Why Does the Majorana Vector Current Vanish?

[gasar8]

Homework Statement

I am trying to show that Majorana vector current vanishes. I am following this article and I am trying to get to the very right hand side of eq. (27).

Homework Equations

$$\psi_M^C = \psi_M,\\ \psi^C_M = C \overline{\psi}_M^T,\\ C^T=-C, \hspace{1cm} C^T\gamma_{\mu}C = -\gamma_{\mu}^T.$$

The Attempt at a Solution

Vector current is
$$\overline{\psi^C}_M \gamma_{\mu} \psi_M^C = C\psi_M^T \gamma_{\mu} C \overline{\psi}_M^T\\ =-\overline{\psi}_M^T C \gamma_{\mu} C \psi_M^T\\ =\overline{\psi}_M^T C^T \gamma_{\mu} C \psi_M^T\\ = -\overline{\psi}_M^T \gamma_{\mu}^T \psi_M^T,$$
where I switched the order of $\psi_M^T$ and $\overline{\psi}_M^T$ from first to second row and then used the last row of relevant equations to the end result.
Is this correct and enough?

 

[mark726]

Your solution looks correct and sufficient to show that the Majorana vector current vanishes. However, it may be helpful to provide a brief explanation of your steps and how they lead to the final result. Additionally, you may want to mention any assumptions or definitions that were used in your solution. Overall, your response provides a clear and concise explanation for the forum post.