 Weyl sigma matrices

For mostly-minus metric signature, the Weyl sigma matrices are ${\sigma }^{\mu }=(I,{\sigma }_1,{\sigma }_2,{\sigma }_3)$ and ${\bar{\sigma }}^{\mu }=(I,-{\sigma }_1,-{\sigma }_2,-{\sigma }_3)$, where ${\sigma }_i$ are the Pauli matrices. The Pauli matrix multiplication law provesA real Lorentz four-vector is equivalently a Hermitian matrix $X=x^{0}I+x^i{\sigma }_i$ with $\det X=(x^0{)}^2-|\mathbf{x}{|}^2$. The transformation $X\mapsto MX{M}^{†}$ for $M\in \mathrm{SL}(2,\mathbb{C})$ preserves this determinant and gives the Lorentz spinor double cover. The indices ${\sigma }_{\alpha \dot{\beta }}^{\mu }$ therefore convert a vector index into one undotted and one dotted Weyl spinor index. They are the invariant tensors in the mixed supercharge anticommutator $\{Q_{\alpha },{\bar{Q}}_{\dot{\beta }}\}=2{\sigma }_{\alpha \dot{\beta }}^{\mu }{P}_{\mu }$.