Table of contents

The author obtains the generalised exponents of the Lie algebra sl(3, c) by a method due to Kostant (1963). Using this result the author proves the hypothesis of Couture and Sharp (1980) about the existence of an integrity basis of the universal enveloping algebra of the sl(3, c) algebra, considered as an sl(3, c) module.

For arbitrary functions f₁, f₂ and f₃ the anharmonic oscillator x+f₁(t)x+f₂(t)x+f₃(t)x³=0 cannot be solved in closed form (i.e. the general solution cannot be expressed as elliptic functions). The authors apply the Painleve test to obtain the constraint on the functions f₁, f₂ and f₃ for which the equation passes the test. The constraint on f₁, f₂ and f₃ (i.e the differential equation which f₁, f₂, and f₃ obey) is discussed and solutions are given.

The authors obtain the general class of ordinary differential equations that can be reduced by a point transformation to the free-particle equation and given the general form of the Lie symmetry generators for these equations. They apply these results to obtain the symmetry generators for the n-dimensional harmonic oscillator and for a particle in a constant magnetic field.

A path-integral approach to the critical behaviour of two-dimensional systems is presented. The method, based on a decoupling change of path-integral variables, enables one to compute critical exponents in a simple and systematic way. Thus, it provides a new tool to be used in the field-theoretical description of certain statistical mechanics models.

Two growing self-avoiding surfaces are introduced as possible models of rapidly polymerising polymer membranes. Monte Carlo simulations indicate that diffusion-limited self-avoiding surfaces have fractal dimension D=2.35+or-0.05, and so are in a different universality class than diffusion-limited aggregates. In contrast, Eden self-avoiding surfaces (1961) appear to be compact, just as the clusters are in the usual Eden model. Both growing surfaces have different fractal dimensions than previously considered models of self-avoiding surfaces in equilibrium.

The influence of a finite concentration of falling particles in random ballistic deposition was studied using a simple two-dimensional model on a square lattice which includes some restructuring. The scaling properties of the surface of the deposit were studied as a function of the concentration. The exponent beta , which describes how the surface thickness grows with the height of the deposit (for an extremely large deposit), varies from 1/4 (for small concentrations) to 1/3 (for large concentrations).

The kinetics of droplet growth in a heterogeneous nucleation process is investigated using Monte Carlo simulations, scaling theory and the Smoluchowski equation (1916). The exponent describing the scaling of the droplet size distribution and the growth law for the mean droplet size are calculated exactly. The simulation data are found to be consistent with the theoretical predictions in d=1, 2, and 3 dimensions.

The authors show that, in an n-vector model with anisotropy affecting the interactions within a finite-dimensional subspace of the spin space, non-analyticity of the free energy can appear in the spherical limit (n= infinity ) even in a finite system. The transition from the paramagnetic state is generally of mean-field type and is accompanied by spontaneous magnetisation within the anisotropy subspace.

The Kleinian theory (1922) of the icosahedral equation is used to investigate a recent conjecture on the locus of zeros for the grand partition function of the hard-hexagon lattice gas model.

The author studies the elasticity and failure characteristics of a system built by 'association in parallel' of links with identical spring constant but random failure thresholds. The occurrence of the first link rupture, which is related to the theory of extreme order statistics, is contrasted with the global failure for which a central limit theorem holds. He also discusses the similarities and differences between the 'democratic' and 'hierarchical' models which differ in the mechanism by which the stress supported by links which have failed is transferred to the other stronger links. This problem gives the simplest model of elastic non-linear behaviour before global rupture occurs.

The dynamics of random sequential absorption on d-dimensional lattices is represented by a simple graphical expansion. Results for the entire filling process, and in particular its saturation limit are easily obtained. It is shown that the large-d limit of the saturation density is ln 2d/2d. The expression gives an accurate approximation for d>or=3.

A (bivariate) anisotropic metric is used to fit theoretical expressions of the electrical resistivity temperature derivative near a critical point. Data chosen are those on TbZn. Effects of temperature and voltage measurement error (or precision) are investigated. Critical amplitudes, the critical exponent and the critical temperature are greatly affected by small uncertainties. The necessity to report such data uncertainties is emphasised. No definite conclusion on parameter values can be made at this time. A method of 'randomly modified data' shows the complementarity between simulation and laboratory work. The method a posteriori indicates how one may observe variations assumed 'constant errors' for standard measurements.

The author integrates the system of nonlinear differential equations x_k=(-1/2)(exp(x_k-1-x_k)+exp(x_k-x_k+1)) and study the behaviour of a system of infinitely many mass points on the line governed by these differential equations. The results are compared with those for the corresponding finite system.

