= Massive induced representation of the Poincare double cover
{title2=$U(\Lambda,a)|p,s,\sigma\rangle=e^{i(\Lambda p)\cdot a}\sum_\tau D^{(s)}_{\tau\sigma}(W(\Lambda,p))|\Lambda p,s,\tau\rangle$}
Fix $m>0$, $k=(m,\mathbf0)$, and a rotationless <Lorentz boost> $L(p)$ taking $k$ to $p$ on the positive-energy <mass shell>. The <little group> on the <Lorentz spinor double cover> is <SU(2)>. Its spin-$s$ irreducible representation induces a unitary representation on $L^2(d^3p/(2p^0))\otimes\mathbb C^{2s+1}$. Here $W(\Lambda,p)=L(\Lambda p)^{-1}\Lambda L(p)$ is the <Wigner rotation>, lifted to <SU(2)> when necessary, and the translation convention is $U(I,a)|p\rangle=e^{ip\cdot a}|p\rangle$. The little-group cocycle proves the group law, and <Wigner's classification> gives irreducibility. Every nonnegative integer or half-integer spin is allowed on the double cover; descent to the ordinary proper orthochronous <Poincaré group> requires integer spin, because its $2\pi$ rotation must act as the identity.
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