Transpose of Grassmanian variables

[ismaili]

Say, $$\psi^1,\ \psi^2$$ are Dirac spinors, and $$M$$ is a matrix composed of Dirac matrices. Is the following equation hold?
$$\bar{\psi^1}M\psi^2 = -\Big(\bar{\psi^1}M\psi^2\Big)^T$$
I'm not quite sure, here is my derivation:
$$\bar{\psi^1}M\psi^2 = \bar{\psi^1}_{i}M_{ij}\psi^2_j = - \psi^2_jM^T_{ji}\psi^1_i = -\Big(\psi^1_iM_{ij}\psi^2_j\Big)^T = -\Big(\psi^1M\psi^2\Big)^T$$
where the last step, i.e. the third equality is what I'm worried about. I originally think that the change of order of the third equality is totally due to the operation of transpose (superscript "T"), so there is no need to change sign in the third equality.
Am I right?

 

[azntoon]

Yes, you are right. The transpose operation does not change the sign of the equation, so the third equality is correct.

 

[denian]

Yes, your derivation is correct. The change of order in the third equality is due to the operation of transpose, so there is no need to change the sign. The transpose of Grassmanian variables follows the same rules as regular matrix transpose, so the equation holds.