Aharonov–Bohm Electrodynamics in Material Media: A Scalar e.m. Field Cannot Cause Dissipation in a Medium

Abstract

In the extension of Maxwell equations based on the Aharonov–Bohm Lagrangian, the e.m. field has an additional degree of freedom, namely, a scalar field generated by charge and currents that are not locally conserved. We analyze the propagation of this scalar field through two different media (a pure dielectric and an ohmic conductor) and study its property over a frequency range where the properties of the media are frequency-independent. We find that an electromagnetic (e.m.) scalar wave cannot propagate in a material medium. If a scalar wave in vacuum impinges on a material medium it is reflected, at most exciting in the medium a pure “potential” wave (which we also call a “gauge” wave) propagating at c, the speed of light in vacuum, with a vector potential whose Fourier amplitude is related to that of the scalar potential by the angular frequency times the vector potential amplitude equal to the wave number times the scalar potential amplitude, where the angular frequency squared equals the speed of light squared times the wave number squared.

Keywords extended electrodynamics · gauge symmetry · fields theory · electromagnetic waves · ohmic conductors · dielectrics

1. Introduction

The extension of Maxwell equations according to the Lagrangian proposed by Aharonov and Bohm has been widely studied in recent years, both theoretically and from the point of view of possible applications. This extension is quite natural and was actually introduced with various motivations and technical approaches, even before the work by Aharonov and Bohm. For recent reviews and developments, see the references.

The extended theory is also called “scalar electrodynamics”, because a crucial role is played in it by a scalar field which is not present in Maxwell theory and represents in fact a further degree of freedom of the e.m. field. The uniqueness of the extended theory, under the usual assumptions of regularity and relativistic invariance, has been proven by Woodside. The Aharonov–Bohm action is characterized by a reduced gauge invariance and has been investigated until now only at the classical level, except, under some restrictive assumptions, in the work by Jimenez and Maroto. Energy and momentum conservation has been studied in detail elsewhere.

The scalar field, which in this work will be denoted by S, can be expressed in terms of the e.m. potentials: it is the vacuum permeability times the vacuum permittivity times the time derivative of the scalar potential, plus the divergence of the vector potential. Due to the reduced gauge invariance, S cannot be set identically to zero like in Maxwell theory, but its occurrence is rare because its source is the “extra-current” — the time derivative of the charge density plus the divergence of the current density — a quantity which is different from zero only where charge is not locally conserved.

For classical sources, within a particle-like description of charge and current where it is always possible to “count” the particles crossing a given surface in a given time, violations of local charge conservation are inconceivable. For quantum sources, however, the concept of exact localization of particles is replaced by the concept of probability expressed through a wavefunction, and the expectation values of physical quantities are subjected to uncertainties, also in macroscopic wavefunctions like those of, e.g., tunnelling Josephson junctions.

Moreover, even though in the solutions of the Schrödinger equation with a local potential and in free quantum field theory the local conservation of probability is guaranteed, several physical systems are described by Schrödinger equations with non-local potentials or by renormalized quantum field theories where anomalies cannot be excluded. Some first-principles numerical calculations of the currents in molecular devices have shown anomalies in local conservation, whose interpretation is still unclear. Certain anomalies might be dismissed as artefacts of the mathematical models, but then, how is it possible in quantum mechanics to speak of a reality beyond mathematical models?

The conclusion is, in our opinion, that the generation of an e.m. scalar field S by quantum systems is rare but cannot be excluded. This process is described by the extended theory in vacuum in a relativistically invariant way and does not involve any violation of causality. A detailed calculation of the radiation field generated by anomalous sources oscillating at high frequency in extended electrodynamics has been performed elsewhere.

