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Here is an interesting paper for the Physics SE community:

On the role of potentials in the Aharonov-Bohm effect. Lev Vaidman. Phys. Rev. A 86 no. 4, 040101 (R) (2012). arXiv:1110.6169 [quant-ph].

You should check it out because it's an amusing read, but I will summarise the argument to whet your appetite, assuming you have basic familiarity with the Aharonov-Bohm (AB) effect. In the traditional AB setup, one considers an electron in a superposition of paths, taking it in two opposite directions around a solenoid treated as a classical source of the electromagnetic field. The observable relative phase acquired between the electron's paths is attributed to the influence of the magnetic vector potential on the electron, which cannot be globally gauged away - despite the absence of a physical field anywhere along the electron's path(s) - due to a topological obstruction.

Instead, Vaidman considers the effect of the electron on the sources of the field, treating the latter as quantum particles. He shows that the relative phase between the two branches of the wavefunction can be considered as arising from the action of the physical field of the electron, which is not zero at the position of the sources. However, Vaidman uses highly contrived gedankenexperiments and completely semi-classical arguments, which begs a pair of concrete and related questions.

1) Can Vaidman's first, electric AB effect gedankenexperiment be described in a fully quantum manner, by solving (at least approximately) the three-particle Schroedinger equation? If not, why not?

2) Is it possible to explain within this formalism the experiments of Tonomura et al. (Phys. Rev. Lett. 56 no. 8, pp. 792-795 (1986)), who used a superconductor to completely shield the magnetic field of the source?

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Here is a really nice explanation of the Ahronov-Bohm effect. Also, here is a paper by Timothy H. Boyer claiming that there is still no evidence confirming the quantum topological effect. –  Dale Jun 12 '13 at 6:18
Vaidman also has a newer paper on the subject, Paradoxes of the Aharonov-Bohm and the Aharonov-Casher effects, arXiv:1301.6153 [quant-ph], to be published in Yakir Aharonov 80th birthday Festschrift. –  Emilio Pisanty Sep 4 '13 at 15:28
After a long time of doing other things, I finally got round to reading Vaidman's papers. It seems to me that any significant process towards your first question is probably valuable, publishable material. –  Emilio Pisanty Sep 4 '13 at 15:33
Related: arXiV:1308.2093. –  Mark Mitchison Apr 8 '14 at 5:03
Perhaps related and this too. –  HolgerFiedler Dec 27 '14 at 20:00

2 Answers 2

On point (1) I can see no reason why it should be impossible, but nobody has done it to the best of my knowledge.

On point (2), there is a paper claiming that the AB experiment is entirely a result of local interactions between fields, and that it does not occur if the field interactions are totally shielded. The author claims there was a flaw in the Tonomura experiment:

Experimentally, no experiments so far have been performed under the condition of perfect shielding of the field interactions. The most ideal one was the experiment performed by Tonomura et al. [10], where the magnetic flux is shielded by a superconductor from the moving electron’s path. Their setup is basically equivalent to Configuration I where the flux is confined in a superconducting shield. Contrary to the analysis for Configuration I, a clear AB phase shift was observed despite the presence of the superconducting shield. In this experiment, however, incident electrons with a speed of about $2.4 × 10^8$m/s were used. In fact, no superconducting material can shield the magnetic field produced by such fast electrons [2], and the ideal shielding analysis in Section IV-A cannot be applied to the experiment in Ref. 10. In other words, the shielding in the experiment of Ref. 10 was only one-sided where the incident electron is moving in a field-free region, whereas shielding of both sides is necessary to eliminate the Aharonov-Bohm effect. The experimental result of Ref. 10 can be fully understood in the framework of the local field interaction between the localized flux and the magnetic field produced by an incident electron.

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In this Comment on Macroscopic Test of the Aharonov-Bohm Effect Tomislav Ivezic wrote "...because only the electric field from the solenoid with steady current exists in the region outside the solenoid and it can locally influence the electron travelling through that region." What he describes is nothing else then the interaction between the flying electron and an electric field from the surface of the edge of a slit or a wire. The result is the diffraction of electrons (or photons) by this field. Then more the field is quantized and the diffraction lead to fringes on the observation screen.

This model is applicable to edges, slits and multi slits and works with single photons (as well as with single electrons) too.

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