Riccardo Giachetti

Three‐dimensional quantum groups from contractions of SU(2)q

Contractions of Lie algebras and of their representations are generalized to define new quantum groups. An explicit and complete exposition is made for the one‐dimensional Heisenberg H(1)q and the two‐dimensional Euclidean quantum group E(2)q obtained by contracting SU(2)q.

The quantum Heisenberg group H(1)q

The structure of the quantum Heisenberg group is studied in the two different frameworks of the Lie algebra deformations and of the quantum matrix pseudogroups. The R‐matrix connecting the two approaches, together with its classical limit r, are explicitly calculated by using the contraction technique and the problems connected with the limiting procedure discussed. Some unusual properties of the quantum enveloping Heisenberg algebra are shown.

The three‐dimensional Euclidean quantum group E(3)q and its R‐matrix

A contraction procedure starting from SO(4)q is used to determine the quantum analog E(3)q of the three‐dimensional Euclidean group and the structure of its representations. A detailed analysis of the contraction of the R‐matrix is then performed and its explicit expression has been found. The classical limit of R is shown to produce an integrable dynamical system. By means of the R‐matrix the pseudogroup of the noncommutative representative functions is considered. It will finally be shown that a further contraction made on E(3)q produces the two‐dimensional Galilei quantum group and this, in turn, can be used to give a new realization of E(3)q and E(2,1)q.

The effective potential and effective Hamiltonian in quantum statistical mechanics

An overview on the theoretic formalism and up to date applications in quantum condensed matter physics of the effective potential and effective Hamiltonian methods is given. The main steps of their unified derivation by the so-called pure quantum self-consistent harmonic approximation (PQSCHA) are reported and explained. What makes this framework attractive is its easy implementation as well as the great simplification in obtaining results for the statistical mechanics of complicated quantum systems. Indeed, for a given quantum system the PQSCHA yields an effective system, i.e. an effective classical Hamiltonian with dependence on h(cross) and beta and classical-like expressions for the averages of observables, that has to be studied by classical methods. Anharmonic single-particle systems are analysed in order to get insight into the physical meaning of the PQSCHA, and its extension to the investigation of realistic many-body systems is pursued afterwards. The power of this approach is demonstrated through a collection of applications in different fields, such as soliton theory, rare gas crystals and magnetism. Eventually, the PQSCHA allows us also to approach quantum dynamical properties.

Quantization of a class of piecewise affine transformations on the torus

We present a unified framework for the quantization of a family of discrete dynamical systems of varying degrees of “chaoticity”. The systems to be quantized are piecewise affine maps on the two-torus, viewed as phase space, and include the automorphisms, translations and skew translations. We then treat some discontinuous transformations such as the Baker map and the sawtooth-like maps. Our approach extends some ideas from geometric quantization and it is both conceptually and calculationally simple.

Nonlinear wave coupling in type IV solar radio bursts

In order to explain a fine structure of parallel ridges in stationary type IV continua, the emission due to the coupling of electrostatic upper hybrid waves and Bernstein waves at the sum frequency of the upper hybrid and harmonics of the gyro frequency has been calculated. If the energy density of these electrostatic waves is of the order of 10-3 times the thermal energy density, then the observed zebra pattern can be emitted by a region with a diameter of ∼ 103 km.

A model for a stable coronal loop

We present here a new plasma-physics model of a stable active-region arch which corresponds to the structure observed in the EUV. Pressure gradients are seen, so that the equilibrium magnetic field must depart from the force-free form valid in the surrounding corona. We take advantage of the data and of the approximate cylindrical symmetry to develop a modified form of the commonly assumed sheared-spiral structure. The dynamic MHD behavior of this new pressure/field model is then evaluated by the Newcomb criterion, taken from controlled-fusion physics, and the results show short-wavelength stability in a specific parameter range. Thus we demonstrate the possibility, for pressure profiles with widths of the order of the magnetic-field scale, that such arches can persist for reasonable periods. Finally, the spatial proportions and magnetic fields of a characteristic stable coronal loop are described.

The structure of coronal magnetic loops

We present here a model, based on observations, for the magnetic-field equilibrium of a cool coronal loop. The pressure structure, taken from the Harvard/Skylab EUV data, is used to modify the usual force-free-field form in quasi-cylindrical symmetry. The resulting field, which has the same direction but different strength, is calculated and its variation displayed. Finally, localized interchange stability is evaluated and discussed, as the first step in a subsequent complete magnetohydrodynamic-stability analysis.

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.

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.