Consider an electromagnetic wave of frequency $\nu$ interacting with a stationary charge placed at point $x$. My question concerns the consistency of two equally valid quantum-mechanical descriptions of the electromagnetic wave. First I will describe the classical picture, then the two quantum mechanical descriptions, then I will ask experts for a conceptual unification of the two quantum mechanical descriptions. For parsimony I'll assume an extremely low frequency wave, but this is not strictly necessary.
Classical description 1: At point $x$ the electromagnetic wave contributes a slowly varying electric field, and a slowly varying magnetic field. The charge at $x$ experiences a force due to the electric field, and begins to move. As the charge begins to move it experiences a force due to the magnetic field. Using the right-hand-rule it is easy to see that the net force on the charge is in the direction of motion of the electromagnetic wave.
Quantum mechanical description 1: In the quantum mechanical description of the same electromagnetic wave, real photons are traveling with momentum $\frac{h\nu}{c}$ (in the direction of motion of the above electromagnetic wave), and are absorbed by the charge at $x$, causing it to recoil in the above electromagnetic wave's direction of motion (due to conservation of momentum).
Quantum mechanical description 2: (I will be assuming that in the quantum mechanical description of the electromagnetic field, the force due to the electric/magnetic fields between two moving charges is due to the exchange of virtual photons). At point $x$ the electromagnetic wave contributes a slowly varying electric field, and a slowly varying magnetic field. The charge at $x$ experiences a force due to the electric field, due to the exchange of virtual photons with the charge that produced the electric field. Similarly for the magnetic field. In other words, there are no real photons -- just the virtual photons mediating between the charge at $x$ and the charge whose movement created the electromagnetic wave in the first place.
Finally, my question: how are descriptions 1 and 2 reconciled? In description 1, the origin of the electric and magnetic fields (a charge), and those fields' descriptions in terms of virtual photon exchange, is completely ignored. On the other hand in description 2 there are no real photons, and the virtual photons are longer-ranged (do they self-interact?). Are the two descriptions equivalent? If so, it must be that a real photon can be written in terms of "virtual-photon" basis states. What is such a decomposition called, and can someone point me to a discussion of it?
Answer
Lubos's answer is 100% correct, but it is missing the subtle error in OP's thinking.
The OP is imagining that if you have two charges that repel according to Coulomb's law, and you slowly shake one, the response of the other is as if it were repelled from the retarded position of the charge. If this were true, then the real photons would just be related to virtual photons, because the actual propagating signal would just be the location of the place to feel a force from.
But this is not at all what happens when you wriggle a charge. The outgoing wave-part is a 1/r field, which is completely separate from the Coulomb repulsion. In Dirac gauge, you can consider the Coulomb repulsion as to the instantaneous current position of the other particle, plus a propagating field which is 1/r. The propagating field fixes up the causality--- this doesn't actually transmit forces faster than light, but the propagating field is not related in a simple way to the Coulomb field.
The two fields, Coulomb and wave field, really are separate even classically, and it is miraculous that Feynman could combine them in quantum mechanics using virtual states. Lubos's answer covers the rest, in particular his discussion of field wave-functionals.
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