= Hermitian matrix representation of Minkowski four-vectors
{c}
{title2=$X=x_\mu\sigma^\mu$}
A real four-vector corresponds to a two-by-two Hermitian matrix with determinant equal to its Minkowski squared norm. The action $X\mapsto NXN^\dagger$ for $N\in SL(2,\mathbb C)$ preserves that determinant and the future cone. Expanding with $\tfrac12\operatorname{Tr}(\bar\sigma^\mu\sigma_\nu)=\delta^\mu{}_{\nu}$ yields $\Lambda^\mu{}_{\nu}(N)=\tfrac12\operatorname{Tr}(\bar\sigma^\mu N\sigma_\nu N^\dagger)$. The map has kernel $\{\pm I\}$ and image $SO^+(1,3)$.
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