# Are current discontinuities in molecular devices experimentally observable?

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 del
Plasma (INFIP), Buenos Aires, Argentina  G. Modanese 222Email address:
<EMAIL_ADDRESS>Free University of Bozen-Bolzano
Faculty of Science and Technology
I-39100 Bolzano, Italy

###### Abstract

An ongoing debate in the first-principles description of conduction in
molecular devices concerns the correct definition of current in the presence
of non-local potentials. If the physical current density ${\bf
j}=(-ie\hbar/2m)(\Psi^{*}\nabla\Psi-\Psi\nabla\Psi^{*})$ is not locally
conserved but can be re-adjusted by a non-local term, which current should be
regarded as real? We prove that the extended Maxwell equations by Aharonov-
Bohm give the e.m. field generated by such currents without any ambiguity. For
an oscillating dipole we show that the radiated electrical field has a
longitudinal component proportional to $\omega\hat{P}$, where $\hat{P}$ is the
anomalous moment $\int\hat{I}(\mathbf{x})\mathbf{x}d^{3}x$ and $\hat{I}$ is
the space-dependent part of the anomaly
$I=\partial_{t}\rho+\nabla\cdot\mathbf{j}$. In the case of a stationary
current in a molecular device, a failure of local current conservation causes
a “missing field” effect that can be experimentally observable, especially if
its entity depends on the total current.

##  I Introduction

There are different opinions in the literature concerning the possible
existence and physical interpretation of quantum systems for which the
electric charge is not locally conserved, i.e., the continuity condition
$\partial_{t}\rho+\nabla\cdot\mathbf{j}=0$ is not valid everywhere.

Wang et al. li2008definition ; zhang2011first  point out explicitly that in
first-principles calculations of the current in molecular devices based on
single-particle non-equilibrium Green’s function (NEGF) and density-functional
theory (DFT) the usual quantum-mechanical current ${\bf
j}=(-ie\hbar/2m)(\Psi^{*}\nabla\Psi-\Psi\nabla\Psi^{*})$ does not satisfy the
continuity condition and must be complemented by a non-local term. The
recourse to the NEGF-DFT formalism is necessary in order to take into account
the contribution of the inner atomic shells. In principle a full quantum field
theory of the system, if it can be formulated in a standard way including all
the internal electrons, would yield a conserved _bare_ current; in practice,
such a formulation does not exist for realistic systems lai2019charge , and
even if it could be achieved, it cannot be excluded that it needs to be
renormalized, giving rise to quantum anomalies (effective breaking of
symmetries and local conservation properties of the bare theory, like for the
ABJ anomaly in condensed matter cheng1984gauge ; parameswaran2014probing ).

An appealing feature of the approach by Wang et al. is that the additional
current they propose is exactly the same as that originating from the extended
Maxwell equations of Aharonov-Bohm without any consideration of the
microscopic/quantum aspects. These equations represent essentially the only
possible covariant extension of Maxwell’s theory which is compatible with the
established standard phenomenology of classical electromagnetism and QED and
is also applicable to currents that are not locally conserved ohmura1956new ;
aharonov1963further ; alicki1978generalised ; cornille1990propagation ;
van2001generalisation ; jimenez2011cosmological ; hively2012toward ;
Modanese2017MPLB ; modanese2017electromagnetic ; arbab2017extended ;
hively2019classical ; reed2020implications .

In a series of papers starting in 2108, Jensen, Garner and collaborators have
analysed in depth the concept of density current in quantum transport,
applying it to specific molecules and obtaining results compatible with the
approach by Wang et al. Ref. cabra2018simulation  sets the general theoretical
framework and notes that besides legitimate physical reasons for the non-
conserving character of local currents, there are technical problems related
to the choice of the basis in the first-principles calculations. The
discussion is illustrated by simulating elastic and inelastic local currents
in a benzenedithiol junction. Simulations show that the local flux does not
necessarily follow molecular bonds, with significant part of the flux going
“through space“.

In Ref. jensen2019current , the current density is investigated in saturated
chains of alkanes, silanes and germanes. The current density is defined in
this context as

$\mathbf{j}(\mathbf{x})=\frac{e\hbar}{4m\pi}\sum_{i,j}\int_{-\infty}^{+\infty}dE\,G^{<}_{ij}(E)[\psi_{i}(\mathbf{x})\nabla\psi_{j}(\mathbf{x})-\psi_{j}(\mathbf{x})\nabla\psi_{i}(\mathbf{x})],$
(1)

where $G^{<}_{ij}(E)$ are the matrix elements of the lesser Green function in
a standard non-orthogonal basis $\\{\psi_{i}\\}$. The authors show that an
enlargement of the eigenbasis for the _ab initio_ calculations does not
substantially improve the conservation of current density in this case (while
it does lead to an improvement in other cases, notably for graphene ribbons
walz2015local ). They then apply the recipe by Wang et al. computing the
secondary currents via a Poisson equation.

In Refs. garner2019helical  and garner2020three  linear carbon wires are
considered. Detailed plots of the integrated local current density are given,
as compared to the (constant) total current.

There are also wave equations in quantum mechanics which do not derive from a
microscopic theory but are proposed as effective models with several important
applications, in which the current is not locally conserved lenzi2008solutions
; lenzi2008fractional ; latora1999superdiffusion ; caspi2000enhanced ;
chamon1997nonlocal ; balantekin1998green ; laskin2002fractional ;
wei2016comment ; modanese2018time . It is important in our opinion to develop
a formalism that allows to compute the electromagnetic field generated by
local currents also in those cases.

In this work, after recalling recent progress in the theory and numerical
solutions of the extended Maxwell-Aharonov-Bohm equations, we derive in Sect.
II the corresponding wave equations for the electric and magnetic field. These
equations allow to compute $\mathbf{E}$ and $\mathbf{B}$ directly from the
physical sources $\rho$ and $\mathbf{j}$, without any reference to the scalar
field $S$ which in the traditional formulation has the role to restore local
conservation through an additional (or “secondary“) charge density
proportional to $\partial_{t}\rho$ and an additional current density
proportional to $\nabla S$. The general solution for localized sources is
written in the form of retarded integrals. In Sect. II.1 we compute for the
first time the electric and magnetic dipole radiation in far-field
approximation. We point out that in general a longitudinal electric radiation
field $\mathbf{E}_{L}$ can be present; this is one of the main predictions of
the extended theory for the case of an oscillating current that is not locally
conserved. On the other hand we note that the magnetic field $\mathbf{B}$ is
simply proportional to the curl $\nabla\times\mathbf{j}$, like in Maxwell’s
theory. Therefore in the limit of stationary currents, in the anomalous case
in which $\nabla\cdot\mathbf{j}$ is not zero everywhere (presence of charge
sinks/sources), $\mathbf{B}$ is insensitive to the secondary current $\nabla
S$, but can reveal the discontinuities in the current simply because to the
“missing links“ of current correspond missing contributions to $\mathbf{B}$ in
the Biot-Savart formula. Such effects are expected to be small, but detectable
with accurate experiments, as briefly discussed in Sect. III. Finally, in
Sect. IV we present some numerical solutions for the case of stationary
currents, obtained not through the direct wave equations derived in this work,
but through the double-retarded integrals written in
modanese2017electromagnetic ; modanese2019design . In this way it is possible
to display explicitly the contributions of the auxiliary field $S$, confirming
that such contributions cancel out and the only consequence for $\mathbf{B}$
is the missing field effect.

