The effects of the Aharonov-Bohm type as tests of
the relativistic interpretation of electrodynamics
(Gianfranco Spavieri, Miguel Rodriguez)*
Abstract: An unsolved problem in special relativity relates to the experimental verification of the electromagnetic force acting on charged particles (open currents) and of the fields generated by them. We consider here some unusual but interesting effects of quantum mechanics, called nonlocal quantum effects of the Aharonov-Bohm type, and show how they could be used to test the relativistic interpretation of classical electrodynamics. Other tests for the force on, and fields of, open currents are presented.
PACS numbers: 03.65.Bz, 03.30.+p, 03.75.Dg.
1 - Introduction
It has been questioned that special relativity theory and quantum mechanics introduce fundamental postulates that defy common reason. For example, in the field of special relativity, several physicists object to the physical meaning of the dependence of the time coordinate t on the space coordinates x, which violates the Galilean principle of simultaneity as it occurs for the Lorentz Transformations (LT). To many it seems more reasonable to use, instead of LT, the Tangherlini Transformations (TT), which preserve simultaneity. Some physicists believe that the TT are experimentally equivalent to the LT (from which supposedly they differ by an arbitrary synchronization parameter) while some others mantain they are not. In any event, the TT have the advantage to possess a form that, in some respects, recalls the Galileo Transformations and, thus, are more intuitive.
Similarly, in quantum mechanics some physicists have objected to the introduction of new postulates or ad hoc techniques such as renormalization that is at the base of the success of quantum electrodynamics.
Even if it may be argued that modern science needs not to be intuitive, there is the risk that an accentuated divorce of abstract formalism from physical reality may lead to a loss of interest in modern science and jeopardize its credibility. Thus, it can be of general interest a discussion with the aim of rescuing the intuitive physical aspects of modern science which seems to have been hidden by mathematical formalisms.
Within this call for return of rationality in modern science, we believe that it would be advantageous for the comprehension of quantum physics if it were possible to point out a closer link between the new quantum results and the well known concepts of classical physics. As an example we consider here some unusual but interesting effects of quantum mechanics, called nonlocal quantum effects of the Aharonov-Bohm type (where no local forces act on the particles), and show how they are related to the last advances of classical electrodynamics.
Another important reason for considering nonlocal quantum effects in Sect. 2 is that the discussion on nonlocality revives the polemic on what are the correct expressions of the em forces on elementary point particles, which may possess a charge, an electric or magnetic dipole. This discussion points out several untested aspects of the relativistic interpretation of classical electrodynamics which assumes the validity of Maxwell's equations in their differential form. The main weakness, which is not known by the majority of physicists, is that the microscopic or differential form of Maxwell's equations, as usually interpreted by special relativity, have not been completely tested, while their integral form (Faraday's Law, Ampere's Law, etc) are considered to have been fully tested.
Moving charges are considered by relativity to form a current regardless of the fact that the forces on currents and their fields have been tested on moving charges or current elements forming closed currents only. Thus, it turns out that, theoretically, there is some arbitrariness in the expression of the microscopic form of the em force on elementary particles. This arbitrariness disappear for macroscopic or integral (e.g. for closed circuits) expressions.
As an example of the relativistic formulation of electrodynamics not yet tested, we consider in Sect. 3 the case of the Lorentz force acting on a charged point particle (open current) and the force on a neutral particle with a magnetic dipole moment. Another example in Sect. 4 refers to the detection of the fields generated by an open current (for example, a moving charge).
If the nonlocal quantum effects of the AB type have ultimately (as some physicists believe) a local origin, the experiments done on these effects represents already an evidence that the standard Lorentz force must be modified.
Other more conclusive tests of the relativistic prediction referring to these examples are within experimental reach with modern techniques and the results of the tests will be helpful and decisive for settling the controversy between the relativistic interpretation of electrodynamics and other non-relativistic interpretations.
