Table of contents

A method is described which enables multiple three-dimensional soliton-like solutions to be found for non-linear Klein-Gordon equations of the type Square Operator phi =F( phi ). The method is extendable to coupled equations Square Operator phi =F( phi , psi ); Square Operator psi =G( phi , psi ).

A systematic procedure for solving the Whitham equation in a two-layer fluid of finite depth is developed. An analytic solution which asymptotically evolves into exactly two solitons is exhibited. The characteristics of these solitons can be quite different from those resulting from the Korteweg-de Vries equation (the shallow water limit of the present theory).

A method is developed for the analysis of critical point singularities of the form f(t) approximately A mod t mod ^q mod ln mod t mod ^p with q known. The d=4, n=0 susceptibility series is extended by two further terms, and is analysed under the assumption of a singularity of the above type with q=-1. It is found that p=0.23+or-0.04, in agreement with the calculation (p=¹/₄) of Larkin and Khmel'nitskii (1969). The connective constant for the model is found to be 6.7720+or-0.0005.

The technique of exact enumeration coupled with series analysis has been used to study the behaviour of the properties of long self-avoiding walks on a square lattice slab as the thickness (D) of the slab is varied. Scaling arguments due to Daoud and de Gennes (1977) predict the variation of mean-square end-to-end distance and of free energy with D. The results obtained are consistent with these scaling predictions for the mean-equare end-to-end distance, but suggest that the free-energy crossover exponent is closer to unity than the value (⁴/₃) predicted by scaling.

A new calculation is presented for the effect of excluded volume interactions on the critical exponent z for polymer dynamics.

A system of two molten polymers of different lengths N_A and N_B is considered when the interaction is repulsive and leads to a segregation below a critical point T_c. It is known that the critical behaviour is of the 'mean field' type if N_A and N_B are comparable. However if N_B=1 (one of the constituents reduced to a small molecule) it is known that the critical exponents are rather of Ising type. The cross-over between these two behaviours is studied. A Flory-Huggins behaviour is found as soon as the small chains are themselves real polymers (i.e. N_B>>1) independently of the ratio N_A/N_B (>1).

The experimental conditions under which the intensity correlation function from a single atom undergoing resonance fluorescence may be observed are studied. As was pointed out by Jakeman et al. (1977), even under optimum conditions the number of fluctuations in the atomic beam prevent a direct observation of photon antibunching under the subsequent normalisation scheme used by Kimble et al. (1977). However, a direct observation of photon antibunching is possible in principle by adopting alternative normalisation schemes of the intensity correlation function.

Solutions to a scalar and an SO(3,1) gauge Minkowskian field theory are presented, which are closely related to the Euclidean instanton solutions and are integrable. They have vanishing energy-momentum tensors and finite actions.

The equation of motion of a spinless quantum particle in an external uniform parallel electromagnetic field is derived from the symmetry of the field by group theoretical techniques. The minimal coupling is obtained directly, without a detour via the free particle equation. This derivation, which is valid in a Galilean as well as in a Poincare framework, is a contribution to the recasting process of quantum mechanics in the spirit of a paper by Levy-Leblond (1975).

A general method of obtaining quasisymmetry (P-symmetry) groups, using the concept of semi-direct products, is established. Using the ideas of little groups and their allowable irreducible representations a new method of association of minor quasisymmetry groups with the irreducible representations of the generator groups is developed. Two concrete examples of diagrammatic representations of objects possessing the symmetry of minor quasisymmetry groups, when viewed as coloured groups, are given.

Well known properties of induced unitary representations are generalised for induced PUA representations. They are applied to obtain an expression for the multiplicities in the Clebsch-Gordan series of induced PUA representations. Finally how these multiplicities can be used for the determination of selection rules is shown.

The topological classification of point and line singularities in the A₁ phase of superfluid ³He is studied in both strong and weak magnetic fields and in both large and small volumes. Particle-like solutions are also discussed briefly.

A method of approximation for solving secular equations is applied to the problem of finding the first few eigenvalues of the quantum mechanical anharmonic oscillator. Numerical results obtained compare favourably with those obtained by other methods.

The problem of generating symmetry adapted wavefunctions, within the framework of the generator coordinate method, is examined. Results for the discretisation technique and the natural state formalism are applied to the quartic anharmonic oscillator.

Generalising the ideas of two previous papers (Lewis (1968), Kruskal (1962)), a method is devised for obtaining exact invariants for time-dependent Hamiltonian systems with one degree-of-freedom. It consists in firstly transforming to a new Hamiltonian which is linear in the momentum variable, and secondly in solving the related Hamilton-Jacobi equation. The Hamiltonian of an oscillator with a supplementary inverse quadratic potential is treated as an illustrative example. After that, a complete application is given to a class of polynomial Hamiltonians, including an interpretation and discussion of the possible extent of the results.