Our final message can be summarized as follows: microscopic models for the
computation of the current density in molecular devices can and should
continue to improve their performance and their precision without worrying
about local conservation of $\mathbf{j}$. The extended Maxwell equations allow
in any case a reliable and efficient computation of the resulting fields. This
investigation may lead to interesting discoveries in those cases where the
e.m. fields generated by molecular currents are strong enough to play a
significant role.

##  II General equations for $\mathbf{E}$ and $\mathbf{B}$ and their radiative
solution

In this section we write the general wave equations for the electric and
magnetic field in extended electrodynamics, in a form in which the auxiliary
field $S$ is completely eliminated, and we find their radiative solution.

The extended Maxwell equations in the Aharonov-Bohm theory are
Modanese2017MPLB ; modanese2017electromagnetic

$\displaystyle\nabla\cdot\mathbf{E}$ $\displaystyle=$ $\displaystyle
4\pi\rho-\frac{1}{c}\frac{\partial S}{\partial t}$ (2)
$\displaystyle\nabla\times\mathbf{E}$ $\displaystyle=$
$\displaystyle-\frac{1}{c}\frac{\partial\mathbf{B}}{\partial t}$ (3)
$\displaystyle\nabla\cdot\mathbf{B}$ $\displaystyle=$ $\displaystyle 0$ (4)
$\displaystyle\nabla\times\mathbf{B}$ $\displaystyle=$
$\displaystyle\frac{4\pi}{c}\mathbf{j}+\frac{1}{c}\frac{\partial\mathbf{E}}{\partial
t}+\nabla S$ (5)

where CGS units have been employed and $S$ is an auxiliary field whose source
is the extra-current $I=\partial_{t}\rho+\nabla\cdot\mathbf{j}$, namely

$\frac{1}{c^{2}}\frac{\partial^{2}S}{\partial
t^{2}}-\nabla^{2}S=\frac{4\pi}{c}\left[\frac{\partial\rho}{\partial
t}+\nabla\cdot\mathbf{j}\right]\equiv\frac{4\pi}{c}I$ (6)

The field $S$ is clearly zero in the pure Maxwell theory, which requires
strict local conservation of the current.

It is possible to interpret the extended equations (2), (5), (6) as involving
some additional or “secondary“ sources, namely a secondary charge density
proportional to $\partial_{t}S$ and a secondary current density proportional
to $\nabla S$. Including these additional charge and current densities gives
total densities which satisfy the continuity equation. Note that the solution
of (6) for a localized extra-source yields $S$ as a retarded integral:

$S\left(\mathbf{x},t\right)=\frac{1}{c}\int\frac{I\left(\mathbf{x}^{\prime},t^{\prime}\right)}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}d^{3}x^{\prime},$

with $t^{\prime}=t-\left|\mathbf{x}-\mathbf{x}^{\prime}\right|/c$. Therefore
$S$ is not localized in the region where the physical sources are present. For
this reason we have called the secondary charge and current in our previous
work “cloud charge” and “cloud current” and we have evaluated them in some
specific cases (see also the numerical simulations in Sect. IV of this work).

The secondary current, in particular, coincides with the additional current
predicted by the Landauer-Büttiker theory for systems with quantum transport
in which local conservation of the current fails li2008definition ;
zhang2011first .

It is possible, however, to write wave equations for $\mathbf{E}$ and
$\mathbf{B}$ in which the field $S$ and the secondary charge and current are
completely absent. This is in some sense reassuring, because it implies that
the physical fields only depend on the localized, physical sources, and that
there is no reason to regard the secondary currents as real and to care, for
instance, about their dissipation properties. (See also an alternative proof
of the independence of the electric and magnetic fields on the scalar source
in Appendix B.)

Although the extended Aharonov-Bohm theory has only a limited gauge
invariance, it is possible to define potentials $\phi$ and $\mathbf{A}$ with
the usual relations to $\mathbf{E}$ and $\mathbf{B}$, namely

$\displaystyle\mathbf{E}$ $\displaystyle=$
$\displaystyle-\nabla\phi-\frac{1}{c}\frac{\partial\mathbf{A}}{\partial t},$
$\displaystyle\mathbf{B}$ $\displaystyle=$
$\displaystyle\nabla\times\mathbf{A}.$

and to write their wave equations as

$\displaystyle\frac{1}{c^{2}}\frac{\partial^{2}\phi}{\partial
t^{2}}-\nabla^{2}\phi$ $\displaystyle=$ $\displaystyle
4\pi\rho-\frac{1}{c}\frac{\partial S}{\partial t},$ (7)
$\displaystyle\frac{1}{c^{2}}\frac{\partial^{2}\mathbf{A}}{\partial
t^{2}}-\nabla^{2}\mathbf{A}$ $\displaystyle=$
$\displaystyle\frac{4\pi}{c}\mathbf{j}+\nabla S$ (8)

where once again the role of the secondary charge and current is evident. The
solutions of (7) and (8) are

$\displaystyle\phi\left(\mathbf{x},t\right)$ $\displaystyle=$
$\displaystyle\int\frac{\rho\left(\mathbf{x}^{\prime},t^{\prime}\right)}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}d^{3}x^{\prime}-\frac{1}{4\pi
c}\int\frac{1}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}\frac{\partial}{\partial
t^{\prime}}S\left(\mathbf{x}^{\prime},t^{\prime}\right)d^{3}x^{\prime},$
$\displaystyle\mathbf{A}\left(\mathbf{x},t\right)$ $\displaystyle=$
$\displaystyle\frac{1}{c}\int\frac{\mathbf{j}\left(\mathbf{x}^{\prime},t^{\prime}\right)}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}d^{3}x^{\prime}+\frac{1}{4\pi}\int\frac{\nabla^{\prime}S\left(\mathbf{x}^{\prime},t^{\prime}\right)}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}d^{3}x^{\prime}.$

By taking the gradient of (7) and adding the time derivative of (8) one
obtains an equation for $\mathbf{E}$ without $S$, while taking the curl of (8)
one obtains an equation for $\mathbf{B}$ without $S$:

$\displaystyle\frac{1}{c^{2}}\frac{\partial^{2}\mathbf{E}}{\partial
t^{2}}-\nabla^{2}\mathbf{E}$ $\displaystyle=$
$\displaystyle=-\frac{4\pi}{c^{2}}\left(\frac{\partial\mathbf{j}}{\partial
t}+c^{2}\nabla\rho\right),$ (9)
$\displaystyle\frac{1}{c^{2}}\frac{\partial^{2}\mathbf{B}}{\partial
t^{2}}-\nabla^{2}\mathbf{B}$ $\displaystyle=$
$\displaystyle\frac{4\pi}{c}\nabla\times\mathbf{j}.$ (10)

These equations have also been derived in hively2019classical , starting from
the extended field equations (2), (5), (6) and using some vector calculus
identities.

