Exclusion statistics,operator algebras and Fock space representations

TL;DR

This paper explores exclusion statistics through operator algebras and Fock space restrictions, analyzing three models of exclusion and their implications for Haldane statistics and counting rules.

Contribution

It introduces a unified framework for different exclusion models in Fock space and discusses limitations of realizing Haldane exclusion statistics within these models.

Findings

Certain exclusion models cannot realize Haldane statistics.

Extended Haldane parameters and counting rules are discussed.

Examples include Calogero-Sutherland, Gentile, and para-Fermi exclusions.

Abstract

We study exclusion statistics within the second quantized approach. We consider operator algebras with positive definite Fock space and restrict them in a such a way that certain state vectors in Fock space are forbidden ab initio.We describe three characteristic examples of such exclusion, namely exclusion on the base space which is characterized by states with specific constraint on quantum numbers belonging to base space M (e.g. Calogero-Sutherland type of exclusion statistics), exclusion in the single-oscillator Fock space, where some states in single oscillator Fock space are forbidden (e.g. the Gentile realization of exclusion statistics) and a combination of these two exclusions (e.g. Green's realization of para-Fermi statistics). For these types of exclusions we discuss extended Haldane statistics parameters g, recently introduced by two of us in Mod.Phys.Lett.A 11, 3081 (1996), and associated counting rules. Within these three types of exclusions in Fock space the original Haldane exclusion statistics cannot be realized.