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.

up vote 3 down vote accepted

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


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.

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