Jordan Schwinger Representations Research Articles

Superintegrability on the Dunkl oscillator model in three-dimensional spaces of constant curvature

In this paper, three-dimensional Dunkl oscillator models are studied in a generalized form of superintegrable Euclidean Hamiltonian systems to curved ones. These models are defined based on curved Hamiltonians, which depend on a deformation parameter of underlying space and involve reflection operators. The corresponding symmetries are obtained by the Jordan–Schwinger representations in the family of the Cayley–Klein orthogonal algebras using the creation and annihilation operators of the dynamical sl−1(2) algebra of the one-dimensional Dunkl oscillator. The resulting algebra is a deformation of soκ1κ2(4) with reflections, which is known as the Jordan–Schwinger–Dunkl algebra jsdκ1κ2(4). Hence, it is shown that this model is maximally superintegrable. On the other hand, the superintegrability of the three-dimensional Dunkl oscillator model is studied from the viewpoint of the factorization approach. The spectrum of this system is derived through the separation of variables in geodesic polar coordinates, and the resulting eigenfunctions are algebraically given in terms of Jacobi polynomials.

The anisotropic Dunkl oscillator problem on the two-dimensional curved spaces

In this paper, we study the two-dimensional (2D) Euclidean anisotropic Dunkl oscillator model in an integrable generalization to curved ones of the 2D sphere [Formula: see text] and the hyperbolic plane [Formula: see text]. This generalized model depends on the deformation parameter [Formula: see text] of underlying space and involves reflection operators [Formula: see text] in such a way that all the results are simultaneously valid for [Formula: see text], [Formula: see text] and [Formula: see text]. It turns out that this system is superintegrable based on the special cases of parameter [Formula: see text], which constant measures the asymmetry of the two frequencies in the 2D Dunkl model. Therefore, the Hamiltonian [Formula: see text] can be interpreted as an anisotropic generalization of the curved Higgs–Dunkl oscillator in the limit [Formula: see text]. When [Formula: see text], the system turns out to be the well-known superintegrable 1:2 Dunkl oscillator on [Formula: see text] and [Formula: see text]. In this way, the integrals of the motion arising from the anisotropic Dunkl oscillator are quadratic in the Dunkl derivatives for the special cases of [Formula: see text]. Moreover, these symmetries obtain by the Jordan–Schwinger representation in the family of the Cayley–Klein orthogonal algebras using the creation and annihilation operators of the dynamical [Formula: see text] algebra of the 1D Dunkl oscillator. The resulting algebra is a deformation of [Formula: see text] with reflections, which is named the Jordan–Schwinger–Dunkl algebra [Formula: see text]. The spectrum of this system is determined by the separation of variables in geodesic polar coordinates, and the resulting eigenfunctions are algebraically given in terms of Jacobi polynomials.

Entanglement entropy in the σ-model with the de Sitter target space

We derive the formula of the entanglement entropy between the left and right oscillating modes of the σ-model with the de Sitter target space. To this end, we study the theory in the cosmological gauge in which the non-vanishing components of the metric on the two-dimensional base space are functions of the expansion parameter of the de Sitter space. The model is embedded in the causal north pole diamond of the Penrose diagram. We argue that the cosmological gauge is natural to the σ-model as it is compatible with the canonical quantization relations. In this gauge, we obtain a new general solution to the equations of motion in terms of time-independent oscillating modes. The constraint structure is adequate for quantization in the Gupta–Bleuler formalism. We construct the space of states as a one-parameter family of Hilbert spaces and give the Bargmann–Fock and Jordan–Schwinger representations of it. Also, we give a simple description of the physical subspace as an infinite product of D12+ in the positive discreet series irreducible representations of the SU(1,1) group. We construct the map generated by the Hamiltonian between states at two different values of time and show how it produces the entanglement of left and right excitations. Next, we derive the formula of the entanglement entropy of the reduced density matrix for the ground state acted upon by the Hamiltonian map. Finally, we determine the asymptotic form of the entanglement entropy of a single mode bi-oscillator in the limit of large values of time.

Geometry of the Quantum States of Light in a Mach-Zehnder Interferometer

A geometric approach to the quantum states of light that is injected into a Mach- Zehnder interferometer is discussed. It is based on the Jordan-Schwinger representation of angular momentum and permits the identification of the photon states with cones in the threedimensional space that are rotated by the action of the optical devices. Some cases of the evolution loops, dynamical processes for which any state evolves cyclically, are presented.

Yang-Baxter relations with orthogonal or symplectic symmetry

Solutions of the Yang-Baxter relations with orthogonal or symplectic symmetry are studied emphasizing the analogies of both cases. Starting from Jordan-Schwinger representations the distinguishing features of the spinor representation and its symplectic analogon are shown. The corresponding L matrix and the spinorial R operator are discussed.

