Quantum Mechanics on a Ring: Continuity versus Gauge Invariance

TL;DR

This paper explores how boundary conditions on a quantum particle on a ring affect gauge invariance and the nature of eigenfunctions, revealing that nonlinear boundary conditions can preserve gauge invariance and allow discontinuous eigenfunctions.

Contribution

It introduces nonlinear boundary conditions that maintain gauge invariance and support discontinuous eigenfunctions, challenging traditional linear boundary assumptions.

Findings

Linear boundary conditions break gauge invariance.

Nonlinear boundary conditions restore gauge invariance.

Discontinuous eigenfunctions are physically plausible.

Abstract

Remarkably we find that for a ring with linear boundary conditions such that the eigenvector and its derivative are continuous, there does not seem to be a way for the well-known de Broglie relation to be gauge invariant. Certain nonlinear boundary conditions assure gauge invariance, and lead to eigenfunctions with a discontinuous but differentiable phase and a continuous spectrum. A discrete subset of this spectrum forms a Hilbert space, while another subset is excluded by the nonlinear boundaries. We conclude that discontinuous momentum eigenfunctions are tenable, and that it is possible that quantum mechanics can have nonlinear boundary conditions in some circumstances.