Quantization of classical integrable systems. Part I: quasi-integrable quantum systems

TL;DR

This paper introduces a new concept of quasi-integrability for quantum systems, inspired by classical noncommutative integrability, by defining conditions on quantum operators based on their properties relative to momenta.

Contribution

It proposes a novel framework for quantum integrability called quasi-integrability, extending classical ideas to quantum operators based on their momentum-related properties.

Findings

Defined a condition for quantum operators as a replacement for classical functional independence

Introduced the concept of quasi-integrable quantum systems based on operator properties

Lays groundwork for further development of quantum integrability theories

Abstract

We propose in this work a concept of integrability for quantum systems, which corresponds to the concept of noncommutative integrability for systems in classical mechanics. We determine a condition for quantum operators which can be a suitable replacement for the condition of functional independence for functions on the classical phase space. This condition is based on the properties of the main parts of the operators with respect to the momenta. We are led in this way to the definition of what we call a "quasi-integrable quantum system". This concept will be further developed in a series of following papers.