# Gauge waves generation and detection in Aharonov-Bohm electrodynamics

F. Minotti 111Email address<EMAIL_ADDRESS>Universidad de Buenos Aires,
Facultad de Ciencias Exactas y Naturales, Departamento de Física, Buenos
Aires, Argentina  CONICET-Universidad de Buenos Aires, Instituto de Física
Interdisciplinaria y Aplicada (INFINA), Buenos Aires, Argentina  G. Modanese
222Email address<EMAIL_ADDRESS>Free University of Bozen-Bolzano
Faculty of Engineering
I-39100 Bolzano, Italy

###### Abstract

The extended Aharonov-Bohm electrodynamics has a simple formal structure and
allows to couple the e.m. field also to currents which are not locally
conserved, like those resulting from certain non-local effective quantum
models of condensed matter. As it often happens in physics and mathematics
when one tries to extend the validity of some equations or operations, new
perspectives emerge in comparison to Maxwell theory, and some “exotic”
phenomena are predicted. For the Aharonov-Bohm theory the main new feature is
that the potentials $A^{\mu}$ become univocally defined and can be measured
with probes in which the “extra-current” $I=\partial_{\mu}j^{\mu}$ is not zero
at some points. As a consequence, it is possible in principle to detect pure
gauge-waves with $\mathbf{E}=\mathbf{B}=0$, which would be regarded as non-
physical in the Maxwell gauge-invariant theory with local current
conservation. We discuss in detail the theoretical aspects of this phenomenon
and propose an experimental realization of the detectors. A full treatment of
wave propagation in anomalous media with extra-currents and of energy-momentum
balance issues is also given.

##  I Introduction

Although classical and quantum electrodynamics are among the most precise and
successful theories in physics, there are reasons to look for extensions of
Maxwell equations in various directions and physical contexts
bandos2020nonlinear ; sorokin2022introductory . In particular, we are
interested into the extension first proposed by Ohmura ohmura1956new  and
often called “Aharonov-Bohm electrodynamics” because of their major
contribution (aharonov1963further ; see also van2001generalisation ;
hively2012toward ; Modanese2017MPLB ; modanese2017electromagnetic ;
arbab2017extended ; hively2019classical ; Hively_2021 ; minotti2021quantum ;
minotti2022electromagnetic ). The logical motivation for this extension, at
least from today’s perspective, is the quest for a Lagrangian density which
has the most general covariant form, in terms of four-potentials $A_{\mu}$ and
that sources $j^{\mu}$ (not necessarily locally conserved). Apart from the
addition of a total four-divergence, this form is

$L_{A.-B.}=\frac{1}{4\mu_{0}}F_{\mu\nu}F^{\mu\nu}-\frac{\lambda}{2\mu_{0}}(\partial_{\mu}A^{\mu})^{2}+A_{\mu}j^{\mu}$
(1)

where a possible additional photon mass term proportional to $A_{\mu}A^{\mu}$
has been excluded based on experimental evidence.

###  I.1 Formal structure of local Aharonov-Bohm electrodynamics

We shall call (1) the “Aharonov-Bohm Lagrangian”, even though it is in fact a
well-known expression contained in textbooks (see e.g. itzykson2012quantum ;
fradkin2021quantum ) and its unicity has been proven by Woodside
woodside2009three  in vector formalism. This Lagrangian is gauge-invariant in
the familiar sense only if $\lambda=0$. Therefore by considering extensions
with non-zero $\lambda$ we are sacrificing gauge invariance to extended
relativistic invariance and especially to the possibility of coupling
$A_{\mu}$ to a current $j^{\mu}$ which is not locally conserved.

Maxwell theory with full gauge invariance can be of course re-obtained from
this extended theory in the special case when $\lambda=0$ and current is
locally conserved everywhere. The power of gauge theories in quantum field
theory and particle physics of course strongly suggests to make this
symmetrical choice whenever possible.

The special choice $\lambda=1$ is known in the literature as “Feynman gauge”
(although it is not a gauge condition in the usual sense) and leads to a
quantization method which is alternative to the Gupta-Bleuler method or to
canonical quantization fradkin2021quantum . As we show in the following, in
the extended Aharonov-Bohm theory this choice is natural and almost mandatory
in the presence of currents which are not locally conserved.

The field equations derived from the Lagrangian (1) are

$\partial^{2}A^{\mu}+(\lambda-1)\partial^{\mu}\partial_{\nu}A^{\nu}=\mu_{0}j^{\mu}$
(2)

with $\partial^{2}=\partial_{\mu}\partial^{\mu}$. From this one has by
contraction

$\partial^{2}(\partial_{\nu}A^{\nu})=\frac{\mu_{0}}{\lambda}I$ (3)

where $I$ is called “extra-current” and defined as

$I=\partial_{\mu}j^{\mu}$ (4)

This allows to rewrite eq. (2), for $\lambda\neq 0$, as

$\partial^{2}A^{\mu}=\mu_{0}j^{\mu}+\mu_{0}\frac{1-\lambda}{\lambda}\partial^{\mu}[(\partial^{2})^{-1}I]$
(5)

Now, on physical grounds it is very tempting to add the condition that in
these equations the fields are “locally forced” by the physical sources
$j^{\mu}$, in the sense that second derivatives of the fields at one point in
spacetime are determined by the values of the sources at that point. All
fundamental theories for massless fields give, at least in their linearized
versions, wave equations for the field which have this property.

In this case, this is accomplished by setting $\lambda=1$. As a consequence,
the four-potentials of Aharonov-Bohm theory resemble those of Maxwell theory
in Lorentz gauge or Feynman gauge, and in fact we shall express them routinely
as retarded integrals over the sources. There is a clear difference, however,
apparent from eq. (3): if charge is not locally conserved everywhere, it is
not possible in general to impose the gauge condition
$\partial_{\nu}A^{\nu}=0$. Actually, there is no gauge freedom in the AB
theory, except for the possibility of reduced gauge transformations of the
form $A^{\mu}\to A^{\mu}+\partial^{\mu}\chi$, where the function $\chi$
satisfies $\partial^{2}\chi=0$.

Therefore in most physical situations where the physical sources are
localized, the potentials are completely determined. As we shall see in the
following, they can even be measured using hypothetical material probes in
which $\partial_{\mu}j^{\mu}\neq 0$ ! All other “standard” probes can only
measure the fields $F^{\mu\nu}$ (i.e., $\mathbf{E}$ and $\mathbf{B}$).

From the discussion above it is also clear that in A.-B. electrodynamics there
is a sort of additional degree of freedom, namely a scalar field
$S=\partial_{\mu}A^{\mu}$ satisfying the equation $\partial^{2}S=\mu_{0}I$. It
is not really an independent degree of freedom because it is expressed in
terms of $A_{\mu}$, but it appears explicitly in the extended field equations.
The scalar field becomes automatically decoupled from the sources if $I=0$.

We do not think that $S$ can have any relevant physical effects in normal
laboratory conditions, because we hypothesize that violations of local charge
conservation can only occur in microscopic quantum sources whose emission is
very weak at the macroscopic scale (the situation may be different at an
astrophysical or cosmological scale, see e.g. jimenez2011cosmological ).
Nevertheless, precise measurements could allow to detect the generation of an
$S$ field modanese2019design ; minotti2021current . Also, in order to check
the general consistency of the theory, we have computed the propagation of $S$
in material media and found that it is completely reflected by ohmic
conductors and pure dielectrics (see minotti2023aharonov , preferably in the
arXiv version).

Local AB electrodynamics at a glance

Fundamental fields | $A^{\mu}$
---|---
Derived fields |  $F^{\mu\nu}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}$; $S=\partial_{\mu}A^{\mu}$
Lagrangian | $\frac{1}{4\mu_{0}}F^{\mu\nu}F_{\mu\nu}-\frac{1}{2\mu_{0}}S^{2}+A_{\mu}j^{\mu}$
Field equations | $\partial^{2}A^{\mu}=\mu_{0}j^{\mu}$
Equation for $S$ |  $\partial^{2}S=\mu_{0}I$, $I=\partial_{\mu}j^{\mu}$
Reduced gauge invariance |  $A^{\mu}\to A^{\mu}+\partial^{\mu}\chi$, $\partial^{2}\chi=0$
Lorenz force 4-density on probe $p$ | $f^{\mu}=F^{\mu}_{\nu}j_{p}^{\nu}-I_{p}A^{\mu}$
Vector Lorenz force and power |  $\mathbf{f}=\rho_{p}\mathbf{E}+\mathbf{j}_{p}\times\mathbf{B}-I_{p}\mathbf{A}$, $w=\mathbf{j}_{p}\cdot\mathbf{E}-I_{p}\phi$

Table 1: Formal framework of Aharonov-Bohm extended electrodynamics with the
choice $\lambda=1$ which ensures local field equations. The “minimalistic”
structure of the theory appears: given the currents $j^{\mu}$, one directly
solves for $A^{\mu}$ and computes the Lorenz four-force density on a probe
with current $j^{\mu}_{p}$ and extra-current $I_{p}$. The four-force density
is given here also in vector form in view of the simple application to a
$\phi$-probe described later; the relation between four- and three-vectors is
$A^{\mu}=(\phi/c,\mathbf{A})$, $j^{\mu}=(c\rho,\mathbf{j})$. For the field
equations for $F^{\mu\nu}$ and a comparison with Maxwell theory in the Lorentz
gauge see Sect. I.3.

###  I.2 Energy conservation. Lorenz force

The conceptual structure of the theory is summarized in Tab. 1 and will be
further discussed in Sect. I.3 in comparison with Maxwell theory. In order to
complete the picture we still need the conservation laws of energy and
momentum of the extended e.m. field and the generalization of the formula for
the Lorenz force.

The conservation laws have been derived in [MM-2021] by coupling the field to
a generic gravitational background. The calculation is complex, but the result
is relatively simple and the energy density $u$, for example, can be written
in vector formalism as

$u=\frac{1}{\mu_{0}}\left(\frac{\mathbf{E}^{2}}{2c^{2}}+\frac{\mathbf{B}^{2}}{2}+\frac{\phi}{c^{2}}\frac{\partial
S}{\partial t}-\mathbf{A}\cdot\nabla S-\frac{S^{2}}{2}\right)$ (6)

Some contributions depending on $S$ and explicitly dependent on the potentials
can be seen here.

For the Lorenz force density and for the power exerted on a probe $p$ with
charge density $\rho_{p}$, current density $\mathbf{j}_{p}$ and extra-current
$I_{p}$ one finds

$\displaystyle\mathbf{f}$ $\displaystyle=$
$\displaystyle\rho_{p}\mathbf{E}+\mathbf{j}_{p}\times\mathbf{B}-I_{p}\mathbf{A}$
(7) $\displaystyle w$ $\displaystyle=$
$\displaystyle\mathbf{j}_{p}\cdot\mathbf{E}-I_{p}\phi$ (8)

This means that anomalous extra-sources not only generate a (usually small)
scalar field contributing to the energy density, but can also feel the effect
of the potentials $\mathbf{A}$, $\phi$, and act as probes for measuring them.
It is thus confirmed that the Aharonov-Bohm potentials should be regarded as
physical. In Sect. III we will study a possible concrete realization of a
$\phi$-detector.

