Why is it interesting?

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• The best explanation can be found here: http://gregnaber.com/wp-content/uploads/GAUGE-FIELDS-AND-GEOMETRY-A-PICTURE-BOOK.pdf at page 18ff

For quite some time it was felt that such phase changes in the wavefunction were of no physical significance since all of the physically measurable quantities associated with the charge q depend only on the squared modulus $|ψ|^2$ and this is the same for $ψ$ and $e^{iqΩ} ψ$. However, in 1959, Aharonov and Bohm [AB] suggested that, while the phase of a single charge may well be unmeasurable, the relative phase of two charged particles that interact should have observable consequences. They proposed an experiment that went roughly as follows: A beam of electrons is split into two partial beams that pass around opposite sides of a solenoid (this is a very tightly wound coil of wire through which a current passes, creating a magnetic field that is confined inside the coil ). Beyond the solenoid the beams are recombined and detected at a screen. The result is a typical interference pattern that manifests itself experimentally as a variation from point to point on the screen of the prob- ability of detecting a particle there. One observes this interference pattern when there is no current flowing through the coil, so that the magnetic field in the solenoid is zero, and then again when there is a current and hence a nonzero magnetic field inside the coil. Since the electrons pass around the coil and the magnetic field is confined inside the coil, any shift in the interference pattern in these two cases cannot be attributed to the magnetic field (which the electrons do not encounter). The vector potential, on the other hand, is generally nonzero outside the solenoid even though the magnetic field in this region is always zero. One could then only conclude that this vector potential induces different phase shifts on the two partial beams before they are recombined and that these relative phase changes account for the altered interference pattern. This experiment has, in fact, been performed (first by R. G. Chambers in 1960) with results that confirmed the expectations of Aharonov and Bohm.page 6 in Topology, Geometry and Gauge Fields: Foundations by Naber

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