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arXiv:math-ph/0606049 · International Journal of Modern Physics A

REPRESENTATIONS AND PROPERTIES OF GENERALIZED Ar STATISTICS

Mohammed Daoud A

Published Sep 10, 2006• 2 citations
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Abstract

A generalization of Ar statistics is proposed and developed. The generalized Ar quantum statistics is completely specified by a set of Jacobson generators satisfying a set of triple algebraic relations. Fock–Hilbert representations and Bargmann–Fock realizations are derived.
DOI
10.1142/s0217751x06031260

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We investigate a generalization of Ar statistics discussed recently in the literature. The explicit complete set of state vectors for the Ar statistics system is given. We consider a Bargmann or an analytic function description of the Fock space corresponding to Ar statistics of bosonic kind. This brings, in a natural way, the so-called Gazeau–Klauder coherent states defined as eigenstates of the Jacobson annihilation operators. The minimization of Robertson uncertainty relation is also considered.
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Generalized quantum statistics, such as paraboson and parafermion statistics, are characterized by triple relations which are related to Lie (super)algebras of type B. The correspondence of the Fock spaces of parabosons, parafermions as well as the Fock space of a system of parafermions and parabosons to irreducible representations of (super)algebras of type B will be pointed out. Example of generalized quantum statistics connected to the basic classical Lie superalgebra B(1|1) ≡ osp(3|2) with interesting physical properties, such as noncommutative coordinates, will be given. Therefore the article focuses on the question, addressed already in 1950 by Wigner: do the equation of motion determine the quantum mechanical commutation relation?