Table of contents

Exact solutions of the equations governing vacuum cylindrical gravitational wave spacetimes and colliding plane electromagnetic and plane gravitational wave spacetimes are presented. Both solutions are found by using the geometric technique of Harrison and Estabrook (1971) to find appropriate similarity variables to reduce partial differential equations to ordinary differential equations. One of the solutions is transformed into a solution of the Ernst equations.

A P, T-invariant theory of massive dually charged particles is proposed. This theory is based on the idea that a particle with multi-spin described by the Dirac-like equation is a magnetically charged particle. The concept of magnetic charge has, in this approach, a pure quantum mechanical nature.

The incremental free energy and magnetisation profile are obtained for a domain wall pinned in the surface of an Ising ferromagnetic half-plane.

A time-dependent real-space renormalisation group is used to show that all the one-dimensional kinetic Ising models with transition rate that depends on nearest neighbours only belong to the same universality glass of the critical dynamics.

Recent field theoretical calculations for lattice animals by Lubensky and Isaacson (see Phys. Rev. A, vol.20, p.2130, 1979) yield d_c=8 as the critical dimension and provide a first order epsilon -expansion for the exponent theta . Support for these predictions may be obtained by extending the previous work of Gaunt and Rushkin (see ibid., vol.11, p.1369, 1978) on the exact enumeration of site and bond animals on a d-dimensional simple hypercubic lattice to arbitrary d.

A large number of derivatives of the free energy with respect to the magnetic field are computed for the two-dimensional spin-¹/₂ Ising model along the phase boundary H=0, T<T_c. This series is found to be divergent, and so the phase boundary is a line of analytic singularities. The same analysis in three dimensions leads to the same conclusion, with lower precision.

The response of a simple cubic lattice of spacing L with a polarisable sphere of polarisability alpha at each vertex to a permanent dipole mu at one vertex deep within the lattice is evaluated.

The Gibbs paradox concerns the entropy of mixing of ideal gases: although the entropy change of mixing samples of different gases is unequal to the entropy change of mixing samples of the same gas, it is not clear within the framework of classical thermodynamics how the identity of the particles effects the analysis. An explicit examination of the concept of indistinguishability can clarify some of the implications of the quantum-mechanical resolution of the Gibbs paradox.

In the framework of the new version of the replica theory, a sequence of approximated solutions is computed for the Sherrington-Kirkpatrick model (see Phys. Rev. Lett., vol.35, p.1972, 1975) of spin glasses.

In an arbitrarily small neighbourhood of any zero of a real-valued bounded continuous atomic wavefunction, the wavefunction has both signs.

An explicit formula is given for evaluating the transmission coefficient of one-dimensional potential barriers corresponding to localised forces. The method is based on a simple discretisation of the Schrodinger equation leading to a three-term recursion relation. A continued fraction is then introduced in a quite natural way. Examples are provided to shed some light on the efficiency of the procedure, as compared to semiclassical approximations.

The concretisation of an abstract isomorphism between two Lie-algebras SO(4,2) leads to the transformation of the Keplerian problem from cartesian coordinates into Delaunay similar elements in the eccentric anomaly.

A self-contained superfield approach to global supersymmetry in anti-de Sitter space (OSp(1,4)) is developed. General transformation laws for OSp(1,4) superfields are established, and all basic elements of the OSp(1,4)-covariant formalism, such as covariant superfield derivatives, invariant integration measures over the superspace OSp(1,4)/O(1,3) in both the real and shifted bases, relations between different parametrisations of superspace, etc., are given explicitly. The reducibility questions are analysed, focusing in particular on the structure of chiral representations of OSp(1,4). The simplest linear OSp(1,4)-invariant models are constructed: the OSp(1,4) analogue of the Wess-Zumino model and OSp(1,4) extension of the Yang-Mills theory. The first model, together with the spontaneous breaking of OSp(1,4), exhibits an effect of the spontaneous violation of P and CP parties with the strength related to the anti-de Sitter radius. The relation of the proposed approach to supergravity is discussed.

Some unknown bilinear relations between the SU(2) Clebsch-Gordan coefficients are given. The derivation is based on the analysis of the triplet P contains/implies L contains/implies SO(3), (Poincare, Lorentz and rotation group). The form of the covariant p_mu -dependent functions with fixed maximal degree with respect to p_mu is given.

