Breaking so(4) symmetry without degeneracy lift

TL;DR

This paper demonstrates that in quantum motion on S^3 with a trigonometric Scarf potential, the so(4) symmetry can be broken by external scales without lifting degeneracy, revealing complex symmetry behavior.

Contribution

It introduces a method to break so(4) symmetry with external parameters while preserving degeneracy, and extends the analysis to higher-dimensional spheres.

Findings

Degeneracy patterns remain intact despite symmetry breaking.

Wave functions do not form irreducible so(4) representations under perturbation.

Explicit expansion of wave functions shows contributions from various angular momenta.

Abstract

We argue that in the quantum motion of a scalar particle of mass "m" on S^3_R perturbed by the trigonometric Scarf potential (Scarf I) with one internal quantized dimensionless parameter, \ell, the 3D orbital angular momentum, and another, an external scale introducing continuous parameter, B, a loss of the geometric hyper-spherical so(4) symmetry of the free motion can occur that leaves intact the unperturbed {\mathcal N}^2-fold degeneracy patterns, with {\mathcal N}=(\ell +n+1) and n denoting the nodes number of the wave function. Our point is that although the number of degenerate states for any {\mathcal N} matches dimensionality of an irreducible so(4) representation space, the corresponding set of wave functions do not transform irreducibly under any so(4). Indeed, in expanding the Scarf I wave functions in the basis of properly identified so(4) representation functions, we find power series in the perturbation parameter, B, where 4D angular momenta K\in [\ell , {\mathcal N}-1] contribute up to the order \left(\frac{2mR^2B}{\hbar^2}\right)^{{\mathcal N}-1-K}. In this fashion, we work out an explicit example on a symmetry breakdown by external scales that retains the degeneracy. The scheme extends to so(d+2) for any d.