= Parity transformation of a Dirac bilinear
{title2=$\bar\psi\psi\mapsto\bar\psi\psi,\quad\bar\psi\gamma^5\psi\mapsto-\bar\psi\gamma^5\psi$}
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 $B\mapsto\gamma^0B\gamma^0$, 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>.
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