= Lorentz spinor double cover
{c}
{title2=$SL(2,\mathbb C)\longrightarrow SO^+(1,3)$}
Identify Minkowski vectors with Hermitian matrices $X=x^0I+x^a\sigma_a$. The action $X\mapsto SXS^\dagger$ preserves $\det X=(x^0)^2-|\mathbf x|^2$. It gives a surjective homomorphism from $SL(2,\mathbb C)$ to the <Proper orthochronous Lorentz group> with kernel $\{I,-I\}$. Polar decomposition retracts the covering group onto <SU(2)>, making it simply connected. This cover permits half-integer spin <group representations> that do not descend to the Lorentz group itself.
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