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Setup

Let $C$ be the charge conjugation operator for spinors and $\gamma$ a Dirac matrix. From this post we conclude that the critical relation between the operator and the Dirac matrices is $$-C(\gamma^\mu)^*C^{-1}=\gamma^\mu.\tag{1}$$ Wikipedia on the other hand states that $$-C(\gamma^\mu)^T C^{-1}=\gamma^\mu.\tag{2}$$

Question

Are these two statements equivalent?

My attempt at showing they are

Let's take $(1)$ and transpose it $$\begin{align*} (\gamma^\mu)^T&= -(C(\gamma^\mu)^*C^{-1})^T =-(C^{-1})^T(\gamma^\mu)^\dagger C^T =-C(\gamma^\mu)^\dagger C^{-1}\\ &=-C\gamma^0\gamma^\mu\gamma^0 C^{-1}, \end{align*}$$ where we have used $C^{-1}=C^T=C^\dagger=-C$ in the Dirac, Majorana and Weyl basis. We now fix the basis to Dirac, so that $C=i\gamma^2\gamma^0$. Inserting this in the previous result we get $$\begin{align*} (\gamma^\mu)^T&=-C\gamma^0\gamma^\mu\gamma^0 C^{-1} = -(i\gamma^2\gamma^0)\gamma^0\gamma^\mu\gamma^0 (-i\gamma^2\gamma^0)= -i\gamma^2\gamma^\mu i\gamma^2\\ &\neq -C^{-1}\gamma^\mu C, \end{align*}$$ where the last line is equivalent to $(2)$. Since the equivalence doesn't hold in the Dirac basis, it shouldn't hold in any other basis either.

So it seems these statements are not equivalent, but in that case I wonder which of the two is true? And why?

joigus
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Sito
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1 Answers1

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The definition of the charge transformation matrix on Wikipedia is correct. Actually, it cannot be easily transformed into the first one, a relation like $\gamma^{\mu\dagger} =\gamma^\mu$ would make that possible. But this is wrong.

For $\gamma$-matrices we have $\gamma^{\mu\dagger} = \gamma^0\gamma^\mu \gamma^0$.

Frederic Thomas
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