Table of contents

We report the calculation of the fourth coefficient in an expansion of the heat kernel of a non-minimal, non-Abelian kinetic operator in an arbitrary background gauge in arbitrary spacetime dimension. The fourth coefficient is shown to bring a non-trivial gauge dependence due to the contribution of the lowest order off-shell gauge-invariant structure.

A Sierpinski gasket resistor network is considered in which the basic microscopically anisotropic resistance distribution has a hierarchical pattern. The system is shown to undergo a transition from macroscopic isotropy to macroscopic anisotropy as a hierarchical parameter R is varied. For , the system may restore macroscopic isotropy, regardless of the degree of anisotropy on the microscopic scale. For , on the other hand, the system remains anisotropic on the macroscopic scale due to the basic anisotropy on the microscopic scale. The degree of macroscopic anisotropy depends on both the hierarchical parameter R and the microscopic anisotropy.

We investigate variational methods for finding approximate solutions to the Fokker - Planck equation, especially in cases lacking detailed balance. These schemes fall into two classes: those in which a Hermitian operator is constructed from the (non-Hermitian) Fokker - Planck operator, and those which are based on soluble approximations to this operator. The various approaches are tested on a simple quantum-mechanical problem, a toy Fokker - Planck equation and finally on the problem of a particle moving in a potential and subject to external non-white noise.

The coupling between order-parameter fluctuations near the wall and depinning fluid interface in the approach to a complete wetting transition is described by a two-field Hamiltonian which improves upon standard capillary-wave models of wetting. We construct a nonlinear renormalization group to study fluctuation effects in d = 3 and show how this elegantly rederives the expression for the renormalized wetting parameter found in earlier (linear) renormalization-group treatments.

We use a `sum-over-paths' method to derive an explicit expression for the Green's function of the diffusion equation on an arbitrary tree, making connection with the conventional Laplace approach but emphasizing the real-time nature of the technique. This provides a general framework for studying both discrete-space and continuous-space diffusion processes on complex one-dimensional topologies.

We analyse the effect of quenched uncorrelated randomness coupling to the local energy density of a model consisting of N coupled two-dimensional Ising models. For N>2 the pure model exhibits a fluctuation-driven first-order transition, characterized by runaway renormalization-group behaviour. We show that the addition of weak randomness acts to stabilize these flows, in such a way that the trajectories ultimately flow back towards the pure decoupled Ising fixed point, with the usual critical exponents , , apart from logarithmic corrections. We also show by examples that, in higher dimensions, such transitions may either become continuous or remain first-order in the presence of randomness.

We introduce a DLA model with two species in order to simulate the growth of alternate clusters. We perform intensive numerical simulations in two dimensions in which the proportions of the two species are varied. A critical point, analogous to a percolation threshold, is found and a new class of critical exponents is obtained for this transition. Within numerical accuracy, the fractal dimension of the clusters is found to be the same as in the usual DLA model, independent of the species concentrations. Possible connections with the growth of two-dimensional ionic crystals are discussed.

We investigate the long-time regime of a zero-temperature kinetic model with glassy behaviour, introduced recently. In this regime, the energy density and its two-time correlation function are found to have simple asymptotic expressions with logarithmic corrections, which we derive exactly. In particular, the energy correlation function exhibits ageing. Its scaling form reads , with a large correction in . We also find that a closely related model, with faster relaxation of the energy, leads to the same asymptotic scaling form for the two-time correlation, with much smaller corrections.

We describe a non-Arrhenius mechanism for the slowing down of dynamics that is inherent to the high dimensionality of the phase space. We show that such a mechanism is at work both in a family of mean-field spin-glass models without any domain structure and in the case of ferromagnetic domain growth. The marginality of spin-glass dynamics, as well as the existence of a `quasi-equilibrium regime' can be understood within this scenario. We discuss the question of ergodicity in an out-of equilibrium situation.

