## 1 Introduction  
The extended Maxwell equations by [PERSON] and [PERSON] [1; 2; 3; 4; 5; 6; 7; 8; 9; 10] are employed for the calculation of electromagnetic fields generated by sources which violate the local charge conservation condition \(\partial_{t}\rho+\
abla\cdot{\bf J}=0\). Barring exceptional situations in cosmology where such violations may occur at the macroscopic level, a possible microscopic failure of local conservation has been predicted in quantum mechanics in the following situations:  
1. In systems described by fractional quantum mechanics [11; 12; 13; 14; 15; 16; 17]. 2. In ordinary quantum mechanics, in the presence of non-local potentials [18; 19; 20; 21; 22; 23; 24; 25; 26], and in particular in first-principles calculations of transport properties using density functional theory and non-equilibrium Green functions [27; 28; 29]. The latter approach has been very successful for the exact description of quantum transport in nano-devices, which is otherwise not viable in terms of local quantum field theories. 3. For the proximity effect in superconductors, especially in thick superconductor-normal-superconductor (SNS) junctions in cuprates, where the Gorkov equation cannot be properly approximated by a local Ginzburg-Landau equation [9; 17; 30; 31]. Concerning Point 2, we recall that the Landauer-Buttiker formula for the current in quantum transport, when applied to wavefunctions in the presence of a non-local potential [27; 28], inevitably leads to the definition of a non-local charge density \(\rho^{ext}\) and current density \({\bf J}^{ext}\) which differ from the usual gauge-invariant expression, and coincide with those of the extended Aharonov-Bohm electrodynamics, namely  
\[\rho^{ext}=\rho+\rho^{non-loc}=\rho-\frac{1}{4\pi c^{2}}\frac{\partial}{\partial t }\int d^{3}y\frac{I(t_{ret},\mathbf{y})}{|\mathbf{x}-\mathbf{y}|} \tag{1}\]  
\[\mathbf{J}^{ext}=\mathbf{J}+\mathbf{J}^{non-loc}=\mathbf{J}+\frac{1}{4\pi c} \
abla\int d^{3}y\frac{I(t_{ret},\mathbf{y})}{|\mathbf{x}-\mathbf{y}|} \tag{2}\]  
where \(t_{ret}=t-c^{-1}|\mathbf{x}-\mathbf{y}|\) and the \"extra-source\" \(I(t,\mathbf{x})\) is the function which quantifies the violation of local current conservation:  
\[I(t,\mathbf{x})=\frac{\partial\rho}{\partial t}+\
abla\cdot\mathbf{J} \tag{3}\]  
\[\rho=|\Psi|^{2};\qquad\mathbf{J}=\frac{-i\hbar}{2m}\left(\Psi^{*}\
abla\Psi- \Psi\
abla\Psi^{*}\right) \tag{4}\]  
The current \(\mathbf{J}\), which can be interpreted as \(\sim\)\(\rho\mathbf{v}\) in a classical limit, is locally non-conserved and has in this case \"sources and sinks\" which are, however, invisible to an electromagnetic probe (this is the so-called \"censorship property\" of Aharonov-Bohm electrodynamics and constitutes a safeguard of the locality of the electromagnetic field). In terms of the extended charge density \(\rho^{ext}\) and extended current \(\mathbf{J}^{ext}\) the Aharonov-Bohm-Maxwell equations in CGS units are then written in the familiar form  
\[\
abla\cdot\mathbf{E}=4\pi\rho^{ext} \tag{5}\]  
\[\
abla\times\mathbf{E}=-\frac{1}{c}\frac{\partial\mathbf{B}}{\partial t} \tag{6}\]  
\[\
abla\cdot\mathbf{B}=0 \tag{7}\]  
\[\
abla\times\mathbf{B}-\frac{1}{c}\frac{\partial\mathbf{E}}{\partial t}=\frac {4\pi}{c}\mathbf{J}^{ext} \tag{8}\]  
The Landauer-Buttiker formula employed in [27] gives the current \(I_{\alpha}\) flowing through a lead \(\alpha\) coupled to another lead \(\beta\) as  
\[I_{\alpha}=\frac{2e}{\hbar}\int dE\left(f_{\alpha}-f_{\beta}\right)\text{Tr} \left[G^{r}\Gamma_{\alpha}G^{\alpha}\Gamma_{\beta}\right] \tag{9}\]  
where \(\Gamma_{\alpha}\), \(\Gamma_{\beta}\) are the linewidths of the leads, \(f_{\alpha}\), \(f_{\beta}\) their Fermi distributions, \(G^{r}\) is the retarded Green function of the scattering region and \(G^{a}\) the corresponding advanced Green function. The authors of [27] prove that the current calculated from the surface integral of \(\mathbf{J}^{ext}\) over the interface between the scattering region and the lead \(\alpha\) is equal to the that obtained from Equation (9).