# Dirac algebra (contraction gamma matrices)

1. Jan 16, 2012

### IRobot

I would like to have a general formula, and I am quite sure it must exist, for: $\gamma^{\mu}_{ab}\gamma_{\mu \,\alpha\beta}$ but I didn't succeed at deriving it, or intuiting it, I am troubled by the fact that it must mix dotted and undotted indices.

2. Jan 16, 2012

### dextercioby

Why does it have different type of indices ?

3. Jan 16, 2012

### IRobot

Because the first gamma matrix is sandwiched between two spinors.

4. Jan 17, 2012

### Bill_K

IRobot, I think what you are referring to are the Fierz identities, which express the product of two Dirac matrices in terms of matrices in the crossed channel. That is, (γμ)abμ)cd expressed in terms of (γμ)adμ)bc. See here for an exhaustive treatment!