The next big question is: what happens when the scalar field encounters some medium and propagates in it? The answer is not simple, because the behavior of a scalar field is markedly different from the familiar behavior of electric and magnetic fields. Moreover, the presence of a scalar field usually implies, in vacuum, the presence of a longitudinal component in propagating electric fields. A simple-minded approach to this issue can lead to paradoxes, as we will show in the following (Section 2.3). For a full description one must take into account the propagation equations of the potentials, and the startling conclusion is the following (Section 2.4): the scalar field cannot enter a medium in which local conservation of charge holds, and correspondingly the longitudinal component of the electric field also vanishes in the medium. This is true both for insulating dielectrics and for conductors in which the current density is the conductivity times the electric field, at least in the range of frequencies where the conductivity can be regarded as independent from the frequency. Only a “pure potential” wave propagates in the medium, with zero scalar field and zero longitudinal electric field. In Section 3 we compute in detail reflection and transmission of all field components at a vacuum-material interface, obtaining the usual Fresnel relations plus the conditions for the potential waves. Finally, Section 4 contains our conclusions and outlook.

2. Aharonov–Bohm Electrodynamics in a Material Medium

In Section 2.1 of this Section we recall the extended field equations in vacuum, in the presence of free charges. The non-Maxwellian scalar field S which appears in the equations can only be generated by charge and currents which are not locally conserved. Therefore we shall suppose that somewhere outside the medium there exist such sources for S. On the other hand, we assume that in the medium which the propagating fields encounter, local conservation does hold true, both for the free charges and currents and for the molecular charges and currents, related to polarization and magnetization in the medium. This assumption allows to reduce the complexity of the problem to an acceptable level. One could, more generally, estimate the higher order corrections due to the possible failure of local conservation also inside the medium; such corrections turn out to be very small.

In Section 2.2, we write the extended field equations in a medium and their general solution in plane waves, which exhibits an apparent paradox concerning the longitudinal electric component. The paradox is resolved in Section 2.4, where we solve the equations for the potentials, distinguishing between the case of a pure dielectric without losses and the case of an ohmic conductor. We show that in both cases, one has in the medium a vanishing scalar field and consequently a vanishing longitudinal electric field.

2.1. Extended Field Equations in Vacuum

In this section, we write the field equations of the extended Aharonov–Bohm electrodynamics in vacuum, in the presence of free charges and currents which do not satisfy everywhere the continuity relation. These equations look like extended Maxwell equations in the sense that they contain a scalar field S giving rise to an additional scalar source, the time derivative of S, for the divergence of the electric field in the first equation, and to an additional vector source, the gradient of S, for the curl of the magnetic field in the fourth equation. From the second and third equations, which are unchanged compared to Maxwell’s formulation, it follows that the electric and magnetic fields can be expressed in terms of the potentials in the usual way. Finally, one defines S as the vacuum permeability times the vacuum permittivity times the time derivative of the scalar potential, plus the divergence of the vector potential. It is understood that the Lagrangian of the field is here the Aharonov–Bohm Lagrangian with reduced gauge invariance, so that, in general, it is not easy to choose the gauge in such a way as to obtain a vanishing S. See Appendix A for the Lagrangian in covariant form and also for covariant expressions of the field equations and of the residual gauge invariance.

(The seven equations of the set labelled 1 in the article are the extended Maxwell system: divergence of E equal to charge density over vacuum permittivity minus the time derivative of S; divergence of B equal to zero; curl of E equal to minus the time derivative of B; curl of B equal to the vacuum permeability times permittivity times the time derivative of E, plus the vacuum permeability times the current density, plus the gradient of S; E as minus the gradient of the scalar potential minus the time derivative of the vector potential; B as the curl of the vector potential; and S as defined above. The exact typeset forms are in the published article.)

The energy–momentum density tensor of the field has been derived previously. The energy density picks up, beyond the usual electric and magnetic terms, three further contributions built from the scalar potential times the time derivative of S, the vector potential dotted into the gradient of S, and the square of S. The corresponding energy flux is the usual Poynting vector, the cross product of E and B divided by the vacuum permeability, corrected by a term in the scalar potential times the gradient of S and a term in the vector potential times the time derivative of S.

