Table of contents

The q-deformed version of a two-dimensional toy interacting boson model (IBM) with the symmetry SU_q(3) supset SU_q(2) supset SO_q(2) is constructed. Energy spectra and transition matrix elements are calculated, the latter being found to be much more sensitive to q-deformation than the former. Arguments in favour of the q-generalization of the full IBM are given.

Explicit wave solutions are found for the electromagnetic field that develops in a family of strongly nonlinear dielectric media. The time frequency of oscillation omega , is shown to be linear in the corresponding 'wavenumber' k, which allows for a standing wave in a finite system. The velocity of propagation along trajectories of constant field in spacetime coordinates is found to be proportional to a power of the amplitude of the field. For the particular oscillatory solutions, this results in wiggly, rather than straight, characteristic lines.

The dynamics of the spin autocorrelation function and the relaxation of the magnetization in the Griffiths phase of the two-dimensional bond-diluted Ising model at the percolation threshold are studied using Monte Carlo techniques. The results resolve a previous ambiguity about the decay law.

The authors present a study of the XY chiral model on a Cayley tree in the presence of an external uniform magnetic field. They calculate the phase diagram at zero temperature from a corresponding 1D mapping. Ferromagnetic commensurate and incommensurate modulated phases as well as chaotic structures are present in the zero temperature limit. In this case they also obtain a devil's staircase and determine the route to chaos, which is in agreement with Feigenbaum's scenario. For finite temperatures, the phase diagrams are obtained from a 2D mapping. The chaotic behaviour is present only at low temperatures.

The critical behaviour of an integrable model of a spin- 1/2 chain with nearest-neighbour XXZ interaction and a competing three-spin interaction involving nearest and next-nearest neighbours is studied. The phase diagram at zero temperature is obtained. Methods from conformal field theory are used to compute the asymptotics of the spin-spin correlation functions.

The method of concatenation (the addition of precomputed shorter chains to the ends of a centrally generated longer chain) has permitted the extension of the exact series for C_N-the number of distinct configurations for self-avoiding walks of length N. The authors report on the leading exponent y and x_c (the reciprocal of the connectivity constant) for the 2D honeycomb lattice (42 terms) 1.3437, 0.541 1968; the 2D square lattice (30 terms) 1.3436, 0.379 0520; the 3D simple cubic lattice (23 terms) 1.161 932, 0.213 4987; the 4D hypercubic (18 terms) y approximately=1, 0.147 60 and the 5D hypercubic lattice (13 terms) y<or=1.025, 0.113 05. In addition they have also evaluated the leading correction terms: honeycomb Delta approximately=1, square Delta approximately=0.85, simple cubic Delta approximately=1.0 and the 4D hypercubic logarithmic correction with delta approximately=0.25.

The dynamics of cellular automata that are homogeneous and symmetric with respect to up-down symmetry is expressed by the probability of the appearance of different neighbourhoods on a lattice. The distribution function found in computer simulations is used to specify the differences in the set of cellular automata. The intrinsic structure of a rule has been proposed to explain the results obtained. The problem of whether or not automata are stable, the length of time needed to reach the stabilization and the type of stabilization, are also discussed.

The authors study d-dimensional lattice model systems whose dynamics consist of competing independent processes which may induce nonequilibrium steady states. Interesting phase transitions and critical phenomena ensue which are influenced, even dominated, by a sort of dynamical bond disorder. The latter, which may occur in natural systems, reminds one of the disorder in more familiar bond-diluted and other impure models, for example, percolation-like phenomena arise in some of the model cases. Exact solutions for d=1, some exact results for d<1, and a comparison with more standard, either quenched or annealed, magnetically diluted models are reported.

A scaling relation between remanent magnetization and excess energy of spin-glasses is tested numerically. Based on a simple domain picture of non-equilibrium states of spin-glasses a power law can be derived, which relates the excess energy to the remanent magnetization Delta E varies as M^x. The exponent x is related to the interface exponent y, which determines whether a spin-glass phase exists for T=0 or not. Therefore equilibrium properties can be determined by measurement of non-equilibrium quantities, x is calculated numerically for different temperatures T and the spatial dimensions d=2 and d=3. The numerical results are consistent with the suggested scaling relation.

A classical scattering system is chaotic if it possesses a fractal set of trapped unstable orbits, resulting in singular deflection functions. A scattering system is regular if it supports only a countable set of trapped unstable orbits. Its deflection functions are piecewise smooth with at most a countable number of scattering singularities caused by the trapped orbits. Despite the simple structure of the deflection functions, the Poincare scattering mapping (PSM) may be regular, hyperbolic or display mixed dynamics. Thus, the degree of chaoticity of the PSM serves as a finer scale for the discussion of the transition to chaotic scattering in the classical domain. In the quantum domain the authors show that the properties of the PSM determine the statistics of the eigenphases of the S-matrix, and that, if the PSM is hyperbolic, the eigenphases follow the statistics predicted by random matrix theory.

