Multi-valued vortex solutions to the Schrödinger equation and radiation

Highlights

• A new explanation for why quantum mechanical wave functions are single-valued.

• Transient multi-valued states might exist that emit radiation.

• Aharonov–Bohm systems should emit radiation dependent on magnetic flux.

• A new and large class of knotted vortex solutions are presented for study.

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.

Introduction

The question whether the Schrödinger equation must always be single-valued occupied a number of the founders of early quantum theory, starting with Schrödinger himself [1], then notably Pauli [2], and subsequently a number of others [3], [4], [5]. The question received renewed interest after the Aharonov–Bohm effect was discovered [6]. The general consensus reached was that even in the Aharonov–Bohm systems the wave function must still be single-valued [4], [5], [7], but there were some dissenters [8]. The subject received a new burst of interest from the insights of Wallstrom regarding the non-obvious motivation for the single-valued constraint in de Broglie–Bohm, hydrodynamic, and stochastic formulations of quantum mechanics [9], [10], [11], [12], [13]. Goldstein also independently commented on multi-valued wave functions in stochastic mechanics [14]. Derakhshani has recently suggested an interesting explanation for the single-valued constraint based on zitterbewegung [15], [16]. Smolin has also considered this issue [17]. Here I take a different approach to this question which is based on electromagnetic radiation.

The prototypical example of a multi-valued wave function is one of the form $\psi \left(x\right)=f\left(x\right)e^{i{l}_z\phi }$, where $f\left(x\right)$ is a single-valued function and where $\phi$ is the azimuthal angle, and $l_z$ is a real constant. The wave function is single-valued only if $l_z=\pm n,n\in \mathbb{Z}$. The single-valued constraint leads to quantization effects. In this simple case, when $l_z$ is not an integer, we can consider the wavefunction to have multiple Riemann sheets that differ from one another by constant phase factors of the form $e^{i{l}_{z}2\pi m}$ for $m$ integer. The fact that the different sheets differ by a constant phase factor which is independent of x, is important, and consequently I shall consider only multi-valued wave functions which have this property here.

Firstly I introduce a broad class of suitable multi-valued wave functions. I do this by borrowing results from the theory of vortex filaments in fluid mechanics. The fluid version of the Biot–Savart law allows one to construct a local potential function which is generated by a vortex filament taking an arbitrary stringlike shape, either closed or open . This includes arbitrary knots and links, and this function is generally multi-valued when analytically continued in $\mathbf{x}$. From these known solutions, I construct multi-valued solutions of the Schrödinger equation. For the usual azimuthal example, the string would be the whole z axis, and be infinitely long. The analysis presented applies to such cases as well as to knots and links. The subject of quantum vortices applied to the Schrödinger equation is not new [18], [19], [20], [21]. Here we generalize these treatments to multi-valued wave functions.

Then I show, by using a nonlinear identity of the Schrödinger equation, that the multi-valued linear Schrödinger equation can be replaced by an equivalent single-valued but nonlinear equation. I then argue from this that if the particle is charged, then the nonlinear term will likely produce spontaneous bremsstrahlung even for a free particle, so that the multi-valued solution may not be stable to radiative decay. Consequently, I argue that the equilibrium state of the system that is approached must be single-valued, and since this could account for the observed fact of single-valuedness, at least for charged particles, that perhaps the single-valued condition of quantum mechanics might not be universally true, and in particular it might be violated for short time periods where, after a collapse of the wave function for example, a transient multi-valued wave function is produced which then subsequently decays into a single-valued one. Although this radiative relaxation to single-valuedness is easiest to understand for charged particles, it might be expected to hold for neutral particles too if they have some multipolar electromagnetic moments, since these too would radiate when accelerated. Neutrons or neutral atoms are examples. So we might expect that multi-valued wave functions if ever created would typically be short-lived for them as well. The exceptions might be neutrinos, or dark matter particles (if they exist), where the lack of any electromagnetic interaction would allow multi-valued wave functions to persist for longer periods of time. This transient multi-valuedness might yield experimentally detectable effects. We discuss one such test below in the vector Aharonov–Bohm discussion. Also, these results might also prove to be useful in the mathematical theory of knots.

When restricted by the single-valued constraint, the vortex solutions here are similar to the quantum vortex solutions in superconductors, and superfluids. Here they are considered as solutions to the single particle Schrödinger equation with or without a potential. I am not aware of the general knot solutions found here having been considered previously in the physics literature, although a number of special cases were elegantly analyzed in [19].