# What is the correct relation between Dirac matrices and Charge conjugation?

### 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?

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 $$\gamma^{\mu\dagger} = \gamma^0\gamma^\mu \gamma^0$$.