Properties of a model quantum scattering system displaying hard classical chaos-'elastic' scattering on a leaky surface of constant negative curvature-are analysed theoretically and serve to interpret previously obtained numerical results. The low energy scattering behaviour is shown to be influenced, in the usual fashion, by a bound state just below the scattering continuum threshold. A connection between the widths of the infinite number of simple poles of the S-matrix and the Lyapunov exponent for classical trajectories is analysed. At high energies, the scattering is characterized by fluctuations in the S-matrix (via its phase) and the time delay. Analytic expressions for the autocorrelation function of the S-matrix (via its phase) and the time delay. Analytic expressions for the autocorrelation function of the S-matrix and of the time delay are obtained using Montgomery's conjecture for the pair correlation function of the celebrated Riemann zeros whose values correspond to the positions of the S-matrix poles in momentum space. The autocorrelation function for the S-matrix is found to be Lorentzian asymptotically (at large energy differences Delta E), that is, to decrease as Delta E^-2, but that for the time delay is not. The distribution of fluctuations of S-matrix phases is likely to be Gaussian.

The authors introduce a pair of canonical-conjugate q-deformed operators D, X and discuss the relations between the deformed operators D, X and q-series. The realizations of some Lie symmetries, Heisenberg and quantum Heisenberg algebras are given for the operators D and X. They show that the q-analogous Hermite polynomials are representations of Heisenberg and quantum Heisenberg algebras realized in this way. When q is a root of unity, the properties of the q-analogous Hermite polynomials are also discussed.

A BCH formula for groups having faithful matrix representations is derived, by a decomposition of the general group element exp(M) into factors, each factor being an exponential involving the infinitesimal generators of the group. This is a generalization of the authors earlier work for SU(3).

The authors give some results on the problem of finding the Lie point-symmetries of autonomous systems of differential equations. In particular, they consider the case in which the nonlinear terms are resonant (in the sense of the Poincare procedure for reducing the system to normal form), and they show that Lie symmetries can be characterized in a useful form.

The authors give a complete expression for the propagator corresponding to the motion of a series of nonlocal harmonic oscillators under the influence of an arbitrary driving force. The method of derivation is based on a direct solution of the corresponding classical equation of motion. The time and end-point parameters are kept completely general throughout.

Some aspects of the contraction process SO₀(1,2) to Poincare are studied. The starting point is the choice of a suitable parametrization for the de Sitterian phase space SO₀(1,2)/SO(2) approximately=SU(1,1)/U(1). The authors show that the contraction to Poincare must be realized by restricting the Fock-Bargmann space to a specific subspace. This constraint is necessary to make the divergent terms disappear. In particular, the classical result according to which the discrete series representation of SU(1,1) contracts onto the Wigner representation P(m) is described at a global level.

A one-fluid, dissipative magnetohydrodynamic model of plasma equilibrium in a torus is considered. The equations include inertial forces, finite resistivity and viscosity, and a particle source which sustains the pressure gradient in the plasma; viscosity is described by the Braginskii operator. Plasma density, resistivity and viscosity coefficients are assumed to be uniform. A boundary-value problem in a general toroidal domain is formulated, no further assumption on the domain being made besides a sufficient regularity of its boundary. The system of equations is reduced to a problem with unknowns p, v, b (p denotes the scalar pressure, v the flow velocity, B the magnetic field). A functional setting of the equations is established and, generalizing the classical mathematical techniques adopted in the theory of viscous incompressible flow to investigate the solvability of the steady-state Navier-Stokes equations, a problem for weak solutions is formulated which is shown to be equivalent to solving a nonlinear equation in a separable Hilbert space.

Polyacetylene, regarded as one of the simplest systems bearing solitons which are coupled to fermions, is investigated at zero temperature. With lambda denoting the soliton size in units of the coherence length, solitons with lambda >>1 are considered. It is found, from the exact results of Nakahara et al (1990) that the soliton creation energy does not possess an expansion in powers of 1/ lambda , as would be expected of a gradient expansion. Rather, the leading terms in a large- lambda expansion are found to be a lambda +b lambda ^-1/2 with constant a and b. Calculations are presented for general soliton profiles. They show that the term linear in lambda is the soliton creation energy to zeroth order in gradients. The term b lambda ^-1/2 is shown to arise purely from bound states of the fermions (electrons) trapped by the soliton. The evaluation of the coefficient b requires the extraction of the finite difference between a divergent sum and integral, a procedure employed in the Casimir effect.

