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?
Best Answer
first of all, the quantum mechanical description 1 in its current form is impossible. A charged particle can't simply absorb a real photon. This is most easily seen in the final particle's rest frame. There is no photon left so the total energy is just the rest mass of the particle, times $c^2$, but the initial state has a higher energy of the particle in this frame, because the particle was moving, and also an additional positive energy of the photon. So the energy couldn't have been conserved in this process.
A charged particle may only absorb a real photon if it emits another one in another direction. So microscopically, it's always a process composed out of the Compton scattering subprocesses. If the number of real photons in the same state is large, they may be described as a classical wave. You may partially quantize the system, keeping the electromagnetic field classical and quantizing just the particle, or otherwise. The reason why this description agrees with the full quantum description in the classical limit is manifest.
Also, I don't quite follow the difference between your classical description and the quantum mechanical description 2. Concerning the usage of virtual photons, well, if you want to study the whole process using the tools of quantum field theory - including the pre-history when no electromagnetic wave existed - then of course the electromagnetic wave had to be created at some point, and some of the photons were absorbed by the charged particle. The charged particle became virtual for a little moment as well, before it emitted another photon which is needed for the energy conservation, as I have explained.
So the photon that was absorbed by the charged particle was virtual - it only existed for a finite amount of time. However, the electromagnetic wave was probably propagating for such a long time that even this photon may be called "real". There is a simple relationship between virtual and real particles - real particles are the virtual ones that happen to sit exactly on the mass shell, so they satisfy $E^2-p^2=m^2$. This identity may be exactly checked only when $E,p$ are measured totally accurately - which means that the particles must exist indefinitely. If they don't exist indefinitely, then they're always "virtual" to some extent, but chances are that if they exist for a long time, you may also imagine that they're "real".
The "virtuality" of a particle may be defined as the difference $E^2-p^2-m^2$ - the distance from the physical mass shell. If the virtuality is low, the virtual particle may exist for a long time and look "real".
Finally, there are no "virtual photon states" in the Hilbert space. The Hilbert space only contains real particles. Virtual particles are an object that appears in the calculation of probability amplitudes for various processes - in the Feynman diagrams. Virtual particles are internal lines of Feynman diagrams, given by propagators that determine the 2-point function (correlator) of a quantum field. But they don't correspond to any physical states. There are no off-shell physical states in the Hilbert space.
So if you have a history in which some particles exist for a finite amount of time, so that they're strictly speaking virtual from the Feynman-diagrammatic viewpoint, it is still true that at every moment, there must exist some real particles that are actually present. However, it is tough, misleading, ambiguous, and unnecessary to calculate the "exact intermediate states" in quantum field theory. Such objects - wave functionals - would also depend on the field redefinitions (of the quantum fields), renormalization schemes, and other things. It's actually very useful to avoid these things when they're not necessary and only talk about the things that can be measured - the cross sections that may be calculated from the scattering amplitudes.
A problem with the "wave functionals" of the intermediate states is that they're only well-defined with respect to a reference frame - but virtually all regularizations we know to calculate the loop diagrams rely on the Lorentz symmetry. Because the Lorentz symmetry is obscure by the foliations of the spacetime, it becomes harder to "regulate" the exact wave functional at the loop level. Of course, at the classical or semiclassical level, one may describe very accurately what's going on.
In your particular situation, there was no real problem because all the photons in the problem were really on-shell, and you may present them as real photons if you wish.
Best wishes Lubos