Remarks on Time-Space Noncommutative Field Theorie

arXiv:hep-th/0103069v19Mar2001CERN-TH/2001-052ZH-TH06/01hep-th/0103069RemarksonTime-SpaceNoncommutativeFieldTheoriesL.Alvarez-Gaum´ea,J.L.F.Barb´ona,1andR.Zwickya,baTheoryDivision,CERNCH1211Geneva23SwitzerlandbInstitutf¨urtheoretischePhysik.UniversitaetZ¨urichWinterthurerstrasse190,8057Z¨urichSwitzerlandAbstractWeproposeaphysicalinterpretationoftheperturbativebreakdownofunitarityintime-likenoncommutativefieldtheoriesintermsofproductionoftachyonicparticles.Theseparticlesmaybeviewedasaremnantofacontinuousspectrumofundecoupledclosed-stringmodes.Inthisway,wegiveaunifiedviewofthestring-theoreticalandthefield-theoreticalno-goargumentsagainsttime-likenoncommutativetheories.Wealsoperformaquantitativestudyofvariouslocalityandcausalitypropertiesofnoncommutativefieldtheoriesatthequantumlevel.CERN-TH/2001-052March20011OnleavefromtheDepartmentodeF´ısicadePart´ıculas.UniversidadedeSantiagodeCompostela,Spain.IntroductionQuantumfieldtheoriesinnoncommutativespaces[1,2]withnoncommutativetimeco-ordinatearenotoriouslyill-defined.Heuristically,noncommutativeparticlesofmomentumpμcanberegardedasrigid,extendeddipolesorientedalongthefour-vectorLμ=θμνpν,andinteractingthroughtheend-points[3].Itisalreadyclearfromthisconsiderationthatnoncommutativetimecoordinatesimplyparticleseffectively‘extendedintime’.Thus,thebreakdownofnaivecriteriaoflocalcausality,intheformof‘advancedeffects’intree-levelscatteringprocesses,shouldnotcomeasasurprise(c.f.[4,5]).Onamoretechnicallevel,thesetheorieshavenostraightforwardHamiltonianquanti-zation(seehowever[6])andaredefined,inanoperationalsense,viatheFeynmandiagramexpansion[7,8].Itisthusnecessarytocheckexplicitlytheunitarityofthetheory.In-deed,inatheorywithbuilt-innonlocalityintime,itishardtoimagineanappropriatenotionofcausalitywhichisatthesametimeusefulandnontrivial.ItismorelikelythatthesetheoriesmustbeinterpretedasS-matrixtheories,inthesamesenseascriticalstringtheory.Thus,fromthisveryfundamentalpointofview,wemayregardtheexistenceofaconsistentS-matrixastheweakestpossiblenotionofcausality.Morespecifically,itwasshownin[9,10]thatthestringyregularizationofsuchsystemsisill-defined.Following[11,12],wecanrecovercertainnoncommutativefieldtheories(NCFTs)asanappropriatelow-energylimitofD-branedynamicsinthebackgroundofaconstantelectromagneticfield.Whilethislimitissmoothinstringperturbationtheoryforthecaseofabackgroundmagneticfield,itisproblematicforabackgroundelectricfield1.Thisistiedtothewell-knowninstabilitiesofopen-stringdynamicsinthepresenceofelectricfields[14],i.e.beyondamaximalvalueoftheelectricfield,theD-branebecomeseffectivelytachyonic.Sincethetime-noncommutativityisdirectlyrelatedtotheelectricfield,andthismustbelarge(instringunits)inthelow-energylimit,onefindsthatthelow-energylimitalwaysliesonthetachyonicregimeofD-branedynamics.Therefore,consistencyofthestringbackgroundrequiresthescaleoftimenoncommutativitytobeofthesameorderofmagnitudeasthestringscale.Thebreakdownofthestringyregularizationfortime-NCFTisastronghintattheinconsistencyofthesemodels.Still,onecanimaginesomecontrivedanalyticcontinuationofthebackgroundparameters,sothattheopen-stringperturbationtheorydoesconvergeintheformallow-energylimittotheseriesofFeynmandiagramsofthetime-NCFT.Inparticular,aformalcontinuationoftheclosedstringcouplinggs→igsandanexchangeoftherolesofspaceandtimeintheplaneoftheelectricfieldwoulddothejob(c.f.[10]).Therefore,itisdesirabletofindaninternalinconsistencyoftheFeynmanexpansionintime-NCFTs.Suchaninconsistencywasfoundin[15],whereaviolationoftheunitaritycuttingruleswasreportedinanumberofexamples.Theauthorsof[15]showedthatscatteringamplitudeshaveextrasingularitiesthatcannotbeunderstoodintermsofunitaritythresholds.Initself,thisdoesnotfurnishano-goargumentfortime-NCFT,sinceFeynmandiagramsinordinarytheoriesareknowntopresenttheso-called‘anomalousthresholds’whoseinterpretationintermsofunitarityrulesisveryproblematic(see[16]1ThemarginalcaseofanullelectromagneticbackgroundE·B=E2−B2=0isalsosmooth,c.f.[13].1forareview).Thus,itiscrucialtointerpretphysicallythenewsingularitiesinordertoevaluatetheviabilityoftime-NCFTs.OneinterestingpossibilitywouldbethatNCFTsmimicopen-stringtheoryinthesensethatunitarityoftheS-matrixrestrictedtoopen-stringinitialandfinalstatesrequirestheintroductionofclosedstringsasstatesintheasymptoticHilbertspace,sincetheycontributesingularitiesinintermediatechannels.Inananalogousfashion,itispossiblethattheS-matrixoftime-NCFTbecomesunitaryonceweaddappropriatenewstatestotheasymptoticHilbertspace.Infact,suchapossibilityishintedatbythesimplestexampleofa‘noncommutativesingularity’,i.e.thecaseofthenormal-orderingcorrectiontothepropagatorofmass-lessφ4∗theoryinfourdimensions.Inthecommutativetheory(oratthelevelofplanardiagramsinthenoncommutativetheory)thenormal-orderingdiagramhasnoanalyticstructure,sinceitcontributesaquadraticallydivergentconstantrenormalizingthemass.Ontheotherhand,thenonplanardiagram,viewedasaone-to-onescatteringamplitude,isgivenbyiA(p)=−iλ·16Zd4q(2π)4ei˜pqiq2+i0=−iλ24π21p◦p+i0,(1)where˜pμ≡pνθνμandp◦p≡−˜p2.Thus,wefindacontributiontotheim