2.2. Extended Field Equations in a Medium

Let us consider a medium with molecular charge density and molecular current density connected to the magnetization field and the polarization field by the usual relations: the molecular charge density is minus the divergence of the polarization, and the molecular current density is the curl of the magnetization plus the time derivative of the polarization. These expressions satisfy by construction local conservation.

We suppose that the molecular parts of the charge density and of the current density are well defined, as distinguished from the free parts, since we are in the absence of ionization processes.

Summarizing, in this subsection we use the following hypothesis: first, validity of Aharonov–Bohm electrodynamics; second, inclusion of the charge and current sources, both molecular and free, using standard textbook relations for linear, homogeneous, isotropic media. We recall, and stress, that those relations imply local charge conservation. From these hypotheses, we are going to derive the equations for the electromagnetic fields and scalar in a material medium.

After including the molecular sources and defining in the familiar way the auxiliary vectors D and H, we obtain the field equations in the medium, plus the wave equation for S — consistent with the local conservation of the molecular sources — whose right-hand side is the vacuum permeability times the extra current, that is, times the sum of the time derivative of the free charge density and the divergence of the free current density.

Using the usual constitutive relations for a homogeneous, isotropic medium, one has the corresponding system for E and B in the medium.

As already discussed at the beginning of this section, we note that the scalar field has sources only where the free charge is not locally conserved. In particular, in the present work, we suppose that S does not have any sources in the material medium considered.

Our approach is consistent with our assumption that S is not generated in the medium but can have sources outside the medium. In principle, if S is non-interacting, then one can suitably re-define the charge and current densities. However, in our approach this is not possible in general, because we admit that S can have other sources, outside the medium, and, therefore, is not free. Therefore it makes sense to prove, as we do in the following, that in the medium S is not only free but actually zero.

2.3. Wave Solution for the Electric Field in a Conductive Medium and the Dissipation Paradox

In this subsection, we further use the standard relation between free current and electric field through the conductivity of the medium, stressing again that this current is conserved — its divergence is associated to a time varying charge density, as dictated by the charge continuity equation — and we look for propagating wave solutions for the electric field.

Let us suppose that there is an ohmic linear relation between current density and electric field, and that local conservation also holds, as previously discussed, for any free charges in the medium. By eliminating the magnetic and scalar fields among these equations, one obtains a relation for the Laplacian of E which couples the transverse and longitudinal parts through the gradient of the divergence of E.

The usual transverse mode, with wave vector perpendicular to the field amplitude, has zero divergence, and the corresponding dispersion relation is the familiar one: the wave number squared equals the permeability times the permittivity times the angular frequency squared, multiplied by one plus the imaginary unit times the conductivity over the permittivity times the angular frequency.

Note that in a good conductor at microwave frequencies the ratio of the conductivity to the permittivity times the angular frequency is large, implying a strong dissipation of the transverse mode. The consideration of microwave frequency here is only for convenience of possible experiments and in order to be in a regime dominated by the electrical conductivity — the opposite limit corresponding to a dielectric. At much larger frequencies, about the electron plasma frequency, the conductivity is strongly frequency-dependent.

By replacing a longitudinal mode with Fourier amplitude proportional to the wave vector, one obtains a relation from which, under the assumption that the longitudinal amplitude does not vanish, would result the dispersion relation for a wave in vacuum: the wave number squared equal to the vacuum permeability times the vacuum permittivity times the angular frequency squared. This result is rather surprising because it shows no decay, while a decay should be present, since the power dissipated per unit volume — the current density dotted into the electric field, that is, the conductivity times the squared magnitude of the electric field — has a positive time average. The answer to this paradox can be obtained considering the equations for the potentials (Section 2.4), which show that there is actually no electric field, so that the assumption of a non-vanishing longitudinal amplitude is incorrect, and no scalar field either, in the longitudinal mode in a material medium. It turns out that this mode is a “pure potential”, or “gauge” wave, with zero fields, and zero power. The same happens in a dielectric without losses; by this, we mean a material with real permittivity, or in other words, the regime considered corresponds to the transparent region of dielectrics.

