Dynamical origin of the $star_theta$-noncommutativ

arXiv:hep-th/0604038v15Apr2006Dynamicaloriginofthe⋆θ-noncommutativityinfieldtheoryfromquantummechanicsMarcosRosenbaum,J.DavidVergaraandL.Rom´anJu´arezInstitutodeCienciasNucleares,UNAM,A.Postal70-543,M´exicoD.F.AbstractWeshowthatintroducinganextendedHeisenbergalgebrainthecontextoftheWeyl-Wigner-Groenewold-Moyalformalismleadstoadeformedproductoftheclas-sicaldynamicalvariablesthatisinheritedtothelevelofquantumfieldtheory,andthatallowsustorelatetheoperatorspacenoncommutativityinquantummechan-icstothequantumgroupinspiredalgebradeformationnoncommutativityinfieldtheory.Keywords:Noncommutativity,star-productsPACS:02.40Gh,11.10.Nx1IntroductionTheoreticalphysicshasprovidedusafairlydeepunderstandingofthemicro-scopicstructureofmatter,butverylittleisknownregardingthemicroscopicstructureofspace-time.Fromamethodologicalpointofview,theuseofanoncommutativestruc-PreprintsubmittedtoElsevierScience1February2008tureforspace-timecoordinateshadalreadybeenproposedintheearlydaysoffieldtheoryasafailedhopeatfindinganeffectiveandLorentzinvariantcutoffneededtocontroltheultravioletdivergencesplaguingthetheory.Fromaconceptualandtheoreticalpointofviewthereisasimpleheuristicargu-ment-basedonHeisenberg’sUncertaintyPrinciple,theEinsteinEquivalencePrincipleandtheSchwarzschildmetric-whichshowsthatthePlancklengthseemstobealowerlimittothepossibleprecisionmeasurementofposition,andthatshorterdistancesdonotappeartohaveanoperationalmeaning[1].ThusQuantumMechanicsandFieldTheory,atdimensionsoftheorderofthePlancklength,oughttoincorporateintheirverystructurethenoncom-mutativityofspace-timebyreplacingtheconceptofaspace-timepointbyacellofadimensiongivenbythePlanckscalearea.Underthesepremisestheveryconceptofmanifoldasanunderlyingmathematicalstructureofphysi-caltheoriesbecomesquestionableandsomepeopleareconvincedthatanewparadigmofgeometricalspaceisneeded.ThenoncommutativegeometryofConnes[2],whichbyresortingtoarbitraryandnoncommutativeC∗-algebrasdualizesgeometryandreplacesitsusualnotionsofmanifoldsandpointsbyanewcalculusbasedonoperatorsinHilbertspaceandtheuseofspectralanalysis,epitomizesthislineofthought.Morerecentlytherehasbeenfurtherevidenceofspace-timenoncommutatitvity[3]comingfromcertainmodelsofstringtheorywhich,althoughwithageometryquitedifferentfromthatofnoncommutativegeometryisnotincompatiblewithit,andhasledtothesameissueofnoncommutativityofspace-timeatshortdistances.Inthenoncommutativequantumfieldtheoryrootedonthephenomenologyofthelowenergyapproximationofstringtheoryinthepresenceofastrongmagneticbackground,thefieldsonatargetspaceofspace-timecanonicalco-ordinatesarereplacedbyaC∗-algebraoffunctionswithadeformedproduct2givenbythesocalledGroenewold-Moyalstar-product:f(x)⋆θg(x)=f(x)e(i2←−∂iθij−→∂j)g(x),(1)wheretheconstantrealandinvertibleanti-symmetrictensorθijhasdimen-sionsoflengthsquared.Oneinterpretation(seee.g.[4])fortheoriginofthisnoncommutativityisbasedonpostulatingthereplacementofthespace-timeargumentofcanonicalcoordinatesxioffieldoperatorsbya“space-time”ofHermitianoperatorsobeyingtheHeisenbergalgebra[ˆxi,ˆxj]=iIθij,i,j=1,...,2d(2)whereIisanidentityoperator.OperatorsO(ˆx),actingonaHilbertspaceofdelta-functionnormalizablefunctionsind-dimensions,arethendefinedintermsofthebasicoperators(2)bymeansoftheWeylbasisg(α,ˆx)=eiαiˆxi.UsingnowtheWeyl-MoyalcorrespondenceO(ˆx)=Zd2dαg(α,ˆx)˜OW(α),(3)where˜OW(α)istheFouriertransformoftheWeylfunctioncorrespondingtoO,itfollows,incompleteanalogytotheresultsderivedfromtheWeyl-Wigner-Groenewold-Moyal(WWGM)formalismofquantummechanics(seethefollowingsection),thattheWeylfunctioncorrespondingtotheoperatorproductO1O2isgivenby(O1)W⋆θ(O2)W.(4)Forareviewofnoncommutativequantumfieldtheorybasedonthesecriteriasee,e.g.,[5].AnalternativeandLorentzinvariant(inthetwistedsymmetrysense)inter-pretationoftheoriginofthestar-product(1)comesfromconsideringthe3twistedcoproductoftheHopfalgebraHoftheuniversalenvelopingU(P)ofthePoincar´ealgebraP.Itcanbeshown(seee.g.[9])thatforacertainDrinfeldtwistingofthecoproductwithaninvertibleF∈U(P)⊗U(P)suchthatF12(Δ⊗id)F=F23(id⊗Δ),(ǫ⊗id)F=1=(id⊗ǫ)F,(5)thiscoproductinducesadeformationintheproduct,m→mF,ofthemod-ulealgebraA=C∞(M)overH,suchthattheactionofHonApreservescovariance,i.e.h⊲mF(a⊗b)=m◦[(F−1(1)⊲a)⊗(F−1(2)⊲b)]=a⋆θb,(6)wherea,b∈Aandh∈H,andwehaveusedtheSweedlernotationthrough-out.Inparticular,consideringthecoordinatesxiaselementsofA,equation(6)impliesthat[xi,xj]⋆θ≡xi⋆θxj−xj⋆θxi=iθij.(7)Note,however,thatalthoughbothoftheabovedescribedrepresentativelinesofthoughtleadtothesamealgebraofoperatorsfornoncommutativequantumfieldtheory,theoriginsofthisnoncommutativityappeartobequitedifferent.Inthelatercase,ashasbeenstressedbyChaichianetal.,theproduct(7)isinheritedfromthetwistoftheoperatorproductofquantumfieldsandnononcommutativityofthecoordinateswasusedinthederivationof(6);whileinthelineofthoughtdescribedin[4]theassumednoncommutativityofthespace-timeoperatorsformsanessentialpartoftheensuingarguments.How-ever,theinferencethatthemultiplicationinthealgebraoffieldsisgivenbythestar-product(6)isanexternalingredientimportedfromthephenomenol