The fundamental problem of the energy and momentum of the self-fields of a moving charge in the classical theory of electromagnetism has not yet been solved to full satisfaction. The widely-held belief that the energy and momentum of the electromagnetic field of a moving charge should behave as components of a 4-vector under a Lorentz transformation, is not borne out by the conventional theory. This apparent anomaly has led to extensive attempts on reinterpretations or even to suggestions for outright modifications of some basic aspects of the classical theory of electromagnetism. The author shows that such drastic steps are not actually needed and that the above mentioned belief is ill-founded. A relativistically consistent picture emerges in the conventional theory when a proper account is taken of all the energy and momentum associated with the electromagnetic phenomenon in the system.

There is a topological structure in the set of the electromagnetic radiation fields (with E.B=0) in vacuum. A subset of them, called the admissible fields, are associated with maps S³(S²) and can be classified in homotopy classes labelled by the value of the corresponding Hopf indexes, which are topological constants of the motion. Moreover, any radiation field can be obtained by patching together admissible fields and is therefore locally equal to one of them. There is, however, an important difference from the global point of view, since the admissible fields obey the topological quantum conditions that the magnetic and the electric helicities are equal to integer numbers n and m times an action constant a which must be introduced because of dimensional reasons, that is integral A B d³r=na, integral C.E d³r=ma, where B and E are the magnetic and electric fields and Del *A=B, Del *C=E. A topological mechanism for the quantization of the electric charge operates in the set of the admissible fields, in such a way that the electric flux through any closed surface around a point charge is always equal to a times an integer number n', equal to the degree of a map S²(S²) corresponding to the existence of a fundamental charge with value q₀a/4 pi . It is argued that results of this kind could help reaching a better understanding of quantum physics.

The authors derive generalizations of the trace formula of Gutzwiller (1967) and Balian and Bloch (1970) that are valid in the presence of a non-Abelian continuous symmetry. The usual trace formula must be modified in such cases because periodic orbits occur in continuous families, whereas the usual trace formula requires that the periodic orbits be isolated at a given energy. These calculations extend the results of a previous paper, in which they considered Abelian continuous symmetries. The most important application of the results is to systems with full three-dimensional rotational symmetry, and they give this case special consideration.

Minimum uncertainty coherent states and annihilation operator coherent states for the Morse oscillator are derived and shown to be equivalent. They reduce, in the limit of small anharmonicity constant, or, equivalently, in the limit of large well depth, to the approximate coherent states derived previously from the use of generalized displacement operator.

In a paper by Fawcett and Bracken (1991), the classical limit of ordinary non-compact quantum systems is considered in terms of the contraction of the underlying kinematical Lie algebra (the Weyl-Heisenberg algebra) and its representations to the Abelian Lie algebra of the same dimension and its representations. Some of those ideas are adapted to discuss in similar terms, the classical limit of compact quantum systems whose underlying kinematical algebras are of the special orthogonal type. The classical dynamical system that results from the classical limit of compact quantum mechanical system will be called a 'compact classical system'. Poisson brackets for such 'compact classical systems' have already been given in the literature and the recovery of the Poisson bracket for so(3) compact classical systems is demonstrated in terms of the contraction limit.

The author presents a systematical study of left-covariant differential calculus on the quantum group GL_p,q(2), a two-parameter deformation of the general linear group. In particular, the author explicitly constructs the most general bicovariant calculus. It depends on a new parameter s. The corresponding Lie algebra of left-coinvariant vector fields is a two-parameter deformation of the classical Lie algebra, p and q appear only in the combination r=pq. In the limit p,q to 1 one obtains non-standard bicovariant differential calculi on the classical Lie group.

A measure of irregularity in a quantum state lies in the surface roughness of the corresponding eigenfunction and shows up as a splitting of contours at successive heights. This phenomenon seems to be absent in integrable systems where the states are regular. The authors carry out an analysis based on the semiclassical theory and extend it by a detailed numerical investigation which reveals that (i) a Gaussian amplitude distribution and the splitting of contours occur together, and (ii) the percentage increase in the number of contours with height is a measure of the degree of surface irregularity. Regular, localized states in chaotic systems for which good quantum numbers can be assigned do, however, show contour splitting.

The authors discuss certain properties of Berry's phases in two quantum systems with supersymmetrically related Hamiltonians and derive an explicit expression for the difference in their Berry's phases. Moreover, they obtain an expression for a topological quantity which can be interpreted in term of holonomy, and which contains the Berry's phase and is invariant in the two supersymmetrically related quantum systems. They take two examples to illustrate their findings. The first one is the example of the Aharonov-Bohm effect in which the Berry's phases of the two supersymmetrically related quantum systems are equal. The second example is a system with a spin-1/2 particle in a time varying magnetic field. They obtain the stated topological quantity and, in their analysis, they discover that this quantity is corresponding to the essential gauge transformation in such systems.