[Wess&Bagger - SUSY&SUGRA] B. Results in Spinor Algebra

 This article is one of Lie Group & Representation contents.

Conventions

$$\eta_{nm}\sim (-1,1,1,1)$$

$$\epsilon_{21}=\epsilon^{12}=1,\quad \epsilon_{12}=\epsilon_{21}=-1,\quad \epsilon_{11}=\epsilon_{22}=0$$

$$\epsilon_{0123}=-1$$

$$\psi^\alpha = \epsilon^{\alpha\beta}\psi_\beta,\quad \psi_\alpha=\epsilon_{\alpha\beta}\psi^\beta$$

$$\psi\chi=\psi^\alpha\chi_\alpha=-\psi_\alpha\chi^\alpha=\chi^\alpha\psi_\alpha=\chi\psi$$

$$\bar{\psi}\bar{\chi}=\bar{\psi}_\dot{\alpha}\bar{\chi}^\dot{\alpha}=-\bar{\psi}^\dot{\alpha}\bar{\chi}_\dot{\alpha}=\bar{\chi}_\dot{\alpha}\bar{\psi}^\dot{\alpha}=\bar{\chi}\bar{\psi}$$

$$(\chi\psi)^\dagger=(\chi^\alpha\psi_\alpha)^\dagger=\bar{\psi}_\dot{\alpha}\bar{\chi}^\dot{\alpha}=\bar{\psi}\bar{\chi}=\bar{\chi}\bar{\psi}$$

$$\Psi_D=\begin{pmatrix}\chi_\alpha\\ \bar{\psi}^\dot{\alpha}\end{pmatrix}$$

$$\gamma^m=\begin{pmatrix}0& \sigma^m\\ \bar{\sigma}^m& 0\end{pmatrix}$$

$$\begin{align}\gamma^5=\gamma^0\gamma^1\gamma^2\gamma^3=\begin{pmatrix}-i& 0\\ 0& i\end{pmatrix}.\end{align}$$

Sigma Matrices

$$\begin{align}\begin{matrix}\sigma^0=& \begin{pmatrix}-1& 0\\ 0& -1\end{pmatrix} & \sigma^1=& \begin{pmatrix}0& 1\\ 1& 0\end{pmatrix}\\ \sigma^2=& \begin{pmatrix}0& -i\\ i& 0\end{pmatrix}& \sigma^3=& \begin{pmatrix}1 & 0\\ 0 & -1\end{pmatrix}.\end{matrix}\end{align}$$

$$\begin{align}\begin{matrix}\bar{\sigma}^{m\dot{\alpha}\alpha}& =& \epsilon^{\dot{\alpha}\dot{\beta}}\epsilon^{\alpha\beta}\sigma_{\beta\dot{\beta}}^m\\ \bar{\sigma}^0& =& \sigma^0\\ \bar{\sigma}^{1,2,3}& =& -\sigma^{1,2,3}.\end{matrix}\end{align}$$

$$\begin{align}\begin{matrix}\mbox{Tr} \sigma^m\bar{\sigma}^n& =& -2\eta^{mn}\\ \sigma_{\alpha\dot{\alpha}}^m\bar{\sigma}_m^{\dot{\beta}\beta}& =& -2\delta_\alpha^\beta\delta^\dot{\beta}_\dot{\alpha}.\end{matrix}\end{align}$$

$$\begin{align}\begin{matrix}(\sigma^m\bar{\sigma}^n+\sigma^n\bar{\sigma}^m)_\alpha^\beta & = & -2\eta^{mn}\delta_\alpha^\beta\\ (\bar{\sigma}^m\sigma^n+\bar{\sigma}^n\sigma^m)^\dot{\alpha}_\dot{\alpha}& =& -2\eta^{mn}\delta^\dot{\alpha}_\dot{\beta},\end{matrix}\end{align}$$

$$\begin{align}\begin{matrix}\sigma^{nm\beta}_\alpha & =&  \frac{1}{4} \left( \sigma^n_{\alpha\dot{\alpha}}\bar{\sigma}^{m\dot{\alpha}\beta} - \sigma^m_{\alpha\dot{\alpha}} \bar{\sigma}^{n\dot{\alpha}\beta} \right)\\ \sigma^{nm\dot{\alpha}}_\dot{\beta} & =&  \frac{1}{4} \left( \bar{\sigma}^{n\dot{\alpha}\alpha}\sigma^m_{\alpha\dot{\beta}} - \bar{\sigma}^{m\dot{\alpha}\alpha} \sigma^n_{\alpha\dot{\beta}} \right).\end{matrix}\end{align}$$

$$\begin{align} \sigma^{mn\alpha}_\alpha &=& 0\\ \sigma^{mn\beta}_\alpha\epsilon_{\beta\gamma}\end{align}$$

$$\begin{align} \epsilon^{abcd}\sigma_{cd} &=& -2i\sigma^{ab}\\ \epsilon^{abcd}\bar{\sigma}_{cd} &=& 2i\bar{\sigma}^{ab}.\end{align}$$

