Quantum Geometry and Quantum Mechanics of Integrable Systems

TL;DR

This paper explores how classical integrable systems can be deformed via quantization to produce quantum systems, revealing the structure of quantum spectra and dynamics through an $ ext{h}$-deformed geometric framework.

Contribution

It introduces a method to deform classical symplectic structures and invariant tori using a quantization parameter, linking classical and quantum integrable systems through explicit $ ext{h}$-deformed action-angle coordinates.

Findings

Quantum tori correspond to the quantum spectrum up to $O( ext{h}^\infty)$

Quantum diffusion over deformed tori is characterized

Explicit $ ext{h}$-deformed action-angle coordinates are constructed

Abstract

Quantum integrable systems and their classical counterparts are considered. We show that the symplectic structure and invariant tori of the classical system can be deformed by a quantization parameter $\hbar$ to produce a new (classical) integrable system. The new tori selected by the $\hbar$-equidistance rule represent the spectrum of the quantum system up to $O(\hbar^\infty)$ and are invariant under quantum dynamics in the long-time range $O(\hbar^{-\infty})$. The quantum diffusion over the deformed tori is described. The analytic apparatus uses quantum action-angle coordinates explicitly constructed by an $\hbar$-deformation of the classical action-angles.