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IV.  Aharonov-Bohm Effect

The main contribution of exterior differential forms approach to Electrodynamics is, however, one of principle. Thus, it is our opinion that the Aharonov-Bohm effect, for instance, calls for an emancipation of Electrodynamics from the constraints of a certain type of accepted experience. In other words, contrary to the currently academically acclaimed opinion, this effect means more to Electrodynamics than to Quantum Mechanics. Indeed, the experience is, first and foremost, an ensemble of experiments. Only in this occurrence can it be affected by technology. In the realm of Electrodynamics our accepted experience is marked by the statement that there are no magnetic charges, which boils down to the equation defining the potential vector:

(32)

For the matters in which we are presently interested this condition is usually written in the integral form

(33)

by coming back to differential forms. The writing in equation (32) helps promoting the reasoning based upon vectors, thus concealing the tremendous role of the differential forms themselves as physical objects. Moreover, passing to (33) also helps promoting the idea that Electrodynamics is all about the integral forms of the electromagnetism. While occasionally this may be the case (Scorgie, 1998), we think that Electrodynamics is not well served by taking this point of view to the extreme. Actually we find a little improper for the present status of the geometrical knowledge, as assisted by the introduction of differential forms, to insist only upon analytical side of these tools, for they might mean more.

The fact seems to have passed largely unnoticed that the writing the basic laws of Electrodynamics by making use of differential forms, opens the question of the very possibility of physical existence of these tools, according to the real experimental conditions they are called to describe. Usually the forms are interpreted in terms of fluxes or circulations, thereby implicitly suggesting the applications of the type of analysis as illustrated above. These differential forms have nevertheless their own algebraic habitat and, sooner or later, we will have to judge the experiments in terms of them, not only in terms of the corresponding vectors. For, while the past classical work seemed to make plainly obvious that, in Electrodynamics, we have to do with vectors, there are situations that cannot be described but only in terms of differential forms. One of these situations is the Aharonov-Bohm effect (Aharonov, Bohm, 1959).

A little digression: we insist in calling it an “effect”, in spite of the fact that we agree without any reserve with Boyer’s conclusion (Boyer, 2000) that the phase lag can be classically explained in detail, and inasmuch as this can be done Boyer’s conclusion is that there is no topological “effect”. Our reason is that Aharonov-Bohm phenomenon is experimentally exhibited through an interference pattern for what we usually consider particles, and associating a wave to a particle is, at least up to this point in history, more of a matter of Wave or Quantum Mechanics rather than Classical Mechanics. And not only because this is the way the phenomenon has been confirmed, but mostly because we think that it adds to the old fundamental problem of legitimacy of associating waves with particles, we prefer to call it “effect”, and continue with the usual literature label. This is, so to speak, the sentimental side of the story. In what follows we will describe the phenomenon as a classical topological matter, so that it might be called effect after all, from each and every point of view.

Let us recall the experimental arrangement with a solenoid. As this is nowadays common place in the specialty 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. This does not mean that the magnetic field is inexistent. It exists, as created by the 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, the argument is true with anything else instead of the solenoid and, we might add, the argument is true even with nothing in the place of solenoid. Personally, we do not think that such a conclusion is too far from truth. Fact is that the presence of magnetic field can be recognized in wave-like behavior of particles and, reciprocally, this presence is decided by the topology of the stream of particles.

Let us translate the experimental set up in mathematical terms as allowed by the differential forms. All that the quantum mechanical description asks from its classical counterpart is the definition of momentum of the particles and indeed this is all that matters in the mathematical description of the phenomenon (Boyer, 1973). Thus the classical definition of the momentum of a free particle

(34)

is here always taken as true. As known, the momentum is represented by a differential 1-form. As a matter of fact the definition in equation (34) involves the vanishing of a 1-form as the expression of the fact that the particles are free in a certain Space region. Indeed, that definition can be rewritten in the form

(35)

and, of course, the electrons should be always characterized by momentum this way, no matter if the field is present or not. However, the presence of a field changes the things. We may rephrase this fact by saying that the electrons are primarily carrying mass; this is the expression of the fact that we consider electrons as particles, in spite of the fact that their pure dynamical properties have been, in the process of their discovery, inferred from experiments involving electro-magnetic properties. However, the facts representing the experimental conditions of definition of particles have to be always present in their definition. When it comes to electrons or to particles carrying charges in general, the formalism related to exterior differential forms gives us a brilliant possibility to account for these conditions. Indeed, a magnetic field is described by the 2-form

(36)

and as we do not apply global considerations, it is irrelevant whether this 2-form is closed or not. Now this 2-form has to be zero over the regions where the free particles move, otherwise these particles would not be free. In this case, there is a theorem of Cartan (Creangă, Luchian, 1963) showing that the 2-form (36) is zero on the set of vectors satisfying equation (35) if, and only if, it can be written in the form

(37)

where A is a conveniently chosen 1-form. This means that

(38)

Thus, the magnetic flux is simply manifest by a plane of action as indeed historically has been the case (Roche, 2000). Now, if there is an action upon electrons in the region where the magnetic field is zero, as the Aharonov-Bohm phenomenon shows, the electrons are not free particles anymore in that region. This means that while the 2-form (36) is zero over the region of Space where the electrons move, the 1-form (35) is not zero over that region by experimental conditions. Then according to another theorem of Cartan, the two factors of (37) are proportional for these conditions

(39)

thus leading to a redefinition of the momentum of particles

(40)

which, incidentally, is all that Quantum Mechanics asks for the description of these particles. So the necessary and sufficient condition that the present magnetic induction form is null over the region of moving particles is that the momentum of particles updates with a term involving a certain convenient vector. The history proves that this vector is the potential vector of Electrodynamics, thus giving identity to the “conveniently chosen form” A. Certainly we cannot deem this situation as non-classical – in the sense of belonging to quantum-mechanical situations – but, also certainly, this way the definition of the momentum is a topological matter. On the other hand, it is the pattern of interference of the waves associated with this momentum which is experimentally under observation here, and what is observed we would prefer to term as a wave-like “effect” exhibited by the streams of particles.

     The argument can be extended in exactly the same manner to any 2-form which is axiomatically introduced, for instance the electric current 2-form from the Hehl-Obukhov approach to Electrodynamics:

(41)

This 2-form must be zero over the ensemble of experiments involving free particles in the mechanical acceptance. Thus, the current must be of the form

(42)

where U is, again, a conveniently chosen 1-form. Now, the existence of Aharonov-Bohm electrostatic effect shows that, in specific experimental arrangements, the electrons are not free (Matteucci, Pozzi, 1985), while the 2-form of electric current is zero over the region where they move, so that we have to update the definition of momentum exactly as before

(43)

Perhaps we should insist a little bit more upon the meaning of the 2-form J, as it receives a whole new meaning in this logical turn. As Matteucci’s and Pozzi’s experimental arrangement shows, the 2-form in question is experimentally represented by some arrangement of electric dipoles. Consequently, it is such an arrangement that has to be described by the current 2-form. This fact may call for a revision of our concept of current, including that of displacement current (Roche, 1998). As far as we are aware, in this connection there is no news in the specialty literature concerning the realization of Aharonov-Bohm electrostatic effect following the Boyer’s suggestion of experiments with electrified cylinders at different potentials (Boyer, 1973). According to the above considerations it may not be possible to realize such an effect.

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