Towards a definition of quantum integrability

TL;DR

This paper explores the challenges of defining quantum integrability by reviewing classical integrability and proposing that a geometric approach to quantum mechanics could offer a solution.

Contribution

It identifies key aspects of classical integrability that should inform a quantum definition and discusses the limitations of naive extensions, suggesting geometry as a potential framework.

Findings

Classical integrability aspects are crucial for quantum definitions

Naive classical-to-quantum extensions are inadequate due to Hilbert space isomorphisms

Geometrical quantum mechanics may help define quantum integrability

Abstract

We briefly review the most relevant aspects of complete integrability for classical systems and identify those aspects which should be present in a definition of quantum integrability. We show that a naive extension of classical concepts to the quantum framework would not work because all infinite dimensional Hilbert spaces are unitarily isomorphic and, as a consequence, it would not be easy to define degrees of freedom. We argue that a geometrical formulation of quantum mechanics might provide a way out.