Runge-Lenz Vector, Accidental SU(2) Symmetry, and Unusual Multiplets for Motion on a Cone

TL;DR

This paper explores the classical and quantum dynamics of a particle on a conical surface with specific potentials, revealing hidden symmetries, accidental degeneracies, and unusual multiplet structures, especially when the cone's deficit angle is rational.

Contribution

It uncovers an accidental SU(2) symmetry and fractional spin multiplets in the quantum system of a particle on a cone, extending understanding of symmetries in curved spaces.

Findings

All bound classical orbits are closed for rational deficit angles.

Quantum system exhibits accidental degeneracies in energy spectrum.

Presence of fractional spin multiplets and unusual degeneracies.

Abstract

We consider a particle moving on a cone and bound to its tip by $1/r$ or harmonic oscillator potentials. When the deficit angle of the cone divided by $2 \pi$ is a rational number, all bound classical orbits are closed. Correspondingly, the quantum system has accidental degeneracies in the discrete energy spectrum. An accidental SU(2) symmetry is generated by the rotations around the tip of the cone as well as by a Runge-Lenz vector. Remarkably, some of the corresponding multiplets have fractional ``spin'' and unusual degeneracies.