An anisotropic integrable spin chain, consisting of spins s = 1 and , is investigated [1]. It is characterized by two real parameters and , the coupling constants of the spin interactions. For the case and the ground-state configuration is obtained by means of thermodynamic Bethe ansatz. Furthermore, the low excitations are calculated. It turns out that apart from free magnon states being the holes in the ground-state rapidity distribution, there exist bound states given by special string solutions of Bethe ansatz equations (BAE) in analogy to [13]. The dispersion law of these excitations is calculated numerically.

A criterion is introduced to find solutions for the mapping of Ising spin systems into generalized percolation models through the extension of the Kasteleyn - Fortuin technique, and specific application to the 2D triangular fully frustrated Ising model is given. The clusters of generalized percolation, selected according to a general scheme, give the way to implement efficient Monte Carlo (MC) cluster dynamics for frustrated Ising spin systems, and the study of their geometric properties suggests the understanding of MC performances.

We study Thomson's problem using a new numerical algorithm, valid for any interacting complex system based on the consideration of simultaneous many-particle transitions to reduce the characteristic slowing down of numerical algorithms when applied to critical or complex systems. We improve or reproduce all previous results on the Thomson problem, using much less computer time than the other numerical algorithms. We report ground-state energies for , and study the stability of the ground state as a function of the number of charges considered. We associate this stability with how well defined are the charges surrounded by five nearest neighbours, whose number always seems to be equal to 12.

The one-dimensional (1D) electron gas model in the strong coupling regime is considered. It is shown that this model is Bethe ansatz solvable and the exact spectrum of the Hamiltonian is obtained. The massive spin elementary excitations are established based on the Bethe ansatz equations. The magnetization near the onset at zero temperature and the low-temperature thermodynamics are calculated.

We consider the L-state cyclic solid-on-solid lattice models under a class of open boundary conditions. The integrable boundary face weights are obtained by solving the reflection equations. Functional relations for the fused transfer matrices are presented for both periodic and open boundary conditions. The eigenspectra of the unfused transfer matrix is obtained from the functional relations using the analytic Bethe ansatz. For a special case of crossing parameter , the finite-size corrections to the eigenspectra of the critical models are obtained, from which the corresponding conformal dimensions follow. The calculation of the surface free energy away from criticality yields two surface specific heat exponents, and , where coprime to L. These results are in agreement with the scaling relations and .

The Hamiltonian for a system of two coupled oscillators with time-dependent coupling parameter and the phase pump is considered. The wavefunction in Schrödinger picture and the Green function are calculated. The squeezing phenomena as well as the Glauber second-order correlation function is discussed. Statistical investigation for the quasi-probability distribution function (P-representation, W-Wigner and Q-functions) are given.

The energy level correlator K(s) and the form factor are calculated for a hypercubic billiard with small hyperspheres placed at random in its interior. Various regimes, characterized by the elastic mean free path l, resulting from the scattering on impurities, are identified. The analysis extends from the ballistic regime, where l is much larger than the size of the system, via intermediate regimes, to the diffusive regime, where l is much smaller than its size. Semiclassical expressions for the density of states of chaotic and integrable systems in terms of classical periodic orbits are used. The diagonal approximation for is made for short times, while non-perturbative methods are used for long times. The analysis makes use of analytic properties of classical dynamical zeta function associated with the Perron - Frobenius operator. The general features are relevant for mesoscopic systems.

A procedure for applications of the pseudospin transformation within the framework of nuclear oscillator shell model is developed. It is valid for operators expressed in terms of single-particle variables and is based on permutation rules for special rotational invariants. The procedure is applied to a number of physical operators including several rotational scalars, the spin and orbital momenta, and the quadrupole moment. An algorithm for generating an approximation to the pseudospin transformation, which gives a simple and accurate expression for dominant parts of required transforms, is also given. The algebras associated with pseudospin transformations are discussed.

