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A spinor identity

  1. Dec 3, 2008 #1
    Actually, the original motivation is to check the closure of SUSY
    [tex]\delta X^\mu = \bar{\epsilon}\psi^\mu[/tex]
    [tex]\delta \psi^\mu = -i\rho^\alpha\partial_\alpha X^\mu\epsilon[/tex]
    where [tex]\rho^\alpha[/tex] is a two dimensional gamma matrix, and [tex]\psi^\mu[/tex] ia s two dimensional Majorana spinor, the index [tex]\mu[/tex] in the two dimensional world is just some label of different fields.
    I try to prove
    [tex][\delta_1,\delta_2]\psi^\mu = 2i\bar{\epsilon}_1\rho^\alpha\epsilon_2\ \partial_\alpha\psi^\mu[/tex]
    The following identity will help me a lot to prove the above formula,
    [tex]\chi_A(\xi\eta) = - \xi_A(\eta\chi) - \eta_A(\chi\xi)\cdots(*)[/tex]
    where [tex]A[/tex] is the spinor index and [tex]\chi,\xi,\eta[/tex] are three spinors.
    My question is, I don't know how to prove (*), and I don't know those spinors in (*) are Majorana spinors or not, moreover, I even don't know those spinors live in what dimension!
    Does anyone know how to prove (*)? or anyone know the reference which treat the algebra of spinors in arbitrary dimension? Thanks a lot!
     
  2. jcsd
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