electromagnetism - Lorenz gauge fixing

Is it always possible to define function $\psi$ satisfying the Lorenz gauge equation $$ \partial_{\mu}\partial^{\mu} \psi + \partial_{\mu}A^{\mu} = 0? $$

Answer

Yes. If you define $f=-\partial_\mu A^\mu$ then you can write the equation in the form $$ \partial_\mu\partial^\mu\psi = f$$ This is the Klein-Gordon equation with a nonzero source ($f$) and can be solved via Green's function methods. Once you have the Klein-Gordon propagator* $G(x)$ (this is derived in any e.g. quantum field theory textbook) appropriate to the boundary conditions the solution can be written as $$ \psi(x)=\int d^4 x' G(x-x') f(x')$$ since Green's functions by definition satisfy $$\partial_\mu\partial^\mu G(x-x')= \delta(x-x')$$ where we take all differentiations to be with respect to x.

*You need the propagator in the position space representation to write this down. It is usually more convenient to write it in momentum space; you can go back and forth using (inverse) Fourier transforms.