A parity-dependent squeezing operator is introduced which imposes different SU(1,1) rotations on the even and odd subspaces of the harmonic oscillator Hilbert space. This operator is used to define parity-dependent squeezed states which exhibit highly non-classical properties such as strong antibunching, quadrature squeezing, strong oscillations in the photon-number distribution, etc. In contrast to the usual squeezed states whose Q and Wigner functions are simply Gaussians, the parity-dependent squeezed states have much more complicated Q and Wigner functions that exhibit an interesting interference in phase space. The generation of these states by parity-dependent quadratic Hamiltonians is also discussed.

Starting from the classical example of the Hénon - Heiles integrable Hamiltonian system, we show that it admits a slightly different formulation from the classical bi-Hamiltonian system. We introduce the general notion of a quasi-bi-Hamiltonian system (QBHS) and study some of its basic properties only in the case of two degrees of freedom; in particular, we give the general form of this structure in the Nijenhuis coordinates which are constructed using the eigenvalues of the Nijenhuis operator of the system and show how they serve to separate the Hamilton - Jacobi equation. In the last section, we look at the problem of the semi-local existence of such structures.

Except for the universe, all quantum systems are open, and according to quantum state diffusion theory, many systems localize to wavepackets in the neighbourhood of phase space points. This is due to decoherence from the interaction with the environment, and makes the quasiclassical limit of such systems both more realistic and simpler in many respects than the more familiar quasiclassical limit for closed systems. A linearized version of this theory leads to the correct classical dynamics in the macroscopic limit, even for nonlinear and chaotic systems. We apply the theory to the forced, damped Duffing oscillator, comparing the numerical results of the full and linearized equations, and argue that this can be used to make explicit calculations in the decoherent histories formalism of quantum mechanics.

The two-mode even and odd coherent Schrödinger cat states are introduced for the generalized stationary and non-stationary harmonic oscillator. The expected values of positions and momenta and their dispersion matrices are calculated for the Schrödinger cat states. The quadrature squeezing and correlation phenomenon are studied and the Wigner and Husimi functions are constructed explicitly for these states. The photon statistics of the two-mode even and odd coherent states of the generalized oscillator is described in terms of the photon distribution function, which is expressed in terms of Hermite polynomials. The photon number, means and dispersions show oscillations which are characteristics of squeezing and correlation phenomenon.

The Rutherford scattering of a classical point charge moving in an attractive field and obeying the Lorentz - Dirac equation is solved. The size of the spatial part of the incoming 4-velocity takes the values 1000, 100 and 0.1, respectively. Asymptotic expansions of physical solutions are derived and used. Results are displayed and discussed. It is shown that all solutions satisfy physical expectations. A condition for treating radiation reaction as a perturbation is applied. Some earlier problems that have led to suggestions of unphysical features of the Lorentz - Dirac equation are explained on a physical basis.

We present a set of tiles consisting of a tetrahedron and an octahedron in two sizes which admits a tile inflation yielding a non-periodic tiling of space.

We study smooth transformations of potentials for which exact bound-state solutions of Schrödinger's equation are known. Eigenvalue approximation formulae are obtained which provide lower or upper energy bounds according to whether the transformation function g is convex or concave. Detailed results are presented for perturbed Coulomb potentials of the form and

The infinite-dimensional representation of the boundary K-operator (the solution of the reflection equation) is considered. The trigonometric K-matrix is studied as a degenerate case of the elliptic operator. A method to construct the elliptic Dunkl operator is proposed, and the relationship with the quantum many-body problem is also discussed.

The dynamics of two atoms coupled to the vacuum radiation field is investigated within the framework of a one-dimensional model in the resonance dipole approximation. The exact solution of the resonance energy transfer problem is obtained. A many-atom generalization of the model is also discussed.

We present a complete set of conjectures for the exact boundary reflection matrix for ade affine Toda field theory defined on a half line with the Neumann boundary condition.

