Classically as well as quantum-mechanically, the source of the Maxwell field is the electron/four-current (Dirac field), so the use of the Green Function propagator for the Maxwell field makes perfect sense: the Maxwell field is inhomogenous in the presence of matter. But what about the source of the Dirac field? Classically, I suppose it's empirically considered as a given (I guess since it's exactly locally conserved, ∂J=0, any initial configuration would be sufficient), but the Dirac equation is still homogenous (if it weren't, ∂J≠0, Right?)...so then why does the Dirac Green Function propagator play such a fundamental part in QED? What is the "source" here -- the Dirac field's version of the Maxwell field's four-current? A naive guess is that the source is the quantum vacuum, or generically "energy", but that seems not only imprecise, but very specific to QED (and therefore somewhat circular; when the inhomogenous form of the Dirac equation was introduced into QED, they surely didn't use QED as a justification) -- and anyway, it seems the same formulation should apply equally well classically. What am I missing -- any thoughts?