Harmonic analysis on Lagrangian manifolds of integrable Hamiltonian systems

TL;DR

This paper develops a representation of the phase space symmetry algebra for integrable Hamiltonian systems using canonical quantization and separation variables, focusing on Lagrangian manifolds and their quantization.

Contribution

It introduces a novel representation of the symmetry algebra over Lagrangian manifolds via canonical quantization in separation variables, highlighting its indecomposable and non-exponentiated nature.

Findings

Representation constructed over Lagrangian manifolds

Representation is indecomposable and non-exponentiated

Uses separation variables for canonical quantization

Abstract

For an integrable Hamiltonian system we construct a representation of the phase space symmetry algebra over the space of functions on a Lagrangian manifold. The representation is a result of the canonical quantization of the integrable system in terms of separation variables. The variables are chosen in such way that a half of them parameterizes the Lagrangian manifold, which coincides with the Liouville torus of the integrable system. The obtained representation is indecomposable and non-exponentiated.