[Wess&Bagger - SUSY&SUGRA] A. Notation and Spinor Algebra

 This article is one of the posts in the Textbook Commentary Project.

We use the metric $\eta_{mn}\sim (-1,1,1,1)$ throughout these lectures. Furthermore, we work with Weyl spinors in the Van der Waerden notation.

To begin, we define $M$ to be a two-by-two matrix of determinant one: $M\in \mbox{SL}(2,C)$. The matrix $M$, its complex conjugate $M^*$, its transpose inverse $(M^T)^{-1}$, and its hermitian conjugate inverse $(M^\dagger)^{-1}$ all represent $\mbox{SL}(2,C)$ ($4$ components). The represent the action of the Lorentz group on two-component Weyl spinors.

Representation of $\mbox{SL}(2,C)$

Two-component spinors with upper or lower dotted or undotted indices transform as follows under $M$: $$\begin{align}\begin{matrix} \psi'_\alpha =& M_\alpha^\beta \psi_\beta & \bar{\psi}'_\dot{\alpha} =& (M^*)^{\dot{\beta}}_\dot{\alpha}\bar{\psi}_\dot{\beta}\\ \psi'^\alpha =& (M^{-1})^{\alpha}_\beta \psi^\beta & \bar{\psi}'^\dot{\alpha} =& (M^{*-1})^{\dot{\alpha}}_\dot{\beta} \bar{\psi}^\dot{\beta}.\end{matrix}\end{align}$$ Spinor are denoted by Greek indices. Those with dotted indices transform under the $(0,\frac{1}{2})$ representation of the Lorentz group, while those with undotted indices transform under the $(\frac{1}{2},0)$ conjugate representation.

$\mbox{SL}(2,C)$ and Lorentz group

The connection between $\mbox{SL}(2,C)$ and the Lorentz group is established through the $\sigma$-matrices $$\begin{align}\begin{matrix} \sigma^0=& \begin{pmatrix}-1&0\\ 0&-1\end{pmatrix} & \sigma^1=& \begin{pmatrix} 0&1\\ 1&0\end{pmatrix}\\ \sigma^2 =& \begin{pmatrix}0&-i\\ i&0\end{pmatrix} & \sigma^3=& \begin{pmatrix}1&0\\ 0&-1\end{pmatrix},\end{matrix}\end{align}$$ in complete analogy to the relation between $\mbox{SU}(2)$ and the rotation group. These matrices form a basis for two-by-two complex matrices: $$\begin{align} P=P_m\sigma^m = \begin{pmatrix} -P_0+P_3 & P_1-iP_2\\ P_1+iP_2 & -P_0-P_3\end{pmatrix}.\end{align}$$ Any hermitian matrix may be expanded with the $P_m$ real.

Lorentz Transformation

From any hermitian matrix $P$, we may always obtain another by the following transformation: $$\begin{align} P'=MPM^\dagger.\end{align}$$ Both $P$ and $P'$ have expansions in $\sigma$, $$\begin{align}\sigma^m P'_m = M\sigma^m P_m M^\dagger.\end{align}$$ Since $M$ is unimodular ($\det M=1$), the coefficitents $P_m$ and $P'_m$ are connected by a Lorentz transforamtion: $$\begin{align} \det [\sigma^m P'_m]=\det [\sigma^m P_m]=P'^2_0-\mathbf{P}'^2 = P_0^2 - \mathbf{P}^2.\end{align}$$ Vectors and tensors are distinguished from spinors by their Latin indices.

From (1) and (5), we see that $\sigma^m$ has the following index structure: $$\begin{align} \sigma_{\alpha \dot{\alpha}}^m.\end{align}$$ With these convensions, $\psi^\alpha \psi_\alpha$, $\bar{\psi}_\dot{\alpha}\bar{\psi}^\dot{\alpha}$, and $\psi^\alpha \sigma_{\alpha\dot{\alpha}}^m \partial_m \bar{\psi}^\dot{\alpha}$ are all Lorentz scalars.

