Weyl sigma matrices
= Weyl sigma matrices
{c}
{title2=$\sigma^\mu_{\alpha\dot\beta}$}
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> proves
$$
\sigma^\mu\bar\sigma^\nu+\sigma^\nu\bar\sigma^\mu=2\eta^{\mu\nu}I.
$$
A real <Lorentz four-vector> is equivalently a <Hermitian matrix> $X=x^0I+x^i\sigma_i$ with $\det X=(x^0)^2-|\mathbf x|^2$. The transformation $X\mapsto MXM^\dagger$ for $M\in\mathrm{SL}(2,\mathbb C)$ preserves this <determinant> and gives the <Lorentz spinor double cover>. The indices $\sigma^\mu_{\alpha\dot\beta}$ therefore convert a vector index into one undotted and one dotted <Weyl spinor> index. They are the invariant tensors in the mixed <supercharge> anticommutator $\{Q_\alpha,\bar Q_{\dot\beta}\}=2\sigma^\mu_{\alpha\dot\beta}P_\mu$.