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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)?

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It's indeed difficult to experimentally realise an infinite solenoid, but this is not the main point. The key point is to have a configuration when $B=0$ whereas $A \neq 0$. This can be done with superconductors, which screen the magnetic field for instance. The experiments are there: –  FraSchelle Apr 25 '13 at 22:26
    
-> A. Tonomura, T. Matsuda, R. Suzuki, A. Fukuhara, N. Osakabe, H. Umezaki, J. Endo, K. Shinagawa, Y. Sugita and H. Fujiwara, Observation of Aharonov-Bohm effect by electron holography, Phys. Rev. Lett. 48(21), 1443-1446, (1982). -> A. Tonomura, N. Osakabe, T. Matsuda, T. Kawasaki, J. Endo, S. Yano and H. Yamada, Evidence for Aharonov-Bohm effect with magnetic field completely shielded from electron wave, Phys. Rev. Lett. 56(8), 792-795, (1986). –  FraSchelle Apr 25 '13 at 22:27
    
-> N. Osakabe, T. Matsuda, T. Kawasaki, J. Endo, A. Tonomura, s. Yano, and H. Yamada, Experimental confirmation of the Aharonov-Bohm effect using a toroidal magnetic field confined by a superconductor, Phys. Rev. A34(2), 815-822, (1986) –  FraSchelle Apr 25 '13 at 22:29
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2 Answers

up vote 3 down vote accepted

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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Thanks. Do you know any reference where I can read more on this practical aspect of A-B effect? –  qoqosz Aug 23 '12 at 14:45
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This is a classic paper on the detection of A-B effect in a metal ring: Observation of h/e Aharonov-Bohm Oscillations in Normal-Metal Rings by Webb, Washburn, et. al. –  Doug Packard Aug 23 '12 at 15:43
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It doesn't require singularities in space. A nonzero magnetic flux through the loop will do. There's absolutely no need for an infinite solenoid.

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The original A-B effect discuss the possibility to observe an interference pattern related to the gauge vector $A$ without magnetic field present along the charge path. So, in this sense, it requires infinite solenoid, since it is the only configuration when $A \neq 0$ but $B=0$. –  FraSchelle Apr 25 '13 at 22:23
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