Multi-valued vortex solutions to the Schr\"odinger equation and radiation

TL;DR

This paper explores multi-valued wave functions in quantum mechanics, proposing they could radiate electromagnetic energy and suggesting radiation might enforce single-valuedness, challenging standard quantum theory.

Contribution

It introduces a class of multi-valued wave functions based on knotted vortex filaments and links their potential radiation to the relaxation to single-valued states.

Findings

Multi-valued wave functions can be constructed from knotted vortex filaments.

Charged multi-valued states may radiate electromagnetic energy.

Single-valued wave functions are likely radiationless.

Abstract

This paper addresses the single-valued requirement for quantum wave functions when they are analytically continued in the spatial coordinates. This is particularly relevant for de Broglie-Bohm, hydrodynamic, or stochastic models of quantum mechanics where the physical basis for single-valuedness has been questioned. It first constructs a large class of multi-valued wave functions based on knotted vortex filaments familiar in fluid mechanics, and then it argues that for free particles, these systems will likely radiate electromagnetic radiation if they are charged or have multipolar moments, and only if they are single-valued will they definitely be radiationless. Thus, it is proposed that electromagnetic radiation is possibly the mechanism that causes quantum wave functions to relax to states of single-valuedness, and that multi-valued states might possibly exist in nature for transient periods of time. If true, this would be a modification to the standard quantum mechanical formalism. A prediction is made that electrons in vector Aharonov-Bohm experiments should radiate energy at a rate dependent on the solenoid's magnetic flux.