I am getting confused by the notation the authors of this book since they define: $$ \bar{\psi}\equiv \psi^\ast \gamma^0 $$ where (I suppose) $^\ast$ means complex conjugate and $\gamma^0$ is one of the Dirac matrices.
To me this looks like a complete nonsense since $\psi$ and $\psi^\ast$ are a column vectors and $\gamma^0$ is a square matrix and I have never seen such a thing as ''column per row'' product.
At first I thought it could siply mean that the author used the notation $^\ast$ instead of $^\dagger$ meaning thus complex conjugate and transposition: $$ \bar{\psi}\equiv \psi^\ast \gamma^0=\psi^\dagger \gamma^0. $$
But then other derivations do not make sense, such as the following: $$ {\psi}^C = U_C\bar{\psi} $$ $$ \bar{\psi}^C\equiv \left[U_C\bar{\psi}\right]^\ast \gamma^0 = \left[U_C(\psi^\ast \gamma^0)\right]^\ast\gamma^0= \left[U_C^\ast(\psi \gamma^{0\ast})\right]\gamma^0 $$ where apparently they really mean just conjugate otherwise they sould have swapped all the stuff in the square bracket: $$ \left[U_C(\psi^\ast \gamma^0)\right]^\dagger\gamma^0= \left[(\gamma^{0\dagger}\psi )U_C^\dagger\right]\gamma^0. $$
Pleas enlighten me!
EDIT: I also found out the authors use the following: $$ \bar{\psi}\left[\gamma (p+eA) + m\right] = 0\\ \left[\tilde{\gamma} (p+eA) + m\right]\bar{\psi}=0 $$ where $\tilde{}$ means transpose. So I guess they just use it as a row or column vector depending on what they need...