Derivation of a gamma matrices identity

Question

While studying Srednicki's book on quantum field theory, I encountered a particular identity that is of interest to me (equation 36.40): $$\mathcal{C}^{-1}\gamma^\mu\mathcal{C}=-(\gamma^\mu)^T$$ where $\mathcal{C}$ is the charge conjugation operator, and $\gamma^\mu$ the well-known gamma matrices. This identity is shown to be true using the chiral/Weyl representation. However, I would like to be able to show it to be true without choosing a representation. Is something like this possible? If yes, could someone outline the procedure for me? Any help would be much appreciated.

This post imported from StackExchange Physics at 2014-03-06 21:53 (UCT), posted by SE-user Danu

Answer 1

Yes, you can show this using only the fact that the Clifford Algebra has a unique representation up to similarity transformation in any dimension. This is shown in the first few pages of

http://arxiv.org/pdf/hep-th/9811101.pdf

Then you observe that if $\gamma^\mu$ obeys the clifford algebra, then so does $-(\gamma^\mu)^T$. $\mathcal{C}$ is then defined as the similiarity transformation between the two representations, whose existence is guaranteed by the uniqueness of the representation of the Clifford algebra.

This post imported from StackExchange Physics at 2014-03-06 21:53 (UCT), posted by SE-user Dan