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A massless particle state with the standard momentum $k^\mu=(\kappa,0,0,\kappa)$ and helicity $\sigma$ is denoted by $\Psi_{k,\sigma}$, Weinberg defines the parity phase $\eta_\sigma$ for the parity operator $P$ \begin{align} U(R_2^{-1}) P \Psi_{k,\sigma} = \eta_\sigma \Psi_{k,-\sigma} \end{align} where $R_2$ is the rotation of the reference frame around 2 axis by angle $\pi$ (or in the "positive" point of view I prefer, rotation of the physical system around 2 axis by angle $-\pi$).

To show how parity acts on states with general momentum, I would have \begin{align} P \Psi_{p,\sigma} &= P U(R(\hat{p}))U(B(\frac{|\vec{p}|}{\kappa}))\Psi_{k,\sigma} = P U(R(\hat{p}))U(B(\frac{|\vec{p}|}{\kappa}))P^{-1}U(R_2^{-1})^{-1}U(R_2^{-1}) P \Psi_{k,\sigma} \\ &=\eta_\sigma P U(R(\hat{p}))U(B(\frac{|\vec{p}|}{\kappa}))P^{-1}U(R_2^{-1})^{-1} \Psi_{k,-\sigma} \\ &=\eta_\sigma U(R(\hat{p}))U(R_2^{-1})^{-1}U(B(\frac{|\vec{p}|}{\kappa})) \Psi_{k,-\sigma} \end{align} where we have used \begin{align} P U(B(\frac{|\vec{p}|}{\kappa})) P^{-1} &= U(R_2^{-1})^{-1} U(B(\frac{|\vec{p}|}{\kappa})) U(R_2^{-1}) \\ P U(R(\hat{p})) P^{-1} &= U(R(\hat{p})) \end{align} Our result differs from the one in the book because $U(R_2^{-1})^{-1}\neq U(R_2)$ or $e^{i 2\pi J_2}\neq 1$ for particles with half-integer spin. Is that a typo of the book and we should define $U(R_2)^{-1} P \Psi_{k,\sigma} = \eta_\sigma \Psi_{k,-\sigma}$?

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That is not a typo. Since $R_2$ is a rotation by $180°$ about the two-axis, $R_2^{-1}$ is a rotation by $180°$ about the two-axis in the opposite sense. So if $U(R_2)=e^{i \pi J_2 }$, then $U(R_2^{-1})=e^{-i \pi J_2}$. This gives $(U(R_2^{-1}))^{-1} = (e^{-i \pi J_2})^{-1}=e^{i \pi J_2} = U(R_2)$.

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