In every physicists training, an electrostatics course will show how to solve Maxwell's equations for different systems, solving for the $E$-field at different points in space.
A separate solution to Maxwell's solution for an accellerating charge shows that there is a wiggling "wave" in the far-field.
Typically to understand the electric-field in a quantum framework, we decide to quantize the fourier coefficients of the solution to Maxwell's equations. I haven't seen a treatment that deals with non-photon-electric fields. (Either by changes in time in the near-field, or static $E$-fields.)
Is there a general framework for what the quantum $E$-field looks like for static electric fields? Do you just grab the zero frequency mode? If so, is this something that has been experimentally measured? This has been asked before, but it seems as though the asker was concerned specifically with reproducing the classical values -- and the accepted answer is very unsatisfying as a result (essentially just saying to use a displacement operator and look at the average..).