For localized sources the corresponding solutions are

$\displaystyle\mathbf{E}\left(\mathbf{x},t\right)$ $\displaystyle=$
$\displaystyle-\frac{1}{c^{2}}\int\left(\frac{\partial\mathbf{j}}{\partial
t^{\prime}}+c^{2}\nabla^{\prime}\rho\right)\frac{d^{3}x^{\prime}}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|},$
(11) $\displaystyle\mathbf{B}\left(\mathbf{x},t\right)$ $\displaystyle=$
$\displaystyle\frac{1}{c}\int\nabla^{\prime}\times\mathbf{j}\frac{d^{3}x^{\prime}}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}.$
(12)

We recall that the current density $\mathbf{j}$ can be in general written as
the sum of an “irrotational” part, which is the gradient of a scalar field and
has zero curl, plus a “solenoidal” part, which is the curl of a vector field
and has zero divergence:

$\mathbf{j}=\mathbf{j}_{irrot}+\mathbf{j}_{solen}\equiv\nabla
F+\nabla\times\mathbf{G}$ (13)

According to eq. (12), the irrotational part has no influence on the magnetic
field. It follows in particular that interruptions in the current inevitably
cause a “missing field” effect, as discussed in Sects. III, IV. (Because the
secondary current $\nabla S$ which restores current conservation in the
extended equations is purely irrotational.) On the other hand, any change in
the solenoidal part (like e.g. in dreyer2018current ) does not affect current
conservation but is reflected in a change in the field.

One can further operate on the expressions (11), (12) considering that

$\int\frac{\partial\mathbf{j}}{\partial
t^{\prime}}\frac{d^{3}x^{\prime}}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}=\int\frac{\partial\mathbf{j}}{\partial
t}\frac{d^{3}x^{\prime}}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}=\frac{\partial}{\partial
t}\int\frac{\mathbf{j}\left(\mathbf{x}^{\prime},t^{\prime}\right)}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}d^{3}x^{\prime},$

and that

$\displaystyle\int\frac{\nabla^{\prime}\rho}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}d^{3}x^{\prime}$
$\displaystyle=$
$\displaystyle\int\nabla^{\prime}\left[\frac{\rho\left(\mathbf{x}^{\prime},t^{\prime}\right)}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}\right]d^{3}x^{\prime}$
$\displaystyle-\int\rho\left(\mathbf{x}^{\prime},t^{\prime}\right)\nabla^{\prime}\frac{1}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}d^{3}x^{\prime}$
$\displaystyle=$
$\displaystyle\oint\frac{\rho\left(\mathbf{x}^{\prime},t^{\prime}\right)}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}d\mathbf{S}^{\prime}+\nabla\frac{\rho\left(\mathbf{x}^{\prime},t^{\prime}\right)}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}d^{3}x^{\prime}$
$\displaystyle=$
$\displaystyle\nabla\int\frac{\rho\left(\mathbf{x}^{\prime},t^{\prime}\right)}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}d^{3}x^{\prime},$

where the surface integral is zero because there is no charge on the external
surface of the volume of the source, and the change from $\nabla^{\prime}$ to
$-\nabla$ in the integral in the second line was done because the function on
which it operates is of argument $\mathbf{x}-\mathbf{x}^{\prime}$.

We have in a similar manner

$\displaystyle\int\frac{\nabla^{\prime}\times\mathbf{j}}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}d^{3}x^{\prime}$
$\displaystyle=$
$\displaystyle\int\nabla^{\prime}\times\left[\frac{\mathbf{j}\left(\mathbf{x}^{\prime},t^{\prime}\right)}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}\right]d^{3}x^{\prime}$
$\displaystyle-\int\nabla^{\prime}\left(\frac{1}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}\right)\times\mathbf{j}\left(\mathbf{x}^{\prime},t^{\prime}\right)d^{3}x^{\prime}$
$\displaystyle=$
$\displaystyle-\oint\frac{\mathbf{j}\left(\mathbf{x}^{\prime},t^{\prime}\right)}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}\times
d\mathbf{S}^{\prime}$
$\displaystyle+\int\nabla\left(\frac{1}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}\right)\times\mathbf{j}\left(\mathbf{x}^{\prime},t^{\prime}\right)d^{3}x^{\prime}$
$\displaystyle=$
$\displaystyle\nabla\times\int\frac{\mathbf{j}\left(\mathbf{x}^{\prime},t^{\prime}\right)}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}d^{3}x^{\prime}.$

The expressions of the fields thus reduce to

$\displaystyle\mathbf{E}\left(\mathbf{x},t\right)$ $\displaystyle=$
$\displaystyle-\frac{1}{c^{2}}\frac{\partial}{\partial
t}\int\frac{\mathbf{j}\left(\mathbf{x}^{\prime},t^{\prime}\right)}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}d^{3}x^{\prime}-\nabla\int\frac{\rho\left(\mathbf{x}^{\prime},t^{\prime}\right)}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}d^{3}x^{\prime},$
(14a) $\displaystyle\mathbf{B}\left(\mathbf{x},t\right)$ $\displaystyle=$
$\displaystyle\frac{1}{c}\nabla\times\int\frac{\mathbf{j}\left(\mathbf{x}^{\prime},t^{\prime}\right)}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}d^{3}x^{\prime}.$
(14b)

These expressions show that the EM fields are obtained from the potentials
evaluated without considering their source terms depending on $S$.

###  II.1 Radiative solution in the dipole approximation

For a pure temporal Fourier mode:
$\mathbf{j}\left(\mathbf{x},t\right)=\widehat{\mathbf{j}}\left(\mathbf{x}\right)\exp\left(-i\omega
t\right)$,
$\rho\left(\mathbf{x},t\right)=\widehat{\rho}\left(\mathbf{x}\right)\exp\left(-i\omega
t\right)$, the general solution for the corresponding scalar potential,
without the source depending on $S$, is of the form
$\phi\left(\mathbf{x},t\right)=\widehat{\phi}\left(\mathbf{x}\right)\exp\left(-i\omega
t\right)$, with

$\widehat{\phi}\left(\mathbf{x}\right)=\int\frac{\widehat{\rho}\left(\mathbf{x}^{\prime}\right)}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}\exp\left(ik\left|\mathbf{x}-\mathbf{x}^{\prime}\right|\right)d^{3}x^{\prime},$

where $k=\omega/c$. For a distant observation point and wavelength large as
compared with the source dimension, the expression is approximated by

$\widehat{\phi}\left(\mathbf{x}\right)=\frac{\exp\left(ikr\right)}{r}\int\widehat{\rho}\left(\mathbf{x}^{\prime}\right)\left(1-ik\mathbf{n}\cdot\mathbf{x}^{\prime}\right)d^{3}x^{\prime},$

with $r=\left|\mathbf{x}\right|$, and $\mathbf{n}=\mathbf{x}/r$.

Even if the charge is not conserved locally, it is conserved globally, so that

$\int\widehat{\rho}\left(\mathbf{x}^{\prime}\right)d^{3}x^{\prime}=0,$

and thus

$\displaystyle\widehat{\phi}\left(\mathbf{x}\right)$ $\displaystyle=$
$\displaystyle-i\frac{k}{r}\exp\left(ikr\right)\mathbf{n}\cdot\int\widehat{\rho}\left(\mathbf{x}^{\prime}\right)\mathbf{x}^{\prime}d^{3}x^{\prime}$
$\displaystyle\equiv$
$\displaystyle-i\frac{k}{r}\exp\left(ikr\right)\mathbf{n}\cdot\widehat{\mathbf{p}},$

where $\widehat{\mathbf{p}}$ is the Fourier amplitude of the usual charge
dipole.

