  In the Aharonov-Bohm (AB) effect, a superposed charge acquires a detectable
phase by enclosing an infinite solenoid, in a region where the solenoid's
electric and magnetic fields are zero. Its generation seems therefore
explainable only by the local action of gauge-dependent potentials, not of
gauge-independent fields. This was recently challenged by Vaidman, who
explained the phase by the solenoid's current interacting with the electron's
field (at the solenoid). Still, his model has a residual non-locality: it does
not explain how the phase, generated at the solenoid, is detectable on the
charge. In this paper we solve this non-locality explicitly, by quantising the
field. We show that the AB phase is mediated locally by the entanglement
between the charge and the photons, like all electromagnetic phases. We also
predict a gauge-invariant value for the phase difference at each point along
the charge's path. We propose a realistic experiment to measure this phase
difference locally, by partial quantum state tomography on the charge, without
closing the interference loop.
