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arXiv:hep-th/9501108v224Jan1995February,2008SAL-TH-94-02,hep-th/9501108MasslessScalarFieldTheoryinaQuantisedSpace-TimeJ.C.Breckenridge,aV.Elias,bandT.G.SteeleaaDepartmentofPhysicsandEngineeringPhysicsandSaskatchewanAcceleratorLaboratory,UniversityofSaskatchewan,Saskatoon,Saskatchewan,CanadaS7N0W0bDepartmentofAppliedMathematics,UniversityofWesternOntario,London,Ontario,CanadaN6A5B9ABSTRACTAmethodisdevelopedtoconstructanon-localmasslessscalarfieldtheoryinaflatquantisedspace-timegeneratedbyanoperatoralgebra.Implicitintheoperatoralgebraisafundamentallengthscaleofthespace-time.Thefundamentaltwo-pointfunctionoffreefieldsisconstructedbyassumingthatthecausalGreenfunctionsstillhavesupportonthelightconeintheoperatoralgebraquantisedspace-time.Incontrasttopreviousstochasticapproaches,themethodintroducedhererequiresnoexplicitaveragingoverspacetimecoordinates.Thetwo-andfour-pointfunctionsofgϕ4theoryarecalculatedtotheone-looplevel,andnoultravioletdivergencesareencountered.ItisalsodemonstratedthattherearenoIRdivergencesintheprocessescon-sidered.Int.Class.forPhysics:0370,1110,04,046021.IntroductionTheexistenceofaminimumobservablelengthinstringtheories[1]andtheevidencesuggestingtheexistenceoffundamentallengthscalesinquantumgravity[2]suggeststhattheconstructionofaquantumfieldtheory(QFT)inspace-timeswhichpossessafundamentalscalebecomesanimportantquestion.Manifestlynon-localactionsforQFTscontainderivativesofinfiniteorderandnecessarilycontainfundamentallengthscaleswhileretainingthecontinuousnatureofspacetime[3].AnalternativeappproachoftheformulationofQFTina(flat)quantisedspace-timewasfirstattemptedbySnyder[4].Thisquantisedspace-timenaturallyintro-ducesafundamentallengthscale,whichaswellasdescribingthespace-timeitself,canalsoregulateUVdivergencesoftheQFT.ThislatteraspectwasthemotivationforSnyder’swork.TheSnyderspace-timeisconstructedessentiallybyaddingnon-commutingoper-atorsforthespace-timecoordinatestotheusualPoincar´egroup.Thespatialcoor-dinateoperatorshavespectrathatareintegermultiplesofafundamentallengtha,whilethetimecoordinateoperatoradmitsacontinuousspectrum.Momentumspaceremainscontinuous,withcommutingmomentumoperators.Thesefeaturesresultinaspace-timestructureviolatingtheBornreciprocityprinciple,orthesymmetrybetweenconfigurationandmomentumspacerepresentationsoffieldtheory.Thislackofreciprocitymakestheconstructionoffieldtheoriesdifficult,duetorelianceonthisfeatureofmostofthepresentlyestablishedformalisms.Thecommonoperationsofanalysisareeitherextremelycumbersome,havingtobecarriedoutintermsofsummations(seeLee[5]),orcannotbecarriedoutatall.Snyder[6]wasabletoformulateaversionoftheclassicalelectromagneticfieldinhisquantisedspace-timebutthereisnoevidencethatthisapproachwaseffectiveforthecaseofaquantumfieldtheory.Gol’fand[7,8],advocatedtheconstructionofaquantumfieldtheoryinamo-mentumspaceofconstantcurvature.ThisisinsomerespectssimilartotheSnyderconstructionsincetheSnyderspacetimeisgeneratedbyanoperatoralgebraformu-latedinanabstractspaceofconstantcurvature.However,inGol’fand’stheory,the3additionofmomentaarenon-commutative,resultinginchangestothelawsofcon-servationofmomentumandenergy.Attemptstorestoresymmetrybetweencoordinateandmomentumspace,thusmaintainingaccesstoestablishedmethodsofQFTarethebasisoftheworkofNam-srai[9–13],NamsraiandDineykhan[14],DineykhanandNamsrai[15–18],andDineykhan,EfimovandNamsrai[19].Theseresearchersusetheconceptofastochas-ticspace-time,inwhichcoordinatesareassumedtoundergoquantumfluctuations.Explicitaveragingoverthesestochasticcoordinatesremovestheasymmetrybetweenconfigurationandmomentumspace.ThisallowsNamsrai[10]tomakeuseoftheexistingformalismoffieldtheory,withcertainchangesintroducedbytheaveraging.Thecombinationofthespace-timestochasticityandtheaveragingresultsinthetheoryhavingamanifestlynon-localnature,asaresultoftheexplicitdependenceoftheGreen’sfunctions,orpropagators,onderivativesofinfiniteorder[3,20].Thisapproachallowsthefamiliarconstructionoffieldtheoriesintermsofanactionprin-ciplewithappropriatelymodifiedpropagators,whichhavetheimportantpropertyofbeingsufficientlyconvergentthattheresultingtheoryisultravioletfinite.Theaveragingoverthespace-time,however,hasthedisadvantageofbeingratheradhoc,andnotcontainedwithinthedynamicsofthetheory.Inthispaperanewmethodwillbeusedtoconstructamasslessscalarfieldinaquantisedspace-timegeneratedbyanoperatoralgebra.Section2developsthealgebraofthespace-time,anditisshownthatthePoincar´ealgebraremainsaninvariantsubgroup.ThemasslessfreescalarfieldsareconsideredinSection3leadingtoamodifiedfreepropagator,whichdependsonthefundamentalscaleofthespace-time.Theone-looptwo-,andfour-pointfunctionsarecalculatedinSection4formasslessscalargϕ4inthequantisedspace-time.ThefreepropagatoriscomparedtothepropagatoroftheNamsraitheoryandfoundtohaveremarkablysimilarbehaviourintheenergyregimep21/a2.Thelargep2asymptoticbehaviour,however,isquitedifferent.Thisdifferenceisperhapsnotsurprisingsincefailuresofthestochasticaveragingatsmallscaleswherethealgebraofthequantizedspacetimebecomescrucialwillleadtodistinctlargep2behaviour.42.Operat