The solution to the corresponding Fourier amplitude of the vector potential
is, with the same approximations,

$\widehat{\mathbf{A}}\left(\mathbf{x}\right)=\frac{1}{cr}\exp\left(ikr\right)\int\widehat{\mathbf{j}}\left(\mathbf{x}^{\prime}\right)d^{3}x^{\prime}.$

Using the identity

$\displaystyle\int\widehat{\mathbf{j}}\left(\mathbf{x}^{\prime}\right)d^{3}x^{\prime}$
$\displaystyle=$
$\displaystyle\oint\mathbf{x}^{\prime}\widehat{\mathbf{j}}\left(\mathbf{x}^{\prime}\right)\cdot
d\mathbf{S}^{\prime}-\int\mathbf{x}^{\prime}\nabla^{\prime}\cdot\widehat{\mathbf{j}}\left(\mathbf{x}^{\prime}\right)d^{3}x^{\prime}$
$\displaystyle=$
$\displaystyle-\int\mathbf{x}^{\prime}\nabla^{\prime}\cdot\widehat{\mathbf{j}}\left(\mathbf{x}^{\prime}\right)d^{3}x^{\prime},$

where in the second line it was used that there is no current leaving or
entering the volume of the source, and also allowing for charge non-
conservation, so that

$\nabla^{\prime}\cdot\widehat{\mathbf{j}}\left(\mathbf{x}^{\prime}\right)=i\omega\widehat{\rho}\left(\mathbf{x}^{\prime}\right)+\widehat{I}\left(\mathbf{x}^{\prime}\right),$

we have

$\displaystyle\widehat{\mathbf{A}}\left(\mathbf{x}\right)$ $\displaystyle=$
$\displaystyle-i\frac{\omega}{cr}\exp\left(ikr\right)\int\mathbf{x}^{\prime}\widehat{\rho}\left(\mathbf{x}^{\prime}\right)d^{3}x^{\prime}-\frac{1}{cr}\exp\left(ikr\right)\int\mathbf{x}^{\prime}\widehat{I}\left(\mathbf{x}^{\prime}\right)d^{3}x^{\prime}$
$\displaystyle=$
$\displaystyle-i\frac{\omega}{cr}\exp\left(ikr\right)\widehat{\mathbf{p}}-\frac{1}{cr}\exp\left(ikr\right)\widehat{\mathbf{P}},$

where the dipolar moment of the Fourier amplitude of the extra source $I$ was
defined.

$\widehat{\mathbf{P}}=\int\mathbf{x}^{\prime}\widehat{I}\left(\mathbf{x}^{\prime}\right)d^{3}x^{\prime}.$

The Fourier amplitude of the radiative electric field is thus

$\displaystyle\widehat{\mathbf{E}}\left(\mathbf{x}\right)$ $\displaystyle=$
$\displaystyle\frac{i\omega}{c}\widehat{\mathbf{A}}\left(\mathbf{x}\right)-\nabla\widehat{\phi}\left(\mathbf{x}\right)$
(15) $\displaystyle=$
$\displaystyle\frac{\omega^{2}}{c^{2}r}\exp\left(ikr\right)\left[\widehat{\mathbf{p}}-\left(\mathbf{n}\cdot\widehat{\mathbf{p}}\right)\mathbf{n}\right]-i\frac{\omega}{c^{2}r}\exp\left(ikr\right)\widehat{\mathbf{P}}$
(16) $\displaystyle=$
$\displaystyle-\frac{\omega^{2}}{c^{2}r}\exp\left(ikr\right)\left[\left(\widehat{\mathbf{p}}\times\mathbf{n}\right)\times\mathbf{n}\right]-i\frac{\omega}{c^{2}r}\exp\left(ikr\right)\widehat{\mathbf{P}}.$
(17)

Analogously,

$\widehat{\mathbf{B}}\left(\mathbf{x}\right)=\nabla\times\widehat{\mathbf{A}}\left(\mathbf{x}\right)=\frac{1}{c^{2}r}\exp\left(ikr\right)\left(-\omega^{2}\widehat{\mathbf{p}}+i\omega\widehat{\mathbf{P}}\right)\times\mathbf{n}.$

Transforming back to the time domain the EM fields are given by

$\displaystyle\mathbf{E}\left(\mathbf{x},t\right)$ $\displaystyle=$
$\displaystyle-\sum_{\omega}\frac{\omega^{2}}{c^{2}r}\exp\left[-i\omega\left(t-r/c\right)\right]\left\\{\left[\widehat{\mathbf{p}}\left(\omega\right)\times\mathbf{n}\right]\times\mathbf{n}\right\\}$
(19)
$\displaystyle-\sum_{\omega}i\frac{\omega}{c^{2}r}\exp\left[-i\omega\left(t-r/c\right)\right]\widehat{\mathbf{P}}\left(\omega\right)$
$\displaystyle=$ $\displaystyle\frac{1}{c^{2}r}\frac{\partial^{2}}{\partial
t^{2}}\sum_{\omega}\exp\left[-i\omega\left(t-r/c\right)\right]\left\\{\left[\widehat{\mathbf{p}}\left(\omega\right)\times\mathbf{n}\right]\times\mathbf{n}\right\\}$
(21) $\displaystyle+\frac{1}{c^{2}r}\frac{\partial}{\partial
t}\sum_{\omega}\exp\left[-i\omega\left(t-r/c\right)\right]\widehat{\mathbf{P}}\left(\omega\right)$
$\displaystyle=$
$\displaystyle\frac{1}{c^{2}r}\left\\{\left[\overset{..}{\mathbf{p}}\left(t-r/c\right)\times\mathbf{n}\right]\times\mathbf{n}+\overset{.}{\mathbf{P}}\left(t-r/c\right)\right\\},$
(22)

and

$\displaystyle\mathbf{B}\left(\mathbf{x},t\right)$ $\displaystyle=$
$\displaystyle\frac{1}{c^{2}r}\sum_{\omega}\exp\left[-i\omega\left(t-r/c\right)\right]\left[-\omega^{2}\widehat{\mathbf{p}}\left(\omega\right)+i\omega\widehat{\mathbf{P}}\left(\omega\right)\right]\times\mathbf{n}$
$\displaystyle=$ $\displaystyle\frac{1}{c^{2}r}\frac{\partial^{2}}{\partial
t^{2}}\sum_{\omega}\exp\left[-i\omega\left(t-r/c\right)\right]\widehat{\mathbf{p}}\left(\omega\right)\times\mathbf{n}$
$\displaystyle-\frac{1}{c^{2}r}\frac{\partial}{\partial
t}\sum_{\omega}\exp\left[-i\omega\left(t-r/c\right)\right]\widehat{\mathbf{P}}\left(\omega\right)\times\mathbf{n}$
$\displaystyle=$
$\displaystyle\frac{1}{c^{2}r}\left[\overset{..}{\mathbf{p}}\left(t-r/c\right)-\overset{.}{\mathbf{P}}\left(t-r/c\right)\right]\times\mathbf{n}.$

The first term of $\mathbf{E}$ vanishes when it is multiplied by $\mathbf{n}$
and therefore represents the transverse component. The longitudinal component
is just $\mathbf{E}\cdot\mathbf{n}$, i.e.