2.4. Equations for the Potentials and Solution of the Paradox

We can derive from the field system the equations satisfied by the potentials, using the expressions of E, B and S in terms of them, and one obtains the wave equations for the scalar and vector potentials with the total — free plus molecular — charge and current densities as sources, as one could have expected.

From these equations, again with the constitutive relations, one readily obtains the corresponding equations in the medium, in which the difference between the vacuum permittivity and the permittivity of the medium appears as a coefficient coupling the scalar potential to the divergence of the vector potential.

If we apply these equations to a dielectric without losses, the amplitudes of the Fourier modes satisfy a pair of coupled algebraic relations. The transverse mode corresponds to an ordinary transverse electromagnetic wave in the gauge with vanishing scalar potential and vanishing divergence of the vector potential, propagating at the speed of light in the medium.

The longitudinal mode has the dispersion relation of a wave in vacuum. However, the remarkable point is that the Fourier amplitude of the scalar — the imaginary unit times the wave vector dotted into the vector potential amplitude, minus the vacuum permeability times permittivity times the angular frequency times the scalar potential amplitude — turns out to be zero for the corresponding longitudinal mode, and, consequently, also the longitudinal electric field is zero.

In the case of a conducting medium with the locally conserved ohmic current, the equations for the Fourier amplitudes acquire the conductivity terms. The corresponding transverse mode is of course the dissipative one found above, and the longitudinal one the previously found paradoxical relation, but again, as in the case of the dielectric, one has a vanishing scalar amplitude, and zero longitudinal electric field — and thus no dissipation, resolving the paradox.

In both, dielectric and conductor, the dispersion relation of the longitudinal mode corresponds to a wave of pure potentials, with zero fields, in which the vector and scalar potentials are related by the angular frequency times the vector potential amplitude equal to the wave number times the scalar potential amplitude, with a correspondingly vanishing scalar field and vanishing longitudinal electric field. We will denote this wave of pure potentials a “gauge” wave.

These results indicate that a scalar field wave cannot propagate in a material medium in which local conservation of charge holds. If such a wave in vacuum converges on a medium it should be reflected, at most exciting in the medium a gauge wave propagating at the speed of light in vacuum. Ordinary transverse waves propagating in the material and in vacuum are, in general, also generated, as studied in some detail in the following Section.

3. Reflection and Transmission of Waves at the Vacuum-Material Interface

3.1. Incident Scalar Wave

We consider a scalar field wave in vacuum incident on a material surface at an angle relative to the external normal. We consider the scalar wave to have scalar and vector potentials, so that for the Fourier mode amplitudes we have the scalar amplitude expressed through them, with a corresponding longitudinal electric field.

At the interface we must consider the presence of the incident wave, reflected ones, and transmitted ones. As discussed above, the longitudinal transmitted one must be a “gauge” wave, with the angular frequency times the transmitted vector potential amplitude equal to the transmitted wave vector times the transmitted scalar potential amplitude.

We must, in general, allow for the generation of usual transverse radiation by the induced currents, free and/or of polarization. We know from the results in the previous Section that the transverse radiation in the medium has only a potential vector. Without loss of generality, we consider the reflected normal radiation also to be determined by only a transverse potential vector. (The most general transverse mode in vacuum can be considered as the superposition of a wave with only a transverse vector potential, and a gauge wave, with a longitudinal vector potential and a scalar potential whose Fourier amplitudes are related as above. The same can be said of any other mode that has the vacuum dispersion relation, since the addition of the gauge wave does not change the corresponding fields — this is the reason of the name chosen for this wave.) The reflected radiation also has a longitudinal component, in which a possible gauge component of the transverse reflected wave can be included.

