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).
The well-known canonical coherent states are expressed as an infinite series in powers of a complex number $z$ together with a positive sequence of real numbers $ρ(m)=m$. In this article, in analogy with the canonical coherent states, we present a class of vector coherent states by replacing the complex variable $z$ by a real Clifford matrix. We also present another class of vector coherent states by simultaneously replacing $z$ by a real Clifford matrix and $ρ(m)$ by a real matrix. As examples, we present vector coherent states on quaternions and octonions with their real matrix representatio…
A class of vector coherent states is derived with multiple of matrices as vectors in a Hilbert space, where the Hilbert space is taken to be the tensor product of several other Hilbert spaces. As examples vector coherent states with multiple of quaternions and octonions are given. The resulting generalized oscillator algebra is briefly discussed. Further, vector coherent states for a tensored Hamiltonian system are obtained by the same method. As particular cases, coherent states are obtained for tensored Jaynes-Cummings type Hamiltonians and for a two-level two-mode generalization of the Jayn…
The behavior of an electron in an external uniform electromagnetic background coupled to a harmonic potential, with noncommuting space coordinates, is considered in this work. The thermodynamics of the system is studied. Matrix vector coherent states (MVCS) as well as quaternionic vector coherent states (QVCS), satisfying required properties, are also constructed and discussed.
We present a class of vector coherent states in the domain $D\times D\times >....\times D$ (n-copies), where $D$ is the complex unit disc, using a specific class of hermitian matrices. Further, as an example, we build vector coherent states in the unit disc by considering the unit disc as the homogeneous space of the group SU(1,1).
We describe a method for constructing vector coherent states for quantum supersymmetric partner Hamiltonians. The method is then applied to such partner Hamiltonians arising from a generalization of the fractional quantum Hall effect. Explicit examples are worked out.
The first part of this work deals with a formalism of vector coherent states construction for a system of $M$ Fermi-type modes associated with $N$ bosonic modes. Then follows a generalization to a Hamiltonian describing the translational motion of the center of mass of a nanoparticle. The latter gives rise to a new mechanism for the electronic energy relaxation in nanocrystals, intensively studied today in condensed matter physics. Finite degeneracies of the involved Hamiltonian systems are also investigated. The defined vector coherent states satisfy relevant mathematical properties of contin…
In this paper we discuss a general strategy to construct vector coherent states of the Gazeau-Klauder type and we use them to built up examples of isospectral hamiltonians. For that we use a general strategy recently proposed by the author and which extends well known facts on intertwining operators. We also discuss the possibility of constructing non-isospectral hamiltonians with related eigenstates.
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.
This work addresses the study of two-dimensional electron gas in a perpendicular magnetic field in the presence of both Rashba and Dresselhaus spin-orbit (SO) interactions, with effective Zeeman coupling. Exact analytical expressions are found for the eigenvalue problem giving the Landau levels and the associated eigenstates, thanks to an appropriate method of parametrization used to quantize the physical system. Such a system exhibits interesting properties of quaternions for which vector coherent states are built. Similarity with the Weyl-Heisenberg group and some relevant properties of thes…
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 $\{ρ(m)\}_{m=0}^\infty$. 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 $n \times n$ matrices. Associated oscillator algebras are discussed with the aid of a generalized ma…
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 in the same spirit. These vector coherent states may have a finite or an infinite number of components. As examples we first apply the technique to construct vector coherent states using the Plancherel isometry for groups and vector coherent states associated to Clifford algebras, in particular quaternions. As physical examples, we discuss vector coherent states for a quantum optical model and fina…
Using a left multiplication defined on a right quaternionic Hilbert space, we shall demonstrate that various classes of coherent states such as the canonical coherent states, pure squeezed states, fermionic coherent states can be defined with all the desired properties on a right quaternionic Hilbert space. Further, we shall also demonstrate squeezed states can be defined on the same Hilbert space, but the noncommutativity of quaternions prevents us in getting the desired results.
This paper deals with an extension of our previous work [J. Phys. A: Math. Theor. {\bf 40} F817] by considering an alternative construction of canonical and deformed vector coherent states (VCSs) of the Gazeau-Klauder type associated with generalized spin-orbit Hamiltonians. We define an annihilation operator which takes into account the finite dimensional space of states induced by the $k$-photon transition processes of the two-level atom interacting with the single-mode radiation field. The class of nonlinear VCSs (NVCSs) corresponding to the action of the annihilation operator is deduced an…
We discuss a general strategy which produces an orthonormal set of vectors, stable under the action of a given set of unitary operators $A_j$, $j=1,2,...,n$, starting from a fixed normalized vector in $\Hil$ and from a set of unitary operators. We discuss several examples of this procedure and, in particular, we show how a set of {\em coherent-like} vectors can be produced and in which condition over the lattice spacing this can be done.
