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Inner product for Dirac Spinors

  1. Mar 12, 2010 #1
    1. The problem statement, all variables and given/known data
    Show that [tex] \psi (\gamma^a\phi)=-(\gamma^a\phi)\psi [/tex]

    2. Relevant equations

    Maybe [tex] \{\gamma^a, \gamma^b\}=\gamma^a\gamma^b+\gamma^b\gamma^a=2\eta^{ab}I [/tex]

    Perhaps also:

    [tex] (\gamma^0)^{\dag}=\gamma^0 [/tex] and [tex] (\gamma^i)^{\dag}=-(\gamma^i) [/tex]

    3. The attempt at a solution
    The gammas are matrices so I guess we start with

    [tex] \psi_{\mu}[(\gamma^a)^{\mu\nu}\phi_{\nu}] [/tex]
    [tex] =\psi_{\mu}[(((\gamma^a)^*)^{\dag})^{\nu\mu}\phi_{\nu}] [/tex]
    [tex] =-[(((\gamma^a)^*))^{\nu\mu}\psi_{\mu}]\phi_{\nu} [/tex]

    Which looks almost correct except the *, and also I'm not sure if I was supposed to assume that a can only refer to spatial indices, not the 0 which is equal to its hermitian conj, not minus it.

    Thanks for any help
  2. jcsd
  3. Mar 14, 2010 #2
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