Discrete Accidental Symmetry for a Particle in a Constant Magnetic Field on a Torus

TL;DR

This paper explores the hidden accidental symmetry of a quantum particle in a magnetic field on a torus, revealing how quantum effects and boundary conditions break classical symmetries and lead to finite degeneracies.

Contribution

It identifies the accidental symmetry algebra of a charged particle in a magnetic field and analyzes how quantum boundary conditions break continuous symmetries to discrete ones.

Findings

Identification of the accidental symmetry algebra.

Quantum boundary conditions break continuous translation symmetry.

Finite degeneracy due to discrete magnetic translation group.

Abstract

A classical particle in a constant magnetic field undergoes cyclotron motion on a circular orbit. At the quantum level, the fact that all classical orbits are closed gives rise to degeneracies in the spectrum. It is well-known that the spectrum of a charged particle in a constant magnetic field consists of infinitely degenerate Landau levels. Just as for the $1/r$ and $r^2$ potentials, one thus expects some hidden accidental symmetry, in this case with infinite-dimensional representations. Indeed, the position of the center of the cyclotron circle plays the role of a Runge-Lenz vector. After identifying the corresponding accidental symmetry algebra, we re-analyze the system in a finite periodic volume. Interestingly, similar to the quantum mechanical breaking of CP invariance due to the $\theta$-vacuum angle in non-Abelian gauge theories, quantum effects due to two self-adjoint extension parameters $\theta_x$ and $\theta_y$ explicitly break the continuous translation invariance of the classical theory. This reduces the symmetry to a discrete magnetic translation group and leads to finite degeneracy. Similar to a particle moving on a cone, a particle in a constant magnetic field shows a very peculiar realization of accidental symmetry in quantum mechanics.