Bohr-Sommerfeld-Heisenberg Theory in Geometric Quantization

TL;DR

This paper extends Bohr-Sommerfeld rules within geometric quantization to develop a full quantization theory similar to Heisenberg's matrix approach, introducing shifting operators and lattice structures.

Contribution

It introduces a novel extension of Bohr-Sommerfeld rules to a comprehensive quantization framework resembling Heisenberg's matrix theory, including the construction of shifting operators.

Findings

Provides a geometric quantization framework with lattice structures.

Defines shifting operators based on Bohr-Sommerfeld rules.

Applies the theory to harmonic oscillator and rotation group orbits.

Abstract

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.