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.

• here is an excellent descrption of virtual particles -in particular, why we should NOT think of them as particles at all. – FrankH Oct 13 '11 at 13:03
• @FrankH: at last some interesting laymen description of 'virtual particles'. (Thank you for the link) – Helder Velez Mar 21 '12 at 23:07

$$J^\mu(x) J^\nu(y) G_{\mu\nu}(x-y)$$