###  I.3 Field equations. Comparison with Maxwell theory

Although the conceptual structure of local Aharonov-Bohm electrodynamics
described so far and summarized in Tab. 1 is consistent and very simple, it is
also useful to look at the equations for the field strengths $F^{\mu\nu}$ and
to compare the theory with the version of Maxwell theory obtained by imposing
the Lorentz gauge condition $\partial_{\mu}A^{\mu}=0$.

The equations for $F^{\mu\nu}$ derived from the Lagrangian (1) have the
following form Modanese2017MPLB , remarkably independent from $\lambda$:

$\partial_{\mu}F^{\mu\nu}=\mu_{0}(j^{\nu}+i^{\nu})$ (9)

where $i^{\nu}$ is an additional four-current defined by
$i^{\nu}=\partial^{\nu}\partial^{-2}I=\partial^{\nu}S/\mu_{0}$. The
antisymmetry of $F^{\mu\nu}$ implies the total conservation
$\partial_{\mu}(j^{\mu}+i^{\mu})=0$. (This is reminiscent of what happens with
the chiral anomaly in Weyl systems, where the anomalous term can be rewritten
as a current that satisfies the classical balance equation morawetz2019chiral
; morawetz2019weyl .) In Sect. II the equations for $F^{\mu\nu}$ are written
explicitly in vector form, in order to apply them to materials with violation
of local conservation in the so-called “$I$-$\gamma$-$j$” model.

In spite of the reduced gauge invariance of the Aharonov-Bohm Lagrangian,
since the homogeneous field equations
$\nabla\times\mathbf{E}=-\partial_{t}\mathbf{B}/c$ and
$\nabla\cdot\mathbf{B}=0$ are the same as in Maxwell theory, it is possible to
write $\mathbf{E}$ and $\mathbf{B}$ in terms of a non-physical, auxiliary
four-potential, whose gauge condition can be freely chosen, as usual in
Maxwell theory. This property is quite useful, since it allows to employ most
traditional techniques also for the extended theory.

At first sight the statement above might appear to lead to a contradiction
with our previous finding that in the extended theory it is impossible in
general to impose the Lorentz gauge. Moreover, since in Maxwell theory the
choice of the Lorentz gauge allows to write the equations for the potentials
just in the desired locally forced form, one might be tempted to conclude that
the local Aharonov-Bohm theory is nothing but Maxwell theory in Lorentz gauge
plus the (apparently arbitrary) assumption that the four-potential is
physical.

This is actually not true, because an auxiliary four-potential in Lorentz
gauge for the field equations (9) in fact exists but satisfies the non-local
equations

$\partial^{2}A^{\mu}_{aux,Lor}=\mu_{0}(j^{\mu}-\partial^{\mu}\partial^{-2}I)$
(10)

and not the equation for the physical four-potential $A^{\mu}$, which is
$\partial^{2}A^{\mu}=\mu_{0}j^{\mu}$ ! Therefore there are no contradictions.
Instead, we can see here two strong connections between Maxwell theory and the
extended AB theory:

(1) In applications where the extra-current $I$ can be disregarded as source
and considered only as probe, the auxiliary potentials $A^{\mu}_{aux,Lor}$ and
the physical potentials $A^{\mu}$ essentially coincide (see Sect. I.5).

(2) If one is only interested into the fields $\mathbf{E}$ and $\mathbf{B}$,
they can be computed using any gauge condition for the auxiliary potentials.

###  I.4 In which physical systems can the extra-current $I$ be non-zero?

The physical systems where a violation of local conservation might occur,
yielding $I\neq 0$ in some regions, are thought to be mainly the following:

  1. 1.

Complex condensed-matter systems described by a quantum field theory, in which
the local conservation of the current operator is spoiled by anomalies
occurring in the renormalization process cheng1984gauge ;
parameswaran2014probing .

  2. 2.

Molecular devices, like e.g. carbon nanotubes and other molecular “wires”, in
which the effect of bound electrons in the inner orbitals upon the conduction
electrons is well modelled through a non-local potential, and the anomaly is
not due to the use of a reduced eigenstates base, but remains at any order in
the computations walz2015local ; jensen2019current ; garner2019helical ;
garner2020three .

  3. 3.

Systems with explicitly non-local wave equations, e.g. fractional quantum
mechanics and other phenomenological models lenzi2008solutions ;
lenzi2008fractional ; latora1999superdiffusion ; caspi2000enhanced ;
chamon1997nonlocal ; balantekin1998green ; laskin2002fractional ;
wei2016comment ; modanese2018time .

  4. 4.

Macroscopic quantum systems like Josephson junctions and superconductors in
general. For a system with macroscopic wavefunction, the charge density and
current density are represented by non-commuting operators. It follows that
the extra-current $I=\partial_{t}\rho+\nabla\cdot\mathbf{j}$, being
essentially a linear combination of these two operators in the frequency-
momentum domain, does not admit eigenstates and has a minimum uncertainty
fixed by the Heisenberg relation $\Delta N\Delta\phi\simeq 1$ which involves
the occupation number and the phase of the wavefunction. In minotti2021quantum
we have given an estimate of the minimum uncertainty in the case of a tunnel
Josephson junction made of Nb.

###  I.5 Definition and properties of gauge waves

We define a gauge wave as a plane wave with potentials

$\displaystyle\phi$ $\displaystyle=$ $\displaystyle\phi_{0}e^{i(\omega
t-\mathbf{k}\mathbf{x})}$ (11) $\displaystyle\mathbf{A}$ $\displaystyle=$
$\displaystyle\mathbf{A}_{0}e^{i(\omega t-\mathbf{k}\mathbf{x})}$ (12)

for which the relation

$\omega\mathbf{A}_{0}=\mathbf{k}\phi_{0}$ (13)

holds. It is therefore a longitudinal wave, with Fourier amplitudes $\phi_{0}$
and $\mathbf{A}_{0}$ connected in absolute value by
$|\phi_{0}|=c|\mathbf{A}_{0}|$, as follows from (13) and from
$\omega/|\mathbf{k}|=c$. It is a solution of the AB equations in vacuum
$\partial^{2}A^{\mu}=0$.

Taking the scalar product by $\mathbf{k}$ in eq. (13) we also find that

$\mathbf{k}\cdot\mathbf{A}_{0}=\frac{\omega}{c^{2}}\phi_{0}$ (14)

This means that the potentials defined by (11), (12) satisfy the equation

$\nabla\cdot\mathbf{A}-\frac{1}{c^{2}}\frac{\partial\phi}{\partial t}=0$ (15)

which is the vector equivalent of the Lorentz gauge condition
$\partial_{\mu}A^{\mu}=0$.

Remembering the definition of the scalar field $S$ as
$S=\partial_{\mu}A^{\mu}$ we thus see that $S=0$ in a gauge wave. It is also
straightforward to prove that in this wave $\mathbf{E}$ and $\mathbf{B}$ are
zero, using the general relations between fields and potentials, namely

$\displaystyle\mathbf{E}$ $\displaystyle=$
$\displaystyle\nabla\phi-\frac{1}{c}\frac{\partial\mathbf{A}}{\partial
t}=i(\mathbf{k}\phi_{0}-\omega\mathbf{A}_{0})e^{i(\omega
t-\mathbf{k}\mathbf{x})}=0$ (16) $\displaystyle\mathbf{B}$ $\displaystyle=$
$\displaystyle\nabla\times\mathbf{A}=i(\mathbf{k}\times\mathbf{A}_{0})e^{i(\omega
t-\mathbf{k}\mathbf{x})}=0$ (17)

This property justifies the name chosen for the gauge wave.

Since the fields $S$, $\mathbf{E}$ and $\mathbf{B}$ are all zero in a gauge
wave, the wave does not carry any energy and momentum, according to the
general expressions in minotti2021quantum  (for the energy flux see also eq.
(50)). In a gauge-invariant theory it would be regarded as non-physical, also
because it would be equivalent, via a gauge transformation, to a four-
potential which is identically zero. We know, however, that in AB
electrodynamics the potentials are physical and interact with an anomalous
matter probe with extra-current $I_{p}$ through the terms $I_{p}\phi$ and
$I_{p}\mathbf{A}$ in the Lorenz force density.

In minotti2023aharonov  we have proven that, consistently with the picture
above, a gauge wave propagates without attenuation in normal media where local
charge conservation holds. In Sect. … of this work the propagation of general
plane waves in anomalous media is analysed using the “$\gamma$-model”
approximation for the anomalous current density.

###  I.6 Generation of gauge waves

Concerning the generation of gauge waves, in minotti2023aharonov  it was shown
that when a plane wave of the $S$ field encounters a normal medium, it cannot
enter it but is totally reflected, possibly generating a gauge wave at the
interface. This process, however, is not the most efficient way of producing
gauge waves, since it requires an incident $S$ wave, which can only be
generated by an anomalous oscillating source (see computation in Sect. IV).

It is thus far more convenient to take advantage of the gauge wave components
which are generated in normal oscillating dipoles and go unnoticed unless they
impinge on a material probe containing an extra-current $I_{p}$.

In fact, the physical AB potentials of an oscillating dipole are (like in
Maxwell theory in Lorentz gauge)

$\displaystyle\phi(\mathbf{x},t)$ $\displaystyle=$
$\displaystyle\frac{\mu_{0}c}{4\pi r}\dot{\mathbf{p}}(t-r/c)\cdot\mathbf{n}$
(18) $\displaystyle\mathbf{A}(\mathbf{x},t)$ $\displaystyle=$
$\displaystyle\frac{\mu_{0}}{4\pi r}\dot{\mathbf{p}}(t-r/c)$ (19)

where $r=|\mathbf{x}|$ and $\mathbf{n}=\mathbf{x}/r$ is the local propagation
direction, such that the spherical wave can be approximated at each point by a
plane wave with vector $\mathbf{k}=|\mathbf{k}|\mathbf{n}$.

The vector $\mathbf{A}$ can be decomposed into a longitudinal component

$\mathbf{A}_{\mathbf{n}}=(\mathbf{A}\cdot\mathbf{n})\mathbf{n}=\frac{\phi}{c}\mathbf{n}$
(20)

satisfying the gauge wave condition $|\mathbf{A}_{\mathbf{n}}|=\phi/c$, and a
transverse component $\mathbf{A}_{T}$.

Along the oscillation axis of the dipole we have that $\mathbf{A}_{T}=0$, so
that the wave is longitudinal and a purely gauge wave. At any other point, the
four-potential can be considered as the sum of a gauge wave
$(\phi,\mathbf{A}_{\mathbf{n}})$ and a transverse vector potential
$(0,\mathbf{A}_{T})$. If the gauge wave encounters a normal conducting medium
it is not affected (supposing the process is linear). The transverse
component, on the other hand, generally interacts with the medium and in
several cases can be strongly dampened, leaving as a result a pure gauge wave
which propagates in the medium and then possibly gets out of the medium again
into free space.

