Attractive electrostatic interaction by exchange of virtual photons? 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?
 A: 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).
A: 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.
A: 
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.
