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In quantum theory, for a system with macroscopic wavefunction, the chargedensity and current density are represented by non-commuting operators. Itfollows that the anomaly $I=\partial_t \rho + \nabla \cdot \mathbf{j}$, beingessentially a linear combination of these two operators in thefrequency-momentum domain, does not admit eigenstates and has a minimumuncertainty fixed by the Heisenberg relation $\Delta N \Delta \phi \simeq 1$which involves the occupation number and the phase of the wavefunction. We givean estimate of the minimum uncertainty in the case of a tunnel Josephsonjunction 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 touse the extended Aharonov-Bohm electrodynamics. After recalling its fieldequations, we compute in general form the energy-momentum tensor and theradiation power flux generated by a localized oscillating source. The physicalrequirements that the total flux be positive, negative or zero yield someconditions on the dipole moment of the anomaly $I$.

Quantum uncertainty and energy flux in extended electrodynamics