Representations and Properties of Generalized $A_r$ Statistics, Coherent States and Robertson Uncertainty Relations

TL;DR

This paper generalizes $A_r$ statistics using Jacobson generators, constructs Fock and Bargmann representations, develops coherent states, and demonstrates their property of minimizing Robertson-Schrödinger uncertainty relations.

Contribution

It introduces a unified framework for generalized $A_r$ statistics, including new coherent states and their properties, expanding the mathematical understanding of these quantum systems.

Findings

Constructed Fock spaces for generalized $A_r$ statistics.

Developed two inequivalent Bargmann representations for bosonic $A_r$ statistics.

Demonstrated that the coherent states minimize Robertson-Schrödinger uncertainty.

Abstract

The generalization of $A_r$ statistics, including bosonic and fermionic sectors, is performed by means of the so-called Jacobson generators. The corresponding Fock spaces are constructed. The Bargmann representations are also considered. For the bosonic $A_r$ statistics, two inequivalent Bargmann realizations are developed. The first (resp. second) realization induces, in a natural way, coherent states recognized as Gazeau-Klauder (resp. Klauder-Perelomov) ones. In the fermionic case, the Bargamnn realization leads to the Klauder-Perelomov coherent states. For each considered realization, the inner product of two analytic functions is defined in respect to a measure explicitly computed. The Jacobson generators are realized as differential operators. It is shown that the obtained coherent states minimize the Robertson-Schr\"odinger uncertainty relation.