Since the frequency is the same for all waves, and all longitudinal, as well as the normal radiation in vacuum, have the same dispersion relation, the matching condition at the interface, which requires the equality of the components parallel to the surface of all wave-vectors, indicates that the angles between wave-vectors and surface normal are all equal for these four waves; that is, the usual law of reflection, and no refraction of the longitudinal transmitted wave. On the other hand, the transverse transmitted wave has the dispersion relation of the medium, so that the matching condition is in this case the usual Snell law.

The continuity of the potentials at the interface requires that the sum of the incident and reflected vector potential amplitudes equals the transmitted ones, and that the sum of the incident and reflected scalar potential amplitudes equals the transmitted scalar potential, which itself equals the speed of light times the transmitted longitudinal vector potential amplitude.

Additional boundary conditions are the continuity of the tangential component of the electric field and the continuity of the normal component of B, which is identically satisfied as it is zero at both sides. Finally, for a dielectric we have the continuity of the normal component of D, and that of the tangential component of H. For a conductor, the same equations indicate that the normal component of E in vacuum must be equal to the surface charge density divided by the vacuum permittivity, and that the tangential component of B in vacuum is equal to the vacuum permeability times the tangential component of the surface current density at the interface. Note that the surface charge density and the tangential surface current density are related by a two-dimensional continuity equation on the interface.

We refer to Figure 1 for the explicit expressions of the boundary conditions. The vector potentials of the longitudinal modes have components in the direction of their respective wave-vectors, and those of the transverse modes have components in the directions explicitly denoted in the figure.

Figure 1. Sketch of the incident, reflected, and transmitted waves at the interface between vacuum (below) and material medium (above). The overline denotes the transverse component parameters.

In the case of a dielectric medium the set of boundary conditions is then written out in full and completed with the Snell law; for the case of a conductor the transmitted transverse amplitude vanishes and the boundary conditions are correspondingly reduced, completed by the relation between the surface charge and the tangential surface current.

The solution of these systems results in all Fourier amplitudes given in terms of those of the incident wave.

For the case of a dielectric we give the expressions of the generated wave amplitudes for the case of high-frequency waves, in particular optical frequencies, so that the magnetization is negligible and the refraction index is the square root of the ratio of the permittivity of the medium to that of vacuum. (The five resulting amplitude relations, labelled 20 in the article, and the four corresponding relations for a conductor, labelled 21, are given there in closed form in terms of the refractive index and the incidence angle.)

The extra source involved in the reflection and transmission of the longitudinal wave can be easily determined from the wave equation of the scalar field, integrated in the conventional pill-box, where the relevant Fourier amplitude is the extra source amplitude times the depth of the region occupied by the non-conserved current. The remarkable result is that for both, dielectric and conductor, the required extra source is zero. This means that the ordinary sources in the medium are sufficient to generate the mode conversion of the incident scalar wave, leading to no propagation of a scalar wave in the bulk of the medium. The only possible longitudinal mode transmitted is a gauge wave.

3.2. Incident Transverse Field Wave

For the usual Maxwell transverse field wave, the fields satisfy the Fresnel relations for reflected and transmitted waves, in the case of a dielectric, and the reflected field laws in case of a conductor. In Aharonov–Bohm electrodynamics, we must also consider the behavior of the corresponding potentials. For this, we must take into account that the potentials of a transverse field wave correspond to a superposition of a pure transverse vector potential and a gauge wave. We thus denote the incident wave potentials Fourier amplitudes by the transverse vector potential amplitude, and by the scalar potential amplitude of the gauge wave, equal to the speed of light times the longitudinal vector potential amplitude. Correspondingly, the reflected transverse field wave has a transverse vector potential amplitude, and a gauge part; in case of a dielectric, there is also a transmitted transverse field wave, with a transverse vector potential and a gauge component.

The polarization of the incident wave must also be taken into account. We thus divide the problem into one with a polarization with magnetic field parallel to the interface, and another with electric field parallel to the interface. See Figure 2 for the conventions used in the case with B parallel to the interface. As before, the transverse component of the vector potentials are in the directions indicated in the figure, while their longitudinal components are in the direction of their corresponding wave-vectors.

