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 $z$ . In the present scheme, the variable $z$ 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).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 $z$. In the present scheme, the variable $z$ 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).

arXiv:math-ph/0305036May 17, 20031 citation

GENERALIZED COHERENT STATES FOR POLYNOMIAL WEYL-HEISENBERG ALGEBRAS

M. Kibler, Mohammed Daoud A

It is the aim of this paper to show how to construct Perelomov and Barut-Girardello coherent states for a polynomial Weyl-Heisenberg algebra. This algebra depends on r parameters. For some special values of the parameter corresponding to r = 1, the algebra covers the cases of the su(1,1) algebra, the su(2) algebra and the ordinary Weyl-Heisenberg or oscillator algebra. For r arbitrary, the generalized Weyl-Heisenberg algebra admits finite or infinite-dimensional representations depending on the values of the parameters. Coherent states of the Perelomov type are derived in finite and infinite dimensions through a Fock-Bargmann approach based on the use of complex variables. The same approach is applied for deriving coherent states of the Barut-Girardello type in infinite dimension. In contrast, the construction of Barut-Girardello coherent states in finite dimension can be achieved solely at the price to replace complex variables by generalized Grassmann variables. Finally, some preliminary developments are given for the study of Bargmann functions associated with some of the coherent states obtained in this work.It is the aim of this paper to show how to construct Perelomov and Barut-Girardello coherent states for a polynomial Weyl-Heisenberg algebra. This algebra depends on r parameters. For some special values of the parameter corresponding to r = 1, the algebra covers the cases of the su(1,1) algebra, the su(2) algebra and the ordinary Weyl-Heisenberg or oscillator algebra. For r arbitrary, the generalized Weyl-Heisenberg algebra admits finite or infinite-dimensional representations depending on the values of the parameters. Coherent states of the Perelomov type are derived in finite and infinite dimensions through a Fock-Bargmann approach based on the use of complex variables. The same approach is applied for deriving coherent states of the Barut-Girardello type in infinite dimension. In contrast, the construction of Barut-Girardello coherent states in finite dimension can be achieved solely at the price to replace complex variables by generalized Grassmann variables. Finally, some preliminary developments are given for the study of Bargmann functions associated with some of the coherent states obtained in this work.

Aug 17, 20160 citations

Generalized coherent states for polynomial Weyl-Heisenberg algebras

M. Kibler, Mohammed Daoud A

It is the aim of this paper to show how to construct á la Perelomov and á la Barut-Girardello coherent states for a polynomial Weyl-Heisenberg algebra. This algebra depends on r parameters. For some special values of the parameter corresponding to r = 1, the algebra covers the cases of the su(1,1) algebra, the su(2) algebra and the ordinary Weyl-Heisenberg or oscillator algebra. For r arbitrary, the generalized Weyl-Heisenberg algebra admits finite or infinite-dimensional representations depending on the values of the parameters. Coherent states of the Perelomov type are derived in finite and infinite dimensions through a Fock-Bargmann approach based on the use of complex variables. The same approach is applied for deriving coherent states of the Barut-Girardello type in infinite dimension. In contrast, the construction of á la Barut-Girardello coherent states in finite dimension can be achieved solely at the price to replace complex variables by generalized Grassmann variables. Finally, some preliminary developments are given for the study of Bargmann functions associated with some of the coherent states obtained in this work.It is the aim of this paper to show how to construct á la Perelomov and á la Barut-Girardello coherent states for a polynomial Weyl-Heisenberg algebra. This algebra depends on r parameters. For some special values of the parameter corresponding to r = 1, the algebra covers the cases of the su(1,1) algebra, the su(2) algebra and the ordinary Weyl-Heisenberg or oscillator algebra. For r arbitrary, the generalized Weyl-Heisenberg algebra admits finite or infinite-dimensional representations depending on the values of the parameters. Coherent states of the Perelomov type are derived in finite and infinite dimensions through a Fock-Bargmann approach based on the use of complex variables. The same approach is applied for deriving coherent states of the Barut-Girardello type in infinite dimension. In contrast, the construction of á la Barut-Girardello coherent states in finite dimension can be achieved solely at the price to replace complex variables by generalized Grassmann variables. Finally, some preliminary developments are given for the study of Bargmann functions associated with some of the coherent states obtained in this work.

arXiv:1201.1811AIP conference proceedingsJan 1, 20121 citation

Bosonic andk-fermionic coherent states for a class of polynomial Weyl–Heisenberg algebras

Mohammed Daoud A, M. Kibler

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 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'.

arXiv:1110.4799Journal of Physics A Mathematical and TheoreticalMay 30, 201222 citations

Vector coherent states from Plancherel's theorem, Clifford algebras and matrix domains

S Twareque Ali, Miroslav Engli, Jean‐Pierre Gazeau

As a substantial generalization of the technique for constructing canonical and the related nonlinear and q -deformed coherent states, we present here a method for constructing vector coherent states (VCS) in the same spirit. These VCS may have a finite or an infinite number of components. The resulting formalism, which involves an assumption on the existence of a resolution of the identity, is broad enough to include all the definitions of coherent states existing in the current literature, subject to this restriction. As examples, we first apply the technique to construct VCS using the Plancherel isometry for groups and VCS associated with Clifford algebras, in particular quaternions. As physical examples, we discuss VCS for a quantum optical model and finally apply the general technique to build VCS over certain matrix domains.As a substantial generalization of the technique for constructing canonical and the related nonlinear and q -deformed coherent states, we present here a method for constructing vector coherent states (VCS) in the same spirit. These VCS may have a finite or an infinite number of components. The resulting formalism, which involves an assumption on the existence of a resolution of the identity, is broad enough to include all the definitions of coherent states existing in the current literature, subject to this restriction. As examples, we first apply the technique to construct VCS using the Plancherel isometry for groups and VCS associated with Clifford algebras, in particular quaternions. As physical examples, we discuss VCS for a quantum optical model and finally apply the general technique to build VCS over certain matrix domains.

arXiv:math-ph/0311042Journal of Physics A Mathematical and GeneralMay 25, 200451 citations

VECTOR COHERENT STATES WITH MATRIX MOMENT PROBLEMS

Abstract. The canonical coherent states can be written as infinite series in powers of a single complex number z and a positive integer ρ(m). The requirement that these states realize a resolution of the identity typically ends up in a moment problem, where the moments form the positive sequence of real numbers {ρ(m)} ∞ m=0. In this paper we obtain new classes of vector coherent states by simultaneously replacing the complex number z and the moments ρ(m) of the canonical coherent states by n × n matrices. A matrix moment problem is proposed and it is solved by introducing a matrix weight. Associated oscillator algebras are discussed. Finally, as a physical application, a generalized version of the Jaynes-Cummings model is discussed. 1.Abstract. The canonical coherent states can be written as infinite series in powers of a single complex number z and a positive integer ρ(m). The requirement that these states realize a resolution of the identity typically ends up in a moment problem, where the moments form the positive sequence of real numbers {ρ(m)} ∞ m=0. In this paper we obtain new classes of vector coherent states by simultaneously replacing the complex number z and the moments ρ(m) of the canonical coherent states by n × n matrices. A matrix moment problem is proposed and it is solved by introducing a matrix weight. Associated oscillator algebras are discussed. Finally, as a physical application, a generalized version of the Jaynes-Cummings model is discussed. 1.

Jan 1, 20030 citations

z

. In the present scheme, the variable

z

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).