For finding a net force on a charge when it is in influence of many charges we simply do vectorical addition of all individual interaction of that charge with others. That's what is principle of superposition. It is an observed experimental fact. The principle of superposition and coulomb's law rule the whole electrostatics and same principle we do use in classical electrodynamics. But QED models how these forces actually work (with the virtual particles and more). So in QED do we still add individual interactions of charges to get a net Electric field at a point or force acting on a particle which is in influence of many, or something strange happens at quantum level?

( Adding more : So,is principle of superposition still an observed experimental fact or do any model keep an opinion about it ? )


In QED, this superposition principle is still valid at least to first order perturbation theory. The deeper reason behind this superposition principle from the QED point of view, is, that photons do not interact with each other. They do not carry charges. They do not "see" each other. So they can be safely superposed without having an effect on each other.

Since the electromagnetic field is described by photons in QED, the same applies for the electromagnetic and therefore also for the electrostatic field. Please be aware that when I just said that "the field is described by photons" this does not mean that you can imagine a field as composed out of photons. The description happens in a highly non-trivial and very abstract manner totally lacking any imaginability.

As ACuriousMind already pointed out, the field is actually an operator-valued distribution in QED which lives in some Hilbertspace (an abstract mathematical space which kind of replaces the imaginability we are used to in classical physics in QM). The field operator at a single point in spacetime has the capability of adding or canceling field quanta ("photons" so to speak) at this point in space and time. This capability of adding and canceling quanta at a certain point in spacetime contributes to a transition amplitude if certain condition (like energy momentum conservation) are fulfilled (to formulate it in rather simple language).

One would not talk about "forces" in QED, by the way. The concept of force is nearly totally absent in QED. It is replaced by the concept of interaction (!!!!and not by the concept of exchange of virtual particles!!!!).

You define a suitable interaction term in the Hamiltonian (= energy) or Lagrangian which contains the fields you would like to interact with each other and compute transition amplitudes with it. Regarding the virtual particles, they only appear when you try to compute the transition amplitude via perturbation theory (which is the only suitable way since exact solutions are not available). So, they are actually kind of artefacts of the approximation scheme we use to evaluate things we can measure at the end.

At the very beginning I said that the superposition principle is valid only to first order perturbation theory. This is true, since when you push the approximation scheme further (to second order), photons indeed can interact with each other. They do not do this directly, but via virtual charged particles. The technical term is "Delbrück scattering". The cross section of this interaction is very, very tiny and does not have an effect in any practical applications.

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QED is rarely concerned with precise forces or fields, although you may calculate e.g. the Coulomb force in a non-relativistic classical limit, see my answer here.

QED is a quantum field theory. As such, the electromagnetic field does not possess a definite "value" at any point, it is to be thought of as an operator-valued distribution and what you may want to compute is e.g. its expectation value. Furthermore, particles in a quantum field theory may be created and destroyed, and it is only at energy/distance scales where this is relevant that QED delivers substantial corrections to the classical electromagnetism, but it is also at this scale that the classical picture of a bunch of charged particles that sit at definite spatial locations is not a good picture of what the theory describes (and presumably how the world is) anymore.

Thus, your question is not really meaningful at the quantum level - we do not think in terms of single charges and the fields produced by them on the level of the quantum field theory.

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  • $\begingroup$ Okay.. What QED will explain the situation when we are talking about three particle system (each with certain charge. Maybe they might not have a definite spatial location but a probabilistic region they might be in) So what will be an expected effect on the third due the presence of two. How QED will try to compute the result due two . $\endgroup$ – user73541 Jul 18 '15 at 12:39
  • $\begingroup$ @NayanTelrandhe: The classical approximation is good enough for that one. QED is mainly concerned with predicting things unexplained in the classical theory, such as the anomalous dipole moment, or the results of high energy scattering experiments. $\endgroup$ – ACuriousMind Jul 18 '15 at 12:42
  • $\begingroup$ Then is Principle of superposition still just an observed experimental fact ? $\endgroup$ – user73541 Aug 19 '15 at 9:32

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