PROBLEMS OF QUANTUM FIELD THEORY

ESITheErwinSchrodingerInternationalBoltzmanngasse9InstituteforMathematicalPhysicsA-1090Wien,AustriaTheThirringModel40YearsLaterN.IlievaW.ThirringVienna,PreprintESI587(1998)August18,1998SupportedbyFederalMinistryofScienceandTransport,AustriaAvailableviaanonymousftporgopherfromFTP.ESI.AC.ATorvia:{5xy{1998July,1998TheThirringModel40YearsLater?N.Ilieva;]andW.ThirringInstitutfurTheoretischePhysikUniversitatWienandErwinSchrodingerInternationalInstituteforMathematicalPhysicsAbstractSolutionstotheThirringmodelareconstructedintheframeworkofalgebraicquantumeldtheory.Itisshownthatforallpositivetemperaturestherearefermionicsolutionsonlyifthecouplingconstantis=q2(2n+1);n2N,otherwisesolutionsareanyons.Dierentanyons(whichareuncountablymany)liveinorthogonalspaces,sothewholeHilbertspacebecomesnon-separableandineachofitssectorsadierentUrgleichungholds.Thisfeaturecertainlycannotbeseenbyanypowerexpansionin.Moreover,ifthestatisticparameteristiedtothecouplingconstantitisclearthatsuchanexpansionisdoomedtofailureandwillneverrevealthetruestructureofthetheory.Onthebasisofthemodelinquestion,itisnotpossibletodecidewhetherfermionsorbosonsaremorefundamentalsincedressedfermionscanbeconstructedeitherfrombarefermionsordirectlyfromthecurrentalgebra.InvitedtalkattheXIInternationalConferencePROBLEMSOFQUANTUMFIELDTHEORYInmemoryofD.I.BlokhintsevJuly1998,Dubna,Russia?Worksupportedinpartby\FondszurForderungderwissenschaftlichenForschunginOsterreichundergrantP11287{PHY;Permanentaddress:InstituteforNuclearResearchandNuclearEnergy,BulgarianAcademyofSciences,Boul.TzarigradskoChaussee72,1784Soa,Bulgaria]E{mailaddress:[e-mail]fasolubleQFT[1]manypeopletriedtondotherexamples;buttosolveanontrivialrelativisticQFTseemedoutofthequestion.TheideathatBethe’sansatz[2]couldbesuccessfullyusedtosolvealsoHeisenberg’s\Ur-gleichung[3]reducedtoonespaceonetimedimensionthenledtoasolublerelativisticeldtheory{theThirringmodel[4].Duringtheyears,thismodelhasnotonlybeenextensivelystudiedbuthasalsobeenactivelyusedforanalysis,testingandillustrationofvariousphenomenaintwo{dimensionaleldtheories.Itisnotourpurposetoreviewtheenormousliteratureonthesubjectbutweratherfocusontheverystartingpoint{Heisenberg’sUrgleichung.Withnobosonspresentinitatall,itrepresentstheultimateversionoftheopinionthatfermionsshouldenterthebasicformalismofthefundamentaltheoryofelementaryparticlesthatisusuallytakenforgranted.Theoppositepointofview,namelythatatheoryincludingonlyobservableelds,necessarilyunchargedbosons,iscapableofdescribingevolutionandsymmetriesofaphysicalsystem,beingthekernelofalgebraicapproachtoQFT[5],alsoenjoysanenthu-siasticsupport.Aswewillsee,thereisnopossibilitytojudgethismatteronthebasisofthemodelinquestion,sincebothformulationscanbeequallywellusedtoconstructthephysicallyrelevantobjects{thedressedfermions.Inanycase,beforeclaimingthatan\Urgleichungofthetype6@(x)=(x)(x)(x)(1.1)determinesthewholeUniverseoneshouldseewhetheritdeterminesanythingmathe-maticallyanditisouraiminthepresentnotetodiscusstheelementsneededtomakeitssolutionwelldened.Infactweshallrstconsideronlyonechiralcomponentandweshallrestrictourselvestothetwo{dimensionalspacetime,sothatthiscomponentdependsonlyononelightconecoordinate.Alsothebose{fermidualitytakesplacethereandwewanttomakeuseofit.Thisphenomenonamountstothefactthatincertainmodelsformalfunctionsoffermieldscanbewrittenthathavevacuumexpectationvaluesandstatisticsofbosonsandviceversa,theequivalencebeingunderstoodwithinperturbationtheory.Thebose{fermidualityisactuallywellestablishedwhentheconstructionofbosonsoutoffermionsisconsidered.Theproblemofrigorousdenitionsofoperator{valueddistributionsandeventuallyoperatorshavingthebasicpropertiesoffermionsbytakingfunctionsofbosoniceldsisrathermoredelicate.Onthelevelofoperatorvalueddistri-butionssolutionshavebeengivenbyDell’Antonioetal.[6]andMandelstam[7]andonthelevelofoperatorsinaHilbertspace|byCareyandcollaborators[8,9]andinaKreinspacebyAcerbi,MorchioandStrocchi[10].2Thusourgoalistogiveaprecisemeaningtothefollowingthreeingredients(a)[(x);(x0)]+=(xx0);[(x);(x0)]+=0CAR(b)1iddx(x)=j(x)(x)Urgleichung(c)j(x)=(x)(x)Current(1.2)Eq.(1.2b)involves(derivativesof)objectswhichareaccordingto(1.2a)ratherdiscon-tinuous.ThereforeitisexpedienttopassrightawaytothelevelofoperatorsinHilbertspacesincethevarietyoftopologiesthereprovidesabettercontroloverthelimitingprocedures.Ingeneralnormconvergencecanhardlybehopedforbutwehavetostriveatleastforstrongconvergencesuchthatthelimitoftheproductistheproductofthelimits.Withf=R11dxf(x)(x),(1.2a)becomes[f;g]+=hfjgi(1.3)forf2L2(R)andh:j:ithescalarproductinL2(R).Thisshowsthatf’sareboundedandformtheC{algebraCAR.Therethetranslationsx!x+tgiveanautomorphismtandweshallusethecorrespondingKMS{states!andtheassociatedrepresentationtoextendCAR.Thoughtherej=1,onecangiveameaningtojasastronglimitinHbysmearing(x)overaregionto(x)anddenejf=Zdxf(x)lim!0((x)(x)!((x)(x)));f:R!RTheselimit