Parity ($\mathcal{P}$)

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

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 $-1$.

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.