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1、1
2、ClassicalSymmetriesTheconceptofsymmetrywillplayacrucialroleinnearlyallaspectsofourdiscussionofweakinteractions.Atthelevelofthedynamics,thefundamentalinteractions(oratleastthatsubsetofthefundamentalinteractionsthatweunderstand)areassociatedwithgauge
3、symmetries".Butmorethanthat,theunderlyingmathematicallanguageofrelativisticquantummechanics
4、quan-tum eldtheory
5、ismucheasiertounderstandifyoumakeuseofallthesymmetryinformationthatisavailable.Inthiscourse,wewilmakeextensiveuseofsymmetryasamathematicalto
6、oltohelpusunderstandthephysics.Inparticular,wemakeuseofthelanguageofrepresentationsofLiealgebras.1.1Noether'sTheorem{ClassicalAttheclassicallevel,symmetriesofanactionwhichisanintegralofalocalLagrangiandensityareassociatedwithconservedcurrents.Consider
7、asetof elds, (x)wherej=1toN,andanactionjZ4S[ ]=dxL( (x)@ (x))(1.1.1)whereListhelocalLagrangiandensity.Theindex,j,iswhatparticlephysicistscalla
avor"index.Di erentvaluesofjlabeldi erenttypes,or
avors",ofthe eld .Thinkofthe eld, ,withoutanyexpliciti
8、ndex,asacolumnvectorin
avorspace.Assume,forsimplicity,thattheLagrangiandependsonlyonthe elds, ,andtheir rstderivatives,@ .TheequationsofmotionareLL@=:(1.1.2)(@ ) Notethat(1.1.2)isavectorequationin
avorspace.Eachsideisarowvector,carryingthe
avor
9、index,j.Asymmetryoftheactionissomein nitesimalchangeinthe elds, ,suchthatS[ + ]=S[ ](1.1.3)orL( + @ +@ )=L( @ )+@V( @ )(1.1.4)whereVissomevectorfunctionoftheorderofthein nitesimal, .Weassumeherethatwecan4throwawaysurfacetermsinthe
10、dxintegralsothattheVtermsmakesnocontributiontotheaction.ButLLL( + @ +@ );L( @ )= +@ (1.1.5) (@ )1WeakInteractions
11、HowardGeorgi
12、draft-February10,1998
13、2because@ =@ .Notethat(1.1.5)isasingleequationwithnojindex.Thetermsontherighthands
14、ideinvolveamatrixmultiplicationin
avorspaceofarowvectorontheleftwithacolumnvectorontheright.From(1.1.2),(1.1.4)and(1.1.5),wehave@N=0(1.1.6)whereLN= ;V:(1.1.7)(@ )Often,wewilbeinterestedinsymmetriesthataresymmetriesoftheLagrangian,notj
