= Parity as odd-mode number parity
{title2=$\mathcal P=e^{i\pi N_{\rm odd}}$}
For a free intrinsic-even <real scalar field>, momentum reflection $Rf(\mathbf p)=f(-\mathbf p)$ is a self-adjoint involution on the one-particle space. Its odd <orthogonal projector> is $(1-R)/2$. The corresponding <second quantization> is $N_{\rm odd}=\tfrac12\int d^3p/(2\pi)^3\,a_{\mathbf p}^\dagger(a_{\mathbf p}-a_{-\mathbf p})$, with integration over all <momentum space>. Conjugation by $e^{i\pi N_{\rm odd}}$ fixes even modes and reverses odd modes, hence sends $a_{\mathbf p}$ to $a_{-\mathbf p}$ and implements the <parity operator>. Its square is one because $N_{\rm odd}$ has integer <eigenvalues>. The equivalent positive expression is $N_{\rm odd}=\tfrac14\int d^3p/(2\pi)^3\,(a_{\mathbf p}^\dagger-a_{-\mathbf p}^\dagger)(a_{\mathbf p}-a_{-\mathbf p})$.
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