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 non-local i.e. the potentials depend on the fields at every point in space. It is this non-locality of classical electrodynamics that is at first instance responsible for the puzzle associated with the Aharonov-Bohm effect: that its interference pattern is affected by fields in a region of space that the electron beam never enters.
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