Parity symmetry in quantum field theory

Parity symmetry means invariance under spatial reflection, together with the intrinsic transformation rules of the fields.

With the mostly-minus Minkowski metric, $|{\eta }_P|=1$ and the usual Dirac adjoint, this transformation makes $\bar{\psi }\psi$ even and $\bar{\psi }{\gamma }^5\psi$ odd under parity. The latter sign follows from anticommutation of ${\gamma }^0$ and ${\gamma }^5$ and does not depend on whether the convention includes an imaginary unit in ${\gamma }^5$.

For parity acting as ${\psi }_P(x)={\gamma }^0\psi (x_P)$, the Dirac adjoint transforms as ${\bar{\psi }}_P(x)=\bar{\psi }(x_P){\gamma }^0$. A bilinear with matrix $B$ therefore transforms by $B\mapsto {\gamma }^{0}B{\gamma }^0$, with its fields evaluated at $x_P$. Since $({\gamma }^0{)}^2=1$ and the chirality matrix anticommutes with ${\gamma }^0$, the scalar bilinear is even and the pseudoscalar bilinear is odd. The same calculation shows that parity exchanges the chiral projectors.

A parity-invariant vacuum satisfies $P|0⟩=|0⟩$ up to an irrelevant phase. Vacuum correlation functions then transform according to the product of the intrinsic parities of their insertions.

A parity-odd scalar interaction is rotationally invariant but changes sign under spatial reflection. It must contain a pseudoscalar contraction, such as a Levi-Civita symbol contracted with three spatial gradients.