Geometric Quantization of Algebraic Reduction

An analogue of geometric quantization of Poisson algebras obtained by algebraic reduction of symmetries is developed. Interpretation of the obtained results and their application to the problem of commutativity of quantization and reduction are givenComment: 22 page

Shifting operators in geometric quantization

In a series of papers on Bohr-Sommerfeld-Heisenberg quantization of completely integrable systems we interpreted shifting operators as quantization of functions ${\mathrm{e}}^{ \pm i{\theta}_j}$ , where $(I_j , {\theta}_j )$ are action angle coordinates. The aim of this paper is to show how these operators occur in geometric quantization.Comment: This is a revised version. It focuses exclusively on shifting operators in Bohr-Sommerfeld theory of quantization of completely integrable Hamiltonian system

Orbits of families of vector fields on subcartesian spaces

Orbits of families of vector fields on a subcartesian space are shown to be smooth manifolds. This allows for a global description of a smooth geometric structure on a family of manifolds in terms of a single object defined on the corresponding family of vector fields. Stratified spaces, Poisson spaces and almost complex spaces are discussed.Comment: 34 page

Singular reduction for nonlinear control systems

We discuss smooth nonlinear control systems with symmetry. For a free and proper action of the symmetry group, the reduction of symmetry gives rise to a reduced smooth nonlinear control system. If the action of the symmetry group is only proper, the reduced nonlinear control system need not be smooth. Using the smooth calculus on nonsmooth spaces, provided by the theory of differential spaces of Sikorski, we prove a generalization of Sussmann's theorem on orbits of families of smooth vector fields.Comment: 13 page

Bohr-Sommerfeld-Heisenberg Theory in Geometric Quantization

In the framework of geometric quantization we extend the Bohr-Sommerfeld rules to a full quantization theory which resembles Heisenberg's matrix theory. This extension is possible because Bohr-Sommerfeld rules not only provide an orthogonal basis in the space of quantum states, but also give a lattice structure to this basis. This permits the definition of appropriate shifting operators. As examples, we discuss the 1-dimensional harmonic oscillator and the coadjoint orbits of the rotation group

Classical and quantum spherical pendulum

This paper extends the Bohr-Sommerfeld quantization of the spherical pendulum to a full quantum theory. This the first application of geometric quantization to a classical system with monodromy

On Bohr-Sommerfeld-Heisenberg Quantization

This paper presents the theory of Bohr-Sommerfeld-Heisenberg quantization of a completely integrable Hamiltonian system in the context of geometric quantization. The theory is illustrated with several examples

Bohr-Sommerfeld-Heisenberg Quantization of the 2-dimensional Harmonic Oscillator

We study geometric quantization of the harmonic oscillator in terms of a singular real polarization given by fibres of the energy momentum map

Bohr-Sommerfeld-Heisenberg Quantization of the Mathematical Pendulum

In this paper we give the Bohr-Sommerfeld-Heisenberg quantization of the mathematical pendulum

Differential spaces in integrable Hamiltonian systems

This paper uses differential spaces to obtain some new results in integrable Hamiltonian system