Antisymmetric Tensor $\epsilon_{\alpha\beta}$

Since $M$ is unmodular, the antisymmetric tensors $\epsilon^{\alpha\beta}$ and $\epsilon_{\alpha\beta}$  ($\epsilon_{21}=\epsilon^{12}=1,\epsilon_{12}=\epsilon^{21}=-1,\epsilon_{11}=\epsilon_{22}=0$) are invariant under Lorentz transformations: $$\begin{align}\begin{matrix} \epsilon_{\alpha\beta} = M_\alpha^\gamma M_\beta^\delta \epsilon_{\gamma\delta}\\ \epsilon^{\alpha\beta}=\epsilon^{\gamma\delta}M_\gamma^\delta M_\delta^\beta.\end{matrix}\end{align}$$ Spinors with upper and lower indices are related through the $\epsilon$-tensor: $$\begin{align} \psi^\alpha = \epsilon^{\alpha\beta} \psi_\beta,\quad \psi_\alpha=\epsilon_{\alpha\beta}\psi^\beta.\end{align}$$ Note that we have defined $\epsilon_{\alpha\beta}$ and $\epsilon^{\alpha\beta}$ such that $\epsilon_{\alpha\beta}\epsilon^{\beta\gamma}=\delta_\alpha^\gamma$. An analogous treatment holds for the $\epsilon$-tensor with dotted indices.

The $\epsilon$-tensor may also be used to raise the indices of the $\sigma$-matrices: $$\begin{align}\bar{\sigma}^{m\dot{\alpha}\alpha}=\epsilon^{\dot{\alpha}\dot{\beta}}\epsilon^{\alpha\beta}\sigma^m_{\beta\dot{\beta}}.\end{align}$$ From the definition of the $\sigma$-matrices, we find $$\begin{align}\begin{matrix}(\sigma^m\bar{\sigma}^n+\sigma^n\bar{\sigma}^m)_\alpha^\beta = -2\eta^{mn}\delta_\alpha^\beta\\ (\bar{\sigma}^m\sigma^n+\bar{\sigma}^n\sigma^m)^\dot{\alpha}_\dot{\alpha}=-2\eta^{mn}\delta^\dot{\alpha}_\dot{\beta},\end{matrix}\end{align}$$ as well as the following completeness relations: $$\begin{align}\begin{matrix} \mbox{Tr} \sigma^m\bar{\sigma}^n=-2\eta^{mn}\\ \sigma_{\alpha\dot{\alpha}}^m\bar{\sigma}_m^{\dot{\beta}\beta}=-2\delta_\alpha^\beta\delta_\dot{\alpha}^\dot{\beta}.\end{matrix}\end{align}$$ These relations may be used to convert a vector to a bispinor and vice versa: $$\begin{align}v_{\alpha\dot{\alpha}}=\sigma_{\alpha\dot{\alpha}}^mv_m,\quad v^m=-\frac{1}{2}\bar{\sigma}^{m\dot{\alpha}\alpha}v_{\alpha\dot{\alpha}}.\end{align}$$

