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Aharonov-Bohm Effect

Obviously with this we leave the realm of the Classical Mechanics, and come towards modern times. From this point of view we can present to our Reader one of the most discussed results of the Theoretical Physics of the last century, the so-called Aharonov-Bohm effect (Aharonov, Bohm, 1959).

Let’s recall the basic facts. As today the discussion on the subject is a common place in the specialty literature, and even in the popularization literature, we do not elaborate on details, but simply say that here we have the situation of a magnetic field null over the regions where the particles move. The argument of Aharonov and Bohm, based on quantum mechanical considerations, is that the particles still feel the magnetic field, in spite of the fact that the classical approach rules out the possibility. The experiment proved them right, thus giving an unprecedented momentum to the wave-mechanical approach. We show here that this is a consequence of the Third Principle of Dynamics as formulated above. First, we have to notice that the magnetic field is not inexistent. It exists, as created by the thin solenoid, but is zero in the space regions where the particles move; this fact should be particularly emphasized. As Ehrenberg and Siday put it, one has to expect

“…wave-optical phenomena to arise which are due to the presence of a magnetic field but not due to the magnetic field itself, i.e. which arise whilst the rays are in field-free regions only” (Ehrenberg, Siday, 1949; our Italics).

Otherwise, as Boyer shows it (Boyer, 1973), the argument is true with everything instead of the solenoid from the original experiment proposed by Aharonov and Bohm. We would like to add that the argument is true even with nothing in the place of solenoid, because the theory describes a characteristic of this class of material points. The fact which has to be reminded here is that the presence of magnetic field can be recognized in the wave behavior of the material points and that this behavior is decided in the topology of the rays of such particles. As Boyer shows, for this we don’t need but the redefinition of the momentum according to equation (13), which makes from the charged particle a material point, part of the general material point defined in the “class of the vector potential”. Therefore the Aharonov-Bohm effect, as well as the whole Electrodynamics, can be presented as consequences of the Third Principle of Classical Dynamics, taken in the form we gave it here.

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