Table of contents

The existence of energy gaps in the electronic spectrum for three-dimensional systems having short-range order but without long-range order, is proved.

Static solitary excitations in a two-dimensional diatomic lattice are studied in detail. The authors' model uses a local anharmonic electron-ion coupling for the polarisability of the anion and harmonic couplings between different chains. This lattice may serve as a model for certain ferro-electric materials. Solitary finite-energy excitations are found and proved to be stable.

A simple practical procedure is described for the calculation of moments of the density of states of 1D random binary systems obeying tight-binding type Hamiltonians. Short-order correlation effects are included. The procedure allows, in principle, the calculation of moments up to an arbitrary order. Explicit results for the 13 first moments are presented.

The largest group of space-time transformations which leaves covariant the free Schrodinger equation is a 12-parameter Schrodinger group S containing the Galilean transformations. The author discusses the covariance of the general Schrodinger equation under group S.

Derives a general expression of the conformally covariant energy-momentum tensor by starting from the general form of the Lagrangian from which the field equations are conformally covariant. The conformal group consists of inhomogeneous Lorentz transformations, dilatation transformations, and special conformal transformations.

If the energy of a molecule coupled to the quantised radiation field is made an extremum with respect to products of arbitrary molecular states and coherent field states, then there results a nonlinear Schrodinger equation for the molecule from which bounds for the energy contribution of the A²-term are derived. It is shown in what sense this contribution can be expected to be small under rather general circumstances.

The Atiyah-Singer index theorem for the Dirac operator on a manifold with boundary involves a non-local term constructed from the eigenvalues of the Dirac operator on the boundary. This eta invariant is evaluated for charged spinors on the left-invariant S³ boundary of the Taub-NUT istanton.

Surface terms in a gravitational action containing the quadratic curvature scalars R², R^{alpha beta} R_{alpha beta} and R^{alpha beta gamma delta} R_{alpha beta gamma delta} are considered and it is shown that a suitable surface term exists only for the combination R²-4R^{alpha beta} R_{alpha beta} +R^{alpha beta gamma delta} R_{alpha beta gamma delta} . The effect that the inclusion of this term in the action has on black hole evaporation is investigated.

Starting from an N-particle Fokker-Planck equation, the results for the concentration autocorrelation function are presented. It is shown that the corresponding memory function has the physical meaning of a (k- omega )-dependent longitudinal viscosity. A numerical mode-mode coupling calculation of the wavevector dependence of the longitudinal viscosity is found to be in good agreement with experimental results. Comparing the Fokker-Planck results with the corresponding Smoluchowski results show that the latter are only valid for small wavevectors or small interaction strength.

The phenomenological renormalisation group is applied to the 1D transverse Ising model with longitudinal imaginary field in order to investigate the Yang-Lee edge singularity (1952) of the equivalent 2D Ising model. The method is extended to calculate directly the exponent sigma of the magnetisation.

Discusses how the cluster size distribution above the percolation threshold is related to the existence of metastable states of the Potts model. The authors thus find that the average number of clusters per site containing n sites is given by C(n) approximately n^{- theta} ' exp(-constant n^1-1/d), where d is the dimensionality of the lattice. The exponent theta ' is given for a range of values of d.

The finite-size renormalisation group technique introduced by Nightingale (1976) is applied to the directed percolation problem. The decay of correlations is anisotropic in this model and finite-size scaling is extended to treat such anisotropy. Precise estimates for critical exponents and percolation probabilities are obtained for site, bond and site-bond percolation on the square lattice with bonds directed along the positive axes. Both free boundary conditions for which the results converge linearly with 1/n as n to infinity , and helical boundary conditions, for which, unexpectedly, the results converge linearly with 1/n³, are considered.

A new Monte Carlo method for studying bond percolation clusters is developed and used to identify new critical quantities associated with the percolation threshold. The bonds in each cluster are partitioned into three distinct connectivity classes, 'red' (singly connected backbone bonds), 'blue' (multiply connected backbone bonds) and 'yellow' (non-backbone bonds, often called dangling ends). Among the new cluster properties studied are the mean number of red bonds, a critical quantity diverging at p_c with exponent gamma _R approximately=1, and the length of the shortest connected path through the cluster which is critical with exponent gamma _min=1.35+or-0.02. For all cluster properties studies, the authors also compute averages over only the largest clusters; the corresponding critical exponents are found to be significantly different from those obtained by averaging over clusters of all sizes.

The critical behaviour of the bond-diluted two-dimensional quantum transverse Ising model is investigated by an approximate cluster decimation transformation. The dependence of the critical field on the concentration shows a discontinuity at the percolation concentration, associated with the existence of long-range order in the percolating cluster at non-zero transverse field, according to Harris' conjecture (1974).

Hamiltonians with Z(2) gauge symmetry are studied on a class of fractal lattices. The lattices considered have Fourier and Hausdorf-Besicovitch dimensions strictly greater than 2, but they have a low degree of connectivity characterised by their connectivity index 2. It is proved that there is no phase transition at non-zero temperatures.

