Deformed Heisenberg algebras, a Fock-space representation and the Calogero model

TL;DR

This paper explores deformed Heisenberg algebras in one dimension, identifying conditions for Fock-space representations, and connects these deformations to the Calogero model and q-deformed oscillators.

Contribution

It provides a general analysis of deformed Heisenberg algebras, deriving conditions for Fock-space representations, and links specific solutions to known models like the Calogero and q-oscillators.

Findings

Identified conditions for Fock-space representations of deformed algebras.

Connected a specific deformation to the Calogero model.

Showed one deformation reduces to harmonic oscillator in certain limits.

Abstract

We describe generally deformed Heisenberg algebras in one dimension. The condition for a generalized Leibniz rule is obtained and solved. We analyze conditions under which deformed quantum-mechanical problems have a Fock-space representation. One solution of these conditions leads to a q-deformed oscillator already studied by Lorek et al., and reduces to the harmonic oscillator only in the infinite-momentum frame. The other solution leads to the Calogero model in ordinary quantum mechanics, but reduces to the harmonic oscillator in the absence of deformation.