$\mathbf{E}_{L}=\frac{1}{c^{2}r}\dot{\mathbf{P}}(t-r/c)\cdot\mathbf{n}$ (23)

To fix the ideas, suppose that the moment $\widehat{\mathbf{P}}$ of the extra-
current is directed along the $z$-axis. Then the component $E_{L}$ at a fixed
distance $r$ is seen to be maximum on the $z$-axis and zero in the $x$-$y$
plane. The opposite happens with the transverse component.

A simple formal example of oscillating extra-current has been introduced in
modanese2019high . Consider a point-like charge $q$ which oscillates between
the positions $(0,0,-a)$ and $(0,0,a)$, without a corresponding current:

$\rho(\mathbf{x},t)=q\cos(\omega
t)[\delta^{3}(\mathbf{x}+\mathbf{a})-\delta^{3}(\mathbf{x}-\mathbf{a})]$ (24)

The moment $\widehat{\mathbf{P}}$ in this case has the only non-zero component
$|\hat{P}_{z}|=2\omega qa$. If we assign the physical parameters $q$, $a$ and
$\omega$ it is straightforward to compute the longitudinal far field and we
have along the $z$-axis $|\hat{E}_{L}|=2\omega^{2}qa/(c^{2}r)$ (in SI units:
$|\hat{E}_{L}|=\mu_{0}\omega^{2}qa/(2\pi r)$). The transverse far field is
vanishing.

More realistically we can suppose that if a local violation of charge
conservation occurs, this will only concern a small fraction $\eta$ of the
oscillating charge, while the rest of the charge will have a corresponding
current and consequently will not generate any longitudinal radiation field.
In that case, the ratio between $E_{L}$ and $E_{T}$ will approximately be
equal to $\eta$.

In modanese2019high  we performed a numerical simulation of a source of this
kind with $a=10^{-7}$ cm, $\omega=2\pi\cdot 10^{9}$ Hz, $\theta=\pi/4$
($\theta$ is the 3D polar coordinate), and we obtained that for a totally
anomalous source (i.e. $\eta=1$, the entire charge oscillates without a
current) the longitudinal field is much larger that the transverse field that
would be generated by a corresponding regular source at the same position. The
field was computed at a relatively small distance ($r$ varied between
$3\lambda$ and $13\lambda$), therefore the result cannot be directly compared
with eq. (23), which holds for larger distances; still it confirms the
presence of a longitudinal field, because even in the near-field range it is
impossible to obtain such big longitudinal components if the source satisfies
local conservation.

In conclusion, if oscillating currents exist that violate even partially the
continuity condition, a sizable longitudinal electric radiation should be
generated which is obviously incompatible with the standard Maxwell equations.

##  III Possible experimental signatures of a missing $\mathbf{B}$ and a
radiative $\mathbf{E}_{L}$

Let us first consider a possible experimental observation of the missing field
effect for stationary currents. As discussed in modanese2019design  and
confirmed in Sects. II and III of this work, if in a linear conductor there
are regions in which a violation of continuity occurs and a fraction $\eta$ of
the current does not flow as “physical current $\rho\mathbf{v}$” but as
“secondary current $\nabla S$”, then the magnetic field generated by the
secondary current is zero, as a consequence of eq. (10). To detect the missing
field, we proposed in modanese2019design  to measure at a fixed distance $r$
the field of a normal conductor carrying a current $i$, and then the field of
the anomalous conductor carrying the same current. A differential measurement
with three wires was devised, in order to minimize errors. The scheme was
especially suited for brief current pulses and for the case when the supposed
anomalous conductor is a superconductor, which can be driven in/out a normal
state by changing its temperature.

Another possible technique, which employs stationary currents and one single
wire, is based on the measurement of the ratio $B/i$ at a fixed distance in
dependence on $i$ (supposing that a variable external bias allows to change
$i$). This would work if the ratio $\eta$ depends on $i$, in which case $B/i$
is expected to change, in open violation of the Maxwell equations. In fact,
the current patterns and local discontinuities observed in simulations of
molecular devices depend in general on the total current.

Possible errors could originate from slight changes in the spatial
distribution of the current at different $i$, if the conductor has a radius
that cannot be disregarded compared to the distance $r$. This radius depends
on how “elementary” the conductor is (for example, a single carbon nanowire
vs. a bundle of nanowires); in turn, that depends on how much current is
needed for the measurement, and thus indirectly on $r$ (because of the size
and sensitivity of the detector).

For example, suppose that a single carbon nanowire can carry a current of
$10^{-10}$ A and the detector can be placed at a distance of $10^{-4}$ m, very
large compared to the radius of the wire. The field would then be of the order
of $10^{-12}$ T, i.e. accessible to a sensitive SQUID. If the ratio $\eta$ is
of the order of $10^{-2}$ and the error on the current is negligible, the
SQUID is required to detect a variation in the field of $10^{-14}$ T.

Turning now to electric fields, the possible generation and detection of a
longitudinal e.m. radiation has rarely been explored in the past decades
giakos1993detection ; monstein2002observation ; monstein2004remarks ;
butterworth2013longitudinal ; umul2018excitation ; simulik2019slightly . A
recent preliminary experiment and its relation to the extended electromagnetic
theory has been described in hively2019classical , including a discussion of
error sources. A distinctive feature of longitudinal electric radiation would
be its ability to penetrate thin layers of good conductors much easier than a
transverse radiation. Even in favourable cases, however, a predominant
transverse component would be present, causing interference and noise. If
future developments can lead to a clean selective detection of the
longitudinal component, the possible technological applications would be
manifold.

There is also much to do, of course, concerning the design of efficient
antennas. First, one would need to identify materials with local non-
conservation that can support a sufficiently large current at high frequency.
Then the most appropriate antenna geometry should be studied. For this
purpose, the general equations developed in this paper constitute a firm
starting point.

##  IV Numerical solutions of the extended equations with the potentials

Consider again the equations for the potentials, written in the form

$\displaystyle\frac{1}{c^{2}}\frac{\partial^{2}\phi}{\partial
t^{2}}-\nabla^{2}\phi=4\pi\rho-\frac{1}{c^{2}}\frac{\partial}{\partial t}\int
d^{3}y\frac{I\left(t_{ret},\textbf{y}\right)}{\left|\textbf{x}-\textbf{y}\right|};$
(25) $\displaystyle\frac{1}{c^{2}}\frac{\partial^{2}\textbf{A}}{\partial
t^{2}}-\nabla^{2}\textbf{A}=\frac{4\pi}{c}\textbf{j}+\frac{1}{c}\nabla\int
d^{3}y\frac{I\left(t_{ret},\textbf{y}\right)}{\left|\textbf{x}-\textbf{y}\right|},$
(26)

where $I(t,\mathbf{x})$ is an assigned extra-current. By solving eqs. (25) and
(26) we obtain for $\phi$ and ${\bf A}$ expressions which contain “double-
retarded integrals” of the extra-current $I$. In modanese2019design  we gave
some numerical computations of these integrals for a slowly varying source, as
recalled below. In this section we consider the stationary case and the
resulting expressions are much simpler. For the numerical integration we
employ a 6D Monte Carlo, which can be quite time consuming but is
straightforward and does not require any analytical approximations.

