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

TL;DR

This paper investigates how certain non-local modifications to quantum mechanics, such as fractional and non-local Schrödinger equations, lead to local non-conservation of probability current and explores their physical implications.

Contribution

It explicitly computes the non-conservation terms in fractional and non-local Schrödinger equations and discusses their significance in electromagnetic interactions and superconductivity.

Findings

Non-local Schrödinger equations break local probability current conservation.

Explicit expressions for non-conservation terms in wave packets are derived.

Implications for electromagnetic coupling and superconductivity are discussed.

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.