# Majorana vector current

Tags:
1. Jan 4, 2019

### gasar8

1. The problem statement, all variables and given/known data
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).

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

3. 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?