Field Theory of Yang-Mills Quantum Mechanics for D

arXiv:0705.1960v3[hep-th]11Jul2007@hep-th/yymmnnnUT-KOMABA/07-6May2007FieldTheoryofYang-MillsQuantumMechanicsforDParticlesTamiakiYoneya∗InstituteofPhysics,UniversityofTokyoKomaba,Meguro-ku,Tokyo153-8902,JapanAbstractWeproposeanewfield-theoreticframeworkforformulatingthenon-relativisticquan-tummechanicsofDparticles(D0branes)inaFockspaceofU(N)Yang-MillstheorieswithalldifferentNsimultaneously.D-particlefieldoperators,whichcreateandannihilateaDparticleandhencechangethevalueofNbyone,aredefined.ThebasespaceoftheseDparticlefieldsisa(complex)vectorspaceofinfinitedimension.ThegaugeinvarianceofYang-Millsquantummechanicsisreinterpretedasaquantum-statisticalsymmetry,whichistakenintoaccountbysettingupanovelalgebraicandprojectivestructureintheformalism.OrdinaryphysicalobservablesofYang-Millstheory,obeyingthestandardalgebra,areexpressedasbilinearformsoftheD-particlefields.Togetherwiththeopen-closedstringduality,ournewformulationsuggestsatrinityofthreedifferentbutdualviewpointsofstringtheory.∗E-mailaddress:tamathep1.c.u-tokyo.ac.jp11.IntroductionTheideaofaquantizedfieldisapreciseexpressionoftheprimordialdualitybetweenwavesandparticlesinquantumtheory.Historically,itstemmedfromthequantizationoftheclassicalelectromagneticfieldonthesideofwaves,whileontheparticlesidefromthesecondquantizationoftheconfiguration-spaceformulationofnon-relativisticparticlequantummechanics.Asaunificationoftwoclassicallydifferentconcepts,thenotionofaquantizedfieldshouldbetakenasmorefundamentalthanitsancestors.Itisthereforenaturalthatstringfieldtheory[1,2,3,4]hasbeendevelopedfromtheearlydaysofstringtheorytowardsitsnon-perturbativeformulation.Indeed,eventhoughduringthefirsttwodecadesoftheirinitialdevelopmentthevariousversionsofstringfieldtheoryrepresentmererewritingsoftheworld-sheetCFTdynamicsofstringsintermsofordinaryFeynmanrules,thesituationhasbeenchanginginrecentyears:Stringfieldtheory,especiallytheversionofopen-stringfieldtheoriesfirstproposedintheseminalworkRef.[2],isnowplayinganimportantroleinthestudiesofsomecrucialaspectsofnon-perturbativestringphysics,suchastachyoncondensation,albeitstillataclassicallevel[5].WiththeadventofD-branes,itbecamerecognizedthatthenaturesofopenandclosedstringfieldtheoriesarequitedifferent:Openstrings,andhenceopen-stringfields,shouldbeunderstoodasthecollectivedegreesoffreedomfordescribingD-branes,whileclosedstringsarethebulkdegreesoffreedomcorrespondingtogravitonsandtheirpartnersforwhichD-branescanberegardedassourcesorassoliton-likeobjects.Inclosedstringfieldtheory,theycanemergeasclassicalsolutions(withorwithoutsources)fortheequationsofmotion.Inprinciple,thesetwodescriptionsmustbeconnectedthroughthedualitybetweenopenandclosedstrings,andconstituteafoundationforgauge-gravitycorrespondence.However,therehasbeenonlylimitedsuccessinformulatingtheopen-closedstringdualityfromthisviewpoint.Forexample,ithasnotbeenclarifiedhowtoextractthebulkdegreesoffreedomofclosedstringsconcretelyfromopen-stringfieldtheoryandviceversa,althoughsomesuggestivepropertieshavebeendiscovered,suchasthoserelatedtolumpsolutions,tobeinterpretedasD-branesintheproposaloftheso-called‘vacuum’stringfieldtheory[6]inbosonicstringtheory.Inordinarylocalquantumfieldtheory,itiswellknownthatthereisananalogousphenomenon,whichprovidesusaclearandrigorousexampleofadualityrelatedto2solitons.Thatistheduality[7]betweenthemassiveThirringmodelandthesine-Gordonmodelintwo-dimensionalspacetime.Inthiscase,thekink(andanti-kink)solitonsofthelatteraredescribedbyamassiveDiracfieldwhoseinteractionisdescribedbythenon-linear4-fermitermintheformer.Thesetwodescriptionscanbetranslatedintoeachotherinbothdirectionsthroughbosonizationorfermionization.ItshouldalsobementionedthatintwodimensionsthemasslessQED,theSchwingermodel[8],providesasuggestiveanalogtotheopen-closedstringduality,inthesensethattheone-loopquantumeffectofthemasslessDiracfieldgivesapolesingularitycorrespondingtoacompositemassivescalarfield.Thisisreminiscentofthephenomenonthattheone-loopamplitudesofopenstringsgiverisetopolesingularitiesassociatedwiththepropagationofaclosedstring.Intheseexamplesoftheformulationsofduality,itiscrucialthatthesolitondegreesoffreedomaretreatedasaquantizedfield,namelyastheDiracfieldforwhichthebosoniza-tiongivesthesine-Gordonfield(orthefreemassivescalarfieldinthecaseoftheSchwingermodel).TheDiracfieldcorrespondstoD-branesdescribedcollectivelybyopenstrings,whiletheelementarybosonicfields,astheanalogofaclosedstringfield,areobtainedfrombilinearformsoftheDiracfield.Theseconsiderationsprovidemotivationforattemptingtoconstructacertainfield-theory-likeformalismfortreatingD-branes,inthehopeofclarifyingthefundamen-talopen-closedstringdualityandtherebyseekingnewlanguagesfornon-perturbativestring/Mphysics.Itseemsworthwhiletoexplorethepossibilityoffieldoperatorscreat-ingandannihilatingD-branes,withinthesetupoftheFockspaceofD-branes.AlthoughsomeofthemathematicalideasusuallyemployedindescribingD-branes,suchastheK-theoryapproach,involvecertainaspectsofchangingthenumberofD-branes(andanti-D-branes)fromtheviewpointoftheirtopologicalproperties,theydonotseemfromaphys