Effective Field Theories on Non-Commutative Space-

arXiv:hep-ph/0305027v225Jun2003hep-ph/0305027CALT-68-2436LMU-TPW2003-02May2,2003EffectiveFieldTheoriesonNon-CommutativeSpace-TimeXavierCalmet1CaliforniaInstituteofTechnology,Pasadena,California91125,USAMichaelWohlgenannt2Ludwig-Maximilians-Universit¨at,Theresienstr.37,D-80333Munich,GermanyAbstractWeconsiderYang-Millstheoriesformulatedonanon-commutativespace-timedescribedbyaspace-timedependentanti-symmetricfieldθμν(x).UsingSeiberg-Wittenmaptechniqueswederivetheleadingorderoperatorsfortheeffectivefieldtheoriesthattakeintoaccounttheeffectsofsuchabackgroundfield.Theseeffec-tivetheoriesarevalidforaweaklynon-commutativespace-time.Itisremarkabletonotethatalreadysimplemodelsforθμν(x)canhelptoloosentheboundsonspace-timenon-commutativitycomingfromlowenergyphysics.Non-commutativegeometryformulatedinourframeworkisapotentialcandidatefornewphysicsbeyondthestandardmodel.toappearinPhys.Rev.D.1email:[e-mail]2email:[e-mail]1IntroductionInthepastyearsaconsiderableprogresstowardsaconsistentformulationoffieldtheoriesonanon-commutativespace-timehasbeenmade.Theideathatspace-timecoordinatesmightnotcommuteatveryshortdistancesisneverthelessnotnewandcanbetracedbacktoHeisenberg[1],Pauli[2]andSnyder[3].Anicehistoricalintroductiontonon-commutativecoordinatesisgivenin[4].Atthattimethemainmotivationwasthehopethattheintroductionofanewfundamentallengthscalecouldhelptogetridofthedivergenciesinquantumfieldtheory.Amoremodernmotivationtostudyaspace-timethatfulfillsthenon-commutativerelation[ˆxμ,ˆxν]≡ˆxμˆxν−ˆxνˆxμ=iθμν,θμν∈C(1)isthatitimpliesanuncertaintyrelationforspace-timecoordinates:ΔxμΔxν≥12|θμν|,(2)whichistheanaloguetothefamousHeisenberguncertaintyrelationsformomentumandspacecoordinates.Notethatθμνisadimensionalfullquantity,dim(θμν)=mass−2.Ifthismassscaleislargeenough,θμνcanbeusedasanexpansionparameterlike~inquantummechanics.Weadopttheusualconvention:avariableorfunctionwithahatisanon-commutativeone.Itshouldbenotedthatrelationsofthetype(1)alsoappearquitenaturallyinstringtheorymodels[5]orinmodelsforquantumgravity[6].Itshouldalsobeclearthatthecanonicalcase(1)isnotthemostgenericcaseandthatotherstructurescanbeconsidered,seee.g.[7]forareview.Inordertoconsiderfieldtheoriesonanon-commutativespace-time,weneedtodefinetheconceptofnon-commutativefunctionsandfields.Non-commutativefunctionsandfieldsaredefinedaselementsofthenon-commutativealgebraˆA=Chhˆxi...ˆxniiR,(3)whereRaretherelationsdefinedineq.(1).ˆAisthealgebraofformalpowerseriesinthecoordinatessubjecttotherelations(1).Wealsoneedtointroducetheconceptofastarproduct.TheMoyal-Weylstarproduct⋆[8]oftwofunctionsf(x)andg(x)withf(x),g(x)∈R4,isdefinedbyaformalpowerseriesexpansion:(f⋆g)(x)=expi2θμν∂∂xμ∂∂yνf(x)g(y)y→x=f·g+i2θμν∂μg·∂νf+O(θ2).(4)1Intuitively,thestarproductcanbeseenasanexpansionoftheproductintermsofthenoncommutativeparameterθ.ThestarproducthasthefollowingpropertyZd4x(f⋆g)(x)=Zd4x(g⋆f)(x)=Zd4xf(x)g(x),(5)ascanbeprovenusingpartialintegrations.Thispropertyisusuallycalledthetraceproperty.Heref(x)andg(x)areordinaryfunctionsonR4.Twodifferentapproachestonon-commutativefieldtheoriescanbefoundintheliterature.Thefirstoneisanon-perturbativeapproach(seee.g.[9]forareview),fieldsareconsideredtobeLiealgebravaluedanditturnsoutthatonlyU(N)structuregroupsareconceivablebecausethecommutator[ˆΛ⋆,ˆΛ′]=12{ˆΛa(x)⋆,ˆΛ′b(x)}[Ta,Tb]+12[ˆΛa(x)⋆,ˆΛ′b(x)]{Ta,Tb}(6)oftwoLiealgebravaluednon-commutativegaugeparametersˆΛ=Λa(x)TaandˆΛ′=Λ′a(x)TaonlyclosesintheLiealgebraifthegaugegroupunderconsiderationisU(N)andifthegaugetransformationsareinthefundamentalrepresentationofthisgroup.But,thisapproachcannotbeusedtodescribeparticlephysicssinceweknowthatSU(N)groupsarerequiredtodescribetheweakandstronginteractions.OratleastthereisnoobviouswayknowntodatetoderivethestandardmodelasalowenergyeffectiveactioncomingfromaU(N)group.FurthermoreitturnsoutthatevenintheU(1)case,chargesarequantized[10,11]anditthusimpossibletodescribequarks.TheotherapproachhasbeendevelopedbyWessandhiscollaborators[12,13,14,15],(seealso[16,17]).Thegoalofthisapproachistoconsiderfieldtheoriesonnon-commutativespacesaseffectivetheories.ThemaindifferencetothemoreconventionalapproachistoconsiderfieldsandgaugetransformationswhicharenotLiealgebravaluedbutwhichareintheenvelopingalgebra:ˆΛ=Λ0a(x)Ta+Λ1ab(x):TaTb:+Λ2abc(x):TaTbTc:+...(7)where::denotessomeappropriateorderingoftheLiealgebragenerators.Onecanchoose,forexample,asymmetricallyorderedbasisoftheenvelopingalgebra,onethenhas:Ta:=Taand:TaTb=12{Ta,Tb}andsoon.Themappingbetweenthenon-commutativefieldtheoryandtheeffectivefieldtheoryonausualcommutativespace-timeisderivedbyrequiringthatthetheoryisinvariantunderbothnon-commutativegaugetransformationsandundertheusual(classical)commutativegaugetransforma-tions.TheserequirementsleadtodifferentialequationswhosesolutionscorrespondtotheSeiberg-Wittenmap[18]thatappearedoriginallyinthecontextofstringtheory.It2shouldbenotedthattheexpansionwhichisperformedinthatapproachisinasensetrivialsinceitcorrespondstoavariablechange.But,itiswellsuitedforaphenomeno-logicalapproachsinceitg