Algebra of one-particle operators for the Calogero model

TL;DR

The paper constructs an infinite-dimensional algebra of symmetric one-particle operators for the Calogero model, independent of interaction parameters, and interprets it as the algebra of observables with a particle statistics parameter.

Contribution

It introduces a new algebra ${ mf extcal{G}}$ of symmetric operators for the Calogero model, independent of the interaction parameter, and links it to particle statistics.

Findings

Algebra ${ mf extcal{G}}$ is independent of $ extlambda$

${ mf extcal{G}}$ is an infinite-dimensional Lie algebra

Parameter $ extlambda$ acts as a statistics parameter

Abstract

An algebra ${\cal G}$ of symmetric {\em one-particle} operators is constructed for the Calogero model. This is an infinite-dimensional Lie-algebra, which is independent of the interaction parameter $\lambda$ of the model. It is constructed in terms of symmetric polynomials of raising and lowering operators which satisfy the commutation relations of the $S_N$-{\em extended} Heisenberg algebra. We interpret ${\cal G}$ as the algebra of observables for a system of identical particles on a line. The parameter $\lambda$, which characterizes (a class of) irreducible representations of the algebra, is interpreted as a statistics parameter for the identical particles.