Does Accidental Degeneracy Imply a Symmetry Group? : Particles and Fields

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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)(p^2+r^2)+λσ・L. For the fixed values of λ=1〓1/2 states. In order to systematically seek the symmetry 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 that 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.
理論物理学刊行会の論文
1984-08-25