Figure 2. Convention used for the case of a dielectric with a transverse incident wave whose polarization corresponds to the magnetic field parallel to the interface, normal to the plane of the figure.

We consider first the case of a dielectric. For a polarization with the magnetic field of the wave parallel to the interface, the conditions of continuity of potentials and of the fields are written out and, for the case of high-frequency waves, resolved into the familiar Fresnel-type ratios for the transverse vector potential amplitudes, together with a non-zero reflected gauge amplitude proportional to the incident transverse amplitude times the square of the refractive index minus one times the sine of the incidence angle, and a transmitted gauge amplitude equal to the sum of the incident and reflected transverse amplitudes.

For a polarization with the electric fields parallel to the interface, we refer to Figure 3 for the conventions used in this case. Here the transmitted gauge amplitude equals the incident longitudinal amplitude, the reflected gauge amplitude vanishes, and the transverse amplitudes again take the familiar Fresnel form.

Figure 3. Convention used for the case of a dielectric with a transverse incident wave whose polarization corresponds to the electric field parallel to the interface, normal to the plane of the figure.

For a conductor, in the case of polarization with the magnetic field parallel to the interface the continuity conditions give a transmitted gauge amplitude equal to the incident longitudinal amplitude plus the incident transverse amplitude times the tangent of the incidence angle, a reflected gauge amplitude equal to the incident transverse amplitude times the tangent of the incidence angle, and a reflected transverse amplitude equal to the incident one. If the polarization of the wave is with the electric fields parallel to the interface, the transmitted gauge amplitude equals the incident longitudinal amplitude, the reflected gauge amplitude vanishes, and the reflected transverse amplitude is minus the incident one.

The transverse vector potential components in the relations obtained in this Subsection correspond to fields that satisfy the above mentioned known laws for the Maxwell transverse waves: Fresnel relations for a dielectric medium, and reflection laws for a conductor. The most remarkable result is the generation of transmitted and reflected gauge waves when a Maxwell transverse wave interacts with a dielectric or a conductor.

4. Conclusions

In this work, we have analyzed with some detail the interaction of different waves, possible in the extended electrodynamics of Aharonov–Bohm, with material media in which the local conservation of current, free and/or molecular, holds. A particularly interesting result is the difficulty of the transmission through those media of the longitudinal modes that, according to that theory, are possible in vacuum. Only the so-called “gauge waves” propagate in the medium, as shown in Section 3.1.

We have regarded the scalar field as a possible additional degree of freedom because, even though it is determined by the scalar and vector potentials, its dynamical equation has as a source an additional scalar quantity, that determines the degree of local non-conservation of charge.

In our treatment of the sources in the material medium, we have used conventional, accepted expressions used in the literature. We thus made clear that we are studying Aharonov–Bohm electrodynamics in conventional media, and that this is only the first, natural step in dealing with a still unconventional theory. In fact, the existence of non-conserved sources is still a controversial matter. We anticipate to study those non-conventional sources in future works. In the present work, we limit ourselves to conventional media in order not to introduce additional hypotheses, still not completely accepted by the scientific community.

It is important to mention that the gauge wave is not just a pictorial representation of the reduced gauge freedom of the theory, as could be used, for instance, to represent the addition of such a wave to another type of wave in order to effect an allowed change of its gauge, without change of the wave fields — electric, magnetic and scalar. According to the theory, a gauge wave can be generated and detected by itself. In other words, the theory allows pure gauge waves, without another accompanying type of wave, as physical entities that can in principle be generated and detected.

As can be seen from the amplitude relations, when such a pure gauge wave reaches a medium — that is, the only incident potentials satisfy the gauge-wave condition — it is fully transmitted without generation of additional waves. This is, of course, consistent with it not interacting with media in which local conservation of charge holds.

