# Derivation of a gamma matrices identity

+ 5 like - 0 dislike
125 views

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

+ 3 like - 0 dislike

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
answered Dec 8, 2013 by (220 points)

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar\varnothing$sicsOverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.