Shifting operators in geometric quantization

TL;DR

This paper explores how shifting operators, interpreted as quantizations of exponential functions of angle variables, naturally arise within the framework of geometric quantization for integrable systems.

Contribution

It demonstrates the emergence of shifting operators in geometric quantization, linking previous interpretations to a geometric framework.

Findings

Shifting operators correspond to quantized exponential functions of angle variables.

The paper connects Bohr-Sommerfeld-Heisenberg quantization with geometric quantization.

Provides a geometric interpretation of shifting operators in integrable systems.

Abstract

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.