Table of contents

A large class of games based on spin models is analysed. For these games, the long-time gain can be calculated exactly. It is shown that the mixing of two losing strategies may lead to a winning one, but also that the mixing of two winning ones may lead to a loss. Also, the mixing of a losing and a winning strategy may give unexpected results. This behaviour is due to two general features: (i) the class of games is such that mixing the playing of two games is equivalent to playing a third one, and (ii) the break-even boundaries for these games are curved.

A specific two-dimensional tiling model, composed of the so-called Wang tiles has been studied at finite temperature using Monte Carlo numerical simulations. In the absence of any thermal bath the Wang tiles provide the opportunity of building a very large number of non-periodic tilings. We can construct a local Hamiltonian such that only perfectly matched tilings are ground states with zero energy. This Hamiltonian has a very large degeneracy. The thermodynamic behaviour of such a system seems to show a continuous phase transition at non-zero temperature. An order parameter with non-trivial features is proposed. Under the critical temperature the model exhibits ageing properties. The fluctuation-dissipation theorem is violated.

We derive a lattice approximation for a class of equilibrium quantum statistics describing the behaviour of any combination and number of bosonic and fermionic particles with any sufficiently binding potential. We then develop an intuitive Monte Carlo algorithm which can be used for the computation of expectation values in canonical and Gaussian ensembles and give lattice observables which will converge to the energy moments in the continuum limit. The focus of the discussion is twofold: in the rigorous treatment of the continuum limit and in the physical meaning of the lattice approximation. In particular, it is shown how the concepts and intuition of classical physics can be applied in this sort of computation of quantum effects. We illustrate the use of Monte Carlo methods by computing canonical energy moments and the Gaussian density of states for charged particles in a quadratic potential.

Detailed mean-field and Monte Carlo studies of the dynamic magnetization-reversal transition in the Ising model in its ordered phase under a competing external magnetic field of finite duration have been presented here. An approximate analytical treatment of the mean-field equations of motion shows the existence of diverging length and time scales across this dynamic transition phase boundary. These are also supported by numerical solutions of the complete mean-field equations of motion and the Monte Carlo study of the system evolving under Glauber dynamics in both two and three dimensions. Classical nucleation theory predicts different mechanisms of domain growth in two regimes marked by the strength of the external field, and the nature of the Monte Carlo phase boundary can be comprehended satisfactorily using the theory. The order of the transition changes from a continuous to a discontinuous one as one crosses over from coalescence regime (stronger field) to a nucleation regime (weaker field). Finite-size scaling theory can be applied in the coalescence regime, where the best-fit estimates of the critical exponents are obtained for two and three dimensions.

For Lindblad's master equation of open quantum systems with the Hamiltonian in a general quadratic form, the propagator of the density matrix is calculated analytically by using path-integral techniques. The time-dependent density matrix is applied to nuclear barrier penetration in heavy ion collisions with inverted oscillator and double-well potentials. The quantum mechanical decoherence of pairs of phase space histories in the propagator is studied and it is shown that the decoherence depends crucially on the transport coefficients.

E(2) is studied as the automorphism group of the Heisenberg algebra H. The basis in the Hilbert space K of functions on H on which the unitary irreducible representations of the group are realized is explicitly constructed. The addition theorem for the Kummer functions is derived.

In this paper we consider the three-particle density correlation function for a fractional quantum Hall liquid. The study of this object is motivated by recent experimental studies of fractional quantum Hall systems using inelastic light scattering and phonon absorption techniques. Symmetry properties of the correlation function are noted. An exact sum rule is derived which this quantity must obey. This sum rule is used to assess the convolution approximation that has been used to estimate the matrix elements for such experiments.

On the basis of an exact formalism, a system of functional equations for the tunnelling parameters of self-similar fractal potentials (SSFPs) is obtained. Three different families of solutions are found for these equations, two of them having one parameter and one being free of parameters. Both one-parameter solutions are shown to be described, in the long-wave limit, by a fractal dimension. At the same time, the third solution yields transfer matrices which are analytical in this region, similar to the case of structures with the `Euclidean geometry'. We have revealed some manifestations of scale invariance in the physical properties of SSFPs. Nevertheless, in the common case these potentials do not possess, strictly speaking, this symmetry. The point is that SSFPs in the common case are specified, in contrast to the Cantor set, by two length scales but not one. A particular case when SSFPs are exactly scale invariant to an electron with well defined energy is found.