Using a complex base with a numerical system of labelling the author calculates symmetrised coupling symbols (the 3-jm symbols) for the icosahedral group. He makes full use of Racah's Lemma and explicitly states all the phase standardisations involved. The 3-jm symbols are chosen in a manner that leads to them possessing reordering symmetries that are almost as straightforward as in the case of SO(3). The basic isoscalars are given and the 3-jm's are used to obtain invariant 6-j symbols.

It is shown that the considered singular integral operator K_sigma has a simple discrete spectrum with asymptotic behaviour.

A semiclassical theory based upon the oscillatory integral representation of the quantal wavefunction is applied to the one-dimensional Schrodinger equations describing motion in linear and parabolic potentials. For more general one and two turning point problems, the oscillatory integral approach would seem to be equivalent to the method of uniform approximation using linear or parabolic comparison functions. The technique yields both continuum and discrete solutions for the parabolic potential-the former corresponding to scattering states (barrier problem) and the latter to resonances or bound states (harmonic oscillator). Oscillatory integral type solutions are globally valid not only in coordinate space but also in parameter space. This enables the general parabolic connection formula relating the asymptotic wavefunction in different parts of the complex coordinate plane to be derived.

Potentials of the r^-(s+2) power type are studied as the superposition of Yukawa potentials. The SL(s,R) group acting on the quadrivector impulsion considered as a quaternion generates a compact self-adjoint operator deduced from the Schrodinger operator by a Fourier-Fock transformation. The operator is approximated by finite rank operators and gives the spectrum of energy as a function of the coupling constant, the angular momentum and the exponent s.

For Schrodinger Hamiltonians with potentials that are rapidly oscillating near infinity, it is shown that the T matrix is in the Hilbert-Schmidt class, implying finiteness of the total scattering cross section. The high-energy behaviour of the cross section is estimated.

A new approach to the problem of constraining the pi pi scattering amplitudes by means of the axiomatically proved properties of unitarity, analyticity, and crossing symmetry is developed. The method is based on the solution of an extremal problem on a convex set of analytic functions and provides a global description of the domain of values taken by any finite number of partial waves at an arbitrary set of unphysical energies, compatible with unitarity, the bounds at complex energies derived from generalised dispersion relations and the crossing integral relations. From this domain the authors obtain new absolute bounds for the amplitudes as well as rigorous correlations between the values of various partial waves.

Some exact solutions are obtained for the non-linear coupled partial differential equations obtained by Lund and Regge (see Phys. Rev. D, vol.14, p.1524, 1976) for a relativistic string in a uniform external field.

Adiabatic regularisation is applied to a scalar field propagating in a Robertson-Walker universe with arbitrary coupling to the scalar curvature. Explicit expressions for the expectation value of the quantum stress tensor in an adiabatic vacuum are obtained. This calculation yields the terms which are to be subtracted from the divergent mode-sum expressions for expectation values of the stress tensor to give a finite, renormalised stress tensor. It is shown that the removal of the infinite terms in this subtraction procedure corresponds to the renormalisation of coupling constants in Einstein's equation. A short description is given of the way in which adiabatic regularisation produces a trace anomaly.

It is shown that certain sets of normalised unity-resolving (e.g. coherent) states in Hilbert space serve to generate upper bounds on the partition function of a given Hamiltonian in that space. These bounds may be viewed as generalisations of bounds derived previously by Golden, Thompson, Hepp and Lieb (see Phys. Rev. A, vol.8, p.2517, 1973). The new bounds are compared to the original Golden-Thompson bound by proving several theorems and by computing explicit examples.

It is shown how series expansions for quantities defined as site and bond valency may be derived. The series are used to obtain a measure of the degree of ramification of the infinite cluster, and techniques for obtaining a free energy series for the infinite cluster in a site-dilute ferromagnet are discussed.

The site percolation on two-dimensional lattices is shown to be related to a boundary problem. This permits a new approach for the site percolation on a Cayley tree; it is shown that there is no percolation on a Cayley tree, if one takes care of the great number of boundary sites; this is consistent with the absence of spontaneous magnetisation on such lattices. However, the usually considered percolation on these lattices is shown to be consistent with the existence of a spontaneous local magnetisation for sites