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Aharonov-Bohm Effect

Could someone (other than Jim Choate) explain the mechanism behind the
Aharonov-Bohm effect, where a quantum field propagating via two distinct
paths finds its wavefunction phase shifted by the amount of magnetic flux
enclosed, even if it is completely shielded from the magnetic field, and
traverses only regions of space where B=0.

This has apparently been experimentally verified, and is a leading
candidate for reading the state of quantum dots.  

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