A hypothetical detection antenna for gauge waves will be described in Sect.
III.

###  I.7 Outline

After this long Introduction, the rest of the work will be organized as
follows. Sect. II is devoted to what we call the “$\gamma$-model”, namely a
simplified one-parameter model for the description of material media
containing currents non locally conserved. The non-dimensional parameter
$\gamma$ quantifies in a well-defined sense the average, macroscopic effect of
the conservation violations which occur at the microscopic level. In Sect.
II.1 the model is applied to free charges, in Sect. II.2 to dielectric media,
where $\gamma$ appears in the relations among $\mathbf{j}$, $\mathbf{P}$
(local polarization vector) and $\mathbf{M}$ (local magnetization vector). As
a consequence, in Sect. II.3 we write the extended AB field equations in a
medium and in Sect. II.4 the wave propagation equations in the medium. Sect.
III contains a discussion of possible observable effects of AB extended
electrodynamics. In Sect. III.1 we describe a basic DC circuit for detection
of gauge-waves. It contains a voltage generator and a resistor made of a
material with conservation violation parameter $\gamma$. The scalar potential
$\phi$ of a gauge-wave can interact with the fraction $\gamma$ of the resistor
current that is not locally conserved. Using one of the fundamental relations
of the $\gamma$-model, namely $\langle\phi
I\rangle=\gamma\langle\mathbf{j}\cdot\nabla\phi\rangle+$ surface term, and
imposing energy conservation and conditions for the surface term to be
negligible, one can prove that an additional voltage $\delta V$ arises in the
resistor, of order $\delta V\sim\gamma\Delta\phi$, where $\Delta\phi$ is the
variation of the potential $\phi$ across the conductor length. Sect. III.2
contains a description of the possible measurable effects of gauge waves. We
first describe the basic scheme of a DC circuit for the detection of a generic
potential gradient $\Delta\phi$ through an extra-current flowing in a
conductor. Then we consider an antenna detector and its response to an
harmonic gauge wave propagating along the direction of the antenna. Sect. IV
contains a general calculation of radiation fields generated by harmonic
currents in anomalous conductors with negligible polarization described by the
$\gamma$ model. Fields are computed in the dipole approximation. An evaluation
of the time-averaged energy flow at large distance from the source closes the
section. Finally, Sect. V contains our Conclusions.

##  II Material media with non-conserved currents: the $\gamma$ model

In order to build up a theory of Aharonov-Bohm (AB) extended electrodynamics
in material media with non-conserved currents we need to reconsider the usual
models for the macroscopic description of fields and sources in those media.

To begin, we start with the AB equations in vacuum

$\displaystyle\nabla\cdot\mathbf{E}$ $\displaystyle=$
$\displaystyle\frac{\rho}{\varepsilon_{0}}-\frac{\partial S}{\partial t},$
(21a) $\displaystyle\nabla\cdot\mathbf{B}$ $\displaystyle=$ $\displaystyle 0,$
(21b) $\displaystyle\nabla\times\mathbf{E}$ $\displaystyle=$
$\displaystyle-\frac{\partial\mathbf{B}}{\partial t},$ (21c)
$\displaystyle\nabla\times\mathbf{B}$ $\displaystyle=$
$\displaystyle\mu_{0}\varepsilon_{0}\frac{\partial\mathbf{E}}{\partial
t}+\mu_{0}\mathbf{j}+\nabla S,$ (21d) $\displaystyle\mathbf{E}$
$\displaystyle=$ $\displaystyle-\nabla\phi-\frac{\partial\mathbf{A}}{\partial
t},$ (21e) $\displaystyle\mathbf{B}$ $\displaystyle=$
$\displaystyle\nabla\times\mathbf{A},$ (21f) $\displaystyle S$
$\displaystyle=$
$\displaystyle\mu_{0}\varepsilon_{0}\frac{\partial\phi}{\partial
t}+\nabla\cdot\mathbf{A},$ (21g)

From Eqs. (21) it results that the extra source $I$, defined as

$I=\frac{\partial\rho}{\partial t}+\nabla\cdot\mathbf{j},$

is the source of the scalar, which satisfies

$\mu_{0}\varepsilon_{0}\frac{\partial^{2}S}{\partial
t^{2}}-\nabla^{2}S=\mu_{0}I.$ (22)

These equations are valid if all sources $\varrho$ and $\mathbf{j}$ are known,
which is the main difficulty in dealing with material media. The usual
approach is then to consider for all magnitudes macroscopic averages
(introduced below), that should average out the small scale, rapid
fluctuations of the sources and fields. Furthermore, the sources are separated
into free and molecular. Molecular sources are those due to microscopic
charges and currents that are highly localized in space, in or around the
molecules they belong to. Free charges, on the other hand, are those that are
not bounded to a particular molecule and can more or less freely move in the
medium.

###  II.1 Free charges

In the model presented in minotti2022electromagnetic , which we refer here as
the $I$-$\gamma$-$j$ model, or simply $\gamma$ model, it is considered that a
likely candidate to account for non-local conservation of charge in material
media is the tunnel effect of electrons. If a microscopic free electron
current of value $i_{n}$ discontinuously crosses from the point
$\mathbf{x}_{n}$ to the point $\mathbf{x}_{n}+\mathbf{d}_{n}$, the
corresponding extra source is

$I_{n}=i_{n}\left[\delta\left(\mathbf{x}-\mathbf{x}_{n}-\mathbf{d}_{n}\right)-\delta\left(\mathbf{x}-\mathbf{x}_{n}\right)\right].$
(23)

This relation results from the definition of the extra source $I$ and the
semiclassical conceptualization of instantaneous tunneling as the
disappearance of the current at $\mathbf{x}_{n}$ with its simultaneous
reappearance at $\mathbf{x}_{n}+\mathbf{d}_{n}$, which corresponds to a
spatial redistribution of charge without an accompanying electric current. It
is to be noted that extra sources with similar opposite peaks appear in the
solution of the Schrödinger equation with non-local interactions
modanese2018time .

Furthermore, in the $I$-$\gamma$-$j$ model the tunneled current $i_{n}$ in the
volume element $d^{3}x=dSdL$ represents a fraction of the current
$\mathbf{j}\cdot d\mathbf{S}$ traversing it, while the tunneled distance is a
fraction of the volume length $dL$, so that
$i_{n}\mathbf{d}_{n}=\gamma\mathbf{j}d^{3}x$, with $\gamma$ the product of
those fractions. In a sense, the $I$-$\gamma$-$j$ model represents the
simplest phenomenological model one can conceive based on a single parameter.

The idea is now to consider a macroscopically small volume $\delta\Omega$
centered at position $\mathbf{x}=\mathbf{X}$, with dimensions much larger than
the characteristic tunneling distance $\left|\mathbf{d}_{n}\right|$, which is
taken as a measure of the microscopic scale. For a generic microscopic field
$F\left(\mathbf{x},t\right)$ one thus defines a macroscopic corresponding
field by spatial averaging (at each time) of the form

$\left\langle
F\right\rangle_{\mathbf{X}}=\frac{1}{\delta\Omega}\int_{\delta\Omega}F\left(\mathbf{x},t\right)d^{3}x,$
(24)

where $\delta\Omega$ is the same for all $\mathbf{X}$.

Immediate properties from the very definition (24) are

$\displaystyle\frac{\partial}{\partial t}\left\langle
F\right\rangle_{\mathbf{X}}$ $\displaystyle=$
$\displaystyle\frac{1}{\delta\Omega}\int_{\delta\Omega}\frac{\partial}{\partial
t}F\left(\mathbf{x},t\right)d^{3}x,$ $\displaystyle\frac{\partial}{\partial
X_{i}}\left\langle F\right\rangle_{\mathbf{X}}$ $\displaystyle=$
$\displaystyle\frac{1}{\delta\Omega}\int_{\delta\Omega}\frac{\partial}{\partial
x_{i}}F\left(\mathbf{x},t\right)d^{3}x.$

Using these properties, since Eqs. (21) are linear their average results in
identical equations, but now applied to the corresponding macroscopic
magnitudes.

Special care must be taken when considering the macroscopic current density,
because, according to the $\gamma$ model, part of charge is redistributed in a
microscopically discontinuous manner. Indeed, if one considers the electric
charge contained in a given elementary volume

$\delta Q=\int_{\delta\Omega}\rho\left(\mathbf{x},t\right)d^{3}x,$

its time variation depends on the net flow of charge across
$\Sigma\left(\delta\Omega\right)$, the boundary of $\delta\Omega$,

$\frac{\partial}{\partial t}\delta
Q=-\oint_{\Sigma\left(\delta\Omega\right)}\mathbf{j}\left(\mathbf{x},t\right)\cdot
d\mathbf{S}+\sum_{n}i_{n},$

where $\mathbf{j}\left(\mathbf{x},t\right)$ represents the flow of charge that
is transported continuously, and $\sum_{n}i_{n}$ represents the sum of the
elementary currents that tunnel across $\Sigma\left(\delta\Omega\right)$ (with
their corresponding sign, according to whether they enter or leave the
volume).

Using Gauss theorem and the average properties we can write

$\displaystyle\frac{1}{\delta\Omega}\oint_{\Sigma\left(\delta\Omega\right)}\mathbf{j}\left(\mathbf{x},t\right)\cdot
d\mathbf{S}$ $\displaystyle=$
$\displaystyle\frac{1}{\delta\Omega}\int_{\delta\Omega}\nabla_{\mathbf{x}}\cdot\mathbf{j}\left(\mathbf{x},t\right)d^{3}x$
$\displaystyle=$
$\displaystyle\nabla_{\mathbf{X}}\cdot\left\langle\mathbf{j}\right\rangle_{\mathbf{X}},$

so that the macroscopic charge density
$\left\langle\rho\right\rangle_{\mathbf{X}}=\delta Q/\delta\Omega$ satisfies
the relation

$\frac{\partial}{\partial
t}\left\langle\rho\right\rangle+\nabla_{\mathbf{X}}\cdot\left\langle\mathbf{j}\right\rangle=\frac{1}{\delta\Omega}\sum_{n}i_{n}.$
(25)

In order to proceed we expand on the $I-\gamma-j$ model. In this model the
tunneled current $i_{n}$ in a given volume element is a fraction of the normal
current flowing through it, and the total tunneled distance is also a fraction
of the length traversed by the normal current in that volume. Furthermore, we
take into consideration that in order that a given $i_{n}$ contributes to the
charge transport across $\Sigma\left(\delta\Omega\right)$ the corresponding
tunneled length $\mathbf{d}_{n}$ must cross the surface (with either
$\mathbf{x}_{n}$ or $\mathbf{x}_{n}+\mathbf{d}_{n}$ outside $\delta\Omega$).
In this way, if normal current crosses a given surface element
($\mathbf{j}\cdot d\mathbf{S}\neq 0$), the probability that tunneled currents
cross the surface element is given by the fraction of tunneled to normal
current, times the fraction of tunneled distance to total distance (the latter
accounts for the probability that $\mathbf{d}_{n}$ crosses the surface). Both
mentioned fractions make up the coefficient $\gamma$, so that the current that
actually tunnels across a given surface element is $\gamma\mathbf{j}\cdot
d\mathbf{S}$. We thus have