The Rayleigh-Schrodinger perturbation series for the cubic anharmonic oscillator can be formally rearranged as the classical Birkhoff series plus quantum corrections proportional to successive powers of h(cross)². Each of the individual quantum corrections is a convergent power series expansion (with the same radius of convergence as the classical series) of the corresponding term in the Jeffreys-Wentzel-Kramers-Brillouin series.

The Hermitian operators behind Wigner's phase space function (1932) are recognised to be simple ordered exponentials of the dynamical variables. This operator basis is highly symmetric; it supplies a perfectly unbiased formulation of operator equations in terms of phase space functions, which is as close to classical physics as it possibly can be. The author demonstrates how the symmetry properties of the basis can be exploited to simplify the computation of Wigner functions enormously. The ordered-operator methods are also applicable to the phase space functions of Kirkwood (1933) and Glauber (1963) type. Both kinds are briefly discussed with emphasis on how they differ from Wigner's description.

Increasing tight upper and lower bounds to the Schrodinger equation eigenvalues are obtained from the Riccati equation for the logarithmic derivative of the wavefunction. The solution of this non-linear equation is written as a Pade approximant and the bounds are given by the roots of a sequence of determinants. Numerical results are shown for the anharmonic and quartic oscillators.

Using the exact solutions of the Schrodinger equation previously given by Froman and Froman (1965) and by Giler et al (1985), the authors calculate the energy of the ground state for polynomial superpotentials with broken supersymmetry and with an arbitrary number of wells. Their method gives a result which, having a general form in accordance with that given by Salomonson and Van Holten (1982) and by Giler et al, is asymptotically precise, revealing features characteristic of supersymmetric potentials.

Recently, relativistic quantum mechanical wave equations involving an extra scalar evolution parameter (analogous to proper time) have been increasingly studied in the literature. Since the physical meaning of this parameter is still very much a matter of discussion the authors aim to give it a clear physical meaning and then to proceed to a stochastic derivation of a nonlinear evolution time Klein-Gordon equation with a view to representing possible vacuum dissipative effects.

Saturating the functional integral directly with instanton-anti-instanton-type fluctuations, the vacuum energies for a supersymmetrical quantum mechanical system with (i) a double-well and (ii) a triple-well potential are studied. In the former, the vacuum energy is raised by these fluctuations, indicating spontaneous breakdown of supersymmetry. In the latter, the vacuum energy stays at zero, indicating that supersymmetry is not broken. In addition, the energy of the next level supersymmetric pair of states has been calculated for the triple-well case.

The author investigates a simple method of constructing potentials which is related to the work of Bhattacharjie and Sudarshan (1962) and for which the Schrodinger equation can be solved in terms of known special functions. It turns out that this method can be related to supersymmetric quantum mechanics and this relationship can help to decide which special functions, satisfying linear homogeneous second-order differential equations, can be solutions of the Schrodinger equation with potentials of the form V(x)=W²(x)-W'(x). The author illustrates this procedure with the example of orthogonal polynomials and obtains explicit expressions of wavefunctions of a wide class of shape-invariant potentials.

A separation between collective and intrinsic degrees of freedom in the Heisenberg model of a ferromagnet at a finite temperature is achieved by reformulating the thermal boson expansion previously obtained by the present authors (1987). In the new approach, all the collective modes (spin waves) have wavenumbers lower than a certain value. The intrinsic energy turns out to be the usual mean-field energy minus a correlation energy due to the excitations of spin waves. The influence of the cutoff in momentum space on the critical temperature is studied and the temperature dependence of the spin-wave renormalisation factor and the magnetisation are calculated.

Using degenerate perturbation theory to all orders the authors derive the Hofstadter cluster rules (1976) for the spectrum of Harper's equation (1955). They also extend their method to calculate the fractal dimension, D_f, of the spectrum and speculate that D_f=1/2 for typical values of the incommensurability parameter phi . The clustering rules are also shown to be valid for a one-dimensional model of quasicrystals.

The finite-size scaling spectra of the six-states quantum chains with D₆ symmetry are studied numerically in the massless phase. At least three values of the central charge c of the Virasoro algebra are found: c=1, 1.25, and 1.3. On the self-dual line, one finds, in the c=1 region, N=2 superconformal invariance. In the c=1.25 region there is a line where N=1 superconformal and Zamolodchikov-Fateev symmetry (1985) is observed. The occurrence of higher symmetries is also discussed.

The relation between scalar field theories with short-range and long-range exchange or correlations is studied. It is shown to all orders in perturbation theory that the critical exponents of fields and composite operators are continuous functions of the parameters alpha ( xi ) characterising the decay rate of the long-range exchange (correlations) with power-like falloff 1/r^{d+2-2 alpha} (1/r^{d-2 xi} ). The scaling law (d-2 xi )v=2, where v is the correlation length exponent, as well as the Harris criteria (1970) are shown to be exact to all orders in perturbation theory. A discrepancy between two widely used approaches to the crossover problem is pointed out.