# Definition of time-ordered product for Dirac spinors

1. Mar 10, 2010

### sith

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:

$$T\left\{\psi(x)\bar{\psi}(x')\right\} = \theta(t - t')\psi(x)\bar{\psi}(x') - \theta(t' - t)\bar{\psi}(x')\psi(x)$$

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

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

or B:

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

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. Mar 10, 2010

### sith

Sorry, I found what I did wrong in the derivations, and now I get it out right with the A definition. :)