In quantum theory, for a system with macroscopic wavefunction, the charge density and current density are represented by non-commuting operators. It follows that the anomaly $I={\partial }_t\rho +\nabla ·\mathbf{j}$, 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 $\mathsf{\Delta }N\mathsf{\Delta }\varphi \simeq 1$, which involves the occupation number and the phase of the wavefunction. We give an estimate of the minimum uncertainty in the case of a tunnel Josephson junction made of Nb. Due to this violation of the local conservation of charge, for the evaluation of the e.m. field generated by the system it is necessary to use the extended Aharonov–Bohm electrodynamics. After recalling its field equations, we compute in general form the energy–momentum tensor and the radiation power flux generated by a localized oscillating source. The physical requirements that the total flux be positive, negative or zero yield some conditions on the dipole moment of the anomaly I.

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 $\mathbf{j}=(-ie\hslash /2m)({\Psi }^{*}\nabla \Psi -\Psi \nabla {\Psi }^{*})$ is not locally conserved but can be re-adjusted by a non-local term, which current should be regarded as real? Situations of this kind have been studied for example, for currents in saturated chains of alkanes, silanes and germanes, and in linear carbon wires. We prove that in any case the extended Maxwell equations by Aharonov-Bohm give the e.m. field generated by such currents without any ambiguity. In fact, the wave equations have the same source terms as in Maxwell theory, but the local non-conservation of charge leads to longitudinal radiative contributions of $\mathbf{E}$, as well as to additional transverse radiative terms in both $\mathbf{E}$ and $\mathbf{B}$. For an oscillating dipole we show that the radiated electrical field has a longitudinal component proportional to $\omega \widehat{P}$, where $\widehat{P}$ is the anomalous moment $\int \widehat{I}\left(\mathbf{x}\right)\mathbf{x}{d}^{3}x$ and $\widehat{I}$ is the space-dependent part of the anomaly $I={\partial }_t\rho +\nabla ·\mathbf{j}$. For example, if a fraction $\eta$ of a charge q oscillating over a distance $2a$ lacks a corresponding current, the predicted maximum longitudinal field (along the oscillation axis) is $E_{L,max}=2\eta {\omega }^{2}qa/\left(c^{2}r\right)$. 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; in this case one should observe at a fixed position changes in the ratio $B/i$ in dependence on i, in contrast with the standard Maxwell equations. The missing field effect is confirmed by numerical solutions of the extended equations, which also show the spatial distribution of the non-local term in the current.

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 $$A^\mu $$A μ become univocally defined and can be measured with probes in which the “extra-current” $$I=\partial _\mu j^\mu $$I = ∂ μ j μ is not zero at some points. As a consequence, it is possible in principle to detect pure gauge-waves with $${\textbf{E}}={\textbf{B}}=0$$E = 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.

In the extension of Maxwell equations based on the Aharonov–Bohm Lagrangian, the e.m. field has an additional degree of freedom, namely, a scalar field generated by charge and currents that are not locally conserved. We analyze the propagation of this scalar field through two different media (a pure dielectric and an ohmic conductor) and study its property over a frequency range where the properties of the media are frequency-independent. We find that an electromagnetic (e.m.) scalar wave cannot propagate in a material medium. If a scalar wave in vacuum impinges on a material medium it is reflected, at most exciting in the medium a pure “potential” wave (which we also call a “gauge” wave) propagating at c, the speed of light in vacuum, with a vector potential whose Fourier amplitude is related to that of the scalar potential by $\omega {\mathbf{A}}_0=\mathbf{k}{\varphi }_0$, where ${\omega }^2=c^2{\left|\mathbf{k}\right|}^2$.

We reconsider some well-known tunneling processes from the point of view of Aharonov-Bohm electrodynamics, a unique extension of Maxwell’s theory which admits charge-current sources that are not locally conserved. In particular we are interested into tunneling phenomena having relatively long range (otherwise the non-Maxwellian effects become irrelevant, especially at high frequency) and involving macroscopic wavefunctions and coherent matter, for which it makes sense to evaluate the classical e.m. field generated by the tunneling particles. For some condensed-matter systems, admitting discontinuities in the probability current is a possible way of formulating phenomenological models. In such cases, the Aharonov-Bohm theory offers a logically consistent approach and allows to derive observable consequences. Typical e.m. signatures of the failure of local conservation are at high frequency the generation of a longitudinal electric radiation field, and at low frequency a small effect of “missing” magnetic field. Possible causes of this failure are instant tunneling and phase slips in superconductors. For macroscopic quantum systems in which the phase-number uncertainty relation $\mathsf{\Delta }N\mathsf{\Delta }\phi \sim 1$ applies, the expectation value of the anomalous source $I={\partial }_t\rho +\nabla ·\mathbf{j}$ has quantum fluctuations, thus becoming a random source of weak non-Maxwellian fields.