S'Darboux coordinates and WKB approximations in deformation quantization

TL;DR

This paper introduces a method using s'Darboux coordinates for calculating joint spectra in quantum integrable systems within deformation quantization, providing explicit corrections to classical quantization rules.

Contribution

It develops a formal power series construction of s'Darboux coordinates for star products, extending Darboux coordinates to quantum deformation contexts.

Findings

Constructed s'Darboux coordinates via formal power series in h-bar.

Derived explicit correction to Bohr-Sommerfeld quantization rule.

Linked s'Darboux coordinates to quantum number operators.

Abstract

We introduce a method for calculating the joint spectra of functions which comprise a quantum integrable system under a deformation quantization star product. The main result involves a construction by formal power series in h-bar of s'Darboux coordinates, a concept we introduce for the star product analogue of Darboux normal coordinates. We will find that underlying the quantum integrable system is a set of s'Darboux coordinates which in turn gives rise to number operators for the system. We present an explicit correction to the lowest order Bohr-Sommerfeld or EBK quantization rule.