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Definition of time-ordered product for Dirac spinors

  1. Mar 10, 2010 #1
    I guess the answer to this question actually should be pretty obvious, but I still have problems getting it right though. I wonder about the definition of the time ordered product for a pair of Dirac spinors. In all the books I've read it simply says:

    [tex]T\left\{\psi(x)\bar{\psi}(x')\right\} = \theta(t - t')\psi(x)\bar{\psi}(x') - \theta(t' - t)\bar{\psi}(x')\psi(x)[/tex]

    The spinor indices are always left out. So should it be A:

    [tex]T\left\{\psi_\alpha(x)\bar{\psi}_\beta(x')\right\} = \theta(t - t')\psi_\alpha(x)\bar{\psi}_\beta(x') - \theta(t' - t)\bar{\psi}_\beta(x')\psi_\alpha(x)[/tex]

    or B:

    [tex]T\left\{\psi_\alpha(x)\bar{\psi}_\beta(x')\right\} = \theta(t - t')\psi_\alpha(x)\bar{\psi}_\beta(x') - \theta(t' - t)\bar{\psi}_\alpha(x')\psi_\beta(x)[/tex]?

    I personally think the A definition feels more natural, but when I use it in my derivations I get strange results. On the other hand, the B definition gives more reasonable results. It could simply be that I've done some mistakes in the derivations, but before I dig into those I want to know if I've got the definition right in the first place.
  2. jcsd
  3. Mar 10, 2010 #2
    Sorry, I found what I did wrong in the derivations, and now I get it out right with the A definition. :)
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