The Role of Type III Factors in Quantum Field Theo

arXiv:math-ph/0411058v24Dec2004TheRoleofTypeIIIFactorsinQuantumFieldTheory∗JakobYngvasonInstitutf¨urTheoretischePhysik,Universit¨atWien,Boltzmanngasse5,1090Vienna,AustriaandErwinSchr¨odingerInstituteforMathematicalPhysics,Boltzmanngasse9,1090Vienna,AustriaEmail:[e-mail]9,2004AbstractOneofvonNeumann’smotivationsfordevelopingthetheoryofoperatoral-gebrasandhisandMurray’s1936classificationoffactorswasthequestionofpossibledecompositionsofquantumsystemsintoindependentparts.Forquan-tumsystemswithafinitenumberofdegreesoffreedomthesimplestpossibility,i.e.,factorsoftypeIintheterminologyofMurrayandvonNeumann,areper-fectlyadequate.Inrelativisticquantumfieldtheory(RQFT),ontheotherhand,factorsoftypeIIIoccurnaturally.Thesameholdstrueinquantumstatisticalmechanicsofinfinitesystems.InthisbriefreviewsomephysicalconsequencesofthetypeIIIpropertyofthevonNeumannalgebrascorrespondingtolocalizedobservablesinRQFTandtheirdifferencefromthetypeIcasewillbediscussed.Thecumulativeeffortofmanypeopleovermorethan30yearshasestablishedaremarkableuniquenessresult:ThelocalalgebrasinRQFTaregenericallyiso-morphictotheunique,hyperfinitetypeIII1factorinConnes’classificationof1973.Specifictheoriesarecharacterizedbythenetstructureofthecollectionoftheseisomorphicalgebrasfordifferentspace-timeregions,i.e.,thewaytheyareembeddedintoeachother.JohnvonNeumannwasthefatheroftheHilbertspaceformulationofquantumme-chanics[1]thathasbeenthebasisofalmostallmathematicallyrigorousinvestigationsofthetheorytothisday.Westartbyrecallingthemainconceptsandexplainingsomenotations.∗LecturegivenatthevonNeumannCentennialConference,Budapest,October15–20,20031TheobservablesofaquantumsystemareselfadjointoperatorsA=A∗onacom-plex,separableHilbertspaceH.Formathematicalconvenienceweconsideronlyob-servablesinthethealgebraB(H)ofboundedoperatorsonH.IfψisanormalizedvectorinH,i.e.,hψ,ψi=1,whereh·,·idenotesthescalarproductonH,thentheexpectationvalueofanobservableAisgivenbyωψ(A)=hψ,Aψi.(1)Thisisapositive,linearfunctionalofAwithωψ(1)=1andωψisreferredtoasthestatedefinedbyψ.Moregeneralstatesaregivenbydensitymatricesρ:ωρ(A)=tr(ρA)=Xiλihψi,Aψii(2)whereλi≥0withPiλi=1aretheeigenvaluesandψithenormalizedeigenvectorsofthepositivetraceclassoperatorρ.IfS⊂B(H)thenitscommutantisdefinedasS′={B∈B(H):[A,B]=0forallA∈S}.(3)ThisisalwaysasubalgebraofB(H)andifSis∗-invariantthensoisS′.Moreover,itisclosedinthetopologydefinedbythestates.AvonNeumannalgebraMisa∗-subalgebraofB(H)thatisequaltoitsdoublecommutant,i.e.,M=M′′.(4)AbasiclemmaofvonNeumannsaysthatthisisequivalenttothealgebrabeingclosedinthetopologydefinedbythestates.Inaseriesoffourpapers[2,3,4,5]MurrayandvonNeumannstudiedspecialvonNeumannalgebras,calledfactors.ThesearethealgebrasMsuchthatM∨M′≡{AB:A∈M,B∈M′}′′=B(H),(5)i.e.,B(H)is“factorized”intoManditscommutant,M′.ThisisequivalenttoM∩M′=C1,(6)i.e.,thecenterofMcontainsonlymultiplesoftheidentityoperator.ThemathematicalproblemaddressedbyMurrayandvonNeumannwastoclassifyallpossibilitiesforsuchalgebras.Thisproblemismotivatedbyquestionsofmathematicalnaturebutalso“severalaspectsofthequantummechanicalformalismstronglysuggesttheelucidationofthissubject”[2].Oneoftheseaspectsisthedivisionofaquantumsystemintotwoinde-pendentsubsystems.Inthesimplestcasethisisachievedasfollows.TheHilbertspaceiswrittenasatensorproduct,H=H1⊗H2.(7)2Theobservablesofonesystemcorrespondtothe(selfadjoint)elementsofM=B(H1)⊗1andtheothertothecommutantM′=1⊗B(H2).Theobservablesofthetotalsystemarethusfactorized:B(H)=B(H1)⊗B(H2).(8)SuchafactorizationoftheobservablealgebraofthetotalsystembasedonatensorproductfactorizationoftheunderlyingHilbertspaceis,intheterminologyofMurrayandvonNeumann,theTypeIcase.ItischaracterizedbytheexistenceofminimalprojectionsinM:Ifψ∈H1andEψ=|ψihψ|isthecorrespondingprojectionontheone-dimensionalsubspaceofH1generatedbyψ,thenE=Eψ⊗1∈M(9)isaminimalprojection,i.e.,ithasnopropersubprojectionsinM.TheotherextremeistheTypeIIIcase.Hereforeverynon-zeroprojectionE∈MthereexistsaW∈MwhichmapsEHisometricallyontoH,i.e.W∗W=1,WW∗=E.(10)VonNeumannhimselfknewonlysporadicexamplesoftypeIIIfactorsandhemay,infact,nothavebeenfullyawareoftheirsignificance.HewasapparentlymoreattractedbytheTypeIIcasethatliesbetweenthetwoextremesjustdescribed:AtypeIIfactorhasnominimalprojections,buteverynon-zeroprojectionEhasasubprojectionFEthatisfiniteinthesensethatF′F,W∗W=F,WW∗=F′impliesF′=F.Thiscase,althoughveryinterestingfromamathematicalpointifview,islessimportantinquantumfieldtheorythantheothertwoandwillnotbediscussedfurtherhere.AfinerclassificationoftypeIIIfactors,basedonTomita-Takesakimodulartheory[6],waspioneeredbyA.Connes[7].Thereisacontinuumofnonequivalenttypes,denotedIIIλwith0≤λ≤1,andthecaseIII1turnsouttobeofparticularimportanceforrelativisticquantumphysics.1ThecharacteristicfeatureoftypeIII1isthatthespectrumoftheTomita-TakesakimodulargroupsisthewholeofR.Amongthefactorsthataregeneratedbyanincreasingfamilyoffinitedimensionalsubalgebras(suchfactorsarecalledhyperfinite)thereis,uptoequivalence,onlyonefactoroftypeIII1.ThisuniquenessresultisduetoU.Haagerup[10].Theo