Abstract
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=∂tρ+∇·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 ΔNΔϕ≃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.
Highlights
The extended electrodynamics theory based on the Aharonov–Bohm Lagrangian has attracted much interest over the last years [1,2,3,4,5,6,7,8,9,10,11]
Unlike the standard Maxwell theory, the extended electrodynamics allows to compute the fields generated by physical systems in which the condition of local conservation of charge is not exactly satisfied
It is worth mentioning that the addition of Λ to Maxwell Lagrangian is used in QED only as a technique to facilitate the renormalization process, at the classical level it results in an extension of Maxwell electrodynamics that allows to include possible violations of the local conservation of charge
Summary
The extended electrodynamics theory based on the Aharonov–Bohm Lagrangian has attracted much interest over the last years [1,2,3,4,5,6,7,8,9,10,11]. Unlike the standard Maxwell theory, the extended electrodynamics allows to compute the fields generated by physical systems in which the condition of local conservation of charge is not exactly satisfied. Such violations of local conservation are quite rare and may occur especially at a microscopic level; the currents involved are usually small, but the associated physical effects are interesting and might lead to useful applications. Cause of local violation, namely the fact that in a macroscopic quantum system charge density and current density are physical quantities represented by non-commuting operators This property has been formally proven in the theory of “quantum circuits” [32,33], but here we give an independent proof in the specific case of a Josephson junction.
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