2 - Locality in the quantum effects of the Aharonov-Bohm type
For decades the Aharonov and Bohm (AB)  effect has attracted the attention of physicists because its quantum interpretation implies that electromagnetic (em) potentials play a fundamental role in quantum mechanics in contrast to their auxiliary role in classical physics. In the AB effect a beam of interfering electrons encircles in their motion a magnetic field B confined within a thin solenoid. When the current of the solenoid is switched on, the observed interference pattern is shifted even though there are no em fields and no forces act locally on the particles. Thus, the quantum interpretation of the AB effect involves the vector potential A which contributes to the phase shift of the quantum wave function.
The difficulty of interpreting this effect in terms of classical em fields has led to suggest the existence of nonlocal topological effects in physics and to ascribe a new physical role to the vector potential . Because of these peculiarities, nonlocal effects have been considered to be of relevance even outside the field of physics. For example, the scientist Roger Penrose has written several articles and books where he shows that nonlocal effects are essential for the understanding of scientific model of human consciousness. For the above reasons, topological nonlocal effects of quantum origin have een considered to be of philosophical and scientific interest and physicists have been motivated to look for other effects f the AB type.
Several em effects have been discussed in the literature since the leading work of AB. The dual effect involving magnetic dipoles was considered by Aharonov and Casher (AC) , and was followed by the Matteucci and Pozzi (MP)  electrostatic effect, the electric dipole effects related to the Röntgen interaction by Spavieri , and few others. Experimental verifications have corroborated several of this effects  .
If the em interaction is time independent, the classical energy H = E is a constant of motion and for free particles coincides with the initial kinetic energy of the particle (1/2)mv2 . The phase of a free particle of classical momentum p = mv and energy E , is j = (p.x - Et)/h' [h' here means h/2p ; for typographical reasons, it has been impossible to reproduce the usual symbol for this constant].
In all these effects the em interaction gives rise to a phase shift d j leading to an observable displacement of the interference pattern. d j is said to be of quantum origin if it is not originated by changes of p and E foreseen by classical theory.
If classical forces act on the particle, p changes by d p and the phase shift is
d j = (1/h')d p.dx .
Since in this case the forces act locally on the particle, the effect is said to be local. If the interaction energy is changed by the time dependent amount d E(t) , then H = E + d E(t) and the phase changes by
d j = -(1/h')d Edt .
Therefore, a classical phase shift reads
d j = (1/h')[ d p.dx - d Edt ] . (1)
The most famous nonlocal effects are the AB and AC effects. In the AB effect, the paths of moving charged particles encircle a solenoid in the absence of forces or energy changes and the interaction with the vector potential originates a path-independent (topological), observable phase shift. Then, the phase of this remarkable nonlocal and topological effect is purely of quantum origin.
The phase j may be related to canonical momentum Q by the expression
j = (1/h')Q.dx
and in turn Q may be related to the em momentum of fields Qem by
Q = ± Qem = ± (1/4p c)(E´ B)d3x' .
For the configuration of fields of the AB effect one finds
Q(x) = Qem = (q/c)A(x) .
In the AC effect a particle possessing a magnetic dipole moment m moves in the presence of an external electric field E produced by a line charge of linear density l . In this effect, the canonical momentum coincides with the so called hidden momentum of the magnetic dipole
Q = Qh = m´ E/c = -Qem . Because of conservation of total momentum, Qh + Qem = 0 , and this explains the minus sign in front of Qem .
The idea that these effects may have instead a classical origin has been considered by several authors , and among them should be considered of relevance the arguments of Boyer  who has interpreted these and the electrostatic experiment by Matteucci and Pozzi  (MP) as classical em-lag effects.
It is worth remarking that the idea that quantum effects may have classical origin is not restricted to the effects of the AB type but it has been claimed to have general validity. Some of the physicists involved in deriving quantum results from classical physics have taken hints from the so-called stochastic electrodynamics and we mention here the cited works of Boyer, Cavalleri and Spavieri , and Salazar and Spavieri .
