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I am reading the BBS, Exercise 5.1

This exercise is nothing but showing that two Majorana spinors $\Theta_1$ and $\Theta_2$ \begin{align} \bar{\Theta}_1 \Gamma_{\mu} \Theta_2 = -\bar{\Theta}_2 \Gamma_\mu \Theta_1 \end{align} In BBS, there are solutions for exercise that I did not fully understand.

First I know the \begin{align} \bar{\Theta}_1 \Gamma_{\mu} \Theta_2 = \Theta_1^{\dagger} \Gamma_0 \Gamma_{\mu} \Theta_2 = \Theta_1^T C \Gamma_{\mu} \Theta_2 \end{align} They said that this can be written in the form \begin{align} - \Theta_2^T \Gamma_{\mu}^T C^T \Theta_1 = - \Theta_2^T C \Gamma_{\mu}\Theta_1 = -\bar{\Theta}_2 \Gamma_{\mu} \Theta_1 \end{align} What i don't understand is why

\begin{align} \Theta_1^T C \Gamma_{\mu} \Theta_2=- \Theta_2^T \Gamma_{\mu}^T C^T \Theta_1 \end{align}

cf) I know that for the gamma matrix for Majorana spinor follows \begin{align} C\Gamma_{\mu}= - \Gamma_\mu^T C \end{align} which is related with above equation.

References:

[BBS] Becker, Becker, Schwarz, "String theory and M-theory: A modern Introduction".

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The relation you ask about is just a reshuffling of the components. Writing out the indices we have $$ \Theta_1^T C \, \Gamma_{\mu} \Theta_2 = (\Theta_1^T)_a C_{ab} \, (\Gamma_{\mu})_{bc} (\Theta_2)_c = - (\Theta_2)_c (\Gamma_{\mu})_{bc} C_{ab} (\Theta_1^T)_a = - (\Theta_2^T)_c (\Gamma_{\mu}^T)_{cb} (C^T)_{ba} (\Theta_1)_a $$ where the minus sign in the second step came from switching the order of the two fermions. Removing the indices again we then have $$ \Theta_1^T C \, \Gamma_{\mu} \Theta_2 = - \Theta_2^T\Gamma_{\mu}^T C^T \Theta_1 $$ which is what you asked about.

Depending on your spinor conventions you might need to be more careful about the placement of the spinor indices than I was above, but the general idea should be the same.

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  • $\begingroup$ Thank your for answer!. It helps me a lot. Now i totally understand!. From your answer, I finally derive the other useful spinors identities! $\endgroup$ – phy_math Jul 3 '15 at 15:33

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