Higgs Algebra Research Articles

On the symmetry algebra of a one-dimensional quantum-mechanical oscillator on a hyperbola

The quantum-mechanical problem of a harmonic oscillator on a hyperbola as a one-dimensional space of constant negative curvature is considered in this article. A generalization to the singular oscillator model in the context of one-dimensional Cayley – Klein geometries is given by the factorization method. The energy spectrum and wave functions of stationary states are found having the curvature of space as a parameter. For the energy levels of the singular oscillator, the effect of non-zero curvature is clearly manifested through a positive or negative term, depending on the sign of the curvature, which is quadratic in the level number. The results obtained are consistent with those previously published. The dynamical symmetry of the problem is shown explicitly as a quadratic Hahn algebra QH(3) or its isomorphic Higgs algebra.

The q-analogue of the Higgs algebra

In this paper, the q-generalization of the Higgs algebra is considered. The realization of this algebra is shown in an explicit form using a nonlinear transformation of the creation-annihilation operators of the q-harmonic oscillator. This transformation is the performance of two operations, namely, a “correction” using a function of the original Hamiltonian, and raising to the fourth power the creation and annihilation operators of a q-harmonic oscillator. The choice of the “correcting” function is justified by the standard form of commutation relations for the operators of the metaplectic realization Uq(SU(1,1)). Further possible directions of research are briefly discussed to summarize the results obtained. The first direction is quite obvious. It is the consideration of the problem when the dimension of the operator space increases or for any value N. The second direction can be associated with the analysis of the relationship between q-generalizations of the Higgs and Hahn algebras.

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.

Pizzetti formula on the Grassmannian of 2-planes

This paper is devoted to the role played by the Higgs algebra $$H_3$$ in the generalisation of classical harmonic analysis from the sphere $$S^{m-1}$$ to the (oriented) Grassmann manifold $${{\text {Gr}}}_o(m,2)$$ of 2-planes. This algebra is identified as the dual partner (in the sense of Howe duality) of the orthogonal group $${\text {SO}}(m)$$ acting on functions on the Grassmannian. This is then used to obtain a Pizzetti formula for integration over this manifold. The resulting formulas are finally compared to formulas obtained earlier for the Pizzetti integration over Stiefel manifolds, using an argument involving symmetry reduction.

Howe duality of Higgs – Hahn algebra for 8D harmonic oscillator

In the light of the Howe duality, two different, but isomorphic representations of one algebra as Higgs algebra and Hahn algebra are considered in this article. The first algebra corresponds to the symmetry algebra of a harmonic oscillator on a 2-sphere and a polynomially deformed algebra SU(2), and the second algebra encodes the bispectral properties of corresponding homogeneous orthogonal polynomials and acts as a symmetry algebra for the Hartmann and certain ring-shaped potentials as well as the singular oscillator in two dimensions. The realization of this algebra is shown in explicit form, on the one hand, as the commutant O(4) ⊕ O(4) of subalgebra U(8) in the oscillator representation of universal algebra U (u(8)) and, on the other hand, as the embedding of the discrete version of the Hahn algebra in the double tensor product SU(1,1) ⊗ SU(1,1). These two realizations reflect the fact that SU(1,1) and U(8) form a dual pair in the state space of the harmonic oscillator in eight dimensions. The N-dimensional, N-fold tensor product SU(1,1)⊗N аnd q-generalizations are briefly discussed.

The q-Higgs and Askey–Wilson algebras

A q-analogue of the Higgs algebra, which describes the symmetry properties of the harmonic oscillator on the 2-sphere, is obtained as the commutant of the oq1/2(2)⊕oq1/2(2) subalgebra of oq1/2(4) in the q-oscillator representation of the quantized universal enveloping algebra Uq(u(4)). This q-Higgs algebra is also found as a specialization of the Askey–Wilson algebra embedded in the tensor product Uq(su(1,1))⊗Uq(su(1,1)). The connection between these two approaches is established on the basis of the Howe duality of the pair (oq1/2(4),Uq(su(1,1))).