Going back to the classical interpretation of the effects of the AB type, as a counter example, a phase of classical origin is that of the MP effect, where charged particles interact with a dipolar charge distribution and there are em forces, of zero average along the path, acting on the particle. The phase shift is originated by the forces producing the change d p in the direction of motion, and d j can also be interpreted as a classical em lag-effect . Since the phase shift is path-independent, this may be defined as a classical, local, and, in the restrictive sense of the word, topological effect.
As already mentioned, topological effects of quantum origin are considered to be of philosophical and scientific interest and, sometimes, subtle considerations are involved in determining whether an em effect may have a quantum, nonlocal character or not.
In the case of the electrostatic effect of MP, Boyer's  approach is transparent and it is clear that this can be considered strictly as an em-lag effect. Nevertheless, in the case of the AB effect the lag has been determined by Boyer by taking into account the change of the em interaction energy which may affect the kinetic energy of the particle, without proof that there is a force on the particle. Furthermore, in the case of the AC effect which involves the interaction between a moving neutral magnetic dipole and the external electric field , the approach of Boyer, has been criticized by Aharonov, Pearle and Vaidman  (APV) for not taking into account the hidden momentum of the magnetic dipole.
As already mentioned, the intriguing aspect of these nonlocal effects is that there is an observable displacement of the interference pattern even though there are no forces acting locally on the beam of particles. Here, the expression of the forces (for example, the Lorentz force) is the one given by the relativistic interpretation of electrodynamics. If the relativistic expression of the force is correct, then these effects are surely nonlocal because the force is zero in the experimental conditions that lead to these effects.
However, if the nature of these effects were not nonlocal and were instead due to a local interaction, then it would imply that the relativistic expression for the force is incorrect so that the actual force on the beam of particles could turn out to be non-null.
In the next sections we consider in detail how, within a nonrelativistic interpretation of classical electrodynamics, the em force on elementary particles may differ from the relativistic expression.
3 - A modified expression of the Lorentz force
We consider here two cases: modification of the Lorentz force due to the hypothetical existence of longitudinal forces in agreement with the integral form of the Faraday Law; test of the validity of the Lorentz force for the differential form of Faraday's Law.
3.1 - longitudinal forces
Making use of the connective derivative d/dt = ¶ t + v.Ñ , from Faraday's Law for a moving loop, the electromotive force (emf) D V may be expressed by means of the effective field Eeff as
D V = Eeff.dl = -(1/c)B.ds = -(1/c)(¶ tA+v´ B).dl . (2)
Using vector identities we may write
-(1/c)(v.Ñ )A.dl = -(1/c)[Ñ (v.A)+v´ B].dl = (1/c)(v´ B).dl , (3)
because the gradient term gives zero contribution to D V in a closed loop.
Equivalently, (2) reads
Eeff.dl = -(1/c)[¶ tA-(v.Ñ )A].dl . (4)
We see from the integral forms (2) and (4) that the velocity dependent term is somewhat arbitrary as it can be defined to within the gradient of a scalar function. The correct expression of the force must be determined by the internal consistency of electrodynamics and by experimental verification.
The velocity dependent effective field term of (2), E^ = (1/c)v´ B , exhibits the correct structure of the force field perpendicular to v as verified experimentally in charged particle accelerators. On the other hand, the parallel component of the velocity dependent effective field term of (4), Eê ê = -(1/c)(v.Ñ )Aê ê = -(1/c)Ñ ê ê (v.A) , exhibits the correct structure of the force field, parallel to v , that can be shown to be consistent with the conservation of energy .
If the charge moves with velocity v in a circuit defined by the element dl = vdt , then
(v.Ñ )A.dl = (v.Ñ )Aê ê .dl = Ñ ê ê (v.A).dl = 0 ,
so that to the standard velocity term of (2) we can add the term Ñ ê ê (v.A) without altering the emf of the Faraday Law in integral form. With the new longitudinal term the velocity terms read
v´ B - Ñ ê ê (v.A) .
