Parity operator ($P$)

The parity operator acts on a spatial wavefunction by $(P\psi )(x)=\psi (-x)$ and satisfies $P^2=I$. For a Hamiltonian operator with an even potential and parity-invariant domain, $HP=PH$. A nondegenerate energy eigenvalue has a one-dimensional eigenspace, so its eigenfunction is an eigenvector of $P$ with eigenvalue $+1$ or $-1$.

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 $N_{\mathrm{odd}}=\frac{1}{2}\int d^{3}p/(2\pi {)}^3{a}_{\mathbf{p}}^{†}(a_{\mathbf{p}}-a_{-\mathbf{p}})$, with integration over all momentum space. Conjugation by $e^{i\pi N_{\mathrm{odd}}}$ 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 $N_{\mathrm{odd}}$ has integer eigenvalues. The equivalent positive expression is $N_{\mathrm{odd}}=\frac{1}{4}\int d^{3}p/(2\pi {)}^3(a_{\mathbf{p}}^{†}-a_{-\mathbf{p}}^{†})(a_{\mathbf{p}}-a_{-\mathbf{p}})$.

Even and odd occupation sectors have eigenvalues $+1$ and $-1$. Pair creation preserves parity, whereas one photon addition or photon subtraction reverses it. An optical squeezed vacuum is even and a squeezed one-photon state is odd.