Does Accidental Degeneracy Imply a Symmetry Group?

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Abstract
The question whether an accidental degeneracy in quantum mechanical system always implies the internal symmetry group of the system is probed by means of the simple model of the three-dimensional harmonic oscillator with a constant spin-orbit potential: H=(1/2)(p2+r2)+λσ·L. For the fixed values of λ=∓1, this model has an interesting degeneracy of remarkable regularity; in particular, the ground state is infinitely degenerate, consisting of all the nodeless j=1±1/2 states. In order to systematically seek the symmetry group, we first investigate the full dynamical group of the system. To our surprise, we find that the dynamical group of our system for an arbitrary value of λ is the graded Sp(6R) — a supergroup extension of the dynamical group Sp(6R) of the three-dimensional harmonic oscillator. We find the graded SU(3) as its subgroup, which is a supergroup extension of the well-known symmetry group SU(3) of the harmonic oscillator. It is further shown that there exists a natural group chain, grSp(6R)⊃Sp(2R)×grO(3), which corresponds to the group chain Sp(6R)⊃Sp(2R)×O(3), where × denotes the direct-product. Next, we examine whether any subgroup of the full dynamical group constitutes the symmetry group of the system responsible for the accidental degeneracy. It is shown that there is no such a symmetry group in the system and that, instead of the symmetry group, there exists a special, simple algebraic structure which is essentially responsible for the accidental degeneracy.

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