E. Sorace

The quantum deformed Dirac equation from the k-Poincaré 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.

Freeq-Schrödinger equation from homogeneous spaces of the 2-dim Euclidean quantum group

After a preliminary review of the definition and the general properties of the homogeneous spaces of quantum groups, the quantum hyperboloidqH and the quantum planeqP are determined as homogeneous spaces ofFq(E(2)). The canonical action ofEq(2) is used to define a naturalq-analog of the free Schrödinger equation, that is studied in the momentum and angular momentum bases. In the first case the eigenfunctions are factorized in terms of products of twoq-exponentials. In the second case we determine the eigenstates of the unitary representation, which, in theqP case, are given in terms of Hahn-Exton functions. Introducing the universalT-matrix forEq(2) we prove that the Hahn-Exton as well as Jacksonq-Bessel functions are also obtained as matrix elements ofT, thus giving the correct extension to quantum groups of well known methods in harmonic analysis.After a preliminary review of the definition and the general properties of the homogeneous spaces of quantum groups, the quantum hyperboloid qH and the quantum plane qP are determined as homogeneous spaces of Fq(E(2)). The canonical action of Eq(2) is used to define a natural q-analog of the free Schrodinger equation, that is studied in the momentum and angular momentum bases. In the first case the eigenfunctions are factorized in terms of products of two q-exponentials. In the second case we determine the eigenstates of the unitary representation, which, in the qP case, are given in terms of Hahn-Exton functions. Introducing the universal T-matrix for Eq(2) we prove that the Hahn-Exton as well as Jackson q-Bessel functions are also obtained as matrix elements of T, thus giving the correct extension to quantum groups of well known methods in harmonic analysis.

Exponential mapping for non-semisimple quantum groups

The concept of a universal T matrix, introduced by Fronsdal and Galindo (1993) in the framework of quantum groups, is discussed here as a generalization of the exponential mapping. New examples related to inhomogeneous quantum groups of physical interest are developed, the duality calculations are explicitly presented and it is found that in some cases the universal T matrix, as for Lie groups, is expressed in terms of usual exponential series.

An R-matrix approach to the quantization of the Euclidean group E(2)

The R-matrices for two different deformations of the Euclidean group E(2), calculated in a two-dimensional representation, are used to determine the deformed Hopf algebra of the representative functions. The duality of the latter with the initial quantum algebras is explicitly proved and the relationship between the two quantum groups is discussed and clarified.

SO q (n+1,n−1) as a real form of SO q (2n,ℂ)

Quantum pseudo-orthogonal groups SOq(n+1,n−1) are defined as real forms of quantum orthogonal groups SOq(n+1,n−1) by means of a suitable antilinear involution. In particular, the casen=2 gives a quantized Lorentz group.

Quantum Galilei group as symmetry of magnons.

Inhomogeneous quantum groups are shown to be an effective algebraic tool in the study of integrable systems and to provide solutions equivalent to the Bethe ansatz. The method is illustrated on the 1D Heisenberg ferromagnet whose symmetry is shown to be the quantum Galilei group Gamma_q(1) here introduced. Both the single magnon and the s=1/2 bound states of n-magnons are completely described by the algebra.

Quantum groups of motion and rotational spectra of heavy nuclei

Abstract A simple model of rotator based on the q -Poincare energy mass relation describes the rotational levels of even-even nuclei. Owing to the presence of the q -deformation, although the rotational velocities result is of the order of 10 −3 c , the behaviour is completely different from the usual rigid rotator. The quantum deformation parameter introduces a time scale that is of the order of the characteristic time of the strong interaction. The second free parameter, the radius of the nucleus, gives the right nuclear dimensions. The two-parameter energy fit is very good, even in comparison with the su q (2) based model. Although describing only single band effects, the model gives the correct experimental values where backbending or upbending occur.

DEFORMATION QUANTIZATION OF THE HEISENBERG GROUP

A *-product compatible with the comultiplication of the Hopf algebra of the functions on the Heisenberg group is determined by deforming a coboundary Lie-Poisson structure defined by a classicalr-matrix satisfying the modified Yang-Baxter equation. The corresponding quantum group is studied and itsR-matrix is explicitly calculated.

Heisenberg XXZ model and quantum Galilei group

The 1D Heisenberg spin model with an anisotropy of the XXZ type is analysed in terms of the symmetry given by the quantum Galilei group Gamma q(1). For a chain with an infinite number of sites the authors show that the magnon excitations and the s=1/2, n-magnon bound states are determined by the algebra. In this case the Gamma q(1) symmetry provides a description naturally compatible with the Bethe ansatz. The recurrence relations determined by Gamma q(1) permit one to express the energy of the n-magnon bound states in a closed form in terms of Tchebischeff polynomials.

Canonical theory of relativistic interactions

SummaryA canonical theory of relativistic interactions is formulated on the basis of Whittaker equations and the physical inconsistency of the no-interaction theorem is shown. Three model Hamiltonians forN + 1 interacting particles are studied. Finally it is briefly considered the case of a charged particle in an external electromagnetic field.RiassuntoSi formula sulla base delle equazioni di Whittaker una teoria canonica per le interazioni relativistiche e si mostra l’inconsistenza fisica del teorema di non interazione. Si studiano tre modelli di hamiltoniana perN + 1 particelle interagenti. Infine si considera brevemente il caso di una particella carica in un campo elettromagnetico esterno.РеэюмеНа основе уравнений Уиттекера формулируется каноническая теория релятивиствких вэаимодействий. Отмечается фиэическая противоречивость теоремы от отсутствии вэаимодействия. Исследуются три модельных Гамильтониана дляN + 1 вэаимодействуюшей частицы. В эаключение рассматривается случай эаряженной частицы во внещнем злектромагнитном поле.