This question is a follow up to the question (Gauge invariant Green's function for electrodynamics). It is not possible to generally solve the eqution \begin{equation} \square A^{\mu}-\partial^{\mu}\left(\partial_{\nu} A^{\nu}\right)=\frac{4 \pi}{c} j^{\mu} \end{equation} However, if we specify the current to the current of a point particle, is there a general solution to \begin{equation} \square A^{\mu}-\partial^{\mu} (\partial_{\nu} A^{\nu})=\frac{4 \pi}{c} \int_{-\infty}^{\infty} d s v^{\mu}(s) \delta^{4}[x-z(s)]~? \end{equation}
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This argument is still valid, for this expression of $j^\mu$ as for any other current distribution.
The reason is that the LHS is invariant under $A^\mu \rightarrow A^\mu +\partial^\mu f$ for any function $f$, so there is no hope of find a general solution without fixing the gauge.
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$\begingroup$ I dont think that hyperlink is what you intended to link to. $\endgroup$– najkimJun 4, 2021 at 16:36
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$\begingroup$ link.springer.com/content/pdf/10.1007/BF01017950.pdf. Originally I also thought this. But e.g. in the paper above, it is claimed that there is such a method $\endgroup$– NicAGJun 4, 2021 at 16:44
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$\begingroup$ In the book 'Introduction to the Classical Theory of Particles and Fields by Boris Kosyakov' in 4.7 he further elaborates on the method but I am not sure if it is fully valid. I don't know if I am allowed to send screenshots. But the second half of the argument is on google book books.google.de/… on page 184. $\endgroup$– NicAGJun 4, 2021 at 16:48
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$\begingroup$ Equation (4) in Kosyakov's paper fixes the gauge (although he doesn't state this explicitly). For EM, you can choose Lorentz gauge $\partial_\mu A^\mu = 0$ and solve using the Green function for the D'Alembertian. $\endgroup$ Jun 4, 2021 at 16:53