# Gamma Matrix Logic

1. Jan 20, 2013

### latentcorpse

I'm reading through some notes on the Clifford algebra and at one point, the author talks about how we can break $$\gamma^{\mu} \gamma^{\nu}$$ into symmetric and antisymmetric pieces and write it as $$\gamma^{\mu} \gamma^{\nu}=\gamma^{\mu \nu} + \eta^{\mu \nu}$$

He then claims underneath that $$\gamma^{\mu \nu \rho} \gamma_{\sigma \tau} = \gamma^{\mu \nu \rho}{}_{\sigma \tau} + 6 \gamma^{[\mu \nu}{}_{[\tau} \delta^{\rho]}{}_{\sigma]} + 6 \gamma^{[\mu} \delta^{\nu}{}_{[\tau} \delta^{\rho]}{}_{\sigma]}$$

Now, he claims that this result can be understood by simple logic rather than having to do an explicit calculation. He offers the following explanation:

"This follows the pattern of first writing the totally antisymmetric Clifford matrix that contains all the indices and then adding terms for all possible index pairings. We write the indices στ in down position to make it easier to indicate the antisymmetry. The second term contains one contraction. One can choose three indices from the first factor and two indices from the second one, which gives the factor 6. For the third term there are also six ways to make two contractions. The δ-functions contract indices that were adjacent, or separated by already contracted indices, so that no minus signs appear."

Can somebody please explain to me what this means. I can kind of see the contractions (but need some clarification) and I don't follow the numerical factors AT ALL! Why would the last term still come with a 6?

Thanks a lot!