A proper formalism developed earlier to study electron tunnelling through a self-similar fractal potential (SSFP) posed on the Cantor set is extended here to describe the SSFP whose levels consist of N fractals of the next level. We have derived a functional equation for the transfer matrix of this potential and found three different solutions. Two of them correspond to SSFP barriers and SSFP wells whose power may be arbitrary. The third one relates to the only SSFP barrier whose power has a definite value. These solutions show that SSFPs, in the general case, are approximately scale invariant in the long- and short-wave regions, and only the limiting SSFP whose fractal dimension is equal to unity should be strictly scale invariant. We have shown that except for the limiting case the tunnelling parameters of SSFPs, with the same fractal dimension depend on N. In addition, we have established a link between the solutions of the functional equation and the power of SSFPs.

We show that the requirements of renormalizability and physical consistency imposed on perturbative interactions of massive vector mesons fix the theory essentially uniquely. In particular, physical consistency demands the presence of at least one additional physical degree of freedom which was not part of the originally required physical particle content. In its simplest realization (probably the only one) these are scalar fields as envisaged by Higgs but in the present formulation without the `symmetry-breaking Higgs condensate'. The final result agrees precisely with the usual quantization of a classical gauge theory by means of the Higgs mechanism. However, the emphasis is shifted: instead of invoking the gauge principle (and the Higgs mechanism) on the local quantum field theory, the principles of local quantum physics restricted by the perturbative renormalizability demand `explains' (via Bohr's correspondence) the classical gauge principle as a selection principle among the many a priori (semi)classical possibilities of coupling vector fields among each other. Our method proves an old conjecture of Cornwall, Levin and Tiktopoulos stating that the renormalization and consistency requirements of spin-1 particles lead to the gauge theory structure (i.e. a kind of inverse of 't Hooft's famous renormalizability proof in quantized gauge theories) which was based on the on-shell unitarity of the S-matrix. Since all known methods of renormalized perturbation theory are off-shell, our proof is different and a bit more involved than the original arguments. We also speculate on a possible future ghost-free formulation which avoids `field coordinates' altogether and is expected to reconcile the on-shell S-matrix point of view with the off-shell field theory structure.

The quantization of the chaotic geodesic motion on Riemann surfaces Σ_g,κ of constant negative curvature with genus g and a finite number of points κ infinitely far away (cusps) describing scattering channels is investigated. It is shown that terms in Selberg's trace formula describing scattering states can be expressed in terms of a renormalized time delay. This quantity is the time delay associated with the surface in question minus the time delay corresponding to the scattering problem on the Poincaré upper half-plane uniformizing our surface. Poles in these quantities give rise to resonances reflecting the chaos of the underlying classical dynamics. Our results are illustrated for the surfaces Σ_1,1 (Gutzwiller's leaky torus), Σ_0,3 (pants), and a class of Σ_g,2 surfaces. The generalization covering the inclusion of an integerB⩾2 magnetic field is also presented. It is shown that the renormalized time delay is not dependent on the magnetic field. This shows that the semiclassical dynamics with an integer magnetic field is the same as the free dynamics.

We consider the coupled system of the higher-order nonlinear Schrödinger equation and Maxwell-Bloch equations with pumping, which governs the nonlinear wave propagation in erbium-doped optical waveguides in the presence of important higher-order effects. We derive the Lax pair with a variable spectral parameter and the exact soliton solution is generated from the Bäcklund transformation.

The necessary condition that a Stäckel-Killing tensor of valence two should be the contracted product of a Killing-Yano tensor of valence two with itself is rederived for a Riemannian manifold. This condition is applied to the generalized Euclidean Taub-NUT metrics which admit a Kepler-type symmetry. It is shown that, in general, the Stäckel-Killing tensors involved in the Runge-Lenz vector cannot be expressed as a product of Killing-Yano tensors. The only exception is the original Taub-NUT metric.

The problem of motion of a rigid body about a fixed point under the action of conservative forces is considered in the case admitting a linear integral but no axis of symmetry - neither in space nor in the body - is present. A simple transformation of the configuration space is used to reduce the problem of motion of the body to another problem concerning the same body under a system of axisymmetric forces. This analogy enables the construction of several new integrable cases of the first problem by transforming certain known ones of the second. The new cases usually involve singular potential terms. Integrals of motion and physical interpretation are given explicitly for one generally integrable case. Other general and conditional cases are pointed out.