Parity ($\mathcal{P}$)

Parity is the eigenvalue $+1$ or $-1$ of a state under spatial reflection.

For a state of integer total angular momentum $J$, natural parity means $\pi =(-1{)}^J$: $0^{+},1^{-},2^{+},3^{-},\dots$. The other sign is called unnatural parity. The terminology is useful when a spin-zero positive-parity parent emits a spin-zero positive-parity particle: angular momentum conservation fixes relative orbital angular momentum $L=J$, and parity conservation then forces natural parity for the daughter.

An interaction commuting with the parity operator preserves total parity. A two-body final state has the product of its particles' intrinsic parities times the orbital factor $(-1{)}^{\ell }$, giving the parity selection rule for a two-body decay.

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 $-1$.

On $L^2(0,a)$, the midpoint parity operator is $(R_a\psi )(x)=\psi (a-x)$. Substitution $y=a-x$ gives $⟨f,R_{a}g⟩=⟨R_{a}f,g⟩$ and $\Vert R_{a}f\Vert =\Vert f\Vert$, so it is a bounded self-adjoint operator and a unitary operator, with $R_a^2=I$. It is the parity operator in the centered coordinate $x-a/2$.

For an infinite square well, ${\psi }_n=\sqrt{2/a}\sin (n\pi x/a)$ obeys $R_a{\psi }_n=(-1{)}^{n+1}{\psi }_n$. Reflection preserves the Dirichlet boundary conditions and commutes with the second derivative, so $R_a$ commutes with the Hamiltonian operator on its operator domain. By contrast, differentiating once gives $P{R}_a=-R_{a}P$ for the momentum operator. This distinction permits conserved midpoint parity with changing mean momentum.

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.

An orbital angular momentum eigenstate with quantum number $\ell$ changes by the factor $(-1{)}^{\ell }$ under spatial reflection. Equivalently,

The parity of a two-particle centre-of-mass state with intrinsic parities ${\eta }_1,{\eta }_2$ and relative orbital angular momentum $\ell$ is

If parity is conserved when a particle of intrinsic parity ${\eta }_i$ decays into two particles of intrinsic parities ${\eta }_1,{\eta }_2$, then their relative partial wave obeys

Spatial reflection reverses every spatial coordinate.

A pseudoscalar is invariant under proper rotations but changes sign under a spatial reflection. In three dimensions the scalar triple product is a pseudoscalar.