If $S_F(x-y)$ is the Green's function for the Dirac operator $(i\gamma^\mu\partial_\mu-m)$, that is, I assume the following matrix equation holds: $$ (i\gamma^\mu\partial_\mu-m)S_F(x-y)=i\delta(x-y) $$
The adjoint dirac equation is: $$ -i\partial_\mu\bar{\psi}\gamma^\mu -m\bar\psi=0 $$
I am looking for the Green's function of the above equation, in terms of $S_F(x-y)$, that is, if $F[S]$ is some function of $S$ (may include complex conjugation, transposing, etc), then I am looking for such an $F[S]$ such that this equation holds: $$ -i\partial_\mu F[S]\gamma^\mu -mF[S]=i\delta(x-y) $$
What I have done so far:
- Dagger this equation: $(i\gamma^\mu\partial_\mu-m)S_F(x-y)=i\delta(x-y)$ and get $$(-i\partial_\mu S_F(x-y)^\dagger \gamma^{\mu\dagger} - S_F(x-y)^\dagger m)=-i\delta(x-y)$$
- Multiply by $\gamma^0$ on the right to get: $$ (-i\partial_\mu [-\bar{S_F}(x-y)] \gamma^{\mu} - [-\bar{S_F}(x-y)] m)=i\delta(x-y)\gamma^0 $$
- So $[-\bar{S_F}(x-y)]$ almost solves this equation, except you get a factor of $\gamma^0$ on the right which I am not sure how to handle.