The Higgs and Hahn algebras from a Howe duality perspective

The Hahn algebra encodes the bispectral properties of the eponymous orthogonal polynomials. In the discrete case, it is isomorphic to the polynomial algebra identified by Higgs as the symmetry algebra of the harmonic oscillator on the 2-sphere. These two algebras are recognized as the commutant of a o(2)⊕o(2) subalgebra of o(4) in the oscillator representation of the universal algebra U(u(4)). This connection is further related to the embedding of the (discrete) Hahn algebra in U(su(1,1))⊗U(su(1,1)) in light of the dual action of the pair (o(4),su(1,1)) on the state vectors of four harmonic oscillators. The two-dimensional singular oscillator is naturally seen by dimensional reduction to have the Higgs algebra as its symmetry algebra.

Statistical Aspects of Coherent States of the Higgs Algebra

We construct and study various aspects of coherent states of a polynomial angular momentum algebra. The coherent states are constructed using a new unitary representation of the nonlinear algebra. The new representation involves a parameter γ that shifts the eigenvalues of the diagonal operator J0.

Quantum superintegrable Zernike system

We consider the differential equation that Zernike proposed to classify aberrations of wavefronts in a circular pupil, whose value at the boundary can be nonzero. On this account, the quantum Zernike system, where that differential equation is seen as a Schrödinger equation with a potential, is special in that it has a potential and a boundary condition that are not standard in quantum mechanics. We project the disk on a half-sphere and there we find that, in addition to polar coordinates, this system separates into two additional coordinate systems (non-orthogonal on the pupil disk), which lead to Schrödinger-type equations with Pöschl-Teller potentials, whose eigen-solutions involve Legendre, Gegenbauer, and Jacobi polynomials. This provides new expressions for separated polynomial solutions of the original Zernike system that are real. The operators which provide the separation constants are found to participate in a superintegrable cubic Higgs algebra.

Superintegrable classical Zernike system

We consider the differential equation that Zernike proposed to classify aberrations of wavefronts in a circular pupil, as if it were a classical Hamiltonian with a non-standard potential. The trajectories turn out to be closed ellipses. We show that this is due to the existence of higher-order invariants that close into a cubic Higgs algebra. The Zernike classical system thus belongs to the class of superintegrable systems. Its Hamilton-Jacobi action separates into three vertical projections of polar coordinates of sphere, polar, and equidistant coordinates on half-hyperboloids, and also in elliptic coordinates on the sphere.

Holstein–Primakoff realization of Higgs algebra and its q-extension

In this paper, Holstein–Primakoff realization of Higgs algebra is obtained by using the linear (or quadratic) deformation of Heisenberg algebra and q-deformed Higgs algebra is proposed. Some applications such as Kepler problem in a two-dimensional curved space and SUSY quantum mechanics are also discussed.

Higgs Algebraic Symmetry of Screened System in a Spherical Geometry

The orbits and the dynamical symmetries for the screened Coulomb potentials and isotropic harmonic oscillators have been studied by Wu and Zeng [Z.B. Wu and J.Y. Zeng, Phys. Rev. A 62 (2000) 032509]. We find similar properties in the corresponding systems in a spherical space, whose dynamical symmetries are described by Higgs algebra. There exist extended Runge—Lenz vector for screened Coulomb potentials and extended quadruple tensor for screened harmonic oscillators. They, together with angular momentum, constitute the generators of the geometrical symmetry group. Moreover, there exist an infinite number of closed orbits for suitable angular momentum values, and we give the equations of the classical orbits. The eigenenergy spectrum and corresponding eigenstates in these systems are derived.

Symmetry and degeneracy of the curved Coulomb potential on the S3 ball

The ‘curved’ Coulomb potential on the S3 ball, whose isometry group is SO(4), takes the form of a cotangent function, and when added to the four-dimensional squared angular momentum operator, one of the so(4) Casimir invariants, a Hamiltonian is obtained which describes a perturbance of the free geodesic motion that results in several peculiar aspects. The spectrum of such a motion has been studied on various occasions and is known to unexpectedly carry so(4) degeneracy patterns despite the non-commutativity of the perturbance with the Casimir operator. We suggest here an explanation for this behavior in designing a set of operators which close the so(4) algebra and whose Casimir invariant coincides with the Hamiltonian of the perturbed motion at the level of the eigenvalue problem. The above operators are related to the canonical geometric SO(4) generators on S3 by a non-unitary similarity transformation of the scaling type. In this fashion, we identify a complementary option to the deformed dynamical so(4) Higgs algebra constructed in terms of the components of the ordinary angular momentum and a related Runge–Lenz vector.

