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The classical (retarded) Lienard-Wiechert scalar and vector potentials describe the electromagnetic field due to an arbitrarily moving electric point charge.

Thus given the motion of electron $A$ one can calculate the EM field at the position of electron $B$. Using the Lorentz force law one can then calculate the force exerted on electron $B$ due to electron $A$.

But should there also be a reaction force exerted back on electron $A$ from electron $B$ ?

The standard view asserts that the momentum gained by electron $B$ is balanced by an opposite momentum in the EM field.

However, consider the following particle interaction picture of the electromagnetic interaction between electron $A$ and electron $B$.

Interacting Electrons

Electron $A$ emits a virtual photon which is absorbed by electron $B$ so that it gains some positive energy-momentum ($\Delta E$,$\Delta p$).

This process is described classically by electron $B$ being acted upon by the retarded Lienard-Wiechert EM field due to electron $A$.

But, according to the Feynman-Stueckelberg interpretation, a virtual photon travelling from $A$ to $B$, delivering energy-momentum ($\Delta E$,$\Delta p$) to electron $B$, is equivalent to a virtual photon travelling from $B$ to $A$, delivering energy-momentum ($-\Delta E$,-$\Delta p$) to electron $A$.

Thus in the particle interaction picture there is also a reaction force ($-\Delta p/\Delta t$) back on electron $A$ due to a virtual photon travelling backwards in time from electron $B$.

The standard classical description due to retarded Lienard-Wiechert potentials does not describe this reaction force back on the source particle.

Thus the standard classical description of the EM interaction between charged particles seems to be incomplete - momentum is not conserved.

It seems that in order to complete the classical description of particle-particle electromagnetic interactions one needs to employ advanced Lienard-Wiechert potentials as well as the retarded potentials.

Does this make sense?

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    $\begingroup$ So, what is your question? $\endgroup$ – Vladimir Kalitvianski Nov 6 '14 at 19:37
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    $\begingroup$ The quantum picture where one says that "virtual particles are exchanged" does not mean that there are actually particles flying through space. Quantumly, you do not describe interaction through forces, and the unholy mixture of classical and quantum thinking in this question leaves me thoroughly confused what you actually want to know. $\endgroup$ – ACuriousMind Nov 6 '14 at 19:39
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The standard classical description due to retarded Lienard-Wiechert potentials does not describe this reaction force back on the source particle.

There is force back on the first particle, but it occurs after a delay necessary for the retarded field of the second particle to get back to the first particle. This delay is due to the choice of retarded fields only. For other choices (advanced field), the force as function of time will be different. However, whatever the choice of the solution to the Maxwell equations, the energy and momentum are always locally conserved.

Thus the standard classical description of the EM interaction between charged particles seems to be incomplete - momentum is not conserved.

No, the change in energy and momentum of particles does not mean total energy or momentum is not conserved (in the common meaning of physics); the change of energy and momentum of particles are accounted for by the opposite changes in EM energy and momentum and their current densities (which are given by the EM stress tensor) so that total momentum and energy are conserved locally (in the sense they move continuously; globally energy or momentum of a system do not need to be conserved because energy and momentum can leak to surrounding space).

It seems that in order to complete the classical description of particle-particle electromagnetic interactions one needs to employ advanced Lienard-Wiechert potentials as well as the retarded potentials. Does this make sense?

There is infinity of different solutions of Mawell's equations, retarded or advanced solutions is just one of them. All solutions are consistent with local conservation of energy and momentum, so there is no way to decide which to use based on such requirement.

The retarded fields are used because they are simple to understand and they have not been falsified by disagreement with experience. Advanced solutions have not been falsified either, but they are strange and do not seem to be necessary.

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No, it does not make any sense. Imagine a heavy charged body and your tiny electron moving in its field. The electron will radiate, but the source body will not. No advanced potentials can save this situation.

The radiation reaction force must be present even here, so its nature is different from "advanced interactions".

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