Unitarity of the charge-conjugation matrix
= Unitarity of the charge-conjugation matrix
{title2=$C^\dagger C=I$}
For $\psi^C=\eta_C C\bar\psi^T$ and $\gamma^{0T}=\gamma^0=-C\gamma^0C^{-1}$, the <Dirac adjoint> transforms as $\bar\psi^C=-\eta_C^*\psi^TC^\dagger$. Anticommuting the components of the <fermion bilinear> gives $(\bar\psi\psi)^C=\bar\psi(C^\dagger C)^T\psi$. Invariance of the scalar bilinear therefore fixes $C^\dagger C=I$.