Understanding Gamma Matrix Logic with Clifford Algebra

[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!

 

[mark726]

I can help clarify the explanation given in the forum post. The author is discussing the properties of the Clifford algebra, which is a mathematical tool used in physics to represent and manipulate objects with multiple indices, such as tensors. In this case, the objects being manipulated are the gamma matrices, which are used in quantum field theory to describe the behavior of spin particles.

The first equation, \gamma^{\mu} \gamma^{\nu}=\gamma^{\mu \nu} + \eta^{\mu \nu}, is a well-known property of the gamma matrices. The gamma matrices are represented by complex matrices, and the notation used here is a shorthand for representing the matrix multiplication. The \gamma^{\mu \nu} term represents the symmetric part of the product, while the \eta^{\mu \nu} term represents the antisymmetric part. This is a general property of matrices, and it can be proven by explicit calculation.

Moving on to the second equation, \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]}, the author is showing how the product of two gamma matrices with four indices can be decomposed into three terms. The first term, \gamma^{\mu \nu \rho}{}_{\sigma \tau}, is the totally antisymmetric Clifford matrix that contains all the indices. This term represents the antisymmetric part of the product.

The second term, 6 \gamma^{[\mu \nu}{}_{[\tau} \delta^{\rho]}{}_{\sigma]}, is the sum of all possible index pairings between the two factors. The brackets [ ] indicate that the indices inside are being antisymmetrized. For example, in the term \gamma^{[\mu \nu}{}_{[\tau} \delta^{\rho]}{}_{\sigma]}, the indices \mu and \nu are being antisymmetrized, as well as the indices \tau and \rho. This term represents the symmetric part of the product.

The third term, 6 \gamma^{[\