Lower bound energy formulae for the N-fermion problem of the type used by Hall (1967) and Post (1956) are considered in terms of a reduced density operator for the interparticle separation. In bounds of this type the N-body energy is bounded below by a sum of eigen-energies of states of relative motion weighted by occupations for the relative states. It is shown that the wavefunctions of relative motion are subject to constraints and in particular must be zero when the particle separation is zero. This modifies the energy spectrum of the reduced Hamiltonian of relative motion and improves the lower bound due to Hall. The improvement is demonstrated for some simple model problems. Furthermore it is shown that the bounds of this type currently known do not exhibit saturation regardless of the interaction details. This leads to some speculation on improved lower bounds.

The general solution of the equation delta R²/ delta g_ij=0 is found under the assumption that the metric has the generic form ds²= Sigma _a=1³e^{2 alpha (t)}(dx^a)²-e^{2 beta (t)}dt². Its relation to the special solution characterised by the constancy of R is examined.

The logic of gravitational field energy localisation for static or quasi-static fields is discussed. A particular form of localisation in the case of spherical symmetry is justified by physical considerations. This form coincides with that general form presented by Moller (1952) for the case of the Schwarzschild constant-matter-density fluid but differs when one considers other equations of state.

In the context of quantum gravity, the asymptotic behaviour of exponential multi-matrix superpropagators is investigated. By varying the dimension of the matrix field, a remarkable property of two-point functions is found, which suggests that quantum gravity is ambiguity free.

Expressions are derived for the free energy of a massless scalar gas confined to a spatial cavity in a static space-time at a finite temperature. A high temperature expansion is presented in terms of the Minakshisundaram coefficients. This gives curvature and boundary corrections to the Planckian form. The regularisation used is the zeta function one, and yields a finite total internal energy. However, it is known that the local energy density diverges in a non-integrable way as the boundary is approached. A 'surface energy' is suggested to reconcile these two facts. Explicit expressions for the total energy inside two infinite rectangular waveguides are obtained.

The author gives a critical analysis of a derivation due to Kubo (1966) of the fluctuation-dissipation theorem for classical systems. It is shown that a basic assumption having the appearance of a causality condition actually should be understood as an incidental corollary of the equations.

The author develops a theory of Brownian motion for interacting spherical particles suspended in a fluid. Due to potential and hydrodynamic interactions the effective diffusion coefficient depends on the concentration of particles. The correction to the bare diffusion coefficient is derived to first order in the concentration.

Operator disentangling, the evaluation of matrix elements in the coordinate representation, and a matrix technique are all useful for calculating ensemble averages of exponential quadratic phonon operators. Examples are presented using all three techniques, and their respective merits are discussed.

A recently proposed asymptotic form, due to Domb (1976), for the total number of bond and site animals of size n, has been investigated numerically. It is found to fit the available data better than simpler forms previously assumed. The critical parameters entering into the asymptotic form are estimated for a number of two- and three-dimensional lattices, and conclusions are drawn about their lattice and dimensional dependence. In particular, the cluster growth parameter lambda is estimated with a higher degree of precision than that previously attained.

The author studies the random bond Ising model on pseudolattices containing triangles. By using the star-triangle transformation, the problem can be mapped onto a Bethe lattice where the solution to the random bond problem is well known. The solution is examined for the dilute ferromagnet in detail. It is shown that the transition temperature goes to zero at the percolation concentration as one would expect.

The authors study a spin-1 Blume-Emery-Griffiths model on the triangular lattice using a generalisation of Niemeijer (1974) and van Leeuwen's (1975) discrete spin renormalisation group transformation which preserves the Potts symmetry of this model. An overall phase diagram is obtained that is basically in agreement with other treatments of this model. The exponents are compared with other renormalisation group studies and some series results.

The critical slowing down, and its dynamic exponent are studied in the one-dimensional Glauber-Ising model. The calculations serve as an example of the application of the ideas of the real space renormalisation group in time-dependent problems.

The stationary point used by Sherrington and Kirkpatrick (1975) in their evaluation of the free energy of a spin glass by the method of steepest descent is examined carefully. It is found that, although this point is a maximum of the integrand at high temperatures, it is not a maximum in the spin glass phase nor in the ferromagnetic phase at low temperatures. The instability persists in the presence of a magnetic field. Results are given for the limit of stability both for a partly ferromagnetic interaction in the absence of an external field and for a purely random interaction in the presence of a field.

The author derives the Feynman rules for the complex phi ⁴ theory using the Euclidean path integral formulation.

Two types of exact solutions for the non-linear field equations for the chiral invariant model of pion dynamics are presented here. One of them is a generalisation of the solutions obtained by Charap (1976). This more general solution includes the soliton solution as a special case.