Return to 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

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

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

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

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

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 Dirac spinors. Here, $\bar{\epsilon}$ means the Dirac adjoint, i.e. $\bar{\epsilon}=\epsilon^{\dagger}\gamma^{0}$.

Are the following identities true?

$$(\bar{\epsilon}\eta)^{\dagger}=\bar{\epsilon}\eta,\quad(\bar{\epsilon}\gamma_{5}\eta)^{\dagger}=\bar{\epsilon}\gamma_{5}\eta,\quad(\bar{\epsilon}\gamma^{\mu}\eta)^{\dagger}=\bar{\epsilon}\gamma^{\mu}\eta.$$

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

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