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1、QuantumYang-MillsTheoryArthurJaffeandEdwardWitten1ThePhysicsofGaugeTheorySincetheearlypartofthetwentiethcentury,ithasbeenunderstoodthatquantummechanicsisneededtodescribenatureatthesubatomicscale.Inquantummechanics,thepositionandvelocityofaparticlearenoncommuting
2、operatorsactingonaHilbertspace,andclassicalnotionssuchas“thetrajectoryofaparticle”donotapply.Butquantummechanicsofparticlesisnotthewholestory.Innineteenthandearlytwentiethcenturyphysics,manyaspectsofnatureweredescribedintermsoffields–theelectricandmagneticfieldst
3、hatenterinMaxwell’sequations,andthegravitationalfieldgovernedbyEinstein’sequations.Itwasclearbythelate1920’sthat,sincefieldsinteractwithparticles,togiveaninternallycoherentaccountofnature,thequantumconceptsmustbeappliedtofieldsaswellasparticles.Whenthisisdone,quan
4、titiessuchasthecomponentsoftheelectricfieldatdifferentpointsinspacetimebecomenoncommutingoperators.WhenoneconstructsaHilbertspaceinwhichtheseoperatorsact,onefindsmanysurprises.Thedistinctionbetweenfieldsandparticlesbreaksdown,sincetheHilbertspaceofaquantumfieldiscon
5、structedintermsofparticle-likeexcitations.Conventionalparticles,suchaselectrons,arereinterpretedasarisingfromthequantizationofafield.Intheprocess,onefindsthepredictionof“antimatter”;foreveryparticle,theremustbeacorrespondingantiparticle,withthesamemassandopposite
6、electriccharge.SoonafterbeingpredictedbyP.A.M.Diraconthebasisofquantumfieldtheory,the“positron”oroppositelychargedantiparticleoftheelectronwasdiscoveredincosmicrays.TheQuantumFieldTheories(QFT’s)thathaveprovedtobemostimportantindescribingelementaryparticlephysic
7、saregaugetheories.Theclassicalexampleofagaugetheoryisthetheoryofelectromagnetism.ThegaugegroupistheabeliangroupU(1).IfAdenotestheU(1)gaugeconnection,whichlocallycanberegardedasaone-formonspacetime,thenthecurvatureorelectromagneticfieldtensoristhetwo-formF=dA,and
8、Maxwell’sequationsread0=dF=d∗F.Here∗istheHodgedualityoperator;Hodgeindeedintroducedhiscelebratedtheoryofharmonicformsasageneralizationtop-formsofMaxwell’sequations.Maxwell’s
