Geometric quantization of the \(1:1\) oscillator
Consider the cotangent bundle of \(\mathbb{R}^2\) with the standard symplectic form and define two smooth functions on \(M\): the Hamiltonian of two dimensional harmonic oscillator and the angular momentum. These functions induce Hamiltonian vector fields on \(M\) which define a generalized distribution. The corresponding torus action is studied: orbit types decomposition and action angle variables are defined. The Bohr-Sommerfeld quantization is carried out explicitly by associating operators to action functions. The Hilbert space of quantization \(\mathfrak{H}\) is spanned by vectors \(\mathbf{e}_{m,n}\) associated to each Bohr-Sommerfeld torus. Using ladder operators it is shown that there is a reducible representation of a finite dimensional Lie algebra on \(\mathfrak{H}\). Then, reductions of \(S^1\) oscillator symmetry give rise to new symplectic manifolds \({S}^2_e\), which are symplectomorphic to coadjoint orbits of \(SU(2)\). The corresponding quantum spaces decompose in two finite dimensional irreducible representations of \(su(2)\). Finally, it is shown that \({S}^2_e\) has a \(\mathbb{Z}_2\)-symmetry whose reduction is a one degree of freedom system, which can be reduced to a space whose quantization is a single finite dimensional irreducible representation of \(su(2)\).
scientific article; zbMATH DE number 7259348