A few questions about spinors and gamma matrices

Question

I am following BUSSTEPP Lectures on Supersymmetry and trying to show that the Wess-Zumino action is invariant under SUSY transformations. I encountered the following questions about spinors and gamma matrices.

Let $\epsilon$ and $\eta$ be any two Grassmann-valued Majorana spinors. Here, $\bar{\epsilon}$ means the Majorana adjoint, i.e. $\bar{\epsilon}=\epsilon^{T}\mathcal{C}$, where $\mathcal{C}$ is the charge conjugation matrix.

I want to prove the following identities

$$\bar{\epsilon}\eta=\bar{\eta}\epsilon,\quad\bar{\epsilon}\gamma_{5}\eta=\bar{\eta}\gamma_{5}\epsilon,\quad\bar{\epsilon}\gamma^{\mu}\eta=-\eta\gamma^{\mu}\epsilon,\quad\bar{\epsilon}\gamma^{\mu}\gamma_{5}\eta=\bar{\eta}\gamma^{\mu}\gamma_{5}\epsilon$$

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I just found that they are equation (3.51) of Supergravity by Daniel Z. Freedman.

Answer 1

Your spinors are real and your Dirac matrices in the Majorana representation are imaginary, so

• $\gamma^0$ is Hermitean A(ntisymmetric)
• $\gamma^i$ are Antihermitean S(ymmetric)
• $\gamma^5$ is Hermitean A(ntisymmetric) $\leadsto$ check this from the above!

Thus $$ \bar{\epsilon}\eta= i\epsilon^T \gamma^0 \eta =-i\eta^T \gamma^{0~~T} \epsilon =i\eta^T \gamma^{0 }\epsilon = \bar{\eta} \epsilon , $$ So now you got your baseline. Make sure you appreciate every step.

The rest follow trivially from the above properties of the γ matrices and the above properties, so supplanting $\gamma^0$, you now have $$(\gamma^0 \gamma^5)^T= \gamma^5 \gamma^0= -\gamma^0 \gamma^5,$$ just like for the scalar, so $\bar{\epsilon}\gamma_{5}\eta=\bar{\eta}\gamma_{5}\epsilon$. (Anti-)likewise, $$ (\gamma^0 \gamma^\mu)^T= \gamma^0 \gamma^\mu, $$ where you can marvel at how the time-like and space like parts synchronize to yield a uniform answer, hence $\bar{\epsilon}\gamma^{\mu}\eta=-\bar{\eta}\gamma^{\mu}\epsilon$. Finally, for the axial, $\gamma^5$ flips the sign to yield $\bar{\epsilon}\gamma^{\mu}\gamma_{5}\eta=\bar{\eta}\gamma^{\mu}\gamma_{5}\epsilon$ .