1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook