Jordan Maps Research Articles

Bosonic Representation of Matrices and Angular Momentum Probabilistic Representation of Cyclic States.

The Jordan-Schwinger map allows us to go from a matrix representation of any arbitrary Lie algebra to an oscillator (bosonic) representation. We show that any Lie algebra can be considered for this map by expressing the algebra generators in terms of the oscillator creation and annihilation operators acting in the Hilbert space of quantum oscillator states. Then, to describe quantum states in the probability representation of quantum oscillator states, we express their density operators in terms of conditional probability distributions (symplectic tomograms) or Husimi-like probability distributions. We illustrate this general scheme by examples of qubit states (spin-1/2 su(2)-group states) and even and odd Schrödinger cat states related to the other representation of su(2)-algebra (spin-j representation). The two-mode coherent-state superpositions associated with cyclic groups are studied, using the Jordan-Schwinger map. This map allows us to visualize and compare different properties of the mentioned states. For this, the su(2) coherent states for different angular momenta j are used to define a Husimi-like Q representation. Some properties of these states are explicitly presented for the cyclic groups C2 and C3. Also, their use in quantum information and computing is mentioned.

A Jordan-Schwinger variant of the spectral theorem for linear operators

In this paper we show variant of the spectral theorem using an algebraic Jordan-Schwinger map. The advantage of this approach is that we don't have restriction of normality on the class of operators we consider.

A Poisson algebra on the Hida Test functions and a quantization using the Cuntz algebra

In this note, we define one more way of quantization of classical systems. The quantization we consider is an analogue of classical Jordan–Schwinger map which has been known and used for a long time by physicists. The difference, compared to Jordan–Schwinger map, is that we use generators of Cuntz algebra mathcal {O}_{infty } (i.e. countable family of mutually orthogonal partial isometries of separable Hilbert space) as a “building blocks” instead of creation–annihilation operators. The resulting scheme satisfies properties similar to Van Hove prequantization, i.e. exact conservation of Lie brackets and linearity.

A new boson realization of fusion polynomial algebras in non-Hermitian quantum mechanics: γ-deformed generators, -symmetry in Fock space and Higgs algebra

A γ-deformed version of algebra with non-Hermitian generators has been obtained from a bi-orthogonal system of vectors in C 2 . The related Jordan–Schwinger map is combined with boson algebras to obtain a hierarchy of fusion polynomial algebras. This makes possible the construction of Higgs algebra of cubic polynomial type which eventually leads to several multi-boson non-Hermitian Hamiltonians. Finally, the notion of global and partial -symmetry have been introduced in a typical Fock space setting. The possibility of -symmetry breaking is also discussed. The deformation parameter γ plays a crucial role in the entire formulation and non-trivially modifies the eigenfunctions under consideration. The symmetry behavior has been conceptualized in terms of a couple of toy models.

Convex topological algebras via linear vector fields and Cuntz algebras

A realization by linear vector fields is constructed for any Lie algebra which admits a biorthogonal system and for its any suitable representation. The embedding into Lie algebras of linear vector fields is in analogue to the classical Jordan—Schwinger map. A number of examples of such Lie algebras of linear vector fields is computed. In particular, we obtain examples of the twisted Heisenberg-Virasoro Lie algebra and the Schrödinger-Virasoro Lie algebras among others. More generally, we construct an embedding of an arbitrary locally convex topological algebra into the Cuntz algebra.

Laser transverse modes of spherical resonators: a review [Invited

The study of structured laser beams has been one of the most active fields of research for decades, particularly in exploring laser beams with orbital angular momentum. The direct generation of structured beams from laser resonators is deeply associated with the formation of transverse modes. The wave representations of transverse modes of spherical cavities are usually categorized into Hermite–Gaussian (HG) and Laguerre–Gaussian (LG) modes for a long time. Enormous experimental results have revealed that the generalized representation for the transverse modes is the Hermite–LG (HLG) modes. We make a detailed overview for the theoretical description of the HLG modes from the representation of the spectral unitary group of order 2 in the Jordan–Schwinger map. Furthermore, we overview how to derive the integral formula for the elliptical modes based on the Gaussian wave-packet state and the inverse Fourier transform. The relationship between the HLG modes and elliptical modes is linked by the quantum Fourier transform. The most striking result is that the HLG modes can be exactly derived as the superposition of the elliptical modes without involving Hermite and Laguerre polynomials. Finally, we discuss the application of the HLG modes in characterizing the propagation evolution of the vortex structures of HG beams transformed by an astigmatic mode converter. This overview certainly provides not only a novel formula for transverse modes, but also a pedagogical insight into quantum physics.

Dynamics of a harmonic oscillator coupled with a Glauber amplifier

A system of a quantum harmonic oscillator bi-linearly coupled with a Glauber amplifier is analysed considering a time-dependent Hamiltonian model. The Hilbert space of this system may be exactly subdivided into invariant finite dimensional subspaces. Resorting to the Jordan–Schwinger map, the dynamical problem within each invariant subspace may be traced back to an effective SU(2) Hamiltonian model expressed in terms of spin variables only. This circumstance allows to analytically solve the dynamical problem and thus to study the exact dynamics of the oscillator-amplifier system under specific time-dependent scenarios. Peculiar physical effects are brought to light by comparing the dynamics of such a system with that of two interacting standard oscillators.

Choquet Order and Jordan Maps

Choquet Order and Jordan Maps

Surjective Jordan maps and Jordan triple maps

Surjective Jordan maps and Jordan triple maps on B(X) are characterized.

Right SUq(2)- and left SUq−1(2)-invariances of the q-Hilbert–Schmidt Scalar products for an adjoint representation of the quantum algebra Ŭq(su2)

The Jordan–Schwinger realization is used to construct tensor operators as the even and odd dimensional irreducible submodules of an adjoint representation of the quantum algebra U ̆ q ( s u 2 ) . All U ̆ q ( s u 2 ) -submodules are equipped with the so-called left and right q -Hilbert–Schmidt scalar products by using the Wigner–Eckart theorem. The bases of all irreducible submodules of the adjoint representation are orthonormal with respect to the left q -Hilbert–Schmidt scalar product, and are orthogonal, but not normalized, with respect to the right one. Consequently, only with respect to the left q -Hilbert–Schmidt scalar product, the adjoint representation of the quantum algebra U ̆ q ( s u 2 ) on the tensor operators is a ∗ -representation. We show that both left and right q -Hilbert–Schmidt scalar products are right S U q ( 2 ) -invariant and left S U q − 1 ( 2 ) -invariant. Moreover, every irreducible submodule of the adjoint representation of the quantum algebra U ̆ q ( s u 2 ) as an associative algebra with unit, is a left quantum space for O ( S U q − 1 ( 2 ) ) and a right quantum space for O ( S U q ( 2 ) ) . Finally, it is shown that there is a natural compatibility between the coproducts and the Haar measures of the quantum groups O ( S U q − 1 ( 2 ) ) and O ( S U q ( 2 ) ) and the definitions of the left and right q -Hilbert–Schmidt scalar products on the tensor operators of the Hopf algebra U ̆ q ( s u 2 ) .

PT symmetry and a dynamical realization of the SU(1, 1) algebra

We show that the elementary modes of the planar harmonic oscillator can be quantized in the framework of quantum mechanics based on pseudo-hermitian Hamiltonians. These quantized modes are demonstrated to act as dynamical structures behind a new Jordan–Schwinger realization of the SU(1, 1) algebra. This analysis complements the conventional Jordan–Schwinger construction of the SU(2) algebra based on hermitian Hamiltonians of a doublet of oscillators.

Hilbert-Schmidt Inner Product for an Adjoint Representation of the Quantum Algebra [formula omitted

The Jordan–Schwinger realization of quantum algebra U ⌣ q ( s u 2 ) is used to construct the irreducible submodule T l of the adjoint representation in two different bases. The two bases are known as types of irreducible tensor operators of rank l which are related to each other by the involution map. The bases of the submodules are equipped with q -analogues of the Hilbert–Schmidt inner product and it is also shown that the adjoint representation corresponding to one of those submodules is a *-representation.

Isomorphisms and automorphisms of discrete multiplier Hopf C*-algebras

We construct Hopf algebra isomorphisms of discrete multiplier Hopf C*-algebras, and Hopf AF C*-algebras (generalized quantum UHF algebras), from \(\displaystyle K\)-theoretical data. Some of the intermediate results are of independent interest, such as a result that Jordan maps of Hopf algebras intertwine antipodes, and the applications to automorphisms of Hopf algebras.

Indecomposable representations and boson realizations of the nonlinear deformed angular momentum algebra of Witten’s first type

In this paper indecomposable representations and boson realizations of the nonlinear angular momentum algebra \(\mathcal{R}_{q,p}^{c_1,c_2,c_3}\) of Witten’s first type are investigated in a purely algebraic manner. Explicit form of the master representation of \(\mathcal{R}_{q,p}^{c_1,c_2,c_3}\) on the space of its universal enveloping algebra is given. Then, from this master representation, other indecomposable representations are obtained in explicit form. Various kinds of single-boson, single inverse boson, and double-boson realizations of \(\mathcal{R}_{q,p}^{c_1,c_2,c_3}\) are respectively obtained by generalizing the Holstein–Primakoff realization, the Dyson realization, and the Jordan–Schwinger realization of the Lie algebras SU(2) and SU(1,1). For each kind, the unitary realization, the nonunitary realization, and their connection by the corresponding similarity transformation are respectively discussed. Using a kind of double-boson realizations, the irreducible representation of \(\mathcal{R}_{q,p}^{c_1,c_2,c_3}\) in the angular momentum basis is given.

