Table of contents

The authors apply the general procedure developed by Kiefer and Rothe (1984) in order to obtain the U matrix in QCD from Dirac brackets. The Coulomb interaction Hamiltonian happens to be that of Christ and Lee (1980).

The authors obtain explicit expressions for the wavefunctions of all the known shape-invariant potentials by using the operator method.

How does one measure static properties in dynamic models? The authors propose a basic approach that should help keep the statics and the dynamics of growth models clearly separated. Their formalism is used to elaborate a large cell renormalisation group scheme which gives very good results for the determination of the fractal dimension of Eden and DLA models. They also propose a dynamic scaling theory for growth models which they verify for both Eden and DLA models.

The time evolution of the irreversible catalytic recombination process A+B to inert is studied both analytically and by computer simulation. A rate equation describing this process is derived. For situations where one of the species poisons the catalyst, the minority species, for low coverages, is found to decay exponentially. Computer simulation results confirm this exponential decay for low coverages. Near the poisoning transition, x_A approximately=x_B, the average relaxation time is found to diverge as zeta varies as (0.5-x_A)^-1 with zeta approximately=1.3. Here x_A and x_B are the compositions of A and B in the gas.

The behaviour of an Ising antiferromagnet which is known to undergo a continuous transition in two dimensions and a first order transition, in three is considered in a slab geometry of finite thickness. The authors note that the phase behaviour is intimately related to the surface boundary conditions which govern wetting. For the interesting case in which wetting phenomena are not important, they find that the bulk behaviour is already manifest when the slab is only eleven layers thick.

The authors have studied rooted spiral trees in dimensions 2, 3 and 4 by exact enumeration and by Monte Carlo methods. Numerical estimates of critical exponents thus obtained support the hypothesis that a dimensional reduction by four occurs in this problem. A possible mechanism of the origin of this dimensional reduction is proposed.

For the critical spherical model in d=r+1 dimensions, finite-size calculations of the spin-spin and energy-energy correlation lengths suggest relations among the critical exponents and the finite-size scaling amplitudes of the correlation lengths similar to those obtained in (1+1) dimensions from conformal invariance. For antiperiodic boundary conditions, the results agree with numerical studies in the (2+1)-dimensional Ising model.

The quantum Ising chain with p equidistant defects is studied. The exact form of the Hamiltonian is found for an infinite number of sites. Using conformal invariance, generalised corner exponents are computed and found to depend continuously on the defect strengths. Realisations of extended conformal algebras in the Hamiltonian spectrum are obtained.

The author considers the dynamics of a polydisperse sol made of randomly branched polymers near the gelation threshold. Assuming that the distribution is that of percolation, the author reviews various possible hydrodynamic interactions, namely the Rouse limit, where no backflow is present, and the Zimm limit, where hydrodynamic interactions are present. In the latter case, the author recovers a conjecture by Kertesz (1983) for the divergence of the viscosity. In the former case, the author recovers the de Gennes-Stephen result (1978). The author discusses the viscoelastic behaviour at short timescales.

A recursive algorithm to determine the backbone of the two-dimensional percolating cluster is adjusted to measure the fractal properties of the backbone of the three-dimensional percolating cluster. The specific properties measured are the fractal dimensionality of the backbone, d^BB_f=1.75+or-0.04, the chemical dimensionality, d_t=1.26+or-0.03 and the minimum path dimensionality d_min=1.39+or-0.03.

Analysis of the mutual arrangements of Delaunay simplices is formulated as a site-correlated percolation problem on the Voronoi network. Typical cluster configurations of simplices of tetrahedral and octahedral form (the main structural elements of simple liquids) are found and percolation thresholds for them are obtained.

The authors propose a mechanism to enhance the learning of an arbitrary but finite number of patterns (marked patterns) without damaging the ability of the network to retrieve the rest of the patterns. They find that the process of forgetting the marked patterns may become continuous, presenting then an unusual behaviour for symmetric neural networks.

Neural networks are only useful as associative memories if an outside observer is able to ascertain whether a particular input corresponds to one of the memory states or not, i.e. the memory must be transparent. The authors design a model system which enters a 2-cycle whenever an input is not within the basin of attraction of some memory state.

The Glauber dynamics of the Little-Hopfield model and the asymmetric SK model is studied with the help of path integrals introduced by Sommers (1987). This dynamical approach confirms the results of Amit et al. (1985). For the fully asymmetric SK model one can calculate the response and autocorrelation functions exactly and observe exponential decay for both functions. The effect of weak correlations between the couplings J_ij and J_ji is investigated.

It is shown that any generator of the infinite-dimensional canonical transformation group in the Hamiltonian formulation of classical mechanics can be obtained as a linear combination of repeated commutators of generators of two finite-dimensional subgroups. This result has some structural similarity with the Ogievetsky theorem (1973) concerning the algebra of the general coordinate transformation group in the theory of relativity.

