Permutation Invariant Algebras, a Fock Space Realization and the Calogero Model

TL;DR

This paper investigates permutation invariant oscillator algebras and their Fock space representations, applying the results to the Calogero model, analyzing Gram matrices, eigenvalues, and conditions for positive norms.

Contribution

It provides a comprehensive analysis of permutation invariant algebras, their Fock space structure, and a detailed application to the Calogero model, including positivity conditions and dual operator construction.

Findings

Eigenvalues and eigenstates of Gram matrices analyzed

Universal critical point for degrees of freedom identified

Conditions for positivity of state norms derived

Abstract

We study permutation invariant oscillator algebras and their Fock space representations using three equivalent techniques, i.e. (i) a normally ordered expansion in creation and annihilation operators, (ii) the action of annihilation operators on monomial states in Fock space and (iii) Gram matrices of inner products in Fock space. We separately discuss permutation invariant algebras which possess hermitean number operators and permutation invariant algebras which possess non-hermitean number operators. The results of a general analysis are applied to the S_M extended Heisenberg algebra, underlying the M-body Calogero model. Particular attention is devoted to the analysis of Gram matrices for the Calogero model. We discuss their structure, eigenvalues and eigenstates. We obtain a general condition for positivity of eigenvalues, meaning that all norms of states in Fock space are positive if this condition is satisfied. We find a universal critical point at which the reduction of the physical degrees of freedom occurs. We construct dual operators, leading to the ordinary Heisenberg algebra of free Bose oscillators. From the Fock-space point of view, we briefly discuss the existence of mapping from the Calogero oscillators to the free Bose oscillators and vice versa.