Metric versions of Herstein's theorems on Jordan maps

Let A be a Banach algebra and let B be an ultraprime Banach algebra. If Φ: A → B is a surjective continuous linear map which tends to satisfy the Jordan multiplicativity condition, then we show that Φ comes near to satisfy either the multiplicativity or the anti-multiplicativity condition. In fact, we give a quantitative estimate of this phenomenon. Furthermore, we estimate how much a continuous linear map Δ: B → B approaches to satisfy the derivation identity in the case when Δ tends to satisfy the Jordan derivation identity

The Two Maps of Israel's Land

Research Article| December 01 2007 The Two Maps of Israel’s Land Rachel Havrelock Rachel Havrelock Search for other works by this author on: This Site Google Journal of Biblical Literature (2007) 126 (4): 649–667. https://doi.org/10.2307/27638460 Cite Icon Cite Share Icon Share Twitter Permissions Search Site Citation Rachel Havrelock; The Two Maps of Israel’s Land. Journal of Biblical Literature 1 January 2007; 126 (4): 649–667. doi: https://doi.org/10.2307/27638460 Download citation file: Zotero Reference Manager EasyBib Bookends Mendeley Papers EndNote RefWorks BibTex toolbar search Search Dropdown Menu toolbar search search input Search input auto suggest filter your search All Scholarly Publishing CollectiveSBL PressJournal of Biblical Literature Search Advanced Search The text of this article is only available as a PDF. Article PDF first page preview Close Modal You do not currently have access to this content.

Two-dimensional isotropic harmonic oscillator approach to classical and quantum Stokes parameters

We show that the well-known Stokes operators, defined as elements of the Jordan–Schwinger map with the Pauli matrices of two independent bosons, are equal to the constants of motion of the two-dimensional isotropic harmonic oscillator. Taking the expectation value of the Stokes operators in a two-mode coherent state, we obtain the corresponding classical Stokes parameters. We show that this classical limit of the Stokes operators, the 2 × 2 unit matrix and the Pauli matrices may be used to expand the polarization matrix. Finally, by means of the constants of motion of the classical two-dimensional isotropic harmonic oscillator, we describe the geometric properties of the polarization ellipse. Our study is restricted to the case of a monochromatic quantized-plane electromagnetic wave that propagates along the z axis.PACS Nos.: 42.50.–p, 42.25.Ja, 11.30.–j

Jordan–Schwinger map, 3D harmonic oscillator constants of motion, and classical and quantum parameters characterizing electromagnetic wave polarization

In this work we introduce a generalization of the Jauch and Rohrlich quantum Stokes operators when the arrival direction from the source is unknown a priori. We define the generalized Stokes operators as the Jordan–Schwinger map of a triplet of harmonic oscillators with the Gell–Mann and Ne'eman matrices of the SU(3) symmetry group. We show that the elements of the Jordan–Schwinger map are the constants of motion of the three-dimensional isotropic harmonic oscillator. Also, we show that the generalized Stokes operators together with the Gell–Mann and Ne'eman matrices may be used to expand the polarization matrix. By taking the expectation value of the Stokes operators in a three-mode coherent state of the electromagnetic field, we obtain the corresponding generalized classical Stokes parameters. Finally, by means of the constants of motion of the classical 3D isotropic harmonic oscillator we describe the geometrical properties of the polarization ellipse.

Classical Hamiltonian systems with sl(2) coalgebra symmetry and their integrable deformations

Several families of classical integrable systems with two degrees of freedom are derived from phase-space realizations of sl(2) Poisson coalgebras. As a remarkable fact, the existence of the N-dimensional integrable generalization of all these systems is always ensured (by construction) due to their underlying dynamical coalgebra symmetry. By following the same approach, different integrable deformations for such systems are obtained from the q-deformed analogues of sl(2). The well-known Jordan-Schwinger realization is also proven to be related to a (non-coassociative) coalgebra structure on sl(2) and the 2 N dimensional integrable Hamiltonian generated by such Jordan-Schwinger representation is obtained. Finally, the relation between complete integrability and the properties of the initial phase-space realization is elucidated through two more examples based on the Heisenberg-Weyl and so(3,2) Poisson coalgebras.

BiHamiltonian quantum systems and Weyl quantization

By using Weyl systems associated with symplectic vector spaces, we show an association between biHamiltonian classical systems and corresponding quantum systems. The correspondence is carried on the example of the harmonic oscillator. We also use the Jordan—Schwinger map to construct alternative realizations of classical Lie algebras.