193 research outputs found
Quadratic Poisson algebras for two dimensional classical superintegrable systems and quadratic associative algebras for quantum superintegrable systems
The integrals of motion of the classical two dimensional superintegrable
systems with quadratic integrals of motion close in a restrained quadratic
Poisson algebra, whose the general form is investigated. Each classical
superintegrable problem has a quantum counterpart, a quantum superintegrable
system. The quadratic Poisson algebra is deformed to a quantum associative
algebra, the finite dimensional representations of this algebra are calculated
by using a deformed parafermion oscillator technique. It is shown that, the
finite dimensional representations of the quadratic algebra are determined by
the energy eigenvalues of the superintegrable system. The calculation of energy
eigenvalues is reduced to the solution of algebraic equations, which are
universal for all two dimensional superintegrable systems with quadratic
integrals of motion.Comment: 28 pages, Late
Finite Dimensional Representations of Quadratic Algebras with Three Generators and Applications
The finite dimensional representations of associative quadratic algebras with
three generators are investigated by using a technique based on the deformed
parafermionic oscillator algebra. One application on the calculation of the
eigenvalues of the two-dimensional superintegrable systems is discussed.Comment: 14 pages, LaTeX 2e, Talk given at the VI International Wigner
Symposium, 16-22 August 1999, Instabul, Turke
Parabosonic and parafermionic algebras. Graded structure and Hopf structures
Parabosonic and parafermionic algebras are
described as quotients of the tensor algebras of suitably choosen vector
spaces. Their (super-) Lie algebraic structure and consequently their (super-)
Hopf structure is shortly discussed. A bosonisation-like construction is
presented, which produces an ordinary Hopf algebra
starting from the super Hopf algebra .Comment: 9 pages. Contribution to the 6th Paanhellenic conference in Algebra
and Number Theory, Aristotle University of Thessaloniki, Thessaloniki, 10-12
June 200
Ternary Poisson algebra for the non degenerate three dimensional Kepler Coulomb potential
In the three dimensional flat space any classical Hamiltonian, which has five
functionally independent integrals of motion, including the Hamiltonian, is
characterized as superintegrable. Kalnins, Kress and Miller have proved that,
in the case of non degenerate potentials, i.e potentials depending linearly on
four parameters, with quadratic symmetries, posses a sixth quadratic integral,
which is linearly independent of the other integrals. The existence of this
sixth integral imply that the integrals of motion form a ternary
parafermionic-like quadratic Poisson algebra with five generators. The Kepler
Coulomb potential that was introduced by Verrier and Evans is a special case of
superintegrable system, having two independent integrals of motion of fourth
order among the remaining quadratic ones. The corresponding Poisson algebra of
integrals is a quadratic one, having the same special form, characteristic to
the non degenerate case of systems with quadratic integrals.Comment: 13 Pages, Contribution to the 4th Workshop on Group Analysis of
Differential Equations and Integrable Systems, Protaras, Cyprus, Oct. 200
Generalized deformed oscillator for vortices in superfluid films
The algebra of observables of a system of two identical vortices in a
superfluid thin film is described as a generalized deformed oscillator with a
structure function containing a linear (harmonic oscillator) term and a
quadratic term. In contrast to the deformed oscillators occuring in other
physical systems (correlated fermion pairs in a single- nuclear shell, Morse
oscillator), this oscillator is not amenable to perturbative treatment and
cannot be approximated by quons. From the mathematical viewpoint, this
oscillator provides a novel boson realization of the algebra su(1,1).Comment: 12 pages, LaTe
Quadratic algebras for three dimensional non degenerate superintegrable systems with quadratic integrals of motion
The three dimensional superintegrable systems with quadratic integrals of
motion have five functionally independent integrals, one among them is the
Hamiltonian. Kalnins, Kress and Miller have proved that in the case of non
degenerate potentials there is a sixth quadratic integral, which is linearly
independent of the other integrals. The existence of this sixth integral imply
that the integrals of motion form a ternary {parafermionic-like} quadratic
Poisson algebra with five generators. We show that in all the non degenerate
cases (with one exception) there are at least two subalgebras of three
integrals having a Poisson quadratic algebra structure, which is similar to the
two dimensional case.Comment: 21 pages, Detailed version of the talk given at the XXVII Colloquium
on Group Theoretical Methods in Physics, Yerevan, Armenia, Aug. 200
Bosonisation and Parastatistics: An Example and an Alternative Approach
Definitions of the parastatistics algebras and known results on their Lie
(super)algebraic structure are reviewed. The notion of super-Hopf algebra is
discussed. The bosonisation technique for switching a Hopf algebra in a braided
category (: a quasitriangular Hopf algebra) into an
ordinary Hopf algebra is presented and it is applied in the case of the
parabosonic algebra. A bosonisation-like construction is also introduced for
the same algebra and the differences are discussed.Comment: 13 pages, Contribution to "AGMF: Algebra, Geometry, and Mathematical
Physics", Baltic-Nordic Workshop: Lund, 12-14 October, 200
Quantum Groups and Their Applications in Nuclear Physics
Quantum algebras are a mathematical tool which provides us with a class of
symmetries wider than that of Lie algebras, which are contained in the former
as a special case. After a self-contained introduction to the necessary
mathematical tools (q-numbers, q-analysis, q-oscillators, q-algebras), the
SUq(2) rotator model and its extensions, the construction of deformed exactly
soluble models (u(3)>so(3) model, Interacting Boson Model, Moszkowski model),
the 3-dimensional q-deformed harmonic oscillator and its relation to the
nuclear shell model, the use of deformed bosons in the description of pairing
correlations, and the symmetries of the anisotropic quantum harmonic oscillator
with rational ratios of frequencies, which underly the structure of
superdeformed and hyperdeformed nuclei, are discussed in some detail. A brief
description of similar applications to the structure of molecules and of atomic
clusters, as well as an outlook are also given.Comment: 82 pages, LaTeX, review articl
Parafermionic and Generalized Parafermionic Algebras
The general properties of the ordinary and generalized parafermionic algebras
are discussed. The generalized parafermionic algebras are proved to be
polynomial algebras. The ordinary parafermionic algebras are shown to be
connected to the Arik-Coon oscillator algebras.Comment: 12 pages, LaTeX. Presented at the International Conference on
Mathematical Physics (Istanbul 1997). To appear in the Proceeding
Quasi-Exactly Soluble Potentials and Deformed Oscillators
It is proved that quasi-exactly soluble potentials corresponding to an
oscillator with harmonic, quartic and sextic terms, for which the lowest
levels of a given parity can be determined exactly, may be approximated by WKB
equivalent potentials corresponding to deformed anharmonic oscillators of
SU(1,1) symmetry, which have been used for the description of vibrational
spectra of diatomic molecules. This connection allows for the immediate
approximate determination of the levels of the same parity lying above the
lowest known levels, as well as of all levels of the opposite parity.
Such connections are not possible in the cases of the q-deformed oscillator,
the Q-deformed oscillator, and the modified P\"oschl-Teller potential with
SU(1,1) symmetry.Comment: 15 pages, LaTeX; to appear in the Proceedings of the 6th Hellenic
Symposium on Nuclear Physics (Piraeus, Greece, 26-27 May 1995
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