Breaking so(4) symmetry without degeneracy lift

• Published in: arXiv.org 2014-02
• Main Authors: Pallares-Rivera, A, F de J Rosales-Aldape, Kirchbach, M
• Format: Article
• Language: English
• Subjects: Angular momentum; Parameters; Perturbation; Power series; Representations; Symmetry; Wave functions
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1305.1281

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.