$\displaystyle\frac{1}{\delta\Omega}\sum_{n}i_{n}$ $\displaystyle=$
$\displaystyle-\frac{1}{\delta\Omega}\oint_{\Sigma\left(\delta\Omega\right)}\gamma\mathbf{j}\left(\mathbf{x},t\right)\cdot
d\mathbf{S}$ (26) $\displaystyle=$
$\displaystyle-\frac{1}{\delta\Omega}\int_{\delta\Omega}\gamma\nabla_{\mathbf{x}}\cdot\mathbf{j}\left(\mathbf{x},t\right)d^{3}x=-\gamma\nabla_{\mathbf{X}}\cdot\left\langle\mathbf{j}\right\rangle.$

Using (25) and (26), the relation between macroscopic free charge and current
densities is finally expressed as

$\frac{\partial}{\partial
t}\left\langle\rho\right\rangle+\left(1+\gamma\right)\nabla_{\mathbf{X}}\cdot\left\langle\mathbf{j}\right\rangle=0.$
(27)

A related important magnitude is the macroscopic extra source

$\left\langle
I\right\rangle_{\mathbf{X}}=\frac{1}{\delta\Omega}\int_{\delta\Omega}\sum_{n\subseteq\delta\Omega}I_{n}d^{3}x,$

which is determined, according to (23), by only those microscopic currents for
which the corresponding tunneled length $\mathbf{d}_{n}$ crosses the surface
with either $\mathbf{x}_{n}$ or $\mathbf{x}_{n}+\mathbf{d}_{n}$ outside
$\delta\Omega$. This gives precisely the same result as (26)

$\left\langle
I\right\rangle_{\mathbf{X}}=-\gamma\nabla_{\mathbf{X}}\cdot\left\langle\mathbf{j}\right\rangle.$
(28)

An important result of the $I-\gamma-j$ model, from (27) and (28), is that for
a stationary case the macroscopic extra source due to free charges is zero:
$\left\langle I\right\rangle_{\mathbf{X}}=0$ if
$\partial\left\langle\rho\right\rangle/\partial t=0$

Finally, it will also be necessary to consider averages of the type

$\left\langle
FI\right\rangle_{\mathbf{X}}=\frac{1}{\delta\Omega}\int_{\delta\Omega}F\left(\mathbf{x}\right)\left(\sum_{n\subseteq\delta\Omega}I_{n}\right)d^{3}x,$

for different fields $F\left(\mathbf{x}\right)$. Using (23) we have

$\displaystyle\left\langle FI\right\rangle_{\mathbf{X}}$ $\displaystyle=$
$\displaystyle\frac{1}{\delta\Omega}\sum_{n\subseteq\delta\Omega}i_{n}\left[F\left(\mathbf{x}_{n}+\mathbf{d}_{n}\right)-F\left(\mathbf{x}_{n}\right)\right]$
$\displaystyle+\frac{1}{\delta\Omega}\sum_{n\subseteq\Sigma\left(\delta\Omega\right)}sg_{n}i_{n}F\left(\mathbf{x}_{n}\right).$

where the second term in the right-hand side accounts for the contribution of
those microscopic currents for which the corresponding tunneled length
$\mathbf{d}_{n}$ crosses the surface with either $\mathbf{x}_{n}$ or
$\mathbf{x}_{n}+\mathbf{d}_{n}$ outside $\delta\Omega$, and $sg_{n}$ is the
corresponding sign, positive when $i_{n}$ enters the volume, and negative when
it leaves it. We now write

$F\left(\mathbf{x}_{n}+\mathbf{d}_{n}\right)-F\left(\mathbf{x}_{n}\right)=\nabla
F\cdot\mathbf{d}_{n},$

and use the $I-\gamma-j$ model as before, which allows us to write the
contributions of the volume and of the surface terms as

$\left\langle
FI\right\rangle_{\mathbf{X}}=\frac{1}{\delta\Omega}\int_{\delta\Omega}\gamma\mathbf{j}\cdot\nabla
Fd^{3}x-\frac{1}{\delta\Omega}\oint_{\Sigma\left(\delta\Omega\right)}\gamma
F\mathbf{j}\cdot d\mathbf{S}.$ (29)

Using Gauss theorem in the surface term we finally obtain

$\left\langle
FI\right\rangle_{\mathbf{X}}=-\frac{1}{\delta\Omega}\int_{\delta\Omega}\gamma
F\nabla\cdot\mathbf{j}d^{3}x=-\gamma\left\langle
F\nabla\cdot\mathbf{j}\right\rangle_{\mathbf{X}}.$ (30)

Care must be taken in the evaluation of averages of this form due to their
being non-linear in the microscopic fields. To proceed we follow the formalism
in Minotti-PRE-2000  in which the averages of the form (24) are complemented
with the definition of the fluctuations, relative to the averages, as

$\delta F\left(\mathbf{x},\mathbf{X},t\right)\equiv
F\left(\mathbf{x},t\right)-\left\langle F\right\rangle_{\mathbf{X}}.$

This particular definition of fluctuations satisfies the relation

$\left\langle\delta F\left(\mathbf{x},\mathbf{X},t\right)\left\langle
G\right\rangle_{\mathbf{X}}\right\rangle_{\mathbf{X}}=0,$ (31)

for any two fields $F\left(\mathbf{x},t\right)$ and
$G\left(\mathbf{x},t\right)$. In this way, by writing

$\displaystyle F\left(\mathbf{x},t\right)$ $\displaystyle=$
$\displaystyle\left\langle F\right\rangle_{\mathbf{X}}+\delta
F\left(\mathbf{x},\mathbf{X},t\right),$
$\displaystyle\nabla\cdot\mathbf{j}\left(\mathbf{x},t\right)$ $\displaystyle=$
$\displaystyle\nabla_{\mathbf{X}}\cdot\left\langle\mathbf{j}\right\rangle_{\mathbf{X}}+\delta\nabla\cdot\mathbf{j}\left(\mathbf{x},\mathbf{X},t\right)$

we have

$\left\langle F\nabla\cdot\mathbf{j}\right\rangle_{\mathbf{X}}=\left\langle
F\right\rangle_{\mathbf{X}}\nabla_{\mathbf{X}}\cdot\left\langle\mathbf{j}\right\rangle_{\mathbf{X}}+\left\langle\delta
F\delta\nabla\cdot\mathbf{j}\right\rangle_{\mathbf{X}}.$

The average of fluctuations in this expression results from two contributions:
that due to fluctuations of scale much smaller than the macroscopic one,
$\delta L$, and that due to fluctuations of scale close to $\delta L$. The
latter contribution can be determined using the method in Minotti-PRE-2000 ,
while the contributions of fluctuations of very small scale need to be modeled
independently. The fluctuations of scale not too small contribute to a generic
average of fluctuations $\left\langle\delta F\delta G\right\rangle$ a term
Minotti-PRE-2000

$\left\langle\delta F\delta G\right\rangle\simeq\frac{\left(\delta
L\right)^{2}}{24}\nabla_{\mathbf{X}}\left\langle
F\right\rangle\cdot\nabla_{\mathbf{X}}\left\langle G\right\rangle,$

and since $\left\langle F\right\rangle$ and $\left\langle G\right\rangle$ are
macroscopic fields their variation over $\delta L$ is small,

$\left|\frac{\left(\delta L\right)^{2}}{24}\nabla_{\mathbf{X}}\left\langle
F\right\rangle\cdot\nabla_{\mathbf{X}}\left\langle
G\right\rangle\right|\ll\left|\left\langle F\right\rangle\left\langle
G\right\rangle\right|,$

so that one can safely neglect this contribution. Concerning the contribution
of the scales much smaller than $\delta L$, we will also neglect it
considering that the level of those fluctuations is not supposed to be highly
excited above their thermal level, in the absence of a mechanisms leading to a
non-linear energy cascade. In this way, we finally obtain

$\left\langle FI\right\rangle_{\mathbf{X}}=-\gamma\left\langle
F\right\rangle_{\mathbf{X}}\nabla_{\mathbf{X}}\cdot\left\langle\mathbf{j}\right\rangle_{\mathbf{X}}.$
(32)

We can now analyze the application of Ohm law. The point is that in
minotti2021quantum  the following expression for the force density on matter
due to the fields was obtained

$\mathbf{f}=\rho\mathbf{E}+\mathbf{j}\times\mathbf{B}-I\mathbf{A}.$ (33)

Using the previous considerations we obtain

$\displaystyle\left\langle I\mathbf{A}\right\rangle_{\mathbf{X}}$
$\displaystyle=$
$\displaystyle-\gamma\left\langle\mathbf{A}\right\rangle_{\mathbf{X}}\nabla_{\mathbf{X}}\cdot\left\langle\mathbf{j}\right\rangle_{\mathbf{X}},$
(34)
$\displaystyle\left\langle\mathbf{j}\times\mathbf{B}\right\rangle_{\mathbf{X}}$
$\displaystyle=$
$\displaystyle\left\langle\mathbf{j}\right\rangle_{\mathbf{X}}\times\left\langle\mathbf{B}\right\rangle_{\mathbf{X}}=\left\langle\mathbf{j}\right\rangle_{\mathbf{X}}\times\left(\nabla_{\mathbf{X}}\times\left\langle\mathbf{A}\right\rangle_{\mathbf{X}}\right).$
(35)

This allows us to estimate the relative order of magnitude of these two terms,
considering that the averaged magnitudes have spatial variations of the same
relative order. This shows that the additional force coming from the non-
conserved currents is a term of small magnitude (for $\gamma$ small) compared
to the Lorentz force. In this way, we take as a reasonable approximation the
usual relation between local electric field and total current to be
$\left\langle\mathbf{j}\right\rangle_{\mathbf{X}}=\sigma\mathbf{E}$, with
$\sigma$ the conductivity of the medium, even if it has to be extended to
include magnetic forces, like in the Hall effect.

###  II.2 Dielectric media

We consider now the macroscopic relations for polarizable media, in which the
molecular sources play a fundamental role.

To begin, the usual expression of averaged molecular charge density (no
explicit distinction between the spatial coordinates of volume centers and
space coordinates would be made from now on)

$\left\langle\rho_{m}\right\rangle=-\nabla\cdot\mathbf{P},$ (36)

expressed in terms of a vector $\mathbf{P}$, is justified considering that the
total molecular charge in a medium without added free charges should be zero,
irrespective of the shape of the medium. Besides, the interpretation of
$\mathbf{P}$ as the volume density of electric polarization results from
expression (36) and the definition of the dipole moment of the medium as
$\int\left\langle\rho_{m}\right\rangle\mathbf{x}d^{3}x$, with the integral
extended to the whole medium.

