716 research outputs found
The Quantum Deformed Dirac Equation from the k-Poincare` Algebra
In this letter we derive a deformed Dirac equation invariant under the
k-Poincare` quantum algebra. A peculiar feature is that the square of the
k-Dirac operator is related to the second Casimir (the k-deformed squared
Pauli-Lubanski vector). The ``spinorial'' realization of the k-Poincare` is
obtained by a contraction of the coproduct of the real form of SO_q(3,2) using
the 4-dimensional representation which results to be, up some scalar factors,
the same of the undeformed algebra in terms of the usual gamma matrices.Comment: 6 pages, Late
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 , for an alternative countable basis . 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 gives rise to the second order
differential equation that defines the corresponding family of orthogonal
polynomials. Thus, the Weyl-Heisenberg algebra with for
Hermite polynomials and with for Laguerre and
Legendre polynomials are obtained.
Starting from the orthogonal polynomials the Lie algebra is extended both to
the whole space of the functions and to the corresponding
Universal Enveloping Algebra and transformation group. Generalized coherent
states from each vector in the space and, in particular,
generalized coherent polynomials are thus obtained.Comment: 11 page
Quantum -Poincare in Any Dimensions
The -deformation of the D-dimensional Poincar\'e algebra
with any signature is given. Further the quadratic Poisson brackets, determined
by the classical -matrix are calculated, and the quantum Poincar\'e group
"with noncommuting parameters" is obtained.Comment: (PLAIN TeX, 10 pp.
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