Wess-Bagger Weyl spinor notation

$${\eta }_{\mu \nu }=\mathrm{diag}(-1,1,1,1)$$$$1={\epsilon }^{12}=-{\epsilon }_{12}$$$${\epsilon }^{0123}=-{\epsilon }_{0123}=1$$$${\psi }^{\alpha }={\epsilon }^{\alpha \beta }{\psi }_{\beta };{\psi }_{\alpha }={\epsilon }_{\alpha \beta }{\psi }^{\beta }$$$${\psi }^{\dot{\alpha }}={\epsilon }^{\dot{\alpha }\dot{\beta }}{\psi }_{\dot{\beta }};{\psi }_{\dot{\alpha }}={\epsilon }_{\dot{\alpha }\dot{\beta }}{\psi }^{\dot{\beta }}$$$$\psi \chi ={\psi }^{\alpha }{\chi }_{\alpha }=-{\psi }_{\alpha }{\chi }^{\alpha }={\chi }^{\alpha }{\psi }_{\alpha }=\chi \psi$$$$\overline{\psi }\overline{\chi }={\overline{\psi }}_{\dot{\alpha }}{\overline{\chi }}^{\dot{\alpha }}=-{\overline{\psi }}^{\dot{\alpha }}{\overline{\chi }}_{\dot{\alpha }}={\overline{\chi }}_{\dot{\alpha }}{\overline{\psi }}^{\dot{\alpha }}=\overline{\chi }\overline{\psi }$$$$(\chi \psi {)}^{†}=({\chi }^{\alpha }{\psi }_{\alpha }{)}^{†}={\overline{\psi }}_{\dot{\alpha }}{\overline{\chi }}^{\dot{\alpha }}=\overline{\psi }\overline{\chi }=\overline{\chi }\overline{\psi }$$$${\mathrm{\Psi }}_D=\left[\begin{array}{c}{\chi }_{\alpha }\\ {\psi }^{\dot{\alpha }}\end{array}\right]$$$${\gamma }^{\mu }=\left[\begin{array}{cc}& {\sigma }^{\mu }\\ {\overline{\sigma }}^{\mu }& \end{array}\right]$$$${\gamma }^{\mu }=\left[\begin{array}{cc}& {\sigma }^{\mu }\\ {\overline{\sigma }}^{\mu }& \end{array}\right]$$$${\sigma }^{\mu }=(-1,\overrightarrow{\sigma });{\overline{\sigma }}^{\mu }=(-1,-\overrightarrow{\sigma })$$$${\overline{\sigma }}^{\mu \dot{\alpha }\alpha }={\epsilon }^{\dot{\alpha }\dot{\beta }}{\epsilon }^{\alpha \beta }{\sigma }_{\beta \dot{\beta }}^{\mu }$$$$\mathrm{tr}{\sigma }^{\mu }{\overline{\sigma }}^{\nu }={\sigma }_{\alpha \dot{\alpha }}^{\mu }{\overline{\sigma }}^{\nu \dot{\alpha }\alpha }=-2{\eta }^{\mu \nu }$$

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