= Parity action on canonical massive spin states
{title2=$P|p,s,\sigma\rangle=\eta_P|p^0,-\mathbf p,s,\sigma\rangle$}
Choose canonical spin states by applying the rotationless <Lorentz boost> $L(p)$ to rest-frame spin states. Spatial inversion $\Pi=\operatorname{diag}(1,-1,-1,-1)$ commutes with rest-frame rotations, so <Schur lemma> makes its action on one irreducible rest spin multiplet a phase $\eta_P$. Since $\Pi L(p)\Pi=L(\Pi p)$, the formula follows. Canonical spin labels are unchanged; <helicity> reverses. With $P^2=I$, the <intrinsic parity> is $\eta_P=\pm1$.
// Target: lie-theory.bigb
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