Dirac algebra (contraction gamma matrices)

[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.

 

[dextercioby]

Why does it have different type of indices ?

 

[IRobot]

Because the first gamma matrix is sandwiched between two spinors.

 

[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, (γ^μ)_ab(γ_μ)_cd expressed in terms of (γ^μ)_ad(γ_μ)_bc. See http://gemma.ujf.cas.cz/~brauner/files/Fierz_transform.pdf for an exhaustive treatment!