Time in quantum mechanics and the local non-conservation of the probability current

Abstract

In relativistic quantum field theory with local interactions, charge is locally conserved. This implies local conservation of probability for the Dirac and Klein-Gordon wavefunctions, as special cases; and then in turn for non-relativistic quantum field theory and for the Schroedinger and Ginzburg-Landau equations, regarded as low energy limits. Quantum mechanics, however, is wider than quantum field theory, as an effective model of reality. For instance, fractional quantum mechanics and Schroedinger equations with non-local terms have been successfully employed in several applications. The non-locality of these formalisms is strictly related to the problem of time in quantum mechanics. We compute explicitly for continuum wave packets the terms of the fractional Schroedinger equation and of the non-local Schroedinger equation by Lenzi et al. which break the local current conservation, and discuss their physical significance. The results are especially relevant for the electromagnetic coupling of these wavefunctions. A connection with the non-local Gorkov equation for superconductors and their proximity effect is also outlined.

19 pages, 5 figures

References in corpus (5)

• Unveiling the link between fractional Schrödinger equation and light propagation in honeycomb lattice
• Comment on "Fractional quantum mechanics" and "Fractional Schroedinger equation"
• Cosmological magnetic fields from inflation in extended electromagnetism
• Generalized Maxwell equations and charge conservation censorship
• Electromagnetic coupling of strongly non-local quantum mechanics

Cited by in corpus (11)

• Gauge waves generation and detection in Aharonov-Bohm electrodynamics
• Design of a test for the electromagnetic coupling of non-local wavefunctions
• Aharonov-Bohm electrodynamics in material media: a scalar e.m. field cannot cause dissipation in a medium
• Stochastic Fractal and Noether's Theorem
• Are current discontinuities in molecular devices experimentally observable?
• High-frequency electromagnetic emission from non-local wavefunctions
• Generalized local charge conservation in many-body quantum mechanics
• Do we need an alternative to local gauge coupling to electromagnetic fields?
• Quantum State Driving along Arbitrary Trajectories
• Quantum uncertainty and energy flux in extended electrodynamics
• Quantum collapse, local conservation of charge, and possible experimental consequences