We thus see that expression (36) can be taken over for the description of
media with non-conserved currents. Furthermore, the phenomenological relation
$\mathbf{P}=\varepsilon_{0}\varkappa\left\langle\mathbf{E}\right\rangle$ for a
linear, isotropic medium, with a polarization coefficient $\varkappa$, should
also be an acceptable model in our case, since it does not conflict with any
assumption in the $\gamma$ model.

We now consider the macroscopic averages $\mathbf{J}_{m}$ of molecular
currents. The condition that molecular currents are highly localized implies
that their flux across any macroscopic cross section of the medium should be
zero, which translates into the relation $\mathbf{J}_{m}=$
$\nabla\times\mathbf{M}$ for a generic vector $\mathbf{M}$. This is valid
under the assumption that the spatial averaging process cancels the rapidly
fluctuating in time, instantaneous unbalances of molecular currents crossing
the surface element. However, when the macroscopic fields are not stationary,
a time variation of a polarization vector $\mathbf{P}$ with non-zero
divergence corresponds to an orderly redistribution of charges, crossing the
boundary of the element, but remaining localized in the molecules that cross
that boundary. This effect gives rise to a contribution to the macroscopic
molecular current of the form $\partial\mathbf{P}/\partial t$. It is thus
customary to include both contributions in the expression

$\mathbf{J}_{m}=\frac{\partial\mathbf{P}}{\partial t}+\nabla\times\mathbf{M}.$

Moreover, for stationary situations, the interpretation of $\mathbf{M}$ as the
volume density of magnetic moment results from the very definition of total
magnetic moment of the medium:
$\frac{1}{2}\int\mathbf{x}\times\mathbf{J}_{m}d^{3}x$, with the integral
extended to the whole medium, and the expression of $\mathbf{J}_{m}$ valid in
this case.

Furthermore, for linear, isotropic media the idea of the orientation of the
molecular magnetic dipoles along the magnetic field leads to a model with a
linear relation between $\mathbf{M}$ and $\mathbf{B}$, usually expressed
through the auxiliary vector
$\mathbf{H}=\left(\left\langle\mathbf{B}\right\rangle-\mathbf{M}\right)/\mu_{0}$
as $\mathbf{M}=\chi\mathbf{H}$, where $\chi$ is the magnetic susceptibility
coefficient. This latter phenomenological model can also be taken over to our
case, since it is fully compatible with the $\gamma$ model.

Note that the macroscopic molecular sources so far considered satisfy the
continuity relation

$\frac{\partial}{\partial
t}\left\langle\rho_{m}\right\rangle+\nabla\cdot\mathbf{J}_{m}=0.$

However, we must take into account that in the spirit of the $\gamma$ model
some of the molecular charge redistribution can take place discontinuously,
even within the molecule in which it is localized. The likely candidate for
this is the redistribution of charge leading to variation of the dipole moment
in each molecule. For this reason, we assume that only a fraction $\alpha$ of
$\partial\mathbf{P}/\partial t$ is due to a continuous charge motion. We thus
consider that an effective fraction $\gamma^{\prime}$ (in general different
from the $\gamma$ for free currents) of the continuous current crosses a given
surface element discontinuously, so that

$\left\langle
I\right\rangle_{\mathbf{X}}=-\gamma^{\prime}\alpha\nabla\cdot\frac{\partial\mathbf{P}}{\partial
t}.$

These considerations lead to a redefinition of the continuous component
$\mathbf{J}_{m}$ as

$\mathbf{J}_{m}=\alpha\frac{\partial\mathbf{P}}{\partial
t}+\nabla\times\mathbf{M},$ (37)

so that now the continuity relation is

$\frac{\partial}{\partial
t}\left\langle\rho_{m}\right\rangle+\nabla\cdot\mathbf{J}_{m}=-\gamma^{\prime}\alpha\nabla\cdot\frac{\partial\mathbf{P}}{\partial
t},$

which using also (36) is written as

$-\frac{\partial}{\partial
t}\nabla\cdot\mathbf{P}+\alpha\nabla\cdot\frac{\partial\mathbf{P}}{\partial
t}=-\gamma^{\prime}\alpha\nabla\cdot\frac{\partial\mathbf{P}}{\partial t},$

which requires that

$\alpha=\frac{1}{1+\gamma^{\prime}}.$

resulting in the macroscopic extra source contributed by the polarization of
the medium to be

$\left\langle
I\right\rangle_{\mathbf{X}}=-\frac{\gamma^{\prime}}{1+\gamma^{\prime}}\nabla\cdot\frac{\partial\mathbf{P}}{\partial
t}.$ (38)

As for the contribution from free charges to the macroscopic extra source,
also that due to molecular sources turns out to be zero in stationary
situations.

###  II.3 Macroscopic AB equations in media with non-conserved currents

We now recall the relations which, according to the considerations in the
previous sections, could serve as a valid model for the macroscopic fields and
sources in a linear, homogeneous medium with non-locally conserved currents.

We first recall that, due to their being linear, the volume average procedure
can be applied directly to Eqs. (21) to yield exactly the same equations, but
where all fields and sources are to be considered as macroscopic ones.
Besides, the sources are to be considered as including both components, free
and molecular.

Furthermore, for easy of notation, from now on all macroscopic fields and
sources are represented without the average symbol
$\left\langle...\right\rangle$. The spatial coordinate $\mathbf{x}$ is also
not distinguished from the center of the volume elements, previously
designated as $\mathbf{X}$.

With these considerations, the molecular sources are given by

$\displaystyle\rho_{m}$ $\displaystyle=$
$\displaystyle-\nabla\cdot\mathbf{P},$ $\displaystyle\mathbf{j}_{m}$
$\displaystyle=$
$\displaystyle\frac{1}{1+\gamma^{\prime}}\frac{\partial\mathbf{P}}{\partial
t}+\nabla\times\mathbf{M},$

where the polarization and magnetization have the usual expressions

$\displaystyle\mathbf{P}$ $\displaystyle=$
$\displaystyle\left(\varepsilon-\varepsilon_{0}\right)\mathbf{E},$
$\displaystyle\mathbf{M}$ $\displaystyle=$
$\displaystyle\left(1/\mu_{0}-1/\mu\right)\mathbf{B}.$

Using the previous relations to express the macroscopic molecular sources we
obtain the equations

$\displaystyle\nabla\cdot\mathbf{E}$ $\displaystyle=$
$\displaystyle\frac{\rho}{\varepsilon}-\frac{\varepsilon_{0}}{\varepsilon}\frac{\partial
S}{\partial t},$ (39a) $\displaystyle\nabla\cdot\mathbf{B}$ $\displaystyle=$
$\displaystyle 0,\;\mathbf{B}=\nabla\times\mathbf{A},$ (39b)
$\displaystyle\nabla\times\mathbf{E}$ $\displaystyle=$
$\displaystyle-\frac{\partial\mathbf{B}}{\partial
t},\;\mathbf{E}=-\nabla\phi-\frac{\partial\mathbf{A}}{\partial t},$ (39c)
$\displaystyle\nabla\times\mathbf{B}$ $\displaystyle=$
$\displaystyle\mu\mathbf{j}+\mu\varepsilon\frac{1+\gamma^{\prime}\frac{\varepsilon_{0}}{\varepsilon}}{1+\gamma^{\prime}}\frac{\partial\mathbf{E}}{\partial
t}+\frac{\mu}{\mu_{0}}\nabla S,$ (39d) $\displaystyle S$ $\displaystyle=$
$\displaystyle\mu_{0}\varepsilon_{0}\frac{\partial\phi}{\partial
t}+\nabla\cdot\mathbf{A}.$ (39e)

The sources appearing explicitly in these relations are only the free sources,
the molecular ones having been included through their expressions in terms of
the fields.

The equations for the potentials can be more simply derived from the original
system (21), which after further averaging and use of the expressions of the
molecular sources in terms of the fields gives

$\displaystyle\mu_{0}\varepsilon_{0}\frac{\varepsilon_{0}}{\varepsilon}\frac{\partial^{2}\phi}{\partial
t^{2}}-\nabla^{2}\phi$ $\displaystyle=$
$\displaystyle\frac{\rho}{\varepsilon}+\left(1-\frac{\varepsilon_{0}}{\varepsilon}\right)\frac{\partial}{\partial
t}\left(\nabla\cdot\mathbf{A}\right),$ (40a)
$\displaystyle\mu\varepsilon\frac{1+\gamma^{\prime}\frac{\varepsilon_{0}}{\varepsilon}}{1+\gamma^{\prime}}\frac{\partial^{2}\mathbf{A}}{\partial
t^{2}}-\nabla^{2}\mathbf{A}$ $\displaystyle=$
$\displaystyle\mu\mathbf{j}+\left(\frac{\mu}{\mu_{0}}-1\right)\nabla\left(\nabla\cdot\mathbf{A}\right)$
(40b)
$\displaystyle+\mu\varepsilon\frac{\frac{\varepsilon_{0}}{\varepsilon}-1}{1+\gamma^{\prime}}\nabla\left(\frac{\partial\phi}{\partial
t}\right).$

The previous equations must be complemented with the relation for free
sources, Eq. (27),

$\frac{\partial\rho}{\partial
t}+\left(1+\gamma\right)\nabla\cdot\mathbf{j}=0,$

where the free current is related to the electric field by the Ohm law

$\mathbf{j}=\sigma\mathbf{E}=-\sigma\left(\nabla\phi+\frac{\partial\mathbf{A}}{\partial
t}\right).$ (41)

###  II.4 Wave propagation

We study in this section the possible plane-wave modes in which all fields and
sources are of the generic form
$a_{0}\exp\left[i\left(\mathbf{k}\cdot\mathbf{x}-\omega t\right)\right]$, with
constant complex amplitude $a_{0}$.

