= Hermiticity of a pseudoscalar Dirac bilinear
{title2=$(\bar\psi\gamma^5\psi)^\dagger=-\bar\psi\gamma^5\psi$}
With the usual <Dirac adjoint>, Hermitian $\gamma^5$ and $\{\gamma^0,\gamma^5\}=0$, taking the adjoint gives $\psi^\dagger\gamma^5\gamma^0\psi=-\psi^\dagger\gamma^0\gamma^5\psi$. Consequently a real pseudoscalar with coupling $-g\phi\bar\psi\gamma^5\psi$ has a Hermitian interaction only if $g^*=-g$. Writing $g=i g_P$ with real $g_P$ gives the familiar interaction $-i g_P\phi\bar\psi\gamma^5\psi$. Formal <Euler-Lagrange field equations> obtained with independent spinor and adjoint variables can still be computed for the original symbolic coupling; it must not silently be identified with a real physical coupling.
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