling,toaphasewhichisidentifiedwiththestripedormatrixphase.Themethodisexpectedtobeapplicablein4dimensions,whereacriticallineisfoundwhichterminatesatanon-trivialpoint,withnonzerocriticalcoupling.Thisprovidesevidenceforanon-trivialfixed-pointforthe4-dimensionalNCφ4model.∗[e-mail]1Introduction12NCscalarfieldsandUV/IRmixing32.1MatrixregularizationofRnθ..........................43Theeigenvaluedistributionofthescalarfield53.1Thefreecase..................................63.2Interactions...................................103.3Angle-eigenvaluecoordinatesinfieldspace..................123.3.1InterpretationoftheorbitsO(φi)....................143.4Relatingthematrixmodeltophysicalobservables..............154Example:theφ4model164.1Phase1(“single-cut”):m′20,orm′20withm′44g′1.........184.2Phase2(“2cuts”):m′20withm′44g′1..................185Weakcouplingandrenormalization195.12dimensions...................................195.24dimensions...................................205.3Higherdimensions...............................216Thephasetransition.216.14dimensions...................................226.22dimensions...................................256.2.1Thefuzzysphere............................267Discussionandoutlook26AppendixA:RegularizationsofR2nθ28ThefuzzytorusT2NandT2N×T2N..........................28Thefuzzysphere...................................30FuzzyCPn......................................32AppendixB:Justificationof(16)341IntroductionTheideaofconsideringquantumfieldtheoryon“quantized”ornon-commutativespaces(NCFT)wasputforwardalongtimeago[1],andhasbeenpursuedvigorouslyinthepastyears;seee.g.[2,3]forareview.Averyintriguingphenomenonwhichwasfoundinthiscontextistheso-calledUV/IRmixing[5],linkingtheusualUVdivergencestonewsin-gularitiesintheIR.Onatechnicallevel,itarisesbecauseofaverydifferentbehaviorofplanarandnon-planardiagrams,whichmustbedistinguishedonNCspaces.Theplanardiagramsareessentiallythesameasinthecommutativecase.Thenon-planardiagrams1howeverleadtooscillatingintegrals,whicharetypicallyfiniteaslongastheexternalmo-mentumpisnon-zero,butbecomedivergentinthelimitp→0.Thisleadstoseriousobstaclestoperturbativerenormalization[5].Furthermore,itappearstosignalanaddi-tionalphasedenotedas“stripedphase”[6–8],whicharisesastheminimumoftheeffectiveactionisnolongeratzeromomentum.BecauseUV/IRmixingissogenericintheNCcase,itisnecessarytocometotermswithitandtofindsuitablyadaptedquantizationmethods.Thefirststepisclearlyasuitableregularizationofthemodels.Thiscanbeachievedbyparametrizingthefieldsintermsoffinitematrices,whichisverynaturalonNCspaces.Severalsuchmethodsareavailablebynow,usinge.g.usingfuzzyspaces,non-commutativetori,etc.Theactionforscalarfieldsisthenafunctionalofahermiteanmatrixφ,wherethepotentialTrV(φ)lookslikea“pure”matrixmodelwithU(N)invariance,whichishoweverbrokenbythekineticterm.TheUV/IRmixingisexpectedtoberecoveredinthecontinuumlimit.Sucharegularizationhasbeenusedrecentlytoconfirmnumericallythenon-trivialphasestructurementionedaboveinthenon-commutativeφ4model[9,10].Therehasalsobeenremarkableprogressontheanalyticalsideusingmatrixtechniques:Amodifiedφ4modelwithanexplicitIRregulatortermintheactionwasshowntobeperturbativelyrenormalizable[11],andcertainself-dualmodelsofNCfieldtheoryweresolvedexactlyusingamatrixmodelformulation[13].Forgaugetheories,theapplicabilityofwell-knowntechniquesfromrandommatrixtheoryhasalsobeenshowninsimplecases[14,15].Fortheφ4model,asimilarapproachusingrandommatrixtheorydoesnotseempossibleatfirstsight,lackingU(N)invariance.Nevertheless,itwasconjecturedin[9]thatthestripedphaseshouldbeidentifiedwitha“matrixphase”forthefuzzysphere,wheretheactionappearstobedominatedbyapurepotentialmodelinthatphase.HenceasimpleanalyticalapproachwhichallowstostudyalsoscalarNCFT’swithnon-trivialphasestructureandUV/IRmixingishighlydesirable.Inparticular,itseemsthattheobviousparallelsbetweenNCFTandpurematrixmodelsduetoUV/IRmixinghavenotyetbeenfullyexploited,apartfromintegrablecases[13].TheaimofthispaperistoshowthatthereisindeedasimplematrixmodeldescriptionwhichcapturesacertainsectorofscalarNCFT,duetoUV/IRmixing.ThissuggestsanewapproachtoscalarNCFTwhichnotonlyprovidesnewinsights,butalsonewtoolstostudytherenormalizationoftheeffectivepotential.Thestartingpointisanappropriateparametrizationforthefields:sinceφisahermitianmatrix,itcanbediagona