From Eqs. (27) and (41) we have for the mode amplitudes of the free sources

$\displaystyle\varrho_{0}$ $\displaystyle=$
$\displaystyle-i\frac{\sigma\left(1+\gamma\right)}{\omega}\left(k^{2}\phi_{0}-\omega\mathbf{k}\cdot\mathbf{A}_{0}\right),$
$\displaystyle\mathbf{j}_{0}$ $\displaystyle=$
$\displaystyle-i\sigma\left(\mathbf{k}\phi_{0}-\omega\mathbf{A}_{0}\right),$

which are used in Eqs. (40) to finally obtain the relations for the potentials
amplitudes

$\displaystyle\left\\{k^{2}\left[1+i\frac{\sigma\left(1+\gamma\right)}{\omega\varepsilon}\right]-\mu_{0}\varepsilon_{0}\frac{\varepsilon_{0}}{\varepsilon}\omega^{2}\right\\}\phi_{0}$
$\displaystyle=$
$\displaystyle\omega\left[1-\frac{\varepsilon_{0}}{\varepsilon}+i\frac{\sigma\left(1+\gamma\right)}{\omega\varepsilon}\right]\mathbf{k}\cdot\mathbf{A}_{0},$
(42a)
$\displaystyle\left(k^{2}-\mu\varepsilon\frac{1+\gamma^{\prime}\frac{\varepsilon_{0}}{\varepsilon}}{1+\gamma^{\prime}}\omega^{2}-i\mu\sigma\omega\right)\mathbf{A}_{0}$
$\displaystyle=$
$\displaystyle\left(1-\frac{\mu}{\mu_{0}}\right)\mathbf{k}\left(\mathbf{k}\cdot\mathbf{A}_{0}\right)$
(42b)
$\displaystyle+\left(\mu\varepsilon\frac{\frac{\varepsilon_{0}}{\varepsilon}-1}{1+\gamma^{\prime}}\omega-i\mu\sigma\right)\mathbf{k}\phi_{0}.$

The transverse mode ($\mathbf{k}\cdot\mathbf{A}_{0}=0$) corresponds to a wave
with $\phi_{0}=0$ and dispersion relation

$k^{2}=\mu\varepsilon\frac{1+\gamma^{\prime}\frac{\varepsilon_{0}}{\varepsilon}}{1+\gamma^{\prime}}\omega^{2}+i\mu\sigma\omega.$

For a dielectric without losses ($\sigma=0$) this is an electromagnetic wave
with phase speed given by

$c=\sqrt{\mu\varepsilon\frac{1+\gamma^{\prime}\frac{\varepsilon_{0}}{\varepsilon}}{1+\gamma^{\prime}}},$

which indicates a slight (for $\gamma^{\prime}\ll 1$) decrease of the usual
speed of light in the medium.

For the case of a conducting medium there is also the usual decay associated
to Joule dissipation, which is very strong in good conductors with
$\frac{\sigma}{\omega\varepsilon}\gg 1$.

For the longitudinal modes ($\mathbf{k}\times\mathbf{A}_{0}=0$) we obtain the
dispersion relation

$\left(\frac{k^{2}}{\mu_{0}\varepsilon_{0}\omega^{2}}-1\right)\left\\{\left(\frac{k^{2}}{\mu_{0}\varepsilon_{0}\omega^{2}}-1\right)\left[1+i\frac{\sigma\left(1+\gamma\right)}{\omega\varepsilon}\right]+\frac{\gamma^{\prime}\left(1-\frac{\varepsilon_{0}}{\varepsilon}\right)}{1+\gamma^{\prime}}+i\frac{\gamma\sigma}{\omega\varepsilon}\right\\}=0.$

We thus see that the branch $k^{2}=\mu_{0}\varepsilon_{0}\omega^{2}$,
corresponding to a wave of pure potentials, or gauge wave, which has zero
fields ($\mathbf{E}$, $\mathbf{B}$ and $S$), is allowed for all values of
$\gamma$, $\gamma^{\prime}$ and $\sigma$.

The other branch has the dispersion relation

$\frac{k^{2}}{\mu_{0}\varepsilon_{0}\omega^{2}}=1-\frac{\gamma^{\prime}\left(1-\frac{\varepsilon_{0}}{\varepsilon}\right)+i\frac{\gamma\sigma}{\omega\varepsilon}\left(1+\gamma^{\prime}\right)}{\left(1+\gamma^{\prime}\right)\left[1+i\frac{\sigma\left(1+\gamma\right)}{\omega\varepsilon}\right]}.$
(43)

When relation (43) is used in expressions (42), the relation between the
amplitudes of the potentials is given for this mode by

$\mathbf{k}\cdot\mathbf{A}_{0}=\frac{\mu_{0}\varepsilon_{0}\omega}{1-\frac{\varepsilon_{0}}{\varepsilon}+i\frac{\sigma\left(1+\gamma\right)}{\omega\varepsilon}}\left(\frac{1-\frac{\varepsilon_{0}}{\varepsilon}}{1+\gamma^{\prime}}+i\frac{\sigma}{\omega\varepsilon}\right)\phi_{0}.$

Consequently, the amplitude of the scalar wave,
$S_{0}=i\left(\mathbf{k}\cdot\mathbf{A}_{0}-\mu_{0}\varepsilon_{0}\omega\phi_{0}\right)$,
has the expression

$S_{0}=i\frac{\mu_{0}\varepsilon_{0}\omega}{\left(1+\gamma^{\prime}\right)}\left[\frac{\gamma^{\prime}\left(\frac{\varepsilon_{0}}{\varepsilon}-1\right)-i\sigma\frac{\gamma}{\varepsilon\omega}\left(1+\gamma^{\prime}\right)}{1-\frac{\varepsilon_{0}}{\varepsilon}+i\frac{\sigma\left(1+\gamma\right)}{\omega\varepsilon}}\right]\phi_{0},$
(44)

and the corresponding longitudinal electric field amplitude satisfies

$\mathbf{k}\cdot\mathbf{E}_{0}=\frac{\omega\frac{\varepsilon_{0}}{\varepsilon}}{1+i\frac{\sigma\left(1+\gamma\right)}{\omega\varepsilon}}S_{0}.$
(45)

For the case of a very good conductor, $\frac{\sigma}{\omega\varepsilon}\gg
1$, relation (43) simplifies to

$\frac{k^{2}}{\mu_{0}\varepsilon_{0}\omega^{2}}=\frac{1}{1+\gamma},$

and (44) to

$S_{0}=-i\frac{\gamma}{1+\gamma}\mu_{0}\varepsilon_{0}\omega\phi_{0}.$

The corresponding longitudinal electric field amplitude is

$E_{0}=-\frac{\gamma\varepsilon_{0}\omega^{2}}{\sigma\left(1+\gamma\right)}\sqrt{\frac{\mu_{0}\varepsilon_{0}}{1+\gamma}}\phi_{0}.$

These relations indicate that a small amplitude (for $\gamma\ll 1$) scalar
wave is possible in this case, propagating without decay in a good conductor.
Note that, consistently, the dissipated power density, $\sigma E_{0}^{2}$,
goes to zero for $\sigma\rightarrow\infty$.

For a dielectric without losses, $\sigma=0$, the dispersion relation is

$\frac{k^{2}}{\mu_{0}\varepsilon_{0}\omega^{2}}=\frac{1+\gamma^{\prime}\frac{\varepsilon_{0}}{\varepsilon}}{1+\gamma^{\prime}},$

and the corresponding amplitudes of the scalar and of the longitudinal
electric field are

$\displaystyle S_{0}$ $\displaystyle=$
$\displaystyle-i\frac{\gamma^{\prime}}{1+\gamma^{\prime}}\mu_{0}\varepsilon_{0}\omega\phi_{0},$
$\displaystyle E_{0}$ $\displaystyle=$
$\displaystyle-i\gamma^{\prime}\frac{\varepsilon_{0}}{\varepsilon}\omega\sqrt{\frac{\mu_{0}\varepsilon_{0}}{\left(1+\gamma^{\prime}\right)\left(1+\gamma^{\prime}\frac{\varepsilon_{0}}{\varepsilon}\right)}}\phi_{0}.$

As in the case of a conductor, a small amplitude (for $\gamma^{\prime}\ll 1$)
scalar wave can propagate also in this case.

##  III Possible measurable effects

The general energy-momentum laws in AB electrodynamics were derived in
minotti2021quantum . In relation to the interaction of fields and matter the
following expression for the power density on matter due to the fields was
obtained

$w=\mathbf{j}\cdot\mathbf{E}-I\phi.$ (46)

This is the microscopic expression that, apart from the usual term found in
Maxwell electrodynamics, includes a term proportional to the extra source $I$.
As a consequence, in media with non-conserved currents it would in principle
be possible to observe the effect of this extra term.

###  III.1 Detection of gauge-waves

As mentioned above, the possibility of material media with non-conserved
currents allows the detection of gauge-waves. This is so because according to
(46) the scalar potential of the wave can interact with any extra source $I$
present in the medium. In the following a basic circuit capable in principle
of this detection is presented.

In the conductor represented in Fig. (1) there is a constant current of value
$i_{0}=V_{0}/\left(R+R_{0}\right)$. The scalar potential $\phi$ of a gauge
wave can then interact with the part of this current that is not locally
conserved.

Figure 1: Possible basic circuit for detection of gauge waves. The conductor
of length $L$ and total electrical resistance $R$ sustains a continuous
current $i_{0}=V_{0}/\left(R+R_{0}\right)$. The scalar potential of a gauge
wave interacts with the not locally conserved part of the current and produces
a varying voltage $\delta V$ at the position denoted by the black dot.

Since the gauge wave does not transport energy, and energy conservation is
already satisfied by the circulating permanent current, the law of
conservation of energy is expressed by (46) applied to the whole circuit,
expressed as

$\int\left[\delta\left(\mathbf{j}\cdot\mathbf{E}\right)-I\phi\right]d^{3}x=0,$

where the integral is extended to the whole volume of the conductor, and
$\delta\left(\mathbf{j}\cdot\mathbf{E}\right)=\delta\mathbf{j}\cdot\mathbf{E}+\mathbf{j}\cdot\delta\mathbf{E}$
represents the small change in the corresponding magnitudes due to the
interaction with the wave (it is assumed that $I\phi$, induced by the wave, is
a small perturbation), while $\mathbf{E}$ and $\mathbf{j}$ are the values of
the magnitudes when no gauge wave is present. In this way, denoting as $\delta
i$ the change of current that arises due to interaction with the gauge wave,
we can write

$\displaystyle\int\delta\mathbf{j}\cdot\mathbf{E}d^{3}x$ $\displaystyle=$
$\displaystyle\delta i\left(-V_{0}+Ri_{0}+R_{0}i_{0}\right),$
$\displaystyle\int\mathbf{j}\cdot\delta\mathbf{E}d^{3}x$ $\displaystyle=$
$\displaystyle i_{0}\left(R\delta i+R_{0}\delta i\right),$

so that

$\int\delta\left(\mathbf{j}\cdot\mathbf{E}\right)d^{3}x=\left[-V_{0}+2\left(R+R_{0}\right)i_{0}\right]\delta
i=V_{0}\delta i.$

In order to evaluate $\int I\phi d^{3}x$ we use the general expression (29)
with $\phi$ in place of $F$, and extend the integration to the whole circuit,
so that

$\int I\phi d^{3}x=\int\gamma\mathbf{j}\cdot\nabla\phi
d^{3}x-\oint\gamma\phi\mathbf{j}\cdot d\mathbf{S}\text{,}$

where $\mathbf{j}$ is the current density in the circuit when there is no
gauge wave. Note that the surface integral in this expression applies if the
current that crosses the surface of the detecting element has a discontinuous
component described by the same $\gamma$ coefficient as that of the bulk of
the detecting medium. In that case, after use of Gauss law we would obtain

$\int I\phi d^{3}x=-\gamma\int\phi\nabla\cdot\mathbf{j}d^{3}x,$

which is zero, since $\nabla\cdot\mathbf{j=0}$ for the DC circuit when no
gauge wave is present. We must thus consider that the detecting medium is
electrically fed by conductors with zero, or a much smaller value of $\gamma$
than that of the detecting element, so as to neglect the surface contribution,
and finally obtain

$\displaystyle V_{0}\delta i$ $\displaystyle=$ $\displaystyle\int I\phi
d^{3}x=\int\gamma\mathbf{j}\cdot\nabla\phi d^{3}x$ $\displaystyle=$
$\displaystyle\gamma i_{0}\int\nabla\phi\cdot\mathbf{n}dl=\gamma
i_{0}\Delta\phi,$

where the integral in the second line is preformed along the length of the
conductor, the unit vector $\mathbf{n}$ indicates the direction of the
circulating current, and the resulting $\Delta\phi$ is the difference of
potential across the conductor length.

