Parity action on canonical massive spin states ($P|p,s,\sigma ⟩={\eta }_P|p^0,-\mathbf{p},s,\sigma ⟩$)

Choose canonical spin states by applying the rotationless Lorentz boost $L(p)$ to rest-frame spin states. Spatial inversion $\Pi =\mathrm{diag}(1,-1,-1,-1)$ 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 ${\eta }_P=\pm 1$.

// Target: lie-theory.bigb