It seems to be a dumb question. But I haven't seen anyone making this connection between QED and Classical EM in a complete fashion. The only example I've seen is the connection between two particle scattering amplitude calculation in QED (Peskin's book), and the amplitude of a particle scattering of a potential in non-relativistic Quantum Mechanics. By making that connection, you can reproduce the equivalent "potential" from QED. In this case, the coulomb potential (see Peskin P121~P125). But what about the full solution to Maxwell's equations. For example, how would one reproduce the Lienard-Wiechert potential, or equivalently the electric field distribution of a randomly moving charge from QED's calculation? It seems to me, in QED there is no real concept of "potential", all are amplitudes. But there must be some way to connect this QED picture to the Classical relativistic potentials that already worked so well. Is there any way to do that? Any reference would be greatly appreciated!