We generalize the concept of coherent states, traditionally defined as special families of vectors on Hilbert spaces, to Hilbert modules. We show that Hilbert modules over $C^*$-algebras are the natural settings for a generalization of coherent states defined on Hilbert spaces. We consider those Hilbert $C^*$-modules which have a natural left action from another $C^*$-algebra say, $\mathcal A$. The coherent states are well defined in this case and they behave well with respect to the left action by $\mathcal A$. Certain classical objects like the Cuntz algebra are related to specific examples …
Several advances have extended the power and versatility of coherent state theory to the extent that it has become a vital tool in the representation theory of Lie groups and their Lie algebras. Representative applications are reviewed and some new developments are introduced. The examples given are chosen to illustrate special features of the scalar and vector coherent state constructions and how they work in practical situations. Comparisons are made with Mackey's theory of induced representations. For simplicity, we focus on square integrable (discrete series) unitary representations althou…
We decouple the Dirac's radial equations in $D+1$ dimensions with Coulomb-type scalar and vector potentials through appropriate transformations. We study each of these uncoupled second-order equations in an algebraic way by using an $su(1,1)$ algebra realization. Based on the theory of irreducible representations, we find the energy spectrum and the radial eigenfunctions. We construct the Perelomov coherent states for the Sturmian basis, which is the basis for the unitary irreducible representation of the $su(1,1)$ Lie algebra. The physical radial coherent states for our problem are obtained b…
In the spirit of some earlier work on the construction of vector coherent states 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 …
In this paper we present results obtained from the unification of $SU(2)$ coherent states with $\mathbb{C}P^N$ sigma models defined on the Riemann sphere having finite actions. The set of coherent states generated by a vector belonging to a carrier space of an irreducible representation of the group gives rise to a map from the sphere into the set of rank-1 Hermitian projectors in that space. The map can be identified with a particular solution of the $\mathbb{C}P^N$ sigma model, where $N+1$ is equal to the dimension of the representation space. In particular a choice of the generating vector …
For applications of group theory in quantum mechanics, one generally needs explicit matrix representations of the spectrum generating algebras that arise in bases that reduce the symmetry group of some Hamiltonian of interest. Here we use vector coherent state techniques to develop an algorithm for constructing the matrices for arbitrary finite-dimensional irreps of the SO(5) Lie algebra in an SO(3) basis. The SO(3) subgroup of SO(5) is defined by regarding SO(5) as linear transformations of the five-dimensional space of an SO(3) irrep of angular momentum two. A need for such irreps arises in …
We determine the most general group of equivalence transformations for a family of differential equations defined by an arbitrary vector field on a manifold. We also find all invariants and differential invariants for this group up to the second order. A result on the characterization of classes of these equations by the invariant functions is also given.
In this paper we show that the position and the derivative operators, $\hat q$ and $\hat D$, can be treated as ladder operators connecting the various vectors of two biorthonormal families, $\mathcal{F}_φ$ and $\mathcal{F}_ψ$. In particular, the vectors in $\mathcal{F}_φ$ are essentially monomials in $x$, $x^k$, while those in $\mathcal{F}_ψ$ are weak derivatives of the Dirac delta distribution, $δ^{(m)}(x)$, times some normalization factor. We also show how bi-coherent states can be constructed for these $\hat q$ and $\hat D$, both as convergent series of elements of $\mathcal{F}_φ$ and $\mat…
Nonlinear realizations of the SO(4,2) group are discussed from the point of view of symmetries. Dynamical symmetry breaking is introduced. One linear and one quadratic model in curvature are constructed. Coherent states of the Klauder-Perelomov type are defined for both cases taking into account the coset geometry. A new spontaneous compactification mechanism is defined in the subspace invariant under the stability subgroup. The physical implications of the symmetry rupture in the context of non-linear realizations and direct gauging are analyzed and briefly discussed.
The generalized vector is defined on an $n$ dimensional manifold. Interior product, Lie derivative acting on generalized $p$-forms, $-1\le p\le n$ are introduced. Generalized commutator of two generalized vectors are defined. Adding a correction term to Cartan's formula the generalized Lie derivative's action on a generalized vector field is defined. We explore various identities of the generalized Lie derivative with respect to generalized vector fields, and discuss an application.
Enhanced quantization offers a different classical/quantum connection than that of canonical quantization in which $\hbar >0$ throughout. This result arises when the only allowed Hilbert space vectors allowed in the quantum action functional are coherent states, which leads to the classical action functional augmented by additional terms of order $\hbar$. Canonical coherent states are defined by unitary transformations of a fixed, fiducial vector. While Gaussian vectors are commonly used as fiducial vectors, they cannot be used for all systems. We focus on choosing fiducial vectors for several…
A new method to describe hyperbolic patterns in two dimensional flows is proposed. The method is based on the Covariant Lyapunov Vectors (CLVs), which have the properties to be covariant with the dynamics, and thus being mapped by the tangent linear operator into another CLVs basis, they are norm independent, invariant under time reversal and can be not orthonormal. CLVs can thus give a more detailed information on the expansion and contraction directions of the flow than the Lyapunov Vector bases, that are instead always orthogonal. We suggest a definition of Hyperbolic Covariant Coherent Str…
