Wess-Bagger Weyl spinor notation

\[\eta_{\mu\nu}=\mathrm{diag}(-1,1,1,1)\]\[1=\varepsilon^{12}=-\varepsilon_{12}\]\[\varepsilon^{0123}=-\varepsilon_{0123}=1\]\[\psi^\alpha=\varepsilon^{\alpha\beta}\psi_\beta;\quad \psi_\alpha=\varepsilon_{\alpha\beta}\psi^\beta\]\[\psi^\dot{\alpha}=\varepsilon^{\dot{\alpha}\dot{\beta}}\psi_\dot{\beta};\quad \psi_\dot{\alpha}=\varepsilon_{\dot{\alpha}\dot{\beta}}\psi^\dot{\beta}\]\[\psi\chi=\psi^\alpha\chi_\alpha=-\psi_\alpha\chi^\alpha=\chi^\alpha\psi_\alpha=\chi\psi\]\[\bar{\psi}\bar{\chi}=\bar{\psi}_\dot{\alpha}\bar{\chi}^\dot{\alpha}=-\bar{\psi}^\dot{\alpha}\bar{\chi}_\dot{\alpha}=\bar{\chi}_\dot{\alpha}\bar{\psi}^\dot{\alpha}=\bar{\chi}\bar{\psi}\]\[(\chi\psi)^\dagger=(\chi^\alpha\psi_\alpha)^\dagger=\bar{\psi}_\dot{\alpha}\bar{\chi}^\dot{\alpha}=\bar{\psi}\bar{\chi}=\bar{\chi}\bar{\psi}\]\[\Psi_D=\begin{bmatrix}\chi_\alpha\\\psi^\dot{\alpha}\end{bmatrix}\]\[\gamma^\mu =\begin{bmatrix}&\sigma^\mu\\\bar{\sigma}^\mu&\end{bmatrix}\]\[\gamma^\mu =\begin{bmatrix}&\sigma^\mu\\\bar{\sigma}^\mu&\end{bmatrix}\]\[\sigma^\mu = (-1,\vec{\sigma});\quad \bar{\sigma}^\mu = (-1,-\vec{\sigma})\]\[\bar{\sigma}^{\mu\dot{\alpha}\alpha}=\varepsilon^{\dot{\alpha}\dot{\beta}}\varepsilon^{\alpha\beta}\sigma^\mu_{\beta\dot{\beta}}\]\[\mathrm{tr}\,\sigma^\mu\bar{\sigma}^\nu = \sigma^\mu_{\alpha\dot{\alpha}} \bar{\sigma}^{\nu\dot{\alpha}\alpha} = -2\eta^{\mu\nu}\]

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