Quantum generalized Calogero-Moser systems from free Hamiltonian reduction

TL;DR

This paper develops a quantum version of the generalized Calogero-Moser system using Hamiltonian reduction, revealing differences in dynamics based on symmetry groups and providing explicit spectra for small particle numbers.

Contribution

It introduces a quantum Hamiltonian derived from free system reduction, capturing symmetry-dependent dynamics and including an additional attractive term absent in standard quantization.

Findings

Derived the generalized quantum Calogero-Moser Hamiltonian

Obtained spectra and wavefunctions for N=2,3 particles

Partially diagonalized the Hamiltonian for general N

Abstract

The one-dimensional system of particles with a $1/x^2$ repulsive potential is known as the Calogero-Moser system. Its classical version can be generalised by substituting the coupling constants with additional degrees of freedom, which span the $\mathfrak{so}(N)$ or $\mathfrak{su}(N)$ algebra with respect to Poisson brackets. We present the quantum version of this generalized model. As the classical generalization is obtained by a symplectic reduction of a free system, we present a method of obtaining a quantum system along similar lines. The reduction of a free quantum system results in a Hamiltonian, which preserves the differences in dynamics of the classical system depending on the underlying, orthogonal or unitary, symmetry group. The orthogonal system is known to be less repulsive than the unitary one, and the reduced free quantum Hamiltonian manifests this trait through an additional attractive term $\sum_{i<j}\frac{-\hbar^2}{(x_i-x_j)^2}$, which is absent when one performs the straightforward Dirac quantization of the considered system. We present a detailed and rigorous derivation of the generalized quantum Calogero-Moser Hamiltonian, we find the spectra and wavefunctions for the number of particles $N=2,3$, and we diagonalize the Hamiltonian partially for a general value of $N$.