We consider the class of three-dimensional finite-range, or similar, potentials , depending on a strength constant . We study the behaviour of the eigenvalue E as a function of , where is the critical value at the transition from 0 1 bound state. For the case, we find , whereas the relationship is linear for . Treating as a continuous parameter in the radial Schrödinger equation, we give the evolution of the power law between and . Besides spherically symmetric scalar potentials, we also discuss the case of a repulsive scalar potential combined with a spin - orbit component of the Thomas form.

We calculated the Shannon entropy of position and momentum for the stationary quantum states of the harmonic oscillator as a function of its energy and determined the corresponding entropic uncertainty relations for them. We found an approximate phenomenological function for the dependence of position and momentum entropies on the large quantum numbers and the corresponding asymptotic entropy - energy relation for the stationary harmonic oscillator. We also studied the time evolution of the position and momentum entropies of the non-stationary harmonic oscillator for the coherent states, squeezed vacuum and Schrödinger cat states.

The adiabatic motion of a charged, spinning, quantum particle in a two-dimensional (i.e. of constant direction) magnetic field is studied. A suitable set of operators generalizing the kinematical momenta and the guiding centre operators of a particle moving in a homogeneous magnetic field is constructed. This allows us to separate the two degrees of freedom of the system into a fast and a slow one which are, in the classical limit, the rapid rotation of the particle around the guiding centre and the slow guiding centre drift. In terms of these operators the Hamiltonian of the system can be rewritten as a power series in the magnetic length , and the fast and slow dynamics separates. The effective guiding centre Hamiltonian is obtained to second order in the adiabatic parameter and reproduces correctly the classical limit.

The paper deals with the real exceptional Lie algebras EIII, EVI, EVII, EIX and their arbitrary irreducible representations. Hermitian forms invariant relative to these representations are considered. The existing signature formulae for these forms are improved.

We study the path integral for a model with a finite number of degrees of freedom and two first-class constraints. To account for the constraints, we construct the appropriate projection operator, and, rather than the resolution of unity, use it at every time slice in the building of the coherent-state path-integral representation of the propagator. The derivation of the projection operator leads to the introduction of bicoherent states and is built by integration over properly-weighted, independent coherent-state bras and kets. The construction of the propagator using bicoherent states leads to a phase space action, which, in general, is complex and has twice as many labels as there are in the standard classical phase space action. The imaginary part of the complex action reduces to a surface term on the classical trajectories. We argue that the projection operator leads to the correct measure in the path-integral representation of the propagator. The measure, which is path dependent, is `modulated' by the imaginary part of the action.

The formulation of Maxwell's equations using exterior differentiation is compared to that involving covariant differentiation. These two formulations are known to be equivalent in a space with a Riemannian connection, and a necessary and sufficient condition is established here for this equivalence to be maintained in the case where the connection is of the most general type, namely a connection with, in general, torsion and non-metricity, in addition to curvature.

A simple method of constructing superoscillatory functions is given. It is based on a zero-shifting principle of functions in Paley - Wiener space.

In this paper all eight-vertex type solutions of the coloured Yang - Baxter equation dependent on spectral as well as colour parameters are given. It is proved that they are composed of three groups of basic solutions, three groups of their degenerate forms and two groups of trivial solutions up to five solution transformations. Moreover, all non-trivial solutions can be classified into two types called Baxter type and free-fermion type.

We comment on the recent work of Alcaraz and Malvezzi on the critical properties of the chain in staggered magnetic field. The method of determining the phase boundary from the finite-size numerical data is also discussed.

We discuss the comments presented by Okamoto and Nomura about the phase diagram of the XXZ chain under the influence of an external staggered magnetic field, reported by Alcaraz and Malvezzi.

In this brief comment we show that, contrary to previous claims (Bartelt and Evans), the asymptotic behaviour of the Kashchiev model of polynuclear growth is trivial in all spatial dimensions, and therefore lies outside the Kardar - Parisi - Zhang universality class.