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Although virtual photons are maybe best considered not to actually exist, it is said that electrostatic attraction and repulsion between two electrically charged particles can be mathematically modeled as being due to the exchange of virtual photons between the particles; in fact, in the (popular version of the) Standard Model (virtual, sometimes?) photons are the force carriers of the electromagnetic force. Can exchange of (virtual) photons really adequately model the electrostatic force? My problem with such an explanation is this (somewhat similar problems have been mentioned on PSE by others):

An electron might send out virtual photons, and another electron might be hit by one & be repelled from the first by conservation of momentum (there are, of course, problems with this involving simultaneous conservation of energy and momentum, which may be solved by not requiring this for virtual photons, [or perhaps by having only partial absorption of the incident photon, with the production of a lower energy one - is this experimentally disconfirmed?]), but what if a proton were hit by one? It would be repelled, also by conservation of momentum, but in actuality it is attracted. This problem is taken care of with the explanation using a classical electrostatic field by the field having a direction at each point, determined by the sign of the charge creating it, and a proton responding differently than an electron to that field because of the difference of the sign of its charge. However, conservation of momentum with the virtual photon would cause the proton to be repelled just as the electron is.

This could be taken care of by having the electron also emit negative mass virtual photons (which might be required anyway for conservation of mass/energy) and the proton interacting with those and so being attracted, but what would cause the proton to interact with the negative mass virtual photon & not the positive mass one? It can't be that protons always interact with negative mass virtual photons instead of positive mass ones, because, assuming protons also emit positive & negative mass virtual photons, another proton would have to interact with the positive mass one in that situation in order to be repelled. This can't be explained by the proton knowing whether the other emitting particle is positively or negatively charged, since under our assumption the interaction is to be explained by local interaction with the incident virtual photon, not by non-local, spooky action-at-a-distance direct interaction with the other particle.

The picture of the electromagnetic interaction via photons becomes more complex when some of the charged particles are accelerating, so that some of the photons are real - doesn't it?

Is it best to entirely give up the interaction via real and virtual photons picture, and just consider the electromagnetic interaction among charged particles as being the interaction of each with the quantized EM field of the others?

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It is always a slippery slope of you are trying to interpret mathematical concepts physically, and it can lead to paradoxes like you mentioned. This holds even for field theory in general, which is merely an attempt to interpret 'action-at-a-distance' type forces as local 'contact' type interactions (even though there is materially nothing present locally to interact with).

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You should really model the electrostatic field due to a point charge as a coherent state. A coherent state is a superposition of an infinite number of terms, each with different numbers of photons. So, you are right that you shouldn't really interpret the electrostatic attraction/repulsion between point charges directly in terms of photon exchange. Nevertheless, there is a rigorous, fully quantum mechanical way to recover the classical picture of an electrostatic field.

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Is it best to entirely give up the interaction via real and virtual photons picture, and just consider the electromagnetic interaction among charged particles as being the interaction of each with the quantized E-M field of the others?

Good point.
To find a solution, we must first consider which parts of the electric and magnetic fields of the proton and electron interact. Let's assume that the magnetic dipoles of the two particles remain constant in value. Their interaction is only that the dipoles of the two particles align with each other as we know from every bar magnet.

What remains is the electric field between them. An empirical fact is that the attraction of an initially resting electron and a proton is accompanied by the emission of photons. We have both acceleration and photon emission, and we have no kinetic energy to begin with. Where then does the energy for the acceleration and the photon emission come from? Only two explanations are possible: from the mass of the charges or from their field. The last explanation is impossible because of the constancy of the electric charge. (The constancy of charge was established in 1909 by Millikan and Fletcher in their famous oil drop experiment and has since been used for isolated electrons as well as for electrons bound to a nucleus). So you have to explain photon emission by a mass defect.

I prefer a model in which the photon emission is caused by the electric field of the charge. The charm of such a model is that the accumulation of protons in the nucleus is not explained by a nuclear force, but by the most extensive loss of electric charge.

Electrostatic interaction by exchange of virtual photons?

For any explanation of field interactions by a deeper model than that of the virtual photon interaction, the electric field must first be modelled. Such a model must include the repulsive effect of charges of the same name and the reduction of the charge field in the case of opposite charges. This is not impossible for the model of the field lines. If the charges are of the same name, the field lines bend according to the principle that where there is one body, there cannot be a second. If the charges are opposite, the field lines contract each other and the released field energy is emitted in the form of photons.

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