Noncommutative Quantization for Noncommutative Fie

arXiv:hep-th/0606183v36Jul2006hep-th/0606183KEK-TH-1092NoncommutativeQuantizationforNoncommutativeFieldTheoryYasumiAbe∗InstituteofParticleandNuclearStudiesHighEnergyAcceleratorResearchOrganization(KEK)Tsukuba305-0801,JapanAbstractWepresentanewprocedureforquantizingfieldtheorymodelsonanon-commutativespacetime.Thenewquantizationdependsonthenoncommu-tativeparameterexplicitlyandreducestothecanonicalquantizationinthecommutativelimit.Itisshownthataquantumfieldtheoryconstructedbythenewquantizationyeildsexactlythesamecorrelationfunctionsasthoseofthecommutativefieldtheory,thatis,thenoncommutativeeffectsdisappearcompletelyafterquantization.Thisimplies,forinstance,thatbyusingthenewquantization,thenoncommutativitycanbeincorporatedintheprocessofquantization,rahterthanintheactionasconventionallydone.∗email:[e-mail]1IntroductionNoncommutativefieldtheoryhasrecentlybeenattractingalotofinterestssincetheseminalworkofSeibergandWitten[1]whichshowedthataparticularlowenergylimitofstringtheoryisdescribedbyagaugetheoryonanoncommutativespacedefinedbythecommutationrelationforcoordinates:[xμ,xν]=iθμν,(1.1)whereθμνisanantisymmetric,constantmatrix.Thenoncommutativerelation(1.1)suggeststhatweshouldtreatcoordinates,orfieldsdependingonthem,asoperatorsratherthanc-numbarquantities.However,ifweintroducetheMoyalstarproductf(x)∗g(x)=expi2θμν∂μ∂′νf(x)g(x′)x′→x,(1.2)andreplacealltheproductsbetweenfieldsbythisstarproduct,wecanstilltreatfieldsonanoncommutativespaceasc-numberfunctions.Indeed,theMoyalstarproductisanoncommutativeproductandreproducesthecommutationrelation(1.1):[xμ,xν]∗=xμ∗xν−xν∗xμ=iθμν.(1.3)Theconventionalprescriptiontodefinetheactionforafieldtheoryonthenoncom-mutativespaceistoreplacealltheproductsintheactionforanordinarycommu-tativefieldtheorybytheMoyalstarproduct(1.2).Quantizedversionsofnoncom-mutativefiedtheoriesarealsoconsidered,andtherehavebeenalargenumberofpapersinvestigatingtheirpropertiessuchasUV/IRmixing,nonrenormalizability,noncommutativestandardmodel,violationofLorentzinvarianceandthechangeofthecausalstructure(forreviews,see,forexample,[2–4]and,forphenomenologicalaspects,seealso[5,6]).Inallsuchstudiesonnoncommutativequantumfieldtheory(QFT),thefieldisquantizedbasedonthecanonicalcommutationrelation.Thebasicreasonforthisseemsthatthecanonicalquantizationgivesanaccuratedescriptionofnatureinlowenergyscaleswhereexperimentaldataareavailable.However,thereisnoevidencetobelievethatthecanonicalquantizationisthecorrectprescriptionuptohighenergyscalessuchwherethenoncommutativefieldtheorywouldberealized.Inprinciple,insuchhighenergyscales,therecouldbeanexoticquantization,otherthanthecanonicalquantization,whichhasthesamelowenergylimitasthatofthecanonicalquantization.Inthispaper,weshowthatsuchaquantizationdoesexistbytheexampleofnoncommutativescalarfieldtheories.Thisnewquantization,whichwecallnoncom-mutativequantization,dependsonthenoncommutativeparameterθμνsuchthatitreducestothestandardcanonicalquantizationwhenθμν→0.WeshallseethatGreen’sfunctionsandtheS-matrixelementsappropriateforthenewquantization1canbedefinedandcalculatedbymeansofperturbativeexpansionsinthesamewayasanordinaryQFT.Tooursurprise,theyturnouttobeequivalenttothoseofthecorrespondingcommutativeQFTinallordersofperturbation.Infact,thisequivalenceholdsevennonperturbatively,andwecanfindanexactmapfromacom-mutativeQFTtoournewQFT.Interestingly,thenewQFTyeidsexactlythesamedynamicsasthoseofthecommutativefieldtheory,eventhoughtheactionincludesnoncommutativeinteractiontermsexplicitly.Thisisbecausethenoncommutativityininteractiontermsandthenoncommutativityintroducedinthequantizationpro-cedurecancelouteachother.Thispeculiarstructureofnewquantizationimplies,forinstance,thatifwestartfromanactionofacommutativefieldtheoryandquan-tizeitbythenoncommutativequantization,thenwegetthedynamicsequivalenttoanoncommutativefieldtheoryquantizedbythecanonicalquantization.Wewillalsoinvestigatethestructureofthiscancellationofthenoncommutativityindetail.Thispaperisorganizedasfollows.Insection2,wepresentthenoncommutativequantizationfornoncommutativescalarfieldtheorieswithpolynomialinteractionterms.Insection3,thisnewquantizationisappliedtoafreefield.Itisshownthatoperatorssimilartotheladdaroperatorscanbeintroduced,withwhicha“Fockspace”representationisconstructed.Section4isdevotedtotheinvestigationofthenoncommutativequantizationforaninteractingfieldtheory.Green’sfunctionsandtheS-matrixaredefinedandtheirequivalencetothoseofthecommutativeQFTisdiscussed.Thestructureofthisequivalenceisarguedinsomedetailinsection5.Ourconclusionanddiscussionaregiveninsection6.2NoncommutativeQuantizationTopresentournewquantizaitonforfieldtheorymodelsonanoncommutativespacewithLorentzianmetric(+,−,−,···,−),weconsider,fordefinitness,wewillconsideronlyd+1dimensionalrealscalarfieldtheorieswhoseinteractiontermsaregivenbythe(noncommutative)polynomialform:L=12h(∂μφ)2−m2φ2i+∞Xn=3λnn!nz}|{φ∗φ∗···∗φ=L0+Lint.(2.1)Forexample,theordinarynoncommutativeφ4theoryisgivenbythecasewhereonlyλ4isnonzero.Toproceedfurther,weimposetime-spacecommutativitycondition,θ0i=0,sothatwear