Using again $V_{0}=i_{0}\left(R+R_{0}\right)$ we thus have

$\delta i=\frac{\gamma}{\left(R+R_{0}\right)}\Delta\phi,$

or, from $\delta V=R_{0}\delta i$,

$\delta V=\gamma\frac{R_{0}}{\left(R+R_{0}\right)}\Delta\phi.$

###  III.2 Antenna detector

We consider as a relatively simple example a linear dipole of length $d$ and
negligible cross dimensions, fed by a source of voltage $V$ of (angular)
frequency $\omega$. Taking the coordinate $z$ along the dipole, with the
origin at the dipole half point (where the power is fed to the antenna), a
good approximation to the current circulating in the dipole is balanis2005

$i\left(z,t\right)=i_{0}\sin\left(\frac{kd}{2}-k\left|z\right|\right)\cos\left(\omega
t\right).$

In terms of the antenna impedance, $Z_{A}=R_{A}+iX_{A}$, and that of the
source and circuit, $Z_{S}=R_{S}+iX_{S}$, the corresponding voltage is

$V=i_{0}\left|Z_{A}+Z_{S}\right|\cos\left(\omega t+\varphi\right),$

where

$\tan\left(\varphi\right)=\left(X_{A}+X_{S}\right)/\left(R_{A}+R_{S}\right).$

We consider that the condition for maximum power delivered to the antenna by
the circuit is fulfilled: $R_{S}=R_{A}$, $X_{S}=-X_{A}$, so that

$V=2R_{A}i_{0}\cos\left(\omega t\right).$

If a gauge wave propagates along the $z$ direction with potential given by

$\phi\left(z,t\right)=\phi_{0}\cos\left(k^{\prime}z-\omega^{\prime}t+\theta\right),$

the instantaneous power exchanged with the antenna is given by (the integral
is extended to the antenna volume)

$\displaystyle-\int I\phi d^{3}x$ $\displaystyle=$
$\displaystyle\gamma\int\phi\nabla\cdot\mathbf{j}d^{3}x$ $\displaystyle=$
$\displaystyle-\gamma ki_{0}\phi_{0}\cos\left(\omega
t\right)\int_{-d/2}^{d/2}sign\left(z\right)\cos\left(\frac{kd}{2}-k\left|z\right|\right)\cos\left(k^{\prime}z-\omega^{\prime}t+\theta\right)dz$
$\displaystyle=$ $\displaystyle 2\gamma
i_{0}\phi_{0}kk^{\prime}\frac{\cos\left(\frac{kd}{2}\right)-\cos\left(\frac{k^{\prime}d}{2}\right)}{\left(k-k^{\prime}\right)\left(k+k^{\prime}\right)}\cos\left(\omega
t\right)\sin\left(\omega^{\prime}t-\theta\right).$

Conservation of energy requires that the additional $\delta i$ generated by
the interaction with the gauge wave satisfies

$-V\delta i+\delta\left(2R_{A}i^{2}\right)-\int I\phi d^{3}x=0,$

which, since $V=2R_{A}i$, reduces to

$V\delta i-\int I\phi d^{3}x=0,$

that determines the additional current in the circuit

$\delta
i=\frac{\gamma\phi_{0}}{R_{A}}kk^{\prime}\frac{\cos\left(\frac{kd}{2}\right)-\cos\left(\frac{k^{\prime}d}{2}\right)}{\left(k-k^{\prime}\right)\left(k+k^{\prime}\right)}\sin\left(\omega^{\prime}t-\theta\right).$

The previous analysis is valid for $\omega^{\prime}$ close to $\omega$ because
the reactive component of the impedances was taken for simplicity as the same
for both frequencies. For the case $\omega^{\prime}=\omega$ the expression of
the additional current simplifies to

$\delta
i=-\frac{\gamma\phi_{0}}{2R_{A}}\frac{kd}{2}\sin\left(\frac{kd}{2}\right)\sin\left(\omega
t-\theta\right).$

It is interesting to note that a half-wavelength dipole, $d=\lambda/2$, would
be the best suited for the detection of a gauge wave at resonant conditions.

##  IV Radiating systems

An important point for possible applications and/or experiments is the
prediction of the fields radiated by a system in the context of the
$I$-$\gamma$-$j$ model. We will consider the fields generated by harmonic in
time currents in conductors in which polarization effects can be neglected.

We first recall that in AB electrodynamics the equations for the potentials
derived from (21) coincide with the corresponding Maxwell equations in the
Lorentz gauge

$\displaystyle\frac{1}{c^{2}}\frac{\partial^{2}\phi}{\partial
t^{2}}-\nabla^{2}\phi$ $\displaystyle=$
$\displaystyle\frac{\varrho}{\varepsilon_{0}},$
$\displaystyle\frac{1}{c^{2}}\frac{\partial^{2}\mathbf{A}}{\partial
t^{2}}-\nabla^{2}\mathbf{A}$ $\displaystyle=$
$\displaystyle\mu_{0}\mathbf{j},$

which after averaging result in equations with the same form for the
macroscopic potentials, but with the sources replaced by the macroscopic ones,
free plus molecular.

The general solutions in free space are given by

$\displaystyle\phi\left(\mathbf{x},t\right)$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi\varepsilon_{0}}\int\frac{\varrho\left(\mathbf{x}^{\prime},t^{\prime}\right)}{\left|\mathbf{x}^{\prime}-\mathbf{x}\right|}d^{3}x^{\prime},$
$\displaystyle\mathbf{A}\left(\mathbf{x},t\right)$ $\displaystyle=$
$\displaystyle\frac{\mu_{0}}{4\pi}\int\frac{\mathbf{j}\left(\mathbf{x}^{\prime},t^{\prime}\right)}{\left|\mathbf{x}^{\prime}-\mathbf{x}\right|}d^{3}x^{\prime},$

where $t^{\prime}=t-\left|\mathbf{x}^{\prime}-\mathbf{x}\right|/c$. For
harmonic in time fields and sources the corresponding Fourier-transformed-in-
time amplitudes satisfy

$\displaystyle\phi\left(\mathbf{x},\omega\right)$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi\varepsilon_{0}}\int\frac{\varrho\left(\mathbf{x}^{\prime},\omega\right)}{R}\exp\left(ikR\right)d^{3}x^{\prime},$
$\displaystyle\mathbf{A}\left(\mathbf{x},\omega\right)$ $\displaystyle=$
$\displaystyle\frac{\mu_{0}}{4\pi}\int\frac{\mathbf{j}\left(\mathbf{x}^{\prime},\omega\right)}{R}\exp\left(ikR\right)d^{3}x^{\prime},$

with $R=\left|\mathbf{x}^{\prime}-\mathbf{x}\right|$, and $k=\omega/c$.

In the non-polarizable conductors considered the relation between the sources
is given by (27), which results for the Fourier amplitudes in

$\varrho\left(\mathbf{x},\omega\right)=-i\frac{1+\gamma}{\omega}\nabla\cdot\mathbf{j}.$

We can thus write

$\displaystyle\phi\left(\mathbf{x},\omega\right)$ $\displaystyle=$
$\displaystyle-i\frac{1+\gamma}{4\pi\varepsilon_{0}\omega}\int\frac{\nabla^{\prime}\cdot\mathbf{j}}{R}\exp\left(ikR\right)d^{3}x^{\prime}$
$\displaystyle=$
$\displaystyle-i\frac{1+\gamma}{4\pi\varepsilon_{0}\omega}\oint\frac{\exp\left(ikR\right)}{R}\mathbf{j}\cdot
d\mathbf{S}^{\prime}$
$\displaystyle+i\frac{1+\gamma}{4\pi\varepsilon_{0}\omega}\int\mathbf{j}\left(\mathbf{x}^{\prime},\omega\right)\cdot\nabla^{\prime}\left[\frac{\exp\left(ikR\right)}{R}\right]d^{3}x^{\prime}.$

Since the volume of integration contains all sources, no current traverses its
boundary, so that the surface integral is zero. Also,
$\nabla^{\prime}R=-\nabla R$, allowing to finally obtain

$\displaystyle\phi\left(\mathbf{x},\omega\right)$ $\displaystyle=$
$\displaystyle-i\frac{1+\gamma}{4\pi\varepsilon_{0}\omega}\nabla\cdot\int\frac{\mathbf{j}\left(\mathbf{x}^{\prime},\omega\right)}{R}\exp\left(ikR\right)d^{3}x^{\prime}$
$\displaystyle=$
$\displaystyle-i\frac{\left(1+\gamma\right)c^{2}}{\omega}\nabla\cdot\mathbf{A}\left(\mathbf{x},\omega\right).$

In this way, all fields outside the sources can be determined from the
solution for the vector potential alone

$\mathbf{A}\left(\mathbf{x},\omega\right)=\frac{\mu_{0}}{4\pi}\int\frac{\mathbf{j}\left(\mathbf{x}^{\prime},\omega\right)}{R}\exp\left(ikR\right)d^{3}x^{\prime},$
(47)

using the fields definitions in terms of the potentials, and the previous
expression of $\phi\left(\mathbf{x},\omega\right)$, to obtain

$\displaystyle\mathbf{B}\left(\mathbf{x},\omega\right)$ $\displaystyle=$
$\displaystyle\nabla\times\mathbf{A}\left(\mathbf{x},\omega\right),$
$\displaystyle\mathbf{E}\left(\mathbf{x},\omega\right)$ $\displaystyle=$
$\displaystyle-\nabla\phi\left(\mathbf{x},\omega\right)+i\omega\mathbf{A}\left(\mathbf{x},\omega\right)$
$\displaystyle=$ $\displaystyle
i\frac{\left(1+\gamma\right)c^{2}}{\omega}\nabla\times\mathbf{B}\left(\mathbf{x},\omega\right)-i\gamma\omega\mathbf{A}\left(\mathbf{x},\omega\right),$
$\displaystyle S\left(\mathbf{x},\omega\right)$ $\displaystyle=$
$\displaystyle-i\frac{\omega}{c^{2}}\phi\left(\mathbf{x},\omega\right)+\nabla\cdot\mathbf{A}\left(\mathbf{x},\omega\right)$
$\displaystyle=$
$\displaystyle-\gamma\nabla\cdot\mathbf{A}\left(\mathbf{x},\omega\right).$