Simple renormalisation group analysis in conjunction with the group-theoretic method is used to prove that the Heisenberg antiferromagnetic chain admits of no phase transition.

A lattice gas model of a solution is described. Its Monte Carlo simulation shows surfactancy and strong evidence for a phase transition in the surface.

A simple method of calculating the force between extended particles reveals the velocity dependence of inter-soliton forces in the sine-Gordon model.

A sufficient condition for nonlinear stability of steady solutions of the quasigeostrophic equations in a multilayer system is found. It proves the stability of given shear flows in a two-layer model that were known to be neutrally stable only.

Renormalised perturbation series are obtained directly (by the hypervirial method) for the perturbed oscillator and hydrogen atom, and indirectly (by series transformation) for the helium atom.

Considers how, in a reasonably motivated manner, to define and then evaluate unique finite differences between individually divergent sums S= Sigma ^infinity f(n) and integrals I= integral ^infinity dnf(n). This is done by first replacing f(n) by f(n mod lambda )=f(n)g(n mod lambda ), where g is a cutoff function (g(n to infinity mod lambda )=0) which obeys the permanence conditions g(n mod lambda to infinity )=1, and then transforming (S-I) into some convenient explicit functional D(f(n mod lambda )) which admits lim_{lambda to infinity} D(f(n mod lambda ))=D(lim_{lambda to infinity} f(n mod lambda ))=D(f(n)), where the final expression converges. There ought but there seems not to exist a convenient yet reasonably general theory for identifying admissible classes of summands and cutoffs and for deriving D(f), even though physicists have long dealt with simple cases ad hoc. Three specialised prescriptions are presented for D(f); one for suitably analytic f and g; another for merely differentiable f and g; and the unconventional ' epsilon -averaging method'. The mutual compatibility of these methods is discussed. Explicit differences are worked out for logarithmic, power-law, and exponential summands.

Burgers' equation is known to have wide application and attempts have been made in the past to solve the non-homogeneous form of the equation. Because of the limitations of the analytical solution for certain values of the parameter nu due to slow convergence the authors discuss a numerical approach which uses the method of lines to solve the homogeneous case and involves finite Fourier series. The method is attractive as it works efficiently and provides good results for nu =1.0 and results with non-increasing 'energy' for nu =0.1.

Representations of the sl(n+1, C) Lie algebras are constructed with the help of canonical (boson) realisations of these algebras. For every weight Lambda on the standard Cartan subalgebra of sl(n+1, C) the authors obtain a representation rho _Lambda ⁽ⁿ⁺¹⁾ (called the maximal representation) which contains an irreducible subrepresentation with Lambda as the highest weight. It is shown that for a major part of the weights Lambda the representations rho _Lambda ⁽ⁿ⁺¹⁾ themselves are irreducible. The standard construction of the highest-weight representations of semi-simple Lie algebras is based on the so-called elementary representations; comparing with them, the authors maximal representations are given in the explicit form.

It is shown that all halving subgroups of a group G which is a weak-direct product of two of its subgroups H and K, can be constructed using halving subgroups of H and K. Similarly, if K is of order two, one can find all subgroups of G via the subgroups of H. Using the former method, all of the 31 families of magnetic axial point groups of arbitrary order are determined. These groups are of interest when ferromagnetic and ferroelectric phases of quasi-one-dimensional systems are considered. Also, it is demonstrated that those among the non-crystallographic magnetic axial point groups which are compatible with ferromagnetism (ferroelectricity), admit magnetisation (polarisation) only along the principle axis of rotation.

Evaluates an integral found by the authors (see Phys. C, vol.6, p.1, 1981) in a meson spectroscopy. The calculations are performed using some properties of the generalised hypergeometric series.

Considers the separation of a system of finite, linear, coupled differential equations. The conditions which govern this separation in the case of a system of two coupled equations are discussed. It is then shown how these results may be extended to the case of a system of a finite number of coupled equations for which a general theorem on the separability is formulated.

The Lax technique can be successfully employed to derive the class of nonlinear differential-difference equations associated with the discrete analogue of the matrix Schrodinger spectral problem, and solvable by the spectral transform.

Shows that the best known nonlinear differential difference equations associated with the discrete Schrodinger spectral problem and also with the discrete Zakharov-Shabat spectral problem can be interpreted as Backlund transformations for some continuous nonlinear evolution equations.

The transformations of the action-angle variables allowed by the definition are described and the arbitrariness in the dependence of the Hamiltonian on the action-angle variables is explained. For Hamiltonian systems with the SU(n) algebra of integrals of motion an inverse relationship of 'actions' to generators of the symmetry group is discussed.

