Scattering theory of space-time non-commutative ab

arXiv:hep-th/0401008v13Jan2004KIAS-P04001hep-th/0401008Scatteringtheoryofspace-timenon-commutativeabeliangaugefieldtheoryChaihoRim1andJaeHyungYee21DepartmentofPhysics,ChonbukNationalUniversityChonju561-756,[e-mail]2InstituteofPhysicsandAppliedPhysics,YonseiUniversitySeoul120-749,[e-mail]-matrixforthespace-timenon-commutativeQEDisconstructedusingthe⋆-timeorderingwhichisneededinthepresenceofderivativeinteractions.BasedonthisS-matrix,perturbationtheoryisformulatedandFeynmanruleispresented.Thegaugeinvarianceisexplicitlycheckedtothelowestorder,usingtheComptonscatteringprocess.Thegaugefixingconditiondependencyoftheclassicalsolutionofthevacuumisalsodiscussed.11Introduction:Non-commutativefieldtheory(NCFT)[1,2]isthefieldtheoryonthenon-commutative(NC)coordinatesspace,[xμ,xν]=iθμν.(1.1)Spacenon-commutativetheory(SSNC)involvesonlythespacenon-commutingcoordi-nates(θ0ν=0),whereasspace-timenon-commutativetheory(STNC)containsthenon-commutingtime(θ0ν6=0).NCFTisconstructedbasedontheWeyl’sidea[3]:Insteadofusingthisnon-commutingcoordinatesdirectly,onemayusethe⋆-productoffieldsovercommutingspace-timecoordinates.The⋆-productencodesallthenon-commutingnatureofthetheoryandfixestheorderingambiguityofnon-commutingcoordinates.WeadopttheMoyalproduct[4]asthe⋆-productrepresentations,f⋆g(x)=ei2∂x∧∂yf(x)g(y)|y=x(1.2)wherea∧b=θμνaμbν.θμνisanantisymmetricc-numberrepresentingthespace-timenon-commutativeness.Usingthisidea,thecommutatorin(1.1)becomesthe⋆-commutator,[xμ⋆,xν]≡xμ⋆xν−xν⋆xμ=iθμν,wherethecoordinatesx’saretreatedasthecommutingones.ThemeritoftheMoyalproductisthatthe⋆-productmaintainstheordinaryformofthekinetictermoftheaction,andallowstheconventionalperturbationwheretheinteractiontermsbecomenon-localreflectingthenon-commutingnatureoftheinteraction.In[5,6]theunitaryS-matrixwasconstructedusingLagrangianformalismofthesecondquantizedoperatorsintheHeisenbergpicture.Herefixingthetime-orderingam-biguityhasthecentralroleinestablishingtheunitaryS-matrix.Thesolutionisgivenastheso-calledminimalrealizationofthetime-orderingstepfunctionand⋆-timeordering.Theproposalby[7]toavoidtheunitarityproblem[8]inSTNCisintherightdirectionbutthetime-orderingsuggestedin[7]needshigherderivativecorrection.Afterthehigherderivativecorrection,therighttime-orderingturnsouttobethe⋆-timeorderingasgivenin[5,6].BasedonthisS-matrix,theperturbationtheoryofSTNCisillustratedusingtherealscalartheoryandFeynmanruleispresented.Inthispaper,wecontinuethisprojecttoinvestigatetheSTNCU(1)gaugefieldtheory.Itiswell-knowninthecommutativegaugetheorythatthepropertime-orderingisthecovarianttime-orderingduetothederivativeinteraction.Inthissense,thetime-orderinginSTNCgaugetheoryneedstobemodifiedfromtheonedefinedintheSTNCscalarfieldtheory,whosecommutativeversioncontainsnoderivativeinteraction.Inaddition,thegaugetheorypossessesthegaugesymmetryanditsquantizedversionshouldmaintainthegaugesymmetrytogetridoftheunphysicalstatesfromtheHilbertspace.Ontheotherhand,itwaspointedoutthatso-calledthetimeorderedperturbationtheory(TOPT)proposedin[9]doesnotpreservethegaugesymmetry.Therefore,fortheSTNCgaugetheorytomakesense,oneneedstoconstructtheS-matrix,notonlyunitarybutalsogaugeinvariant.Insection2,unitaryS-matrixofNCabeliangaugetheoryispresented,withthepropertime-ordering.Duetothepresenceofthederivativeinteraction,thecovariant2time-orderingisnecessaryinadditiontotheminimalrealizationofthe⋆-time-orderinginscalartheories.Insection3,Feynmanruleispresentedinthemomentumspace.Insection4,STNCquantumelectrodynamicsisconsideredandFeynmanruleispresented.Insection5,thegaugeinvarianceisexplicitlycheckedtothelowestorderusingtheComptonscatteringamplitude.Thisnon-trivialcheckprovideshowthe⋆-timeorderingcuresthedefectspresentintheTOPTin[9].Insection6,classicalvacuumsolutionsisre-analyzedfortheSTNCpuregaugetheoryusingthevariousgaugefixingconditions,andhowthegaugefixingconditionsaffectthevacuumsolution.Section7istheconclusion.2S-matrixofNCabeliangaugetheoryTheNCU(1)actioninD-dimensionalspace-timeisgivenasS=−14ZdDxFμν⋆Fμν(2.1)whereFμνisthefieldstrength,Fμν=∂μAν−∂νAμ−ig[Aμ⋆,Aν]withthegaugecouplingconstantg.Theactionisgaugeinvariantunderthegaugetransformation,ieA′ν(x)=U(x)⋆∂μ−igAμ(x)⋆¯U(x).Thisshowsthatthefieldstrengthisgauge-covariantratherthangauge-invariant.Forthein-comingorout-goingphoton,however,thefieldstrengthisgaugeinvariant,sinceaccordingtothefundamentalansatzofthefieldtheory,thein-orout-photonisassumedtosubjecttothefreetheorywhichisthecommutativefieldtheory.Asnotedinthescalartheorycase,thenon-commutativenatureofspaceandtimedoesnotallowtheunitarytransformationofthequantumfieldintheintermediatetimeintofreetheory.Nevertheless,S-matrixcanbedefined,whichrelatestheout-goingfieldtothein-comingfield.Therefore,toconstructS-matrixwedonotneedtotransformthefieldstrengthinNCFTintotheabeliancommutativefieldstrengthasinSSNCgaugetheory[1].WiththeLorentzgaugefixing,theactionismodifiedasS=−ZdDx(14Fμν⋆Fμν+λ2(∂μAμ)⋆(∂νAν)).(2.2)Thisactioncanberewrittenintermsofthestar-operationF,S=ZdDx(K(x)+FxV(x))(2.3)wher