Quantum Canonical Transformations and Integrability: Beyond Unitary Transformations

TL;DR

This paper explores quantum canonical transformations, including non-unitary ones, proposing a new definition of quantum integrability that encompasses systems with fewer integrals of motion than degrees of freedom.

Contribution

It introduces a generalized framework for quantum canonical transformations and a novel definition of quantum integrability that accounts for non-unitary transformations.

Findings

Non-unitary transformations play a crucial role in quantum integrability.

A new definition of quantum integrability is proposed that extends traditional concepts.

Quantum canonical transformations are not limited to unitary operations.

Abstract

Quantum canonical transformations are defined in analogy to classical canonical transformations as changes of the phase space variables which preserve the Dirac bracket structure. In themselves, they are neither unitary nor non-unitary. A definition of quantum integrability in terms of canonical transformations is proposed which includes systems which have fewer commuting integrals of motion than degrees of freedom. The important role of non-unitary transformations in integrability is discussed.