Generalized Weyl-Heisenberg Algebra Research Articles

Generalized Weyl-Heisenberg Algebra, Qudit Systems and Entanglement Measure of Symmetric States via Spin Coherent States.

A relation is established in the present paper between Dicke states in a d-dimensional space and vectors in the representation space of a generalized Weyl–Heisenberg algebra of finite dimension d. This provides a natural way to deal with the separable and entangled states of a system of symmetric qubit states. Using the decomposition property of Dicke states, it is shown that the separable states coincide with the Perelomov coherent states associated with the generalized Weyl–Heisenberg algebra considered in this paper. In the so-called Majorana scheme, the qudit (d-level) states are represented by N points on the Bloch sphere; roughly speaking, it can be said that a qudit (in a d-dimensional space) is describable by a N-qubit vector (in a N-dimensional space). In such a scheme, the permanent of the matrix describing the overlap between the N qubits makes it possible to measure the entanglement between the N qubits forming the qudit. This is confirmed by a Fubini–Study metric analysis. A new parameter, proportional to the permanent and called perma-concurrence, is introduced for characterizing the entanglement of a symmetric qudit arising from N qubits. For (), this parameter constitutes an alternative to the concurrence for two qubits. Other examples are given for and 5. A connection between Majorana stars and zeros of a Bargmmann function for qudits closes this article.

Generalized Grassmann variables for quantum kit (k-level) systems and Barut–Girardello coherent states for su(r + 1) algebras

This paper concerns the construction of $su(r+1)$ Barut--Girardello coherent states in term of generalized Grassmann variables. We first introduce a generalized Weyl-Heisenberg algebra ${\cal A}(r)$ ($r \geq 1$) generated by $r$ pairs of creation and annihilation operators. This algebra provides a useful framework to describe qubit and qukit ($k$-level) systems. It includes the usual Weyl-Heisenberg and $su(2)$ algebras. We investigate the corresponding Fock representation space. The generalized Grassmann variables are introduced as variables spanning the Fock--Bargmann space associated with the algebra ${\cal A}(r)$. The Barut--Girardello coherent states for $su(r+1)$ algebras are explicitly derived and their over--completion properties are discussed.

A polynomial class of u(2) algebras

A r-parameter u{κ1,κ2,…,κr}(2) algebra is introduced. Finite unitary representations are investigated. This polynomial algebra reduces via a contraction procedure to the generalized Weyl–Heisenberg algebra 𝒜{κ1,κ2,…,κr} [M. Daoud and M. Kibler, J. Phys. A: Math. Theor.45 (2012) 244036]. A pair of nonlinear (quadratic) bosons of type 𝒜κ ≡ 𝒜{κ1=κ,κ2=0,…,κr=0} is used to construct, à la Schwinger, a one parameter family of (cubic) uκ(2) algebra. The corresponding Hilbert space is constructed. The analytical Bargmann representation is also presented.

Extended Lorentz Transformations in Clifford Space Relativity Theory

Some novel physical consequences of the Extended Relativity Theory in C-spaces (Clifford spaces) were explored recently. In particular, generalized photon dispersion relations allowed for energy-dependent speeds of propagation while still retaining the Lorentz symmetry in ordinary spacetimes, but breaking the extended Lorentz symmetry in C-spaces. In this work we analyze in further detail the extended Lorentz transformations in Clifford Space and their physical implications. Based on the notion of “extended events” one finds a very different physical explanation of the phenomenon of “relativity of locality” than the one described by the Doubly Special Relativity (DSR) framework. A generalized Weyl-Heisenberg algebra, involving polyvector-valued coordinates and momenta operators, furnishes a realization of an extended Poincare algebra in C-spaces. In addition to the Planck constant ħ, one finds that the commutator of the Clifford scalar components of the Weyl-Heisenberg algebra requires the introduction of a dimensionless parameter which is expressed in terms of the ratio of two length scales : the Planck and Hubble scales. We finalize by discussing the concept of “photons”, null intervals, effective temporal variables and the addition/subtraction laws of generalized velocities in C-space.

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

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 h4 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’.

Symplectic fluctuations for electromagnetic excitations of Hall droplets

We show that the integer quantum Hall effect systems in a plane, sphere or disc can beformulated in terms of an algebraic unified scheme. This can be achieved by making use ofa generalized Weyl–Heisenberg algebra and investigating its basic features. We study theelectromagnetic excitation and derive the Hamiltonian for droplets of fermions on atwo-dimensional Bargmann space (phase space). This excitation is introduced through adeformation (perturbation) of the symplectic structure of the phase space. We show themajor role of Moser’s lemma in a dressing procedure, which allows us to eliminatethe fluctuations of the symplectic structure. We discuss the emergence of theSeiberg–Witten map and the generation of an Abelian noncommutative gaugefield in the theory. As an illustration of our model, we give the action describingthe electromagnetic excitation of a quantum Hall droplet in a two-dimensionalmanifold.

Miscellaneous Applications of Quons

This paper deals with quon algebras or deformed oscillator algebras, for which the deformation parameter is a root of unity. We motivate why such algebras are interesting for fractional supersymmetric quantum mechanics, angular momentum theory and quantum information. More precisely, quon algebras are used for (i) a realization of a generalized Weyl-Heisenberg algebra from which it is possible to associate a fractional supersymmetric dynamical system, (ii) a polar decomposition of SU2 and (iii) a construction of mutually unbiased bases in Hilbert spaces of prime dimension. We also briefly discuss (symmetric informationally complete) positive operator valued measures in the spirit of (iii).

Fractional supersymmetry and hierarchy of shape invariant potentials

Fractional supersymmetric quantum mechanics is developed from a generalized Weyl-Heisenberg algebra. The Hamiltonian and the supercharges of fractional supersymmetric dynamical systems are built in terms of the generators of this algebra. The Hamiltonian gives rise to a hierarchy of isospectral Hamiltonians. Special cases of the algebra lead to dynamical systems for which the isospectral supersymmetric partner Hamiltonians are connected by a (translational or cyclic) shape invariance condition.

Two approaches to fractional supersymmetrical quantum mechanics

AbstractTwo complementary approaches of 𝒩 = 2 fractional supersymmetrical quantum mechanics of order k are studied in this article. The first, based on a generalized Weyl–Heisenberg algebra Wk (which comprises the affine quantum algebra Uq(sl2) with qk = 1 as a special case), apparently contains solely one bosonic degree of freedom. The second uses generalized bosonic and k‐fermionic degrees of freedom. As an illustration, a particular emphasis is put on the fractional supersymmetrical oscillator of order k. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003