Thus, in this case, the standard Lorentz force
[ r E + (1/c)J´ B ]dt ,
may be implemented by an extra term that takes the form
f = -(1/c)Ñ ê ê (J.A)dt . (5)
In the standard interpretation of the AB effect the fields E and B of the solenoid are zero at the position of the moving charge q and there are no forces acting on the charge. In the standard, relativistic interpretation, the AB effect is a nonlocal effect.
A last remark, that may link the present discussion on quantum effects with the description of classical electromagnetism by means of LT or TT, concerns the improbable but not impossible case where even the AB effect may have classical origin. If the AB effect is a local effect there must be a force acting on the particle that produce the lag effect proposed by Boyer. This force could be the modified Lorentz force (5) proposed by Spavieri et al. . This modification is unacceptable within the framework of SR but is not impossible within the framework of modern ether theories based on TT or other generalized Galileo Transformations.
The argument leading to the modification (5) of the Lorentz force is based on a general definition of force as the time derivative of the electromagnetic momentum Qem . As shown above, in a nonrelativistic interpretation, the term (5) indicates that a force
-(q/c)Ñ ê ê (v.A) = -(q/c)(v.Ñ )Aê ê
may act on q in the direction of motion. This longitudinal force produces an em-lag effect that precisely accounts for the AB effect. If the new longitudinal term exists, then the AB effect is a local effect. Assuming that the effects of the AB type are actually local effects, the tests performed corroborating the displacement of the interference pattern in the AB effect are a strong experimental evidence of the existence of longitudinal forces.
3.2 test of the differential form of Faraday's Law
Faraday's Law has been tested only on closed circuits where the emf induced in a closed loop can be measured with a voltmeter. There is no test of the force on an isolated charge which would corroborate Faraday's Law in differential form.
One problem related to the Faraday Law in differential form is the one pointed out by Shockley and James who noticed that when the current in a solenoid varies with time a force
qE = -(q/c)Am
acts on the stationary charge but no opposite force acts on the solenoid.
It is convenient to reconcile, within the standard relativistic interpretation, the force on the charge with the expression of the opposite force on the solenoid or line of magnetic dipoles of moment m .
When both q and m are at rest we have a stationary configuration, and the em momentum in can be written, without approximations, as
Pem = -Ph = (1/c)r Adt = (1/c2)F Jdt = (1/c)(E´ M)dt =
= -(1/c) m´ E = -(q/c) m´ r/r3 = (q/c)Am , (6)
where Ph is the hidden momentum or momentum due to the internal stresses, , F is the scalar electric potential, M the magnetization density and r = xm - xq the distance between m and q. In (6) E has to be evaluated at the position xm of the dipole and the vector potential
Am = -m´ (xm-xq)/ç xm-xqç 3
of the dipole has to be evaluated at the position xq of the charge. Using the result that for stationary system the total momentum is zero ,
P = Pem + Ph = 0
we conclude, as mentioned above, that the total system cannot acquire a net mechanical momentum when the current is switched off, and find that in correspondence of the action force on q
fq = -Pem = -(1/c)r Adt = -(q/c)Am ,
when the current of the dipole varies with time, there is an equal and opposite reaction force fm = -fq ,
fm = -Ph = (1/c2)F Jdt = -(1/c)(m´ E) (7)
acting on m , that solves paradox of Shockley and James.
The force (7) is not contained explicitly in the standard Lorentz expression which turns out to be zero in this case, but it is obtained from a more general expression that accounts for the internal structure of the magnetic dipole. Thus, the standard Lorentz expression must be implemented by (7) in order to account for fm .
