358 research outputs found

### Coherent Orthogonal Polynomials

We discuss as a fundamental characteristic of orthogonal polynomials like the
existence of a Lie algebra behind them, can be added to their other relevant
aspects. At the basis of the complete framework for orthogonal polynomials we
put thus --in addition to differential equations, recurrence relations, Hilbert
spaces and square integrable functions-- Lie algebra theory.
We start here from the square integrable functions on the open connected
subset of the real line whose bases are related to orthogonal polynomials. All
these one-dimensional continuous spaces allow, besides the standard uncountable
basis ${|x>}$, for an alternative countable basis ${|n>}$. The matrix elements
that relate these two bases are essentially the orthogonal polynomials: Hermite
polynomials for the line and Laguerre and Legendre polynomials for the
half-line and the line interval, respectively.
Differential recurrence relations of orthogonal polynomials allow us to
realize that they determine a unitary representation of a non-compact Lie
algebra, whose second order Casimir ${\cal C}$ gives rise to the second order
differential equation that defines the corresponding family of orthogonal
polynomials. Thus, the Weyl-Heisenberg algebra $h(1)$ with ${\cal C}=0$ for
Hermite polynomials and $su(1,1)$ with ${\cal C}=-1/4$ for Laguerre and
Legendre polynomials are obtained.
Starting from the orthogonal polynomials the Lie algebra is extended both to
the whole space of the ${\cal L}^2$ functions and to the corresponding
Universal Enveloping Algebra and transformation group. Generalized coherent
states from each vector in the space ${\cal L}^2$ and, in particular,
generalized coherent polynomials are thus obtained.Comment: 11 page

### Inomogeneous Quantum Groups as Symmetries of Phonons

The quantum deformed (1+1) Poincare' algebra is shown to be the kinematical
symmetry of the harmonic chain, whose spacing is given by the deformation
parameter. Phonons with their symmetries as well as multiphonon processes are
derived from the quantum group structure. Inhomogeneous quantum groups are thus
proposed as kinematical invariance of discrete systems.Comment: 5 pags. 0 fig

### Heisenberg XXZ Model and Quantum Galilei Group

The 1D Heisenberg spin chain with anisotropy of the XXZ type is analyzed in
terms of the symmetry given by the quantum Galilei group Gamma_q(1). We show
that the magnon excitations and the s=1/2, n-magnon bound states are determined
by the algebra. Thus the Gamma_q(1) symmetry provides a description that
naturally induces the Bethe Ansatz. The recurrence relations determined by
Gamma_q(1) permit to express the energy of the n-magnon bound states in a
closed form in terms of Tchebischeff polynomials.Comment: (pag. 10

### Bases in Lie and Quantum Algebras

Applications of algebras in physics are related to the connection of
measurable observables to relevant elements of the algebras, usually the
generators. However, in the determination of the generators in Lie algebras
there is place for some arbitrary conventions. The situation is much more
involved in the context of quantum algebras, where inside the quantum universal
enveloping algebra, we have not enough primitive elements that allow for a
privileged set of generators and all basic sets are equivalent. In this paper
we discuss how the Drinfeld double structure underlying every simple Lie
bialgebra characterizes uniquely a particular basis without any freedom,
completing the Cartan program on simple algebras. By means of a perturbative
construction, a distinguished deformed basis (we call it the analytical basis)
is obtained for every quantum group as the analytical prolongation of the above
defined Lie basis of the corresponding Lie bialgebra. It turns out that the
whole construction is unique, so to each quantum universal enveloping algebra
is associated one and only one bialgebra. In this way the problem of the
classification of quantum algebras is moved to the classification of
bialgebras. In order to make this procedure more clear, we discuss in detail
the simple cases of su(2) and su_q(2).Comment: 16 pages, Proceedings of the 5th International Symposium on Quantum
Theory and Symmetries QTS5 (July 22-28, 2007, Valladolid (Spain)

### Identical Particles and Permutation Group

Second quantization is revisited and creation and annihilation operators
areshown to be related, on the same footing both to the algebra h(1), and to
the superalgebra osp(1|2) that are shown to be both compatible with Bose and
Fermi statistics.
The two algebras are completely equivalent in the one-mode sector but,
because of grading of osp(1|2), differ in the many-particle case.
The same scheme is straightforwardly extended to the quantum case h_q(1) and
osp_q(1|2).Comment: 8 pages, standard TEX, DFF 205/5/94 Firenz

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