Algebra of one-particle operators for the Calogero model

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 $λ$ 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 $λ$, which characterizes (a class of) irreducible representations of the algebra, is interpreted as a statistics parameter for the identical particles.