Concerning the generation of a gauge wave, from the results for a transverse wave incident in a medium, we see that a normal transverse wave can generate a gauge wave when it interacts with that medium. Consider, for instance, the potentials of a normal transverse wave generated by an oscillating elementary dipole, assumed at the origin of coordinates. These potentials have been computed previously: the scalar potential is the vacuum permeability times the speed of light, divided by four pi times the distance, times the retarded time derivative of the dipole moment projected on the unit radial vector; the vector potential is the vacuum permeability divided by four pi times the distance, times the retarded time derivative of the dipole moment.

As discussed previously, unlike in the Maxwell theory, the wave equations for the potentials are uniquely determined in the Aharonov–Bohm theory. The residual gauge freedom of the theory (see Appendix A) allows to add sourceless wave solutions to satisfy boundary conditions when it is more practical to work in terms of these conditions than in terms of the actual sources that give rise to the potentials. The conclusion is that no actual gauge freedom exists in Aharonov–Bohm theory if the sources are fully known. The limited gauge freedom left is in fact a flexibility of the theory that allows to work in terms of boundary conditions when, from a practical point of view, the actual sources are difficult to determine.

From the previous equations, we see that for a given propagation direction there is a field-transverse wave consisting in the superposition of a transverse vector potential — the retarded dipole derivative projected onto the plane perpendicular to the propagation direction — and a gauge wave, whose longitudinal vector potential equals the scalar potential divided by the speed of light.

In particular, we see that in the direction of the dipole, a pure gauge wave is emitted, while at right angles, what is emitted is a pure transverse vector potential. For intermediate angles a mixture of components is involved. Interaction of the waves emitted at these intermediate angles with a medium would thus generate gauge waves, additional to those emitted in the direction of the dipole.

In this context, we would like to point out again that what one denotes as the gauge freedom of Aharonov–Bohm electrodynamics has a different meaning from that in Maxwell’s. In the latter theory, even if all electromagnetic sources are known, one is still free to select different gauges. In the former theory, however, if all sources are known — in the example above the only source is an elementary dipole — the gauge is completely determined. The reduced gauge freedom allows only to include “incoming” potentials, as boundary conditions, generated by sources other than those in the region considered. In this way, the gauge wave components determined in the example of the dipole are unique, a consequence of the theory that lends further support to the “reality” of the gauge wave.

As for the detection of a pure gauge wave, from the earlier results one concludes that this type of wave can only interact with media in which the extra source is not zero, resulting in a power per unit volume equal to minus the extra source times the scalar potential, exchanged between the wave and the medium supporting the extra current, that could be detected as changes in currents and voltages in the circuit. Collective quantum effects are likely required for this.

Since specific examples of such a circuit require detailed models of media in which charge is not locally conserved, an area in which much theoretical and experimental work is needed, we postpone the discussion of the matter to a future work.

Author Contributions

Conceptualization: F.M., G.M.; formal analysis: F.M., G.M.; writing: F.M., G.M. All authors have read and agreed to the published version of the manuscript.

Funding

This work was supported by the Open Access Publishing Fund of the Free University of Bozen-Bolzano.

Data Availability Statement

Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

Conflicts of Interest

The authors declare no conflict of interest.

Appendix A. Covariant form of the Aharonov–Bohm Lagrangian and of the Field Equations

The Aharonov–Bohm Lagrangian in SI units is, including an interaction term, the usual Maxwell term built from the field-strength tensor, minus a term proportional to the square of the four-divergence of the four-potential divided by twice the vacuum permeability, plus the interaction of the four-potential with the four-current.

The reduced gauge transformations are of the form of a shift of the four-potential by the four-gradient of a scalar function, subject to the condition that the scalar function satisfies the wave equation.

The covariant field equations can be written as the four-divergence of the field-strength tensor equal to the vacuum permeability times the sum of the four-current and an induced four-current, the latter being minus the four-gradient of the inverse d’Alembert operator applied to the four-divergence of the current. Notice that the field equations do not depend on the parameter multiplying the extra Lagrangian term.

In summary, the Aharonov–Bohm Lagrangian has a reduced gauge symmetry compared to the Maxwell Lagrangian, but fully preserves the relativistic covariance.