We can now use the usual approximations for the far field, or radiation
region, in the solution of (47), with sources located around the origin of
coordinates, and
$R\simeq\left|\mathbf{x}\right|-\mathbf{x}^{\prime}\cdot\mathbf{n}$, with
$\mathbf{n}=\mathbf{x}/\left|\mathbf{x}\right|$. By further denoting
$r\equiv\left|\mathbf{x}\right|$ the far-field term is

$\mathbf{A}\left(\mathbf{x},\omega\right)=\frac{\mu_{0}}{4\pi
r}\exp\left(ikr\right)\int\mathbf{j}\left(\mathbf{x}^{\prime},\omega\right)\exp\left(-ik\mathbf{x}^{\prime}\cdot\mathbf{n}\right)d^{3}x^{\prime}.$
(48)

Since space derivatives of $r^{-1}$ and of $\mathbf{n}$ generate additional
factors $r^{-1}$, the only space derivatives of (48) that contribute in the
radiation region are those of $\exp\left(ikr\right)$, so that the general
expression of the fields in the radiation region are

$\displaystyle\mathbf{B}\left(\mathbf{x},\omega\right)$ $\displaystyle=$
$\displaystyle ik\mathbf{n}\times\mathbf{A}\left(\mathbf{x},\omega\right),$
(49a) $\displaystyle\mathbf{E}\left(\mathbf{x},\omega\right)$ $\displaystyle=$
$\displaystyle-i\omega\left(1+\gamma\right)\mathbf{n}\times\left[\mathbf{n}\times\mathbf{A}\left(\mathbf{x},\omega\right)\right]-i\gamma\omega\mathbf{A}\left(\mathbf{x},\omega\right)$
(49b) $\displaystyle=$ $\displaystyle
i\omega\mathbf{A}_{\perp}\left(\mathbf{x},\omega\right)-i\gamma\omega\mathbf{A}_{\parallel}\left(\mathbf{x},\omega\right),$
$\displaystyle S\left(\mathbf{x},\omega\right)$ $\displaystyle=$
$\displaystyle-
ik\gamma\mathbf{n}\cdot\mathbf{A}\left(\mathbf{x},\omega\right),$ (49c)
$\displaystyle\phi\left(\mathbf{x},\omega\right)$ $\displaystyle=$
$\displaystyle\left(1+\gamma\right)c\mathbf{n}\cdot\mathbf{A}\left(\mathbf{x},\omega\right)$
(49d)

with $\mathbf{A}$ given by (48), and where
$\mathbf{A}_{\parallel}=\left(\mathbf{n}\cdot\mathbf{A}\right)\mathbf{n}$,
$\mathbf{A}_{\perp}=\mathbf{A}-\mathbf{A}_{\parallel}$. Relation $\omega=kc$
was also used.

We thus see that the transverse components of the electromagnetic radiation
field generated by a given current distribution coincide with those predicted
by Maxwell electrodynamics. The additional terms are the longitudinal
component of the electric field and the scalar field.

The instantaneous energy flow has in AB electrodynamics the expression
minotti2021quantum

$\mathbf{S}_{u}=\frac{1}{\mu_{0}}\left(\mathbf{E}\times\mathbf{B}-\phi\nabla
S+\mathbf{A}\frac{\partial S}{\partial t}\right).$ (50)

In terms of the Fourier amplitudes the time-averaged energy flow is thus
expressed as (the asterisk indicates complex conjugation)

$\overline{\mathbf{S}}_{u}=\frac{1}{4\mu_{0}}\left[\mathbf{E}^{\ast}\times\mathbf{B}+\mathbf{E}\times\mathbf{B}^{\ast}+ik\left(\phi
S^{\ast}-\phi^{\ast}S\right)+i\omega\left(\mathbf{A}S^{\ast}-\mathbf{A}^{\ast}S\right)\right].$

Using expressions (49) we readily obtain

$\overline{\mathbf{S}}_{u}=\frac{\omega^{2}}{2\mu_{0}c}\left[\mathbf{A}_{\perp}\cdot\mathbf{A}_{\perp}^{\ast}-\gamma\left(1+\gamma\right)\mathbf{A}_{\parallel}\cdot\mathbf{A}_{\parallel}^{\ast}\right]\mathbf{n}.$

The term that contains the perpendicular component $\mathbf{A}_{\perp}$
coincides with the usual Poynting vector in Maxwell electrodynamics. The
interesting point is that the term containing the parallel component
$\mathbf{A}_{\parallel}$ represents an ”incoming” energy flow. This indicates
that the tunneling currents lead to a ”gain” of energy for the radiating
system.

As an example we take the finite-length dipole antenna previously considered,
for which

$\mathbf{A}\left(\mathbf{x},\omega\right)=\mathbf{e}_{z}\frac{\mu_{0}i_{0}}{4\pi
r}\exp\left(ikr\right)\int_{-d/2}^{d/2}\sin\left(\frac{kd}{2}-k\left|z^{\prime}\right|\right)\exp\left(-ikz^{\prime}\cos\theta\right)dz^{\prime},$

where $\theta$ is the angle between $\mathbf{x}$ and the $z$ axis
($\mathbf{e}_{z}$ is the unit vector along the $z$ axis). An elementary
integration gives

$\mathbf{A}\left(\mathbf{x},\omega\right)=\frac{\mu_{0}i_{0}}{2\pi
kr}\exp\left(ikr\right)\frac{\cos\left(\frac{kd}{2}\cos\theta\right)-\cos\left(\frac{kd}{2}\right)}{\sin^{2}\theta}\mathbf{e}_{z}.$

The resulting magnetic far field has only azimuthal component

$\mathbf{B}\left(\mathbf{x},\omega\right)=i\frac{\mu_{0}i_{0}}{2\pi
r}\exp\left(ikr\right)\frac{\cos\left(\frac{kd}{2}\right)-\cos\left(\frac{kd}{2}\cos\theta\right)}{\sin\theta}\mathbf{e}_{\varphi},$

which is transverse to the propagation direction $\mathbf{n}$.

The electric far field has the transverse $\theta$-component

$E_{\theta}\left(\mathbf{x},\omega\right)=i\frac{i_{0}}{2\pi\varepsilon_{0}cr}\exp\left(ikr\right)\frac{\cos\left(\frac{kd}{2}\right)-\cos\left(\frac{kd}{2}\cos\theta\right)}{\sin\theta},$

and the radial, longitudinal component

$E_{r}\left(\mathbf{x},\omega\right)=i\gamma\frac{i_{0}}{2\pi\varepsilon_{0}cr}\exp\left(ikr\right)\frac{\cos\left(\frac{kd}{2}\right)-\cos\left(\frac{kd}{2}\cos\theta\right)}{\sin^{2}\theta}\cos\theta.$
(51)

The scalar far field is

$S\left(\mathbf{x},\omega\right)=i\gamma\frac{\mu_{0}i_{0}}{2\pi
r}\exp\left(ikr\right)\frac{\cos\left(\frac{kd}{2}\right)-\cos\left(\frac{kd}{2}\cos\theta\right)}{\sin^{2}\theta}\cos\theta.$
(52)

The time-averaged energy flow is

$\overline{\mathbf{S}}_{u}=\frac{\mu_{0}i_{0}^{2}c}{8\pi^{2}r^{2}}\left[\frac{\cos\left(\frac{kd}{2}\cos\theta\right)-\cos\left(\frac{kd}{2}\right)}{\sin^{2}\theta}\right]^{2}\left[\sin^{2}\theta-\gamma\left(1+\gamma\right)\cos^{2}\theta\right]\mathbf{n}.$

##  V Conclusions

Building upon previous work, we have proven that the Aharonov-Bohm extended
electrodynamics with the parameter choice $\lambda=1$ (corresponding to the
so-called Feynman gauge Lagrangian in Maxwell theory) is the only theory of
the electromagnetic field which allows a coupling to non-conserved currents
and preserves the locality of field equations. In this theory the potentials
$A^{\mu}$ are fixed, apart from a small class of residual gauge
transformations, and can in principle be physically measured using anomalous
conducting probes (i.e., probes made of materials in which the continuity
$\partial_{\mu}j^{\mu}=0$ is violated at some points). As briefly recalled in
Sect. I.4, such anomalies may exist, at the microscopic and quantum level, in
various physical systems.

We can thus conclude that, although the exact local conservation of the
current and the associated gauge symmetry are important and “desirable”
properties, they do not constitute an absolute necessity, because the
equations of the electromagnetic field make perfectly sense without them, and
even become logically simpler.

On the other hand, we have proven that since it is still possible to introduce
auxiliary, non-physical potentials with the familiar gauge-fixing choices, the
extended theory is well compatible with Maxwell theory and inherits from it
most of the usual computational techniques.

In order to demonstrate the consistency of the extended theory we have
elaborated in detail on a phenomenological model of anomalous media, in which
the continuity equation is replaced by the relation (derived from some
microscopic fundamental hypotheses)
$\partial_{t}\langle\rho\rangle+(1+\gamma)\nabla\cdot\langle\textbf{j}\rangle=0$.
Here $\gamma$ is in general an adimensional parameter much smaller than 1;
however, calculations of radiation emission in Sect. IV indicate that $\gamma$
can approach 1 from below without giving rise to any singularity. In Sect.
II.4 the propagation of plane waves in anomalous media has been computed.

Possible physical realizations of anomalous conductors are represented by
molecular devices walz2015local ; jensen2019current ; garner2019helical ;
garner2020three . We would like to point out that the parts of Ref.
jensen2019current  most in line with our ideas are the Supplementary
Materials, where the method by Wang et al. for the calculations of the non-
local part of the current li2008definition  is applied. The authors find that
a part of the current “does not follow the bonds” (i.e., the wavefunctions)
also when the basis of states employed for the first-principles calculations
is much enlarged. It should be stressed that first-principles calculations
give excellent results for observable quantities like energy spectra etc., so
they appear to produce a good picture of the physics of the system, probably
equivalent to that of an hypothetical renormalized quantum field theory, in
which anomalies can play a role.

Another kind of macro-molecule exhibiting high conductivity and a charge
transmission mechanism which is still not completely understood is graphene.
Due to the chiral nature of quasi-particles in single and bi-layer graphene,
quantum tunneling in these materials becomes highly anisotropic and markedly
different from the case of normal, non-relativistic electrons. It is in fact a
condensed-matter realization of the Klein tunneling, which raises a paradox
concerning local charge conservation katsnelson2006chiral ; allain2011klein .
The solutions of the paradox that have been proposed go beyond single-particle
wavefunctions and involve multi-particle quantum field theory processes (see
e.g. alkhateeb2021relativistic  and refs.). However, it is well known that
chiral symmetry is often broken at the quantum level.

For this reason we believe that in the detection circuits for gauge waves
proposed in Sect. III it might be convenient to use graphene or graphite as
possible anomalous medium with non-zero extra-current $I$. However, a detailed
discussion of those detectors concerning signal/noise sensitivity and possible
error sources is beyond the scope of this paper and will be the subject of
future work.

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