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

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

2006-06-20

arxiv.org/abs/math-ph/0606050v1

Physics

Mathematical Physics

16 pages, published in JPA

Scientific paper

10.1088/0305-4470/39/4/010

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.

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