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arXiv:math-ph/0409038 · Journal of Physics A Mathematical and General

On the construction of generalized Grassmann representatives of state vectors

M El Baz, Y. Hassouni

Published Mar 29, 2004• 2 citations
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Abstract

Generalized Z k -graded Grassmann variables are used to label coherent states related to the nilpotent representation of the q -oscillator of Biedenharn and Macfarlane when the deformation parameter is a root of unity. These states are then used to construct generalized Grassmann representatives of state vectors.
DOI
10.1088/0305-4470/37/15/005

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published in J. Phys. A37 (2004) 4361 Generalized Zk-graded Grassmann variables are used to label coherent states related to the nilpotent representation of the q-oscillator of Biedenharn and Macfarlane when the deformation parameter is a root of unity. These states are then used to construct generalized Grassmann representatives of state vectors. Recently in [1] we have constructed coherent states for k-fermions using Kerner’s [2] Z3-graded extension of the Grassmann variables [3]. These results were obtained in the case where the deformation parameter is a primitive cubic root of unity, i.e., k = 3. In order to obtain similar results in the generic case (by generic we mean here that q, the deformation parameter, is an arbitrary k th root of unity, i.e., for an arbitrary positive integer k) one should use Zk-graded generalizations of the Grassmann variables. Unfortunately, up to our knowledge, such structures have not been yet constructed in the spirit of Kerner’s variables. There exist however in the literature another point of view and other generalized Zk-graded Grassmann variables [4]. In this letter we investigate on the use of these latter variables for the description of k-fermions. Namely, we will construct k-fermionic coherent states labeled by Zk-graded Grassmann variables. The coherent states will be used to derive a space of (Zk-graded) Grassmann representatives in which state vectors are represented as ”holomorphic ” functions of the Grassmann variable. There exist many deformations of the harmonic oscillator which, for some values of the deformation parameter, give rise to k-fermions. In this letter we will illustrate the construction using the oscillator deformation of Biedenharn [5] and Macfarlane [6]. This q-oscillator is described by the operators {N, a, a +} with the following relations
A general scheme is proposed for constructing vector coherent states, in analogy with the well-known canonical coherent states, and their deformed versions, when these latter are expressed as infinite series in powers of a complex variable zzz. In the present scheme, the variable zzz is replaced by a matrix valued function over appropriate domains. As particular examples, we analyze the quaternionic extensions of the the canonical coherent states and the Gilmore-Perelomov and Barut-Girardello coherent states arising from representations of SU(1,1).
The aim of this paper is to construct coherent states à la Perelomov and à la Barut–Girardello for a polynomial Weyl–Heisenberg algebra. This generalized Weyl–Heisenberg algebra, denoted by , depends on r real parameters and is an extension of the one-parameter algebra (Daoud and Kibler 2010 J. Phys. A: Math. Theor. 43 115303) which covers the cases of the su (1, 1) algebra (for κ > 0), the su (2) algebra (for κ < 0) and the h 4 ordinary Weyl–Heisenberg algebra (for κ = 0). For finite-dimensional representations of and , where is a truncation of order s of in the sense of Pegg–Barnett, a connection is established with k -fermionic algebras (or quon algebras). This connection makes it possible to use generalized Grassmann variables for constructing certain coherent states. Coherent states of the Perelomov type are derived for infinite-dimensional representations of and for finite-dimensional representations of and through a Fock–Bargmann analytical approach based on the use of complex (or bosonic) variables. The same approach is applied for deriving coherent states of the Barut–Girardello type in the case of infinite-dimensional representations of . In contrast, the construction of coherent states à la Barut–Girardello for finite-dimensional representations of and can be achieved solely at the price of replacing complex variables by generalized Grassmann (or k -fermionic) variables. Some of the results are applied to su (2), su (1, 1) and the harmonic oscillator (in a truncated or not truncated form). This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Coherent states: mathematical and physical aspects'.
The purpose of this paper is to generalize fermionic coherent states for two-level systems described by pseudo-Hermitian Hamiltonian (Cherbal et al 2007 J. Phys. A: Math. Theor. 40 1835 ) to n -level systems. Central to this task is the expression of the coherent states in terms of generalized Grassmann variables. These kinds of Grassmann coherent states satisfy the bi-overcompleteness condition instead of the overcompleteness one, as it is reasonably expected because of the biorthonormality of the system. Choosing an appropriate Grassmann weight function, the resolution of identity is examined. Moreover Grassmannian coherent and squeezed states of the deformed group SU q (2) for the three-level pseudo-Hermitian system are presented.