 Solution

Write ${\psi }^C=\mathcal{C}{\bar{\psi }}^T$ and ${\bar{\psi }}^C={\psi }^T\mathcal{C}$. Anticommuting the spinor fields and using ${\mathcal{C}}^{-1}({\gamma }^{\mu }{)}^T\mathcal{C}=-{\gamma }^{\mu }$ givesThe free Dirac terms are invariant up to the total derivative used to move the derivative between the two anticommuting fields. The QED interaction $-e\bar{\psi }{\gamma }^{\mu }\psi A_{\mu }$ is also invariant because both the Dirac current and $A_{\mu }$ are odd. Together with $F_{\mu \nu }{F}^{\mu \nu }$ invariance, this proves charge-conjugation invariance of quantum electrodynamics.