A New Type of String Field Theory

arXiv:hep-th/0111240v127Nov2001hep-th/0111240NBI-HE-01-11YITP-01-79November2001YukawaInstituteKyotoANewTypeofStringFieldTheory∗HolgerB.NielsenNielsBohrInstitute,Blegdamsvej17,CopenhagenøDenmarkandMasaoNinomiyaYukawaInstituteforTheoreticalPhysics,KyotoUniversity,Kyoto,606-8502,JapanAbstractWeproposeanewwayofsecondquantizingstringtheory.ThemethodisbasedonconsideringtheFockspaceofstringsdescribedbyconstituentswhichmakeuptheXμRandtheXμLi.e.therightandleftmovermodesseparately.AstatewithanynumberofstringsgetrepresentedbytheCartesianproductoftwofreeparticleFockspaces,oneforrightmoverdegreesoffreedom,andoneforleft.Theresultingstringfieldtheoryisafreetheory.∗ToappearintheProceedingsofThe10thInternationalSymposiumonStrings,July3-7,Fukuoka,Japan,2001(AIPPress)1.IntroductionThereexistalreadyseveralvariantsofstringfieldtheoriesalongthelineoftheKaku-Kikkawa’sone[1]whichhasforanystatesingle-stringcreationandannihi-lationoperatorssothatvariousnumbersofstringscanbepresentinthedifferentsingle-stringstates.Inmodelsofthiskindofsecondquantizedstringtheoriesonecandistinguishtwotwo-stringstateswhicharedenotedas|1iand|2)i(seeFigure1)althoughtheylooksomewhatsimilarinthefollowingway:1)Thestate|1)iisatwo-particlestateinwhichtwoopenstringsarepresentinsuchaconfigurationthatthetwostringsliejustalongthesamecurveforapiecesomewhereinthemiddleofthestrings.2)Thestate|2)iisacorrespondingtwo-stringstatetotheoneunder1),butthetwostringsfolloweachothersomewhereinthemiddlebypermutingsotospeakthe“tails”ofthetwostringsintheFock-spacestate|1)i.Thatistosaythatthetwo-particlestate|2)iintheFockspacedescribestwostringsoneofwhich“half”coincideswithapieceofstringnumberoneinFockstate|1)iwhiletheother“half”insteadcoincideswiththe“tail”partofthesecondstringinFockstate|1)i.ThetwoFockstates|1)iand|2)ihavesomestringmaterialpresent–insingleordoubleamounts–injustthesamecurvepiecesinspace,sothattheycanonlybedistinguishedifonecanfindouthowthestringpieceshangtogether.NeverthelessstringfieldtheoriessuchasKaku-Kikkawa’sone[1],Kyotogroup’s(HIKKO’s)one[2],Witten’scubicone[3],andZwiebach’sone[4]havetwo-particlestates|1)iand|2)iasmentionedthatarecountedasquitedifferent,distinguishableFockspacestates.ItisthepurposeofthepresentworktopresentideastomakeastringfieldtheorymodelrepresentingtheclassoftheoriesinwhichtheFockstates|1)iand|2)iarenotdistinguishablebutratherrepresentthesamephysicalstate.ThisclassofmodelshasnotbeenmuchstudiedunlessonecountsthatthestringsofQCDaswellasthestringsofmatrixmodels[5,6]arereallyofthistype.InQCDyouhaveonlylocalfieldstodescribewhereastringispresentanditwouldseemveryhardtoseehowtwoQCD-stringslyingontopofeachotheralongapieceofcurvecouldgettheir“heads”and“tails”associatedwitheach–2–otherbyQCDdegreesoffreedominformation.SoitseemsmuchmorelikelythatQCDdevelopsthetypeofstringswherethestates|1)iand|2)ijustdescribedabovemustbeidentified.ThisconclusionbecomesevenmoreobviousifweuseastrongcouplingapproximationasthemethodforimplementingthestringsintothelatticeQCDorjustYangMillstheories.Then,thestringsbecomefluxquantaofcolorelectricfluxandthereisnowaytokeeptrackoforidentifypartsofthesamestring.QCDorYangMillstheoriesaswellasalsomatrixmodelsprovidesmodelsofstringfieldtheoriesofthesametypeaswearegoingtoproposeinthepresentarticle.Itis,however,ourgoaltomakestringfieldtheorymodelthatdoesnotneedaveryhardandnon-linearcalculationtoconnecttothestringpictureasQCDneeds.Weshallindeedseethatourmodelisinspiredbyaninfinitesetofseeminglyconservedquantitiesnoticeableinclassical(i.e.nonquantummechanical)stringtheory,asweshallexplaininthefollowingsectionII.Thenweshallstartthede-scriptionofourstringfieldtheoryinsectionIII.Acrucialcomplicationofourmodelisthatitneedsaconstraintensuringthateach“constituent”inXR-orXL-spacehasasuccessorconstituentasshallbedescribedinsectionIV.Sinceourmodelhasatfirstsomebadfeaturesbecausetoomanystateshavebeenmadeidenticalitisfarfromobviousthatourmodelisindeedanacceptablestringfieldtheory.ItisthereforeabsolutelycrucialthatitcouldbeusedtodeducetheVeneziano-modelscatteringamplitude.ThatweshallbrieflysketchinsectionV.FinallyinsectionVIweshallresumeandconcludeamongotherthingsthatourmodelisafreethe-oryandthatitthusbecomesimportantforjudgingthevalidityofstringtheoryasamodelfornature,ifreallyafreetheorycouldbethemodelfornature.2.InspirationbytheconservationofrightandleftmovingpatternsThecrucialobservationthathasinspiredourproposalforstringfieldtheoryoriginatesfromconsideringclassicalstring“scattering”whichtakesplacebyacoupleofpermutingtheir“tails”whenstringstouchinonepoint.Herebyweunderstandthat,saytwostringscomealonginsuchawaythatinamomentoftimetheyhaveonepointincommon,butthatthenafterthismomentthestringsdevelopasiftheywereadifferentpairofstrings.Namelytheoneobtainedbycombiningthefirstpartofthestringnumber1withthesecondpartofstringnumber2,andviceversa.Wecallthatthestringsgettheirtailspermutedwhenthebeginningsandendsoftheoriginalstringsarecombinedwiththeendsinandifferentway.–3–Consider–inclassicalapproximation–twostringsdescribedbeforethecollisionby:Thefirststring:XμI(σ,τ)=XμRI(σ,τ)+Xμ