The idea is to take, instead of a source with a slow temporal variation, a
truly stationary source. Then the retarded integrals become simple space
integrals. We recall that the typical time scale of the source in the work
modanese2019design  was $\tau\simeq 10^{-5}$ s, chosen because (1) it
corresponds to the proposed experimental conditions, being the characteristic
discharge time of the circuit; (2) it allows to disregard certain phenomena
occurring only at high frequency, like the intervention of stray capacitance
and temporary charge accumulation.

In modanese2019design  we considered a case of local non-conservation
involving a point-like sink, where current partially disappears, and a point-
like source where the current reappears. More precisely, the sizes of sink and
source were given by a regulator $\varepsilon$ of the order of $10^{-7}$ cm.
The regulator was applied after the first 3D integration (analytical) in
$d^{3}z$. Then the second numerical integration was made (in $d^{3}y$, over
the extended secondary cloud current decreasing like $1/|{\bf y}|^{3}$), using
the command NIntegrate of Mathematica and checking the results by comparison
with a Monte Carlo integration in 3D.

The disadvantage of that procedure is the complication at the formal-algebraic
level. The expressions obtained after the first integration are quite bulky.
At the same time, the assumption of point-like sources is limiting, because
when the failure of local conservation occurs in a quantum wavefunction, the
regions where $\partial_{t}\rho+\nabla\cdot{\bf j}\neq 0$ are not pointlike
but extended, with a shape more similar to that of a couple of disks.

Therefore we model here the extra-current source with a double Gaussian (Fig.
1), and the method actually applies to any geometrical shape:

$I({\bf x})=\frac{I_{0}}{\sqrt{(2\pi)^{3}}\varepsilon d^{2}}\cdot$ (27)
$\left\\{\exp\left[-\frac{1}{2}\left(\frac{x_{1}^{2}}{d^{2}}+\frac{x_{2}^{2}}{d^{2}}+\frac{(x_{3}-a)^{2}}{\varepsilon^{2}}\right)\right]-\exp\left[-\frac{1}{2}\left(\frac{x_{1}^{2}}{d^{2}}+\frac{x_{2}^{2}}{d^{2}}+\frac{(x_{3}+a)^{2}}{\varepsilon^{2}}\right)\right]\right\\}$

The auxiliary anomalous vector potential written as double integral of the
extra-current is

${\bf A}({\bf x})=\frac{k}{4\pi}\int d^{3}y\frac{1}{|{\bf x}-{\bf
y}|}\nabla_{\bf y}\int d^{3}z\frac{I({\bf z})}{|{\bf y}-{\bf z}|}$ (28)

When we compute the magnetic field at the position ${\bf x}=(r,0,0)$, its only
non-zero component is ${\bf B}_{2}$:

${\bf B}_{2}({\bf x})=\frac{\partial A_{1}^{s}}{\partial x_{3}}-\frac{\partial
A_{3}^{s}}{\partial x_{1}}=\frac{k}{4\pi}\int d^{3}y\int
d^{3}z\left(\frac{\partial}{\partial x_{3}}\frac{1}{|{\bf x}-{\bf
y}|}\right)\left(\frac{\partial}{\partial y_{1}}\frac{1}{|{\bf y}-{\bf
z}|}\right)I({\bf z})-(1\leftrightarrow 3)$ (29)

The derivative of the first term is

$\frac{\partial}{\partial
x_{3}}\frac{1}{\sqrt{(x_{1}-y_{1})^{2}+(x_{2}-y_{2})^{2}+(x_{3}-y_{3})^{2}}}=\frac{y_{3}-x_{3}}{\left[(x_{1}-y_{1})^{2}+(x_{2}-y_{2})^{2}+(x_{3}-y_{3})^{2}\right]^{3/2}}$

The other derivatives are similar, and in total we obtain, also considering
that $x_{3}=0$ in this particular configuration:

${\bf B}_{2}({\bf x})=\frac{k}{4\pi}\int d^{3}y\int
d^{3}z\frac{y_{3}z_{1}-x_{1}y_{3}+x_{1}z_{3}-y_{1}z_{3}}{|{\bf x}-{\bf
y}|^{3}|{\bf y}-{\bf z}|^{3}}I({\bf z})$ (30)

This expression is simpler and more direct than those obtained in
modanese2019design  and suitable for a 6D Monte Carlo algorithm which first
generates random values of $\mathbf{z}$ in a quite narrow range, corresponding
to the support of the Gaussian (IV), i.e., inside the primary source. The main
computational difficulty is (as before, actually) that for ${\bf y}$ it is
necessary to sample a much wider region. Typically for each value of ${\bf z}$
we need to generate tens of values of ${\bf y}$ in the sampling.

While we sample the integrand in order to evaluate the total integral, we also
make a map of the function resulting from the partial integration in
$\mathbf{z}$. For this purpose we divide the 3D cube of integration in
$\mathbf{y}$ (with variable side $2R_{y}$) into 100 cells along each
direction. The contribution to the magnetic field from each cell can be
interpreted as being generated by a secondary current, or “cloud” current,
proportional to the gradient of the field $S$. As seen from Tables 1 and 2 and
from Figures 2-5, the regions with the largest secondary current density are
those between the current source and sink and close to them, but the large
cloud that extends around them and whose density decreases slowly with
distance also gives a relevant contribution to the total integral; in the end,
the latter contribution exactly cancels that of the localized sources. In
order to evaluate the far contributions we sample the integrand over cubes of
$\mathbf{y}$ of increasing size, excluding each time a cubic core equal to the
previous cube. This is necessary because the number of sampling points cannot
be increased beyond a practical limit of the order or $10^{10}$, and if we
would sample the cores together with the periphery, the cores would produce
too much noise. All the physical parameters have been chosen of the same
magnitude order as in modanese2019design .

Figure 1: Parameters and location of the Gaussian extra-sources employed in the numerical simulations. (Orange: current sink. Blue: current source.)  range $R_{y}$ (units $10^{-7}$) | contribution to $B^{s}/B^{0}$
---|---
$[0-10]$ | $0.338$
$[10-50]$ | $-0.005$
$[50-100]$ | $-0.001$
$[100-200]$ | $-0.004$
$[200-400]$ | $-0.001$
$[400-800]$ | $-0.036\pm 0.003$
$[800-1100]$ | $-0.09\pm 0.01$
$[1100-1600]$ | $-0.133$
$[1600-3200]$ | $-0.046$
$[3200-6400]$ | $-0.006$
Total | $0.016\pm 0.014$
Table 1: Contributions of the secondary current $\nabla S$, in various regions of the $y$-space, to the ratio $B^{s}/B^{0}$. $B^{s}$ is the anomalous field generated by the secondary current. $B^{0}$ is the Biot-Savart field that the same total current would generate if flowing as a primary local current through the gap of length $2a$. The field is computed at a distance $r=10^{-4}$ cm from the origin, on the $x_{1}$ axis. The shape of the current sink and source is spherical ($\varepsilon=D=0.5\cdot 10^{-7}$ cm). The integration range in $z_{1}$ and $z_{2}$ is $R=2\cdot 10^{-7}$ cm, in $z_{3}$ is $R_{3}=5\cdot 10^{-7}$ cm. In the $y$ range $[800-1100]\cdot 10^{-7}$ cm, which contains the point where $B^{s}$ is computed, the integration is further split into two parts in order to reduce the noise due to the factor $1/|\mathbf{x}-\mathbf{y}|$. As seen from the last row of the table, the total anomalous field is zero within errors (“missing field” effect).  range $R_{y}$ (units $10^{-7}$) | contribution to $B^{s}/B^{0}$
---|---
$[0-10]$ | $0.324$
$[10-50]$ | $0.009$
$[50-100]$ | $-0.000$
$[100-200]$ | $-0.003$
$[200-400]$ | $-0.014$
$[400-800]$ | $-0.042\pm 0.003$
$[800-1100]$ | $-0.088\pm 0.005$
$[1100-1600]$ | $-0.133$
$[1600-3200]$ | $-0.046$
$[3200-6400]$ | $X$
Total | $0.007+X\pm 0.010$
Table 2: Same as in Tab. 1, but for a sink and source with the shape of a
disk/ellipsoid ($\varepsilon=0.5\cdot 10^{-7}$ cm, $D=2.5\cdot 10^{-7}$ cm).
The integration range in $z_{1}$ and $z_{2}$ changes accordingly: here
$R=10\cdot 10^{-7}$ cm. The total anomalous field is again zero within errors;
the single contributions differ significantly from the case of spherical sink
and source only at small $R_{y}$ range. The data for a wider disk ($D=10\cdot
10^{-7}$ cm, not shown here) reveal a similar behavior: in the first two $y$
ranges we have respectively contributions 0.185 and 0.143, with no substantial
differences in the other ranges.