Work exploring the connection between SO(2,1) Lie algebra and the Jacobi-matrix method for scattering is extended to the Morse potential. Regular and irregular solutions of the three-term recursion associated with the Jacobi matrix are specified by their asymptotic behaviour and closed-form expressions in terms of generalised hypergeometric functions are obtained. This example further illustrates the connection between scattering theory and orthogonal polynomials.

A generalisation of the procedure for constructing ladder representations of finite and infinite-dimensional Lie algebras is considered. Such a general procedure is further generalised and provides a method for affinisation of Lie algebras. A general form of the central term for arbitrary Lie algebra is obtained.

A rather powerful method, based upon the theory of distributions and the Hill-Wheeler ansatz, is introduced in order to deal with the eigensolutions of the most general one-dimensional Hamiltonian H=p²/2m+V(x), and illustrated with reference to two special (but important) examples.

An exact expression for the single-eigenvalue probability density function for any dimension of a Gaussian orthogonal ensemble is found by analytic continuation. It is shown that the form can be used to derive a number of exact results in a simple way. In odd dimensions it is shown that the single-eigenvalue probability density function can be written as a sum of two densities.

An infinite sequence of new representations having Froman and Froman's form (1974) has been found for the solutions to the one-dimensional Schrodinger equation (SE). This enables the author to construct, in compact WKBJ-like form, approximate solutions to SE. The new approximate solutions have the property that their asymptotic expansions in h(cross) coincide with the corresponding asymptotic expansions of the exact solutions to any, a priori given, order in h(cross). They generalise the well known WKBJ formula, being in this scheme the zeroth-order approximation. The generalised WKBJ formulae are then tested by the quantisation of the simplest quantum mechanical systems of the harmonic and anharmonic (quartic) oscillators as well as a more complicated one of a double-well potential. The approximate quantisation rules (a generalisation of the well known Bohr-Sommerfeld quantisation conditions) for the general polynomial potential is finally obtained and the formulae for calculating the necessary corrections are derived. The limiting formal solutions to SE having the WKBJ form are also presented. Their asymptotic expansions in h(cross) reproduce those found by Froman (1966).

The factorisation of the L operator in a Lax pair (L,P) has played an important role in the theory of integrable systems. For example, the Miura transformation can be immediately obtained from the factorisation of the Schrodinger operator; - delta _x²+u-k²=(- delta _x+a)( delta _x+b) requires that u=k²-a_x+a² and a=b. For zero curvature representations this procedure is not available, however. The author attempts to produce an analogous theory of factorisations for zero-curvature representations.

The derivative non-linear Schrodinger equation exhibiting non-ultralocal canonical structure is investigated to obtain various relationships including classical r-s matrices and the Yang-Baxter relation modified by the non-ultralocality. The complete integrability of the system is established through explicit action-angle canonical variables. An attempt has been made to solve the corresponding quantum field model in the semiclassical approximation. A possibility of eliminating non-ultralocality of the model, necessary for exact quantum inverse scattering treatment, is demonstrated.

The analytic properties of energy levels as the functions of coupling parameters in one-dimensional quantum mechanical models described by the potentials of delta-function type are investigated. The following models are considered: (i) the rectangular potential well with delta-function barrier, (ii) two delta-function wells in the unbounded space, (iii) the Kronig-Penney model with periodic delta-function potential, and (iv) the Bose gas with delta-function interaction between particles.

The quantum theory of a massless scalar field in a box with one chaotically moving wall is presented. The time dependence of normally ordered coherence functions is found to be chaotic, whereas the number of quanta present is insensitive to the details of the chaotic motion.

The authors analyse the scaling limit of discretised random paths with curvature-dependent action. For finite values of the curvature coupling constant the theory belongs to the universality class of simple random walks. It is possible to define a non-trivial scaling limit if the curvature coupling tends to infinity. They compute exactly the two-point function in this limit and discuss the relevance of their results for random surfaces and string theories.

The authors introduce the short-time transition probability associated with the most general Boltzmann-Fokker-Planck equation in cartesian coordinates, thereby extending Chandrasekhar's results (1960). They exploit the covariant formulation of the short-time propagator for the Fokker-Planck equation as introduced by Graham and Deininghauss (1979). The case of a locally isotropic velocity background distribution is investigated and the conditions for thermodynamic equilibrium are found. The results are a direct generalisation of Chandrasekhar's, with the covariant drift introduced by Graham (1977) replacing the ordinary drift in Chandrasekhar's simpler approach.

The authors develop the short-time propagator for the Boltzmann-Fokker-Planck equation in arbitrary coordinates thereby extending their previous result which was carried out in special coordinates. This treatment uses the approach developed by Graham (1977) which uses the diffusion matrix as the metric tensor. They examine the connection between the covariant formalisms of Graham and that of Rosenbluth et al, (1957) showing that both formulations lead to the same form of the invariant partial differential equation for a scalar density distribution function. However, the Graham formulation has a major advantage over that of Rosenbluth et al, in that it is the only one which has been able to set up a functional integral for the general form of the Fokker-Planck equation. They have here extended Graham's results for the functional integral expression of the short time in general coordinates for the Boltzmann-Fokker-Planck equation and then illustrated the use of this formulation in setting up the short-time propagator for the problem of globular clusters.