A macroscopical experimental test of this force is here considered. In the AB effect there exists an em momentum Qem = (q/c)Am ¹ 0 , and the modified Lorentz force
fq = -¶ tQem = -(q/c)¶ tAm (8)
represents the differential statement of Faraday's law, which has been tested for closed loops only but not on a single stationary charge. The test of this force has the same importance of the experimental verification of basic interaction forces such as those of Newton's and Coulomb's laws.
The following experiment may be used to test the quantum interpretation of the AB effect vs. the alternative classical interpretation, or else, the standard force expression vs. the nonstandard. A long solenoid or toroid carrying a current i is placed near a macroscopic charge distribution Q. Both the solenoid and the charge may be kept stationary and: a) the current may be switched off; or: b) the solenoid may be removed with velocity v . In both cases an impulse on the charge,
fdt = -Qem ,
is predicted according to Eq. (8) and can be measured.
In the first case a), the impulse
QEmdt = Qem = QAm
is due to the radiation field Em = -(1/c)¶ tAm , and a non-null result of the experiment represents a test of the Faraday's law in its differential form. An equal and opposite force acts on the solenoid and solves the Shockley-James paradox (). Notice that there is no Shockley-James paradox for the usual emf induced in closed circuits because these are neutral.
In the second case b), the standard Lorentz force is zero because, according to SR, there are no fields outside the moving solenoid which becomes also electrically polarized. However, the nonstandard expression, which in a nonrelativistic formulation does not foresee electric polarization, predicts an impulse
fdt = -QAm ,
in the direction of v , of the same order of magnitude of that due to the force QEm . Here, a non-null result favors the nonrelativistic interpretation.
4 - Test of the fields produced by an open current
As mentioned in the Introduction, all the existing tests of the fields produced by moving charges refer to charges moving in closed circuits (closed currents). The magnetic field produced by an isolated moving charge (open current) has never been tested.
If a charge is placed in the laboratory reference frame it produces a static electric field in that frame. According to relativity a frame moving with respect to the laboratory will experience also a magnetic field. In fact, for relativity the moving charge forms a current and the charge and current density transform as a four-vector. Thus, by Maxwell's equations the current generates a magnetic field even though there is only an electric field in the rest frame of the charge. In a nonrelativistic interpretation of electrodynamics, a current may be defined as due to charges moving in a closed circuit. Isolated charges may not form a true current even if they move with respect to some observer (open current).
To many physicists the idea that open currents are not true currents generating a magnetic field may seem to be untenable. However, we recall that the physical behavior of charges moving in a closed circuit is quite different than that of open currents. For example, the correct (for relativity) expression of the force on a magnetic dipole has to take into account the fact that the moving charges forming the dipole current are linked to the closed loop forming the dipole m . There are internal stresses contributing to the correct force on m that do not exist for open currents. Thus it is not unreasonable to think that the forces acting on, or the fields generated by, open currents may differ from those due to closed currents. Furthermore, the motion of charges in a closed circuit is a kind of absolute motion (with respect to the frame at rest with the closed circuit) that implies acceleration. On the contrary, open currents do not require absolute but only relative motion and no accelerations are involved.
To test the existence of a magnetic field generated (according to relativity) by open currents, we have proposed in previous articles to perform the inverse of Rowland's experiment. This would consist in moving a magnetic needle through the electric field generated by a charge distribution at rest in the laboratory. Since this experiment is difficult to be realized, we propose here the following alternative experiment that exploits Faraday's Law.
A parallel plate condenser is charged at a high voltage and a coil is made to move at constant speed through the plates. In the frame of the coil, according to relativity, a time-varying magnetic field is created and by Faraday's Law it induces a current in the coil. This current should be observable with the help of a sensitive electrometer.
On the other hand, according to a nonrelativistic interpretation of electrodynamics, open currents do not generate a magnetic field. Since there is no time varying magnetic field, there is no reason why an emf should be induced in the moving coil.