(a)

(b)

Figure 2: (a) Contributions of the “cloud” of secondary current to the
anomalous field $B^{s}$ for spherical sink and source (see Tab. 1), in the $y$
range $[0-10]\cdot 10^{-7}$ cm. In the 6D Monte Carlo integration, the 3D
region of the $y$ variable has been subdivided into $100^{3}$ cells; for each
cell, the contribution of sampling points falling into it is displayed. The
numerical values on the color scale must be normalized to compute the field,
and are therefore not meaningful as absolute value, but their sign and
relative values are of interest. In this figure we have $y_{3}=0$, i.e., we
are observing the cloud on the plane equidistant from source and sink. All
contributions are positive, since we are between the sources (compare Fig. 3).
(b) Here we have $y_{1}=0$, therefore we are observing the cloud in the plane
that cuts sink and source. Note that sink and source both give positive
contributions on the inner side, and negative contributions on the outer side.
From Tab. 1 we deduce that the total contribution of the region shown in this
figure is positive.  Figure 3: Same sources and cutting plane as in Fig. 2,
(b), but in the $y$ range $[10-50]\cdot 10^{-7}$ cm. (The central region
$[0-10]$ is not sampled because it would give a strong noise, compared to the
larger region.) Note that the pattern of Fig. 2, (b) is confirmed concerning
the inner/outer regions with positive/negative contributions respectively.
Figure 4: This figure can be compared with Fig. 2, (a), with the only
difference that sink and source are here Gaussian disks/ellypsoids of diameter
$D=10\cdot 10^{-7}$ cm instead of $D=0.5\cdot 10^{-7}$ cm. ($D$ corresponds to
the $\sigma$ of the Gaussian density distribution.) As in Fig. 2, (a), the
contributions shown lie on the plane between source and sink, with $y_{3}=0$.
Figure 5: This figure can be compared with Fig. 4, but with source-sink having
diameter $D=10\cdot 10^{-7}$ cm.

##  V Conclusion

In summary, we have shown on general grounds that for any charge density
$\rho$ and current density $\mathbf{j}$ given by a microscopic model, the
generated e.m. field is given by eqs. (9) and (10), no matter if the
continuity equation is satisfied or not.

In the case of oscillating charges and currents, the radiated longitudinal
electric field is given by eq. (17) or (22). For example, if a fraction $\eta$
of a charge $q$ oscillating along the $z$-axis over a distance $2a$ lacks a
corresponding current, the predicted maximum longitudinal field (along $z$) is
$E_{L,max}=2\eta\omega^{2}qa/(c^{2}r)$. It is tempting to speculate that this
kind of oscillations could happen in graphene-based nanojunctions or carbon
nano-wires wang2013time ; yu2014current ; walz2015local ; pohl2019imaging ;
garner2020three .

The equation for the magnetic field (10) indicates that it is possible in
principle to observe experimentally if a fraction $\eta$ of a stationary
current flows in a conductor with microscopic spatial “interruptions”
affecting a volume that is a fraction $\chi$ of the total volume: in that
case, the average Biot-Savart magnetic field generated will be
$(1-\eta\chi)B_{0}$, where $B_{0}$ is the field generated by the same total
current flowing in a conductor where $\nabla\cdot\mathbf{j}=0$ everywhere. The
microscopic models cited in this work allow in principle to estimate the
fractions $\eta$ and $\chi$. On the experimental side, a possible strategy to
prove some anomalies (though not to disprove them) consists of looking for
changes in the ratio $B/i$ in dependence on $i$. We have briefly discussed the
possible uncertainties associated with this measurement.

Finally we note that the control of magnetic field patterns in molecular
devices may be interesting for applications to high-density memory storage
consisting of single molecular magnets (see nozaki2017current  and refs.). It
has been further suggested that NMR-type experiments can be performed in order
to observe the spatial fluctuations of magnetic fields generated by DC current
flow walz2014current .

##  Appendix A Equations in SI units

In SI units the equations for the potentials and the auxiliary field $S$ read

$\displaystyle\frac{1}{c^{2}}\frac{\partial^{2}\phi}{\partial
t^{2}}-\nabla^{2}\phi$ $\displaystyle=$
$\displaystyle\frac{\rho}{\varepsilon_{0}}-\frac{\partial S}{\partial t},$
$\displaystyle\frac{1}{c^{2}}\frac{\partial^{2}\mathbf{A}}{\partial
t^{2}}-\nabla^{2}\mathbf{A}$ $\displaystyle=$
$\displaystyle\mu_{0}\mathbf{j}+\nabla S,$
$\displaystyle\frac{1}{c^{2}}\frac{\partial^{2}S}{\partial t^{2}}-\nabla^{2}S$
$\displaystyle=$ $\displaystyle\mu_{0}\left[\frac{\partial\rho}{\partial
t}+\nabla\cdot\mathbf{j}\right]\equiv\mu_{0}I,$

with $\varepsilon_{0}$ and $\mu_{0}$ the permittivity and permeability of free
space, respectively.