$$\begin{align}\begin{matrix} \sigma_{\alpha\dot{\alpha}}^n\sigma_{\beta\dot{\beta}}^m-\sigma_{\alpha\dot{\alpha}}^m\sigma_{\beta\dot{\beta}}^n&=& 2\left[ (\sigma^{nm}\epsilon)_{\alpha\beta}\epsilon_{\dot{\alpha}\dot{\beta}}+(\epsilon\bar{\sigma}^{nm})_{\dot{\alpha}\dot{\beta}}\epsilon_{\alpha\beta}\right] \\ \sigma_{\alpha\dot{\alpha}}^n\sigma^m_{\beta\dot{\beta}}+\sigma_{\alpha\dot{\alpha}}^m\sigma_{\beta\dot{\beta}}^n&=& -\eta^{nm}\epsilon_{\alpha\beta}\epsilon_{\dot{\alpha}\dot{\beta}}+4(\sigma^{ln}\epsilon)_{\alpha\beta}(\epsilon\bar{\sigma}^{lm})_{\dot{\alpha}\dot{\beta}}.\end{matrix}\end{align}$$

$$\begin{align}\begin{matrix}\sigma^a\bar{\sigma}^b\sigma^c+\sigma^c\bar{\sigma}^b\sigma^a=2(\eta^{ac}\sigma^b-\eta^{bc}\sigma^a-\eta^{ab}\sigma^c)\\ \bar{\sigma}^a\sigma^c\bar{\sigma}^c+\bar{\sigma}^c\sigma^b\bar{\sigma}^a=2(\eta^{ac}\bar{\sigma}^b-\eta^{bc}\bar{\sigma}^a-\eta^{ab}\bar{\sigma}^c),\end{matrix}\end{align}$$

$$\begin{align}\mbox{Tr}\sigma^{mn}\sigma^{kl}=-\frac{1}{2}(\eta^{mk}\eta^{nl}-\eta^{ml}\eta^{nk})-\frac{i}{2}\epsilon^{mnkl}.\end{align}$$

$$\begin{align}\begin{matrix}\bar{\sigma}^a\sigma^b\bar{\sigma}^c-\bar{\sigma}^c\sigma^b\bar{\sigma}^a=-2i\epsilon^{abcd}\bar{\sigma}_d\\ \sigma^a\bar{\sigma}^b\sigma^c-\sigma^c\bar{\sigma}^b\sigma^a=2i\epsilon^{abcd}\sigma_d,\end{matrix}\end{align}$$

Spinor Algebra

$$\begin{align}\theta^\alpha \theta^\beta=-\frac{1}{2}\epsilon^{\alpha\beta}\theta\theta\end{align}$$

$$\begin{align}\theta_\alpha\theta_\beta = \frac{1}{2}\epsilon_{\alpha\beta}\theta\theta\end{align}$$

$$\begin{align}\bar{\theta}^\dot{\alpha}\bar{\theta}^\dot{\beta}=-\frac{1}{2}\epsilon_{\dot{\alpha}\dot{\beta}}\bar{\theta}\bar{\theta}.\end{align}$$

$$\begin{align}\theta\sigma^m\bar{\theta}\theta\sigma^n\bar{\sigma}=-\frac{1}{2}\theta\theta\bar{\theta}\bar{\theta}\eta^{mn}.\end{align}$$

$$\begin{align}\begin{matrix}(\theta\phi)(\theta\psi)&=& -\frac{1}{2}(\phi\psi)(\theta\theta)\\ (\bar{\theta}\bar{\theta})(\bar{\theta}\bar{\psi})&=& -\frac{1}{2}(\bar{\phi}\bar{\psi})(\bar{\theta}\bar{\theta}).\end{matrix}\end{align}$$

$$\begin{align}\epsilon^{\alpha\beta}\frac{\partial}{\partial\theta^\beta}=-\frac{\partial}{\partial \theta_\alpha}.\end{align}$$

$$\begin{align}\begin{matrix}\epsilon^{\alpha\beta}\frac{\partial}{\partial\theta^\alpha}\frac{\partial}{\partial\theta^\beta}\theta\theta=4\\ \epsilon_{\dot{\alpha}\dot{\beta}}\frac{\theta}{\partial\bar{\theta}_\dot{\alpha}}\frac{\partial}{\partial\bar{\theta}_\dot{\beta}}\bar{\theta}\bar{\theta}=4.\end{matrix}\end{align}$$

$$\begin{align}\begin{matrix}\chi\sigma^n\bar{\psi} &=& -\bar{\psi}\bar{\sigma}^n\chi, & (\chi\sigma^m\bar{\psi})^\dagger &=& \psi\sigma^m\bar{\chi}\\ \chi\sigma^m\bar{\sigma}^n\psi &=& \psi\sigma^n\bar{\sigma}^m\chi, & (\chi \sigma^m\bar{\sigma}^n\psi)^\dagger &=& \bar{\psi}\bar{\sigma}^n\sigma^m\bar{\chi}.\end{matrix}\end{align}$$

$$\begin{align}(\psi\phi)\bar{\chi}_\dot{\beta} = -\frac{1}{2} (\phi\sigma^m \bar{\chi}) (\psi \sigma^m)_\dot{\beta}.\end{align}$$