Hidden algebra of the $N$-body Calogero problem

TL;DR

This paper constructs a new algebraic framework involving a generalized algebra related to $gl(N,\mathbb{R})$ with permutation operators, providing an alternative representation of the Calogero model's Hamiltonian that reveals invariant subspaces.

Contribution

It introduces a novel algebraic structure dependent on permutation operators and demonstrates its application to representing the Calogero Hamiltonian.

Findings

Hamiltonian expressed as a polynomial in algebra generators

Existence of invariant subspaces with explicit bases

Alternative to Bargmann-Fock representation

Abstract

A certain generalization of the algebra $gl(N,{\bf R})$ of first-order differential operators acting on a space of inhomogeneous polynomials in ${\bf R}^{N-1}$ is constructed. The generators of this (non)Lie algebra depend on permutation operators. It is shown that the Hamiltonian of the $N$-body Calogero model can be represented as a second-order polynomial in the generators of this algebra. Given representation implies that the Calogero Hamiltonian possesses infinitely-many, finite-dimensional invariant subspaces with explicit bases, which are closely related to the finite-dimensional representations of above algebra. This representation is an alternative to the standard representation of the Bargmann-Fock type in terms of creation and annihilation operators.