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arXiv:math-ph/0311042 · Journal of Physics A Mathematical and General

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

S Twareque Ali, Miroslav Engli, Jean‐Pierre Gazeau

Published May 25, 2004• 51 citations
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Abstract

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.

DOI
10.1088/0305-4470/37/23/008

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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.

•Jan 1, 2003•0 citations
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.
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 matrix valued functions over appropriate domains. As particular examples, we analyze the quaternionic extensions of the canonical coherent states and the Gilmore–Perelomov and Barut–Girardello coherent states arising from representations of SU(1,1). Possible physical applications are indicated.
In the spirit of some earlier work on the construction of vector coherent states (VCS) over matrix domains, we compute here such states associated to some physical Hamiltonians. In particular, we construct vector coherent states of the Gazeau–Klauder type. As a related problem, we also suggest a way to handle degeneracies in the Hamiltonian for building coherent states. Specific physical Hamiltonians studied include a single photon mode interacting with a pair of fermions, a Hamiltonian involving a single boson and a single fermion, a charged particle in a three-dimensional harmonic force field and the case of a two-dimensional electron placed in a constant magnetic field, orthogonal to the plane which contains the electron. In this last example, which is related to the fractional quantum Hall effect, an interesting modular structure emerges for two underlying von Neumann algebras, related to opposite directions of the magnetic field. This leads to the existence of coherent states built out of Kubo-Martin-Schwinger (KMS) states for the system.
Vector coherent states (VCS) viewed as a generalization of ordinary coherent states for higher rank tensor Hilbert spaces are investigated. We consider a systematic way of generating classes of VCS which are solvable (i.e., in the present context, normalizable states satisfying a resolution of the identity) on the Hilbert space of 2D and 3D harmonic oscillators. Thanks to the type of construction, these VCS are classified according to specific criteria. Furthermore, in many cases, the found classes of VCS are continuously deformable one onto another, still remaining solvable.
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).
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 results 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. Associated oscillator algebras are discussed with the aid of a generalized matrix factorial. Two physical examples are discussed. In the first example coherent states are obtained for the Jaynes–Cummings model in the weak coupling limit and some physical properties are discussed in terms of the constructed coherent states. In the second example coherent states are obtained for a conditionally exactly solvable supersymmetric radial harmonic oscillator.
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.