Lorentz Group Generator

The generators of the Lorentz group in the spinor representation are given by $$\begin{align}\begin{matrix}\sigma^{nm\beta}_\alpha = \frac{1}{4} \left( \sigma^n_{\alpha\dot{\alpha}}\bar{\sigma}^{m\dot{\alpha}\beta} - \sigma^m_{\alpha\dot{\alpha}} \bar{\sigma}^{n\dot{\alpha}\beta} \right)\\ \sigma^{nm\dot{\alpha}}_\dot{\beta} = \frac{1}{4} \left( \bar{\sigma}^{n\dot{\alpha}\alpha}\sigma^m_{\alpha\dot{\beta}} - \bar{\sigma}^{m\dot{\alpha}\alpha} \sigma^n_{\alpha\dot{\beta}} \right).\end{matrix}\end{align}$$ Other useful relations involving the $\sigma$-matrices are $$\begin{align}\begin{matrix}\bar{\sigma}^a\sigma^b\bar{\sigma}^c-\bar{\sigma}^c\sigma^b\bar{\sigma}^a=-2i\epsilon^{abcd}\bar{\sigma}_d\\ \sigma^a\bar{\sigma}^b\sigma^c-\sigma^c\bar{\sigma}^b\sigma^a=2i\epsilon^{abcd}\sigma_d,\end{matrix}\end{align}$$ where $\epsilon_{0123}=-1$, as well as $$\begin{align}\begin{matrix}\sigma^a\bar{\sigma}^b\sigma^c+\sigma^c\bar{\sigma}^b\sigma^a=2(\eta^{ac}\sigma^b-\eta^{bc}\sigma^a-\eta^{ab}\sigma^c)\\ \bar{\sigma}^a\sigma^c\bar{\sigma}^c+\bar{\sigma}^c\sigma^b\bar{\sigma}^a=2(\eta^{ac}\bar{\sigma}^b-\eta^{bc}\bar{\sigma}^a-\eta^{ab}\bar{\sigma}^c),\end{matrix}\end{align}$$ and $$\begin{align}\begin{matrix} \sigma_{\alpha\dot{\alpha}}^n\sigma_{\beta\dot{\beta}}^m-\sigma_{\alpha\dot{\alpha}}^m\sigma_{\beta\dot{\beta}}^n&=& 2\left[ (\sigma^{nm}\epsilon)_{\alpha\beta}\epsilon_{\dot{\alpha}\dot{\beta}}+(\epsilon\bar{\sigma}^{nm})_{\dot{\alpha}\dot{\beta}}\epsilon_{\alpha\beta}\right] \\ \sigma_{\alpha\dot{\alpha}}^n\sigma^m_{\beta\dot{\beta}}+\sigma_{\alpha\dot{\alpha}}^m\sigma_{\beta\dot{\beta}}^n&=& -\eta^{nm}\epsilon_{\alpha\beta}\epsilon_{\dot{\alpha}\dot{\beta}}+4(\sigma^{ln}\epsilon)_{\alpha\beta}(\epsilon\bar{\sigma}^{lm})_{\dot{\alpha}\dot{\beta}}.\end{matrix}\end{align}$$

Weyl Spinor

The equations (11) make it easy to relate two-component to four-component spinors. This is done through the following realization of the Dirac $\gamma$-matrices: $$\begin{align} \gamma^m=\begin{pmatrix}0& \sigma^m\\ \bar{\sigma}^m &0\end{pmatrix}.\end{align}$$ We shall this the Weyl basis. In this basis, Dirac spinors contain two Weyl spinors, $$\begin{align} \Psi_D = \begin{pmatrix}\chi_\alpha \\ \bar{\psi}^\dot{\alpha}\end{pmatrix},\end{align}$$ while Majorana spinors contain only one: $$\begin{align} \Psi_M = \begin{pmatrix}\chi_\alpha \\ \bar{\chi}^\dot{\alpha}\end{pmatrix}.\end{align}$$

Throughour these lecture we shall use the following spinor summation convention: $$\begin{align} \begin{matrix} \psi\chi = \psi^\alpha\chi_\alpha=-\psi_\alpha\chi^\alpha = \chi^\alpha\chi_\alpha=\chi\psi\\ \bar{\psi}\bar{\chi}=\bar{\psi}_\dot{\alpha}\bar{\chi}^\dot{\alpha}=-\bar{\psi}^\dot{\alpha}\bar{\chi}_\dot{\alpha}=\bar{\chi}_\dot{\alpha}\bar{\psi}^\dot{\alpha}=\bar{\chi}\bar{\psi}.\end{matrix}\end{align}$$ Here we have assumed, as always, that spinors anticommute. The definition of $\bar{\psi}\bar{\chi}$ is chosen in such a way that $$\begin{align} (\chi\psi)^\dagger=(\chi^\alpha\psi_\alpha)^\dagger=\bar{\psi}_\dot{\alpha}\bar{\chi}^\dot{\alpha}=\bar{\psi}\bar{\chi}=\bar{\chi}\bar{\psi}.\end{align}$$ Note that conjugation reverses the order of the spinors.