Search for a Universal Symmetry Group in Two Dimensions

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Abstract
Recently several authors have proposed a universal symmetry group and demonstrated the classical validity of the concept. Supposedly, an SU(n) symmetry group could be constructed for whatsoever system of n degrees of freedom. The claim is assuredly valid for all classically degenerate systems, but is in contradiction with most of the well‐known and widely accepted solutions of Schrödinger's equation. We examine the reasons for this discrepancy on the quantum‐mechanical level. The construction of the universal symmetry group requires ladder operators, which in most cases are the ladder operators of Infeld and Hill. Complications which owe their origin entirely to numerical relationships imposed by quantization prevent these operators from forming a von Neumann algebra and, in turn, an SU(n) group of constants of the motion. Two important effects are those imposed by anisotropy, wherein not all quanta have the same size, and by non‐Cartesian coordinates, wherein quanta in some dimensions are restricted in size by those in other dimensions. These effects can be seen quite clearly in two‐dimensional systems: anisotropy in the Cartesian anisotropic harmonic oscillator, and conflict between dimensions in the polar form of the isotropic oscillator. Further complications arise when the two effects are combined, as in the harmonic oscillator or hydrogen atom with ``excess'' angular momentum. Enough residue of the universal symmetry concept remains that many similarities between the hydrogen atom and harmonic oscillator may be understood, including the fact that some levels of the hydrogen atom, which ordinarily transform according to an orthogonal group, may form irreducible representations of the unitary group.

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