Quantization in terms of symplectic groups: The harmonic oscillator as a generic example

Abstract

The conventional quantization of the harmonic oscillator in terms of operators Q and P can be implemented with the help of irreducible unitary representations of the Heisenberg-Weyl group which acts transitively and effectively on the simply connected classical phase space Sq,p ≊ ℝ2. In the description of the harmonic oscillator in terms of angle and action variables ϕ and I the associated phase space Sϕ,I corresponds to the multiply connected punctured plane ℝ2 – {0}, on which the 3-dimensional symplectic group Sp(2, ℝ) acts transitively, leaving the origin invariant. As this group contains the compact subgroup U(1) it has infinitely many covering groups. In the here relevant irreducible unitary representations (positive discrete series) the self-adjoint generator Ko of U(l) represents the classical action variable I. It has the possible spectra n + k, n = 0,1,…; k > 0, where k depends on the covering group. This implies different possible spectra for the action variable Hamiltonian ℏωK0 of the harmonic oscillator. On the other hand, expressing the operators Q and P (non-linearly) in terms of the three generators K0 etc. of Sp(2, ℝ) leads to the usual framework. Possible physical (experimental) implications and generalizations to higher dimensions are discussed briefly.