Aharonov–Bohm effect in brief is due to some singularities in space. In books it's infinite solenoid most of the time, which makes some regions of space not simply connected.
What intrigues me is the fact that in real experiment we can't use infinite solenoid. So even if we use one and say that locally it's good approximation it doesn't change the fact that whole space is still simply connected. But the fact is that the effect was experimentally observed.
So the question arises - how one should describe this effect in more rigorous manner (or maybe not rigorous but possible in real world)?
Answer
The solenoid might not be infinite in space, but in general neither is the configuration space for electrons. For example, in experiments that test the A-B effect, electrons are typically confined to a wire or ribbon of metal, with a hole that a (finite) solenoid can be placed through. In this case the electrons' position space is not simply-connected by construction, but the A-B effect is that you get path-dependent phases from the gauge field, regardless of how the non-simply-connectedness of your space came about.
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