Dynamical symmetry algebra of the Calogero model

TL;DR

This paper investigates the dynamical symmetry algebra of the N-body Calogero model, revealing its polynomial nature and exploring its structure across different particle numbers and statistical parameters.

Contribution

It characterizes the polynomial structure of the dynamical symmetry algebra for the Calogero model and constructs explicit generators for various particle numbers and parameters.

Findings

Algebra is intrinsically polynomial.

Explicit generators found for N=4 particles.

Finite algebra structure demonstrated for non-zero statistical parameter.

Abstract

We study the dynamical symmetry algebra of the N-body Calogero model describing the structure of degenerate levels and demonstrate that the algebra is intrisically polynomial. We discuss some general properties of an algebra of S_N-symmetric operators acting on the S_N-symmetric subspace of the Fock space for any statistical parameter \nu. In the bosonic case (\nu=0) we find the algebra of generators for every N. For \nu\neq 0, we explicitly reproduce the finite algebra for the 4-particle model, demonstrating some general features of our construction.