Quantum vs Classical Integrability in Calogero-Moser Systems

TL;DR

This paper explores the relationship between classical and quantum integrability in Calogero-Moser systems, revealing surprising quantization of classical quantities and suggesting deeper connections between the two regimes.

Contribution

It provides analytical and numerical analysis of classical data in Calogero-Moser systems, uncovering their quantized nature and highlighting the link to quantum integrability.

Findings

Classical quantities are often integers or polynomials with integer coefficients.

Most classical data at equilibrium are quantized or appear as integers.

The relationship between quantum and classical integrability is deeper than previously understood.

Abstract

Calogero-Moser systems are classical and quantum integrable multi-particle dynamics defined for any root system $\Delta$. The {\em quantum} Calogero systems having $1/q^2$ potential and a confining $q^2$ potential and the Sutherland systems with $1/\sin^2q$ potentials have "integer" energy spectra characterised by the root system $\Delta$. Various quantities of the corresponding {\em classical} systems, {\em e.g.} minimum energy, frequencies of small oscillations, the eigenvalues of the classical Lax pair matrices, etc. at the equilibrium point of the potential are investigated analytically as well as numerically for all root systems. To our surprise, most of these classical data are also "integers", or they appear to be "quantised". To be more precise, these quantities are polynomials of the coupling constant(s) with integer coefficients. The close relationship between quantum and classical integrability in Calogero-Moser systems deserves fuller analytical treatment, which would lead to better understanding of these systems and of integrable systems in general.