Under the vector addition of an arbitrary number of different angular momenta, each equal to s₁, s₂, ..., s_r, the resulting angular momentum j occurs with some multiplicity. The recurrence relations and the generating functions are obtained together with the general exact and asymptotic formulae for these multiplicities, which provide a complete solution to the problem. A comparison with the exact multiplicities is given.

Uses the non-relativistic limit of the integro-differential formulation of the Dirac equation in order to find an effective Schrodinger equation which, solved in the well known non-relativistic free-particle basis, gives approximately the same results (the probability amplitudes) as the Dirac equation for non-relativistic states written in terms of the free-particle spinors. Effective Hamiltonians for the electromagnetic and pseudoscalar interactions are derived. Comparison with the results of the Foldy-Wouthuysen method (1950) for the electromagnetic case is also achieved.

For a wavepacket representing a particle subject to a conservative force consideration is given to the evolution in time of the mean position and momentum and to the evolution of the spread of these quantities as measured by the mean square deviation from the mean. A closed system of equations involving these spreads is obtained by expanding the potential in powers about the mean position and by neglecting terms of third order or higher in the deviations from the mean, an approximation appropriate when the force does not vary too much over the width of the packet. These equations are solved in terms of the trajectories of a classical time-dependent oscillator. These trajectories can be found by differentiation of the trajectories for the force under consideration. In more than one dimension, or for more than one particle, the appropriate generalisation of the spread is the set of second-order correlations.

Gives an overview of the derivation and properties of exact solutions of the Einstein field equations which are spatially inhomogeneous with the source assumed to be an irrotational perfect fluid. It is shown that the known such spatially inhomogeneous solutions either admit a group of isometries with 2D orbits or are algebraically special. The solutions are related to a classification scheme which is based on the intrinsic and extrinsic geometry of the hypersurfaces orthogonal to the fluid flow.

It is proved, without imposing translation invariance, that the only stationary states for the infinite two-dimensional stochastic lattice-gas (or binary alloy) model are the canonical Gibbs states, and that these dynamics describe a strong return to equilibrium.

The critical behaviour of the square-lattice Ising model, with nearest and next-nearest neighbour interactions of either sign, has been investigated by means of high-temperature series. lhe location of the critical lines in the coupling constant plane has been accurately determined. Along the critical line which corresponds to transitions to the layered or superantiferromagnetic state a breakdown of universality is observed and explicit numerical estimates obtained for the exponent of the ordering susceptibility.

Using a generalised duality transformation, symmetry considerations and assuming criticality to be continuous in the system parameters, the authors obtain the phase diagram for two-dimensional Z(N) spin models (four-dimensional gauge Z(N) models). Besides the phases characterised by the spontaneous breakdown of Z(N) symmetries for spin systems (the behaviour of the Wilson loop for gauge systems), the authors predict the existence of a soft phase characterised by the vanishing of all powers like the perimeter). For the spin system phases with non-vanishing order and disorder parameters are forbidden when those parameters obey non-trivial commutation relations. For gauge systems all combinations of Wilson and 't Hooft loops decaying as the area and the perimeter are allowed. Duality relations for three-dimensional gauge plus Higgs system are given.

Considering relativistic wave equations in the first-order form, wherein the transformation of the wavefunction under the proper Lorentz group involves a certain number of inequivalent irreducible representations (IIRS) repeated an arbitrary number of times, the authors note some of the restrictions (on the skeleton matrix of the matrices beta ^mu occurring in the equation and on the spin blocks of beta ⁰) which arise from the requirements that the equation be barnacle free, and that there be solutions corresponding to a single spin and single mass only (without any degeneracy). If the number of IIRS is just two, these restrictions permit only two types of equations with no repeated IRS in either case. The authors also consider equations involving three IIRS with arbitrary multiplicity, carry out a reduction of the skeleton matrix, and analyse the implications of the mentioned requirements with regard to the possible existence of equations in which the multiplicity of one of the IRS is the sum of the multiplicities of the other two. Nothing is assumed about the specific IRS involved, except that they are linked.

A class of SU(2), classical, n instanton configurations is constructed over space-times which consists of S⁴ wrapped round itself n times.

Generalised Bose operators are constructed and applied to the linearisation of the equations of motion describing three- and four-wave mixing, resulting in integral equations for the temporal behaviour of the various fields. Some mathematical properties of these integral equations are studied, thus establishing the equivalence between approaches differing in the manner in which the linearisation is carried out. The integral equations are solved analytically in terms of jacobian elliptic functions.

From a semi-empirical model for the relaxation time of the heat flux and from a generalised Maxwell relation, the authors obtain the equations of state of an extended Gibbs equation which describes the dependence of a non-equilibrium entropy on the internal energy and the heat flux. By integration of the Gibbs equation, an explicit expression is obtained for the corresponding non-equilibrium entropy.

The equations obeyed by the density operator in a scattering problem are derived. In Liouville space notation, these resemble the Lippmann-Schwinger equations. Expressions for the transition rate for one- and two-potential scatterings are also derived.