Hermitian matrix representation of Minkowski four-vectors ($X=x_{\mu }{\sigma }^{\mu }$)

A real four-vector corresponds to a two-by-two Hermitian matrix with determinant equal to its Minkowski squared norm. The action $X\mapsto NX{N}^{†}$ for $N\in SL(2,\mathbb{C})$ preserves that determinant and the future cone. Expanding with $\frac{1}{2}\mathrm{Tr}({\bar{\sigma }}^{\mu }{\sigma }_{\nu })={\delta }^{\mu }{}_{\nu }$ yields ${\Lambda }^{\mu }{}_{\nu }(N)=\frac{1}{2}\mathrm{Tr}({\bar{\sigma }}^{\mu }N{\sigma }_{\nu }{N}^{†})$. The map has kernel $\{\pm I\}$ and image $S{O}^{+}(1,3)$.