(Following the calculations done in 'Quantum Field Theory in a Nutshell' [Second Edition] by Zee, Page 101)
The Dirac equation in the presence of an electromagnetic field is given by:
$$ [i \gamma^{\mu} (\partial_{\mu} - i e A_{\mu}) - m]\psi = 0 $$
where $\gamma^{\mu}$ are the gamma matrices, $A_{\mu}$ is the gauge field, $e$ is charge, $m$ is mass and $\psi$ is a spinor.
The complex conjugate of this equation is given by:
$$ [-i \gamma^{\mu *} (\partial_{\mu} + i e A_{\mu}) - m]\psi^{*} = 0 $$
Zee defines:
$$ -\gamma^{\mu *} = (C \gamma^{0})^{-1} \gamma^{\mu} (C \gamma^{0}) $$
where $C$ is the charge conjugation operator. Zee states that you can plug this into the complex conjugated Dirac equation to get:
$$ [i \gamma^{\mu} (\partial_{\mu} + i e A_{\mu}) - m]\psi_{c} = 0 $$
where $\psi_{c} = C \gamma^{0} \psi^{*}$. When I try to do this I get the following:
$$ [i (C \gamma^{0})^{-1} \gamma^{\mu} (C \gamma^{0}) (\partial_{\mu} + i e A_{\mu}) - m]\psi^{*} = 0 $$
$$ (C \gamma^{0}) \cdot [i (C \gamma^{0})^{-1} \gamma^{\mu} (C \gamma^{0}) (\partial_{\mu} + i e A_{\mu}) - m]\psi^{*} = (C \gamma^{0}) \cdot 0 $$
$$ [-i (C \gamma^{0})(C \gamma^{0})^{-1} \gamma^{\mu} (C \gamma^{0}) (\partial_{\mu} + i e A_{\mu}) - m(C \gamma^{0})]\psi^{*} = 0 $$
$$ [-i (C \gamma^{0})(C \gamma^{0})^{-1} \gamma^{\mu} (\partial_{\mu} - i e A_{\mu})(C \gamma^{0}) - m(C \gamma^{0})]\psi^{*} = 0 $$
$$ [-i (C \gamma^{0})(C \gamma^{0})^{-1} \gamma^{\mu} (\partial_{\mu} - i e A_{\mu}) - m]C \gamma^{0}\psi^{*} = 0 $$
$$ [i (C \gamma^{0})(C \gamma^{0})^{-1} \gamma^{\mu} (-\partial_{\mu} + i e A_{\mu}) - m]\psi_{c} = 0 $$
where I used the antilinear property $Ci = -iC$.
Now I have run into two issues. I am unsure of how to treat $(C \gamma^{0})(C \gamma^{0})^{-1}$ and also the sign of the $\partial_{\mu}$ term does not match the sign in Zee's expression. Can anyone point me in the right direction?
