As a disclaimer, this is somewhat similar to this unanswered question, but not entirely.
In standard QED theory, it is frequently demonstrated that the derivation of the Coulomb Potential can be found by equating the Born Approximation with the scattering amplitude between two electrons using only the lowest-order single photon exchange diagram (example here).
One thing I don't understand is why this first order approximation is sufficient for the Coulomb potential and, as it follows, the electromagnetic wave equation.
Of course, one is able to perform higher order corrections to the coulomb potential by the inclusion of higher order diagrams. Notably for example, if you include one of the next highest order diagrams: the two-photon exchange which is associated with the Casimir-Polder potential (see Larry Spruch, Physics Today). We can consider this as the first correction to the Coulomb potential (neglecting self-energy corrections):
$$ V \approx \frac{e^2}{r} + \frac{\hbar e^4}{c^3m^2r^3} $$
Other examples of include the coulomb force with a correction term due to dressing. From this, it seems that you can derive a correction to the electromagnetic wave equation (though I haven't worked this out yet).
Then I think there is a potential paradox to this: the quantized electromagnetic field that is assumed is based on quantizing classical EM wave solutions, not ones with higher order corrections. Is this just a one of many options that one can take? For example, is it possible to described the photon’s field in terms of quantizing a EM wave solution that includes higher order corrections, and just be as valid?
Thank you in advanced.