Beyond fuzzy spheres

We study polynomial deformations of the fuzzy sphere, specifically given by the cubic or the Higgs algebra. We derive the Higgs algebra by quantizing the Poisson structure on a surface in . We find that several surfaces, differing by constants, are described by the Higgs algebra at the fuzzy level. Some of these surfaces have a singularity and we overcome this by quantizing this manifold using coherent states for this nonlinear algebra. This is seen in the measure constructed from these coherent states. We also find the star product for this non-commutative algebra as a first step in constructing field theories on such fuzzy spaces.

Higgs algebraic symmetry in the two-dimensional Dirac equation

The dynamical symmetry algebra of the two-dimensional Dirac Hamiltonian with equal scalar and vector Smorodinsky-Winternitz potentials is constructed. It is the Higgs algebra, a cubic polynomial generalization of SU(2). With the help of the Casimir operators, the energy levels are derived algebraically.

Geometric Phase in a Time-Dependent System with Higgs Algebra Structure

By using of the invariant theory, we have studied the geometric phase in a time-dependent system with Higgs algebra structure, the dynamical and geometric phases are given, respectively. The Aharonov-Anandan phase is also obtained under the cyclical evolution. The disappearing condition of the geometric phase is given.

Dynamical symmetries of the Klein–Gordon equation

The dynamical symmetries of the two-dimensional Klein–Gordon equations with equal scalar and vector potentials (ESVPs) are studied. The dynamical symmetries are considered in the plane and the sphere, respectively. The generators of the SO(3) group corresponding to the Coulomb potential and the SU(2) group corresponding to the harmonic oscillator potential are derived. Moreover, the generators in the sphere construct the Higgs algebra. With the help of the Casimir operators, the energy levels of the Klein–Gordon systems are yielded naturally.

Generalized rotational Hamiltonians from nonlinear angular momentum algebras

Higgs algebras are used to construct rotational Hamiltonians. The correspondence between the spectrum of a triaxial rotor and the spectrum of a cubic Higgs algebra is demonstrated. It is shown that a suitable choice of the parameters of the polynomial algebra allows for a precise identification of rotational properties. The harmonic limit is obtained by a contraction of the algebra, leading to a linear symmetry.

Two-boson Realizations of the Polynomial Angular Momentum Algebra and Some Applications

In this paper two kinds of two-boson realizations of the polynomial angular momentum algebra are obtained by generalizing the well known Jordan–Schwinger realizations of the SU(2) and SU(1,1) algebras. Especially, for the Higgs algebra, an unitary realization and two nonunitary realizations, together with the properties of their respective acting spaces are discussed in detail. Furthermore, similarity transformations, which connect the nonunitary realizations with the unitary ones, are gained by solving the corresponding unitarization equations. As applications, the dynamical symmetry of the Kepler system in a two-dimensional curved space is studied and phase operators of the Higgs algebra are constructed.

Boson realizations of the polynomial angular momentum algebras with arbitrary powers and their unitarization

The inhomogeneous single-, two- and three-boson realizations of the more general polynomial angular momentum algebra SU Λ(2) are obtained from the Fock representations of SU Λ(2) that correspond to the indecomposable representation on the space of universal enveloping algebra U( SU Λ(2)) and to the induced representations on the quotient spaces U( SU Λ(2))/I i , with I i being some left ideals with respect to U( SU Λ(2)) . Then these boson realizations are unitarized by the corresponding similarity transformations. As examples, the explicit expressions for the similarity transformations used to unitarize the inhomogeneous single-boson realizations of SU 2(2) and the Higgs algebra are respectively calculated.