Lectures on Gamma Matrix and Supersymmetry

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1、LecturesonGammaMatrixandSupersymmetryJeong-HyuckPark∗PhysicsDepartment,SungkyunkwanUniversityChunchun-dong,Jangan-gu,Suwon440-746,KoreaAbstractThislecturenotesurveystheGammamatrixingeneraldimensionswitharbitrarysignature,thestudyofwhichisessentialtos

2、tudysupersymmetryinthedimension.∗Electroniccorrespondence:jhp@newton.skku.ac.kr1GammaMatrixWestartwiththefollowingwellknownLemma.LemmaAnymatrix,M,satisfyingM2=λ26=0,λ∈Cisdiagonalizable,andfurthermoreifthereisanotherinvertiblematrix,N,whichanti-commut

3、eswithM,{N,M}=0,thenMis2n×2nmatrixoftheform!λ0−1M=SS.(1.1)0−λInparticular,trM=0.1.1InEvenDimensionsInevend=t+sdimensions,withmetricµνη=diag(++···+−−···−),(1.2)

4、{z}

5、{z}tsgammamatrices,γµ,satisfytheCliffordalgebraµννµµνγγ+γγ=2η.(1.3)With1µ1µ2···µm[µ1µ2µ

6、m]γ=γγ···γ,(1.4)wedefineΓM,M=1,2,···2dbyassigningnumberstoindependentγµ1µ2···µm,e.g.imposingµ1<µ2<···<µm,Mµµνµ1µ2···µm12···dΓ=(1,γ,γ,···,γ,···,γ).(1.5)Then{ΓM}/Zformsagroup2ΓMΓN=ΩMNΓL,ΩMN=±1,(1.6)whereLisafuctionofM,NandΩMN=±1doesnotdependonthespecific

7、choiceofrepresentationofthegammamatrices.Lemma(1.1)implies1MNMNtr(ΓΓ)=ΩMNδ,(1.7)2n1“[]”meansthestandardanti-symmetrizationwith“strengthone”.1whichshowsthelinearindependenceof{ΓM}sothatanygammamatrixshouldnotbesmallerthan2d/2×2d/2.Intwo-dimensions,one

8、cantakethePaulisigmamatrices,σ1,σ2asgammamatriceswithapossiblefactor,i,dependingonthesignature.Ingeneral,onecanconstructd+2dimensionalgammamatricesfromddimensionalgammamatricesbytakingtensorproductsasµ123(γ⊗σ,1⊗σ,1⊗σ):uptoafactori.(1.8)Thus,thesmalle

9、stsizeofirreduciblerepresentationsis2d/2×2d/2and{ΓM}formsabasisof2d/2×2d/2matrices.Byinductiononthedimensions,fromeq.(1.8),wemayrequiregammamatricestosatisfythehermiticityconditionµ†γ=γµ.(1.9)Withthischoiceofgammamatriceswedefineγ(d+1)asq(d+1)t−s12dγ=

10、(−1)2γγ···γ,(1.10)satisfyingγ(d+1)=(γ(d+1))−1=γ(d+1)†,(1.11){γµ,γ(d+1)}=0.Fortwosetsofirreduciblegammamatrices,γµ,γ0µwhicharen×n,n0×n0respectively,weconsideramatrixX0MM−1S=ΓT(Γ),(1.12)MwhereT,isanarbitrary2n0×2nmatrix.ThismatrixsatisfiesforanyNfromeq.