Geometric quantization of the \(1:1\) oscillator
type: Item
instance of: scholarly article
title: Geometric quantization of the \(1:1\) oscillator
zbMATH Open document ID: 1452.81120
author: Richard Cushman
author: J. Śniatycki
published in: Qualitative Theory of Dynamical Systems
publication date: 2020-10-12T00:00:00Z
review text: 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 \(su(2)\).
reviewed by: Andrea Galasso
Mathematics Subject Classification ID: 81Q70
Mathematics Subject Classification ID: 70H05
Mathematics Subject Classification ID: 81S08
Mathematics Subject Classification ID: 17B08
Mathematics Subject Classification ID: 81S10
zbMATH DE Number: 7259348
zbMATH Keywords: 2d-harmonic oscillator
zbMATH Keywords: Bohr-Sommerfeld-Heisenberg quantization
zbMATH Keywords: \(\mathrm{SU}(2)\) representation
MaRDI profile type: Publication
full work available at URL: s12346-020-00417-x
OpenAlex ID: W3082012093
cites work: Global Aspects of Classical Integrable Systems
cites work: On Bohr-Sommerfeld-Heisenberg quantization
cites work: Geometric quantization and quantum mechanics
cites work: Differential geometry of singular spaces and reduction of symmetry
DOI: 10.1007/S12346-020-00417-X
description: scientific article; zbMATH DE number 7259348