Role of the nonlocality of the vector potential in the Aharonov-Bohm effect

When the electromagnetic potentials are expressed in the Coulomb gauge in terms of the electric and magnetic fields rather than the sources responsible for these fields they have a simple form that is nonlocal (i.e., the potentials depend on the fields at every point in space). It is this nonlocality of classical electrodynamics that is primarily responsible for the puzzle associated with the static Aharonov-Bohm effect: that its interference pattern is affected by fields in a region of space that the electron beam never enters.