What's the relation between virtual photons and electromagnetic potentials? Given that:
1) virtual photons mediate the electric and magnetic force fields
2) the magnetic field is the curl of the magnetic vector potential
3) the electric field is the negative gradient of the electric scalar potential
How do we understand the vector and scalar potentials in terms of virtual photons? 
Specifically, I mean curl-free magnetic vector and gradient-free electric scalar potential fields, where no electromagnetic waves or physical forces are present. 
For instance, what's actually happening outside of the solenoid in the Aharonov-Bohm experiment that alters the phase of the passing electron? Are virtual particles involved there? This question has really stumped me because I only have an undergrad level of understanding in physics. Thanks for any help.
 A: The electric and magnetic potentials are the temporal and spatial parts, respectively, of a four-vector which play the role of connection in a U(1) gauge theory. The quantized connection provides the interpretation of what you call "virtual photons" which mediate (in a sense) electromagnetic interactions. Yes, I guess these virtual photons are responsible for the AB effect but the electric and magnetic fields associated with these photons outside the internal region of the solenoid never come to existence namely, they never go 'on shell'. This means that no physical electromagnetic field can be detected in that region whatsoever.
A: Virtual photon clouds are responsible for potentials, not electric and magnetic fields, and this is what makes the explanation of forces in terms of photon exchange somewhat difficult for a newcomer. The photon propagation is not gauge invariant, and the Feynman gauge is the usual one for getting the forces to come out from particle exchange. In another useful gauge, Dirac's, the photons are physical, and the electrostatic force is instantaneous.
When you have a solenoid, the photons are generated by the currents in the solenoid, and a charge moving through this virtual photon cloud has an altered energy and canonical momentum according to the distribution of the photons at any point in space. The effect can be understood from the current-current form of the interaction:
$$ J^\mu(x) J^\nu(y) G_{\mu\nu}(x-y)$$
Where G is the propagation function, and the current J is the probability amplitude for emitting/absorbing a photon. The propagation function reproduces the vector potential from a source J, as it acts on another source J at another point.
There is no difference between classical sources producing photons and classical currents producing a vector potential--- they are the same. The electric and magnetic field description is not fundamental, and the gauge dependence of the photon propagator is just something you have to live with.
