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arXiv:math-ph/0606050 · Journal of Physics A Mathematical and General

Representations and properties of generalizedArstatistics, coherent states and Robertson uncertainty relations

Mohammed Daoud A

Published Jan 11, 2006• 12 citations
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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 with 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–Schrödinger uncertainty relation.

DOI
10.1088/0305-4470/39/4/010

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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 with 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–Schrödinger uncertainty relation.
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.
Generalized intelligent states (coherent and squeezed states) are derived for an arbitrary quantum system by using the minimization of the so-called Robertson-Schrödinger uncertainty relation. The Fock-Bargmann representation is also considered. As a direct illustration of our construction, the Pöschl-Teller potentials of trigonometric type will be chosen. We will show the advantage of the Fock-Bargmann representation in obtaining the generalized intelligent states in an analytical way. Many properties of these states are studied.
The so-called Gazeau–Klauder and Perelomov coherent states are introduced for an arbitrary quantum system. We give also the general framework to construct the generalized intelligent states which minimize the Robertson–Schrödinger uncertainty relation. As illustration, the Pöschl–Teller potentials of trigonometric type will be chosen. We show the advantage of the analytical representations of Gazeau–Klauder and Perelomov coherent states in obtaining the generalized intelligent states in analytical way.
The Clebsch–Gordan problems of the Barut–Girardello, and Perelomov coherent states of SL(2,R) are studied using the associated Hilbert spaces as the respective carrier spaces of the representations of the group. For the Barut–Girardello coherent states this Hilbert space is a subspace of the Bargmann–Segal Hilbert space B(C2) called the ‘‘reduced Bargmann space.’’ The generators of the group in this realization are essentially the boson operators of Holman and Biedenharn which provide a convenient starting point of the problem. For the Perelomov coherent states the associated Hilbert space turns out to be Bargmann’s canonical carrier space for the realization of the discrete series of representations, namely, the Hilbert space of functions analytic inside the open unit disc. The scalar product, the principal vector, and a complete orthonormal set in these Hilbert spaces are constructed and used for the explicit evaluation of the Clebsch–Gordan coefficients. For each of the coherent state systems the product state turns out to be the principal vector and, therefore, the coupled state itself is the Clebsch–Gordan coefficient. For the Barut–Girardello coherent states this is, apart from normalization, the product of a Bessel function and d-function. For the Perelomov coherent states, on the other hand, this closely resembles the Clebsch–Gordan coefficient of the SU(2) coherent states.
We construct representations of the Lie algebra using representations of the momentum and position operators satisfying the R -deformed Heisenberg relations, in which the fractional dimension d and angular momentum ℓ appear as parameters. The Bargmann index κ, which characterizes representations of the positive discrete series of , can take any positive value. We construct coherent states in fractional dimensions, in particular we extend the two well-known analytic representations of coherent states for , Perelomov and Barut–Girardello states, from dimension one to any dimension d . We generalize this construction to time-dependent coherent states by means of the symmetries of the quantum time-dependent harmonic oscillator in fractional dimensions. We investigate the uncertainty relations of the momentum and position operators with respect to these coherent states, and their dependence on the dimension.