The EM fields are expressed as

$\displaystyle\mathbf{E}$ $\displaystyle=$
$\displaystyle-\nabla\phi-\frac{\partial\mathbf{A}}{\partial t},$
$\displaystyle\mathbf{B}$ $\displaystyle=$
$\displaystyle\nabla\times\mathbf{A}.$

For a localized extra-source $I$, the solution for $S$ is

$S\left(\mathbf{x},t\right)=\frac{\mu_{0}}{4\pi}\int\frac{I\left(\mathbf{x}^{\prime},t^{\prime}\right)}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}d^{3}x^{\prime},$

with $t^{\prime}=t-\left|\mathbf{x}-\mathbf{x}^{\prime}\right|/c$

The wave equations for the electric and magnetic fields are:

$\displaystyle\frac{1}{c^{2}}\frac{\partial^{2}\mathbf{E}}{\partial
t^{2}}-\nabla^{2}\mathbf{E}$ $\displaystyle=$
$\displaystyle-\mu_{0}\frac{\partial\mathbf{j}}{\partial
t}-\frac{\nabla\rho}{\varepsilon_{0}}=-\mu_{0}\left(\frac{\partial\mathbf{j}}{\partial
t}+c^{2}\nabla\rho\right),$
$\displaystyle\frac{1}{c^{2}}\frac{\partial^{2}\mathbf{B}}{\partial
t^{2}}-\nabla^{2}\mathbf{B}$ $\displaystyle=$
$\displaystyle\mu_{0}\nabla\times\mathbf{j}.$

For localized sources the corresponding solutions are

$\displaystyle\mathbf{E}\left(\mathbf{x},t\right)$ $\displaystyle=$
$\displaystyle-\frac{\mu_{0}}{4\pi}\int\left(\frac{\partial\mathbf{j}}{\partial
t^{\prime}}+c^{2}\nabla^{\prime}\rho\right)\frac{d^{3}x^{\prime}}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|},$
$\displaystyle\mathbf{B}\left(\mathbf{x},t\right)$ $\displaystyle=$
$\displaystyle\frac{\mu_{0}}{4\pi}\int\nabla^{\prime}\times\mathbf{j}\frac{d^{3}x^{\prime}}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}.$

##  Appendix B Alternative proof of the independence of the EM fields on the
scalar source.

The contribution to the potentials given by the scalar source is

$\displaystyle\phi\left(\mathbf{x},t\right)$ $\displaystyle=$
$\displaystyle-\frac{1}{4\pi c}\int\frac{\partial
S\left(\mathbf{x}^{\prime},t^{\prime}\right)}{\partial
t^{\prime}}\frac{1}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}d^{3}x^{\prime},$
$\displaystyle\mathbf{A}\left(\mathbf{x},t\right)$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\int\nabla^{\prime}S\left(\mathbf{x}^{\prime},t^{\prime}\right)\frac{\mathbf{1}}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}d^{3}x^{\prime},$

where it is assumed that at infinity no scalar field exists (no incoming
scalar field from large distances in the remote past, and local additional
sources acting during finite time intervals).

Considering that

$\displaystyle\nabla\phi\left(\mathbf{x},t\right)$ $\displaystyle=$
$\displaystyle-\frac{1}{4\pi c}\int\frac{\partial
S\left(\mathbf{x}^{\prime},t^{\prime}\right)}{\partial
t^{\prime}}\nabla\left(\frac{1}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}\right)d^{3}x^{\prime}$
$\displaystyle=$ $\displaystyle\frac{1}{4\pi c}\int\frac{\partial
S\left(\mathbf{x}^{\prime},t^{\prime}\right)}{\partial
t^{\prime}}\nabla^{\prime}\left(\frac{1}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}\right)d^{3}x^{\prime}$
$\displaystyle=$ $\displaystyle\frac{1}{4\pi
c}\int\nabla^{\prime}\left(\frac{\partial
S\left(\mathbf{x}^{\prime},t^{\prime}\right)}{\partial
t^{\prime}}\frac{1}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}\right)d^{3}x^{\prime}$
$\displaystyle-\frac{1}{4\pi
c}\int\frac{1}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}\nabla^{\prime}\left(\frac{\partial
S\left(\mathbf{x}^{\prime},t^{\prime}\right)}{\partial
t^{\prime}}\right)d^{3}x^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi c}\oint\frac{\partial
S\left(\mathbf{x}^{\prime},t^{\prime}\right)}{\partial
t^{\prime}}\frac{1}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}d\mathbf{S}^{\prime}$
$\displaystyle-\frac{1}{4\pi
c}\int\frac{1}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}\nabla^{\prime}\frac{\partial
S\left(\mathbf{x}^{\prime},t^{\prime}\right)}{\partial
t^{\prime}}d^{3}x^{\prime},$

that

$\displaystyle\frac{\partial\mathbf{A}\left(\mathbf{x},t\right)}{\partial t}$
$\displaystyle=$ $\displaystyle\frac{1}{4\pi}\int\frac{\partial}{\partial
t}\nabla^{\prime}S\left(\mathbf{x}^{\prime},t^{\prime}\right)\frac{\mathbf{1}}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}d^{3}x^{\prime}$
$\displaystyle=$
$\displaystyle\frac{1}{4\pi}\int\frac{\mathbf{1}}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}\frac{\partial}{\partial
t^{\prime}}\nabla^{\prime}S\left(\mathbf{x}^{\prime},t^{\prime}\right)d^{3}x^{\prime},$

and that

$\displaystyle\nabla\times\mathbf{A}\left(\mathbf{x},t\right)$
$\displaystyle=$
$\displaystyle\frac{1}{4\pi}\int\nabla^{\prime}S\left(\mathbf{x}^{\prime},t^{\prime}\right)\nabla\times\left(\frac{\mathbf{1}}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}\right)d^{3}x^{\prime}$
$\displaystyle=$
$\displaystyle-\frac{1}{4\pi}\int\nabla^{\prime}S\left(\mathbf{x}^{\prime},t^{\prime}\right)\nabla^{\prime}\times\left(\frac{\mathbf{1}}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}\right)d^{3}x^{\prime}$
$\displaystyle=$
$\displaystyle\frac{1}{4\pi}\int\nabla^{\prime}\times\left[\nabla^{\prime}S\left(\mathbf{x}^{\prime},t^{\prime}\right)\frac{\mathbf{1}}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}\right]d^{3}x^{\prime}$
$\displaystyle-\frac{1}{4\pi}\int\frac{\mathbf{1}}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}\nabla^{\prime}\times\left[\nabla^{\prime}S\left(\mathbf{x}^{\prime},t^{\prime}\right)\right]d^{3}x^{\prime},$
$\displaystyle=$
$\displaystyle-\frac{1}{4\pi}\oint\frac{\mathbf{1}}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}\nabla^{\prime}S\left(\mathbf{x}^{\prime},t^{\prime}\right)\times
d\mathbf{S}^{\prime},$

we have for the EM fields

$\displaystyle\mathbf{E}\left(\mathbf{x},t\right)$ $\displaystyle=$
$\displaystyle-\nabla\phi-\frac{1}{c}\frac{\partial\mathbf{A}}{\partial
t}=-\frac{1}{4\pi
c}\oint\frac{1}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}\frac{\partial
S\left(\mathbf{x}^{\prime},t^{\prime}\right)}{\partial
t^{\prime}}d\mathbf{S}^{\prime},$
$\displaystyle\mathbf{B}\left(\mathbf{x},t\right)$ $\displaystyle=$
$\displaystyle\nabla\times\mathbf{A}=-\frac{1}{4\pi}\oint\frac{\mathbf{1}}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}\nabla^{\prime}S\left(\mathbf{x}^{\prime},t^{\prime}\right)\times
d\mathbf{S}^{\prime}.$

Since the scalar field is assumed to originate from a localized source that
was not acting in the infinite past, the surface integrals at infinity are
zero for finite $\left(\mathbf{x},t\right)$. This is so because the
contributions from $\left|\mathbf{x}^{\prime}\right|\rightarrow\infty$ to the
fields at finite values of $\left(\mathbf{x},t\right)$, come from the scalar
at $t^{\prime}\rightarrow-\infty$, which is zero by hypothesis.

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