Since a null result rules out the relativistic interpretation, this test should be able to discriminate between the relativistic and other interpretations of electrodynamics.
5 - Conclusions
We have shown that the effects of the AB type may be used as tests for the Lorentz force acting on a charged particle. The test proposed consists in verifying the interaction force between a solenoid and a charge when they are stationary (and the current is switched off) or when they are in relative motion.
When they are stationary the existence of an interaction force would corroborate the differential form of Faraday's Law that has never been tested.
When they are in relative motion, there should be no interaction force according to relativity. If a force is detected, the standard Lorentz force is disproved and these quantum effects have a local origin.
To test the fields generated by open currents we simply propose to detect the induced emf when a coil moves through a charged condenser. Relativity is confirmed only if an emf is induced.
All these tests are within experimental reach by modern techniques and should be able to settle the controversy between the relativistic and other nonstandard interpretations of electrodynamics.
 Y. Aharonov and D. Bohm, Phys. Rev. 115, 485 (1959).
 See for example, R. P. Feynman, R. B. Leighton and M. Sands, The Feynman Lectures on Physics, Vol. II, 15-5, (Addison-Wesley, Reading, Mass., 1964).
 Y. Aharonov and A. Casher, Phys. Rev. Lett., 53, 319 (1984).
 G. Matteucci and G. Pozzi, Phys. Rev. Lett., 54, 2469 (1985).
 G. Spavieri, Phys. Rev. Lett., 81, 1593 (1998); Phys. Rev. A (April 1999); Phys. Rev. A, 59, 3194 (1999); Phys. Rev. Lett., 82, 3932 (1999).
 For the AB effect see: A. Tomonura et al., Phys. Rev. Lett., 56, 792 (1986), and therein cited references. For the AC effect see: A. Cimmino et al., Phys. Rev. Lett., 63, 380 (1989).
 Y. Aharonov, P. Pearle, and L. Vaidman, Phys. Rev. A, 37, 4052 (1988). See also G.Spavieri, Nuovo Cimento, 109 B, 45-57 (1994).
 See for example, X. Zhu and W.C. Henneberger, J. Phys. A, 23, 3983 (1990); S.M. Al-Jaber, X. Zhu and W.C. Henneberger, Eur. J. Phys., 12, 268 (1991).
 T. H. Boyer, Nuovo Cimento B, 100, 685 (1987); Phys. Rev. D, 8, 1667 (1973); Phys. Rev. D, 8, 1680 (1973); Phys. Rev. A, 36, 5083 (1987).
 G. Cavalleri and G Spavieri, Nuovo Cimento, 95B, 194 (1986); Nuovo Cimento, 101A, 213 (1989) and references therein.
 J.J. Salazar and G. Spavieri, Nuovo Cimento, 92B, 157 (1986).
 G. Spavieri, R. Angulo and O. Rodriguez, Hadronic J., 20, 621 (1997).
[This research was made possible by a grant from the CDCHT, ULA, Mérida, Venezuela]
*Centro de Astrofísica Teórica, Facultad de Ciencias,
Universidad de Los Andes, Mérida, Venezuela
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Gianfranco Spavieri was born in Italy in 1942 where he received a Doctor degree in Engineering at the Polytechnic of Milan and, later, he specialized in Theoretical Physics at the University of California. He carried out research activities in California, Arizona, at the International Center for Theoretical Physics of Trieste and the National Research Council of Rome. He is presently a faculty member of the Center for Theoretical Astrophysics of the University of The Andes in Venezuela. Several of his scientific contributions on the foundations of physics have appeared in international journals, together with some philosophical essays on science and humanities. His book Fragments of the Rainbow (Frammenti d'arcobaleno), on the relationship between science and myth, has been published by Edizioni Arcipelago (Milan, 1998). As a member of the graduate program of Fundamental Physics of the Center of Theoretical Astrophysics, he is directing some thesis and Miguel Rodriguez is one of his graduate students.