= Algebraic charge conjugation reverses chirality
{title2=$\gamma_5\psi=\eta\psi\Longrightarrow\gamma_5\psi^c=-\eta\psi^c$}
The <Dirac adjoint> of a chiral field has opposite projector placement because $\gamma_5$ anticommutes with $\gamma^0$: $\bar\psi\gamma_5=-\eta\bar\psi$. Since $C\gamma_5^TC^{-1}=\gamma_5$, the algebraic charge conjugate satisfies $\gamma_5 C\bar\psi^T=C(\bar\psi\gamma_5)^T=-\eta\psi^c$. This is conjugation of a projected spinor, distinguished from <operator conjugation of a chiral field component>, where the numerical projector is kept outside the conjugation operator.
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