The dynamic scaling of the 2D chain-chain aggregation model with a diffusion coefficient varying with the size k gamma of the chains as k^y is investigated by Monte Carlo simulations. Assuming that the sticking probability for two chains of size k is proportional to k^{- phi} , the mean chain size grows like t^z with z=(1+ phi - gamma )^-1.

For pt.I, see ibid., vol.21, p.773, (1988). The authors present exact enumerations of trails (self-intersecting but non-overlapping lattice walks) tabulated according to their length (l), number of intersection (I) and end-to-end distance for the square lattice (up to l=21, I=7) and for the simple cubic lattice (up to l=15, I=5). They introduce a fugacity for intersection to explore the transition to a collapsed phase through a novel tricritical point which is unaccessible by the renormalisation group approach. The existence of tricritical points in both lattices is manifested by the divergence of the specific heat. The values of the tricritical couplings and exponents are extracted by a DlogPade analysis. They find for the three-dimensional lattice nu _t approximately=0.48, and gamma _t approximately=0.43 ( nu _t approximately=0.52 and gamma _t approximately=1.25 for the square lattice).

The Lee-Yang formulae for the free energy and the magnetisation of the Ising model on a square lattice in an imaginary magnetic field H=1/2(iπkT) are generalised to a checkerboard lattice.

The mean-field renormalisation group method is used to study the critical temperature of the three-state Potts antiferromagnet in 3D lattices, as well as the three-state Potts model on the square lattice with ferromagnetic (along x) and antiferromagnetic (along y) interactions, and its quantum version, the 1D Potts antiferromagnet in a transverse field. Comparison is made with other results.

The authors present the explicit pre-gelation solution of the discrete coagulation equation for the general bilinear kernel and arbitrary initial conditions. For the pure additional kernel, they prove that gelation never takes place even if the second moment of the initial particle distribution is infinite.

A binary uncharged isotopic mixture at liquid density is described by the state variables M₁ and M₂, the component masses, volume V and temperature T. To these, in an extended irreversible thermodynamic treatment, is added the diffusion flow J_d. In the framework of this approach, a kinetic equation relates J_d to the driving force, - Del _T( mu ₁- mu ₂), where the mu _i are chemical potentials and T is kept constant in computing the gradient. To calculate the coefficient gamma of this force, a microscopic model is adduced which expresses gamma in terms of P_xx, the pressure tensor. Evaluation of gamma and application of reciprocity permits the calculation of Phi , the thermodynamic force associated with J_d. Once Phi is known, integrability conditions for the Helmholtz function yield the O(J_d²) terms in the pressure P and the mu _i. From these results and an estimate from computer simulations of the relaxation time for J_d, the authors obtain D₂ in the diffusion coefficient, D=D₀+D₂J_d². Estimates for a hard-sphere model representing a mixture of ³⁶Ar and ⁴⁰Ar predict that all the J_d² terms in D,P, mu _i are negligible unless mod J_d mod >or approximately=10⁴ kg m^-2 S^-1 at liquid density.

The non-relativistic theory of the Lorentz gauge is developed for the region of a perfectly conducting surface using the Gupta-Bleuler indefinite metric and a non-orthogonal set of basis vectors. The Lorentz gauge is transformed into the Coulomb gauge by a unitary transformation and in so doing expressions are obtained for the Coulomb energy and its special case the image force. The two non-physical photons of the Lorentz gauge are found to exist also in the Coulomb gauge.

The author identifies the spin connection of four-dimensional spacetime with a subgroup of a simple gauge group G. The remaining subgroups of G are the usual internal symmetry groups. Using the work of Weinberg (1983) to relate gauge constants to the circumferences of compact spaces and identifying the known spin of elementary particles with the eigenvalues of the spin connection allows us to set the scale of G. The author can then calculate the remaining gauge coupling constants numerically. The author shows how this mechanism works for the sample calculation where G is SU₄, the spin connection is the usual SU₂*SU₂ and the internal gauge symmetry is U₁. The author finds the electromagnetic fine-structure constant to be approximately 1/100.

A real-space renormalisation group method is applied to viscous fingering to analyse the fractal nature at a finite viscosity ratio. The renormalisation group transformation is obtained for the permeability of the surface layer which plays the role of coupling constant in the renormalisation group of the phase transition. A stable fixed point is found as a function of the viscosity ratio. The growth probabilities on the perimeter bonds of the injected fluid are evaluated at the fixed point. The multifractal scaling properties are found to be described by the generalised dimensions and the alpha -f spectra. It is shown that at a finite viscosity ratio the displaced area is compact with a surface fractal dimension between 1 and the DLA result with increasing viscosity ratio. This result is consistent with that of King's numerical simulation (1987).

The author shows that the two-dimensional N-colour Ashkin-Teller model exhibits an Ising-like continuous transition line in the region where the model has two phase transitions. The result is also valid for the antiferromagnetic ordering. In addition they present results which support the conjecture of Grest and Widom (1981) about the location of the disorder point.