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On the construction of generalized Grassmann representatives of state vectors

M. El Baz, Y. Hassouni

Published Sep 13, 2013• 3 citations

Abstract

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

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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

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'.

Nonlinear fermions of degree n (n-fermions) are introduced as particles with creation and annihilation operators obeying the simple nonlinear anticommutation relation AA† + A†nAn = 1. The (n + 1)th-order nilpotency of these operators follows from the existence of unique A-vacuum. Supposing appropriate (n + 1)th-order nilpotent para-Grassmann variables and integration rules the sets of n-fermion number states, 'right' and 'left' ladder operator coherent states (CS) and displacement-operator-like CS are constructed. The (n + 1) × (n + 1) matrix realization of the related para-Grassmann algebra is provided. General (n + 1)th-order nilpotent ladder operators of finite-dimensional systems are expressed as polynomials in terms of n-fermion operators. Overcomplete sets of (normalized) 'right' and 'left' eigenstates of such general ladder operators are constructed and their properties are briefly discussed.

Nonlinear fermions of degree n ( n -fermions) are introduced as particles with creation and annihilation operators obeying the simple nonlinear anticommutation relation AA † + A † n A n = 1. The ( n + 1)th-order nilpotency of these operators follows from the existence of unique A -vacuum. Supposing appropriate ( n + 1)th-order nilpotent para-Grassmann variables and integration rules the sets of n -fermion number states, 'right' and 'left' ladder operator coherent states (CS) and displacement-operator-like CS are constructed. The ( n + 1) × ( n + 1) matrix realization of the related para-Grassmann algebra is provided. General ( n + 1)th-order nilpotent ladder operators of finite-dimensional systems are expressed as polynomials in terms of n -fermion operators. Overcomplete sets of (normalized) 'right' and 'left' eigenstates of such general ladder operators are constructed and their properties are briefly discussed. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Coherent states: mathematical and physical aspects'.

A set of operators, the so-called k-fermion operators, that interpolate between boson and fermion operators are introduced through the consideration of an algebra arising from two non-commuting quon algebras. The deformation parameters q and 1/q for these quon algebras are roots of unity with q to the power k being equal to 1. The case k = 2 corresponds to fermions and the case k going to infinity to bosons. Generalized coherent states (connected to the k-fermionic states) and super-coherent states (involving a k-fermionic sector and a purely bosonic sector) are investigated.

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.