Wigner–Eckart theorem for the non-compact algebra 𝔰𝔩(2, ℝ)

The Wigner–Eckart theorem is a well known result for tensor operators of 𝔰𝔲(2) and, more generally, any compact Lie algebra. In this paper, the theorem will be generalized to the particular non-compact case of 𝔰𝔩(2, ℝ). In order to do so, recoupling theory between representations that are not necessarily unitary will be studied, namely, between finite-dimensional and infinite-dimensional representations. As an application, the Wigner–Eckart theorem will be used to construct an analogue of the Jordan–Schwinger representation, previously known only for representations in the discrete class, which also covers the continuous class.

Jordan-Schwinger Representations and Factorised Yang-Baxter Operators

The construction elements of the factorised form of the Yang-Baxter R operator acting on generic representations of q-deformed sl(n+1) are studied. We rely on the iterative construction of such representations by the restricted class of Jordan-Schwinger representations. The latter are formulated explicitly. On this basis the parameter exchange and intertwining operators are derived.

Iterative Construction of U q (s ℓ(n+1)) Representations and Lax Matrix Factorisation

The iterative construction of a generic representation of gl(n + 1) or of the trigonomentric deformation of its enveloping algebra is conveniently formulated in terms of Lax matrices. The Lax matrix of the constructed representation factorises into parts determined by the Lax matrix of a generic representation of the algebra with reduced rank and others appearing in the factorised expression of the Lax matrix of the special Jordan–Schwinger representation.

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.

A natural basis for spinor and vector fields on the noncommutative sphere

The product of two Heisenberg–Weil algebras contains the Jordan–Schwinger representation of su(2). This algebra is quotiented by the square-root of the Casimir to produce a nonassociative algebra denoted by Ψ. This algebra may be viewed as the right-module over one of its associative subalgebras which corresponds to the algebra of scalar fields on the noncommutative sphere. It is now possible to interpret other subspaces as the space of spinor or vector fields on the noncommutative sphere. A natural basis of Ψ is given which may be interpreted as the deformed entries in the rotation matrices of SU(2).

Quantum stereographic projection and the homographic oscillator.

The quantum deformation created by the stereographic mapping from ${\mathit{S}}_{2}$ to C is studied. It is shown that the resulting algebra is locally isomorphic to su(2) and is an unconventional deformation of which the undeformed limit is a contraction onto the harmonic oscillator algebra. The deformation parameter is given naturally by the central invariant of the embedding su(2). The deformed algebra is identified as a member of a larger class of quartic q oscillators. We next study the deformations in the corresponding Jordan-Schwinger representation of two independent deformed oscillators and solve for the deforming transformation. The invertibility of this transformation guarantees an implicit coproduct law which is also discussed. Finally we discuss the analogy between Poincar\'e's geometric interpretation of the quantum Stokes parameters of polarization and the stereographic projection as an important physical application of the latter. \textcopyright{} 1996 The American Physical Society.

The Jordan–Schwinger representations of Cayley–Klein groups. I. The orthogonal groups

The Cayley–Klein groups are defined as the groups that are obtained by the contractions and analytical continuations of the orthogonal groups. The Jordan–Schwinger representations of Cayley–Klein groups are discussed based on the mixed sets of creation and annihilation operators of boson or fermion type. The matrix elements of finite group transformations are obtained in the bases of coherent and Fock states.

The Jordan–Schwinger representations of Cayley–Klein groups. III. The symplectic groups

The symplectic Cayley–Klein groups are defined as the groups that are obtained by the contractions and analytical continuations of the classical symplectic groups. The Jordan–Schwinger representations of the groups under consideration are discussed based on the mixed sets of creation and annihilation operators of boson or fermion type. The matrix elements of finite group transformations are obtained in the bases of coherent states.

The Jordan–Schwinger representations of Cayley–Klein groups. II. The unitary groups

The unitary Cayley–Klein groups are defined as the groups that are obtained by the contractions and analytical continuations of the special unitary groups. The Jordan–Schwinger representations of the groups under consideration are discussed based on the mixed sets of creation and annihilation operators of boson or fermion type. The matrix elements of finite group transformations are obtained in the bases of coherent states.

Theorems on the Jordan–Schwinger representations of Lie algebras

The Jordan–Schwinger representations of Lie algebras are discussed based on the mixed sets of creation and annihilation operators of boson or fermion type. When the representation is Hermitian, the Lie algebra is shown to be unitary for the fermion case and pseudounitary for the boson case. It is also shown that the representation of a Lie algebra leads to a projective representation of the Lie group corresponding to the Lie algebra.