[Physics] Property of Charge Conjugation Operator

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In class, we have defined the Charge Conjugation Operator ($C$) such that:
\begin{equation}
C \left(\gamma^\mu\right)^T C^{-1} = – \gamma ^\mu ,
\end{equation}

\begin{equation}
\psi^C \equiv C\,\overline{\psi}^T ,
\end{equation}

\begin{equation}
(\psi^C)^C = \psi,
\end{equation}

with $\overline{\psi} = \psi^\dagger \gamma ^0$.

I would like to demonstrate that $C^T = – C$ in every representation, but I am not sure how to proceed.

What I tried: Applying the first equation twice we get:
\begin{eqnarray}
\gamma ^\mu &\; = C \left[ C (\gamma^\mu)^T C^{-1} \right]^T C^{-1} \\
&= C (C^{-1})^T \gamma ^\mu C^T C^{-1}.
\end{eqnarray}

This can only be true if
\begin{equation}
C^T = \pm C.
\end{equation}

Now I have to restrict C to be antisymmetric. I tried using the third equation, but I've got to:
\begin{equation}
(\psi ^C)^C = \psi = C (\gamma ^0 )^T C^* (\gamma ^0)^T \gamma^0 \psi \gamma ^0,
\end{equation}
Now I am not sure how to proceed. Any ideas?

Best Answer

I'll post the answer n case anybody needs it.

We have that $$\psi ^C = C \;\overline{\psi} ^T = C \left( \psi ^\dagger \gamma ^0\right)^T = C (\gamma ^0 )^T \psi^*,$$

using the first property in the question: $$C (\gamma ^0) ^T C^{-1} = - \gamma ^0 \implies C (\gamma^0)^T = - \gamma ^0 C$$

we have $$\psi = (\psi ^C)^C = -\left( \gamma ^0 C \psi ^* \right)^C = - C \left(\overline{\gamma ^0 C \psi ^*}\right)^T. $$

Since $\overline{\gamma}^0 = \gamma ^0$ and $\overline{C} = \gamma^0 C ^\dagger \gamma ^0$, then $$\psi= - C \left( \psi^T \gamma ^0 \gamma^0 C^\dagger \gamma^0 \gamma ^0 \right)^T = - C C^* \psi.$$

For this to be true, we must have $$- C C ^* = \mathbb{1} \implies - C^\dagger C^T = \mathbb{1}. $$

Since it is true that $C C^\dagger = \mathbb{1}$, we have that $C^T = - C$.

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