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Action of the parity operator on a state is not determined only by intrinsic parities of the particles at hand, but also by the state itself. For example, if you have a one-particle state $\Psi_p$ with impulse $p$, then $P\Psi_p=\pm\Psi_{-p}$, and such a state does not even have a definite parity (it is not an eigenstate). You probably know that if there is ...