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 Schrödinger 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 Schrödinger 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 Schrödinger equation and of the non-local Schrödinger 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. *

Implications of Gauge-Free Extended Electrodynamics

“…Spirichev [141] proposed a version of EED (by analogy to elasticity theory), finding: (i) A µ is a physically measurable field, as discussed above; (ii) C and the SLW exist; and (iii) PAPF create a particle-antiparticle plasma, which is compressible (elastic), facilitating SLW propagation. Modanese [69][70][71][72][73][74][75][76][77][78] has explored non-local, quantum wave-functions via violation of local charge conservation in condensed matter. A high-frequency dipole oscillation at 10 GHz leads to double-retarded integrals [69] from EED; the E L magnitude then exceeds the E T magnitude by a factor of 10 2 -10 3 in the far-field.…”

High-Frequency Electromagnetic Emission from Non-Local Wavefunctions

“…In order to vary the shape of the sources, we change the parameter d in the Gaussian charge density Ansatz (17); this parameter fixes the size of the source in the directions z 1 and z 2 , i.e. transversally with respect to the oscillation direction of the dipole.…”

“…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].…”

“…3. For the proximity effect in superconductors, especially in thick SNS junctions in cuprates, where the Gorkov equation cannot be properly approximated by a local Ginzburg-Landau equation [9,17,30,31].…”

“…This choice is based on Ref. [17], where we have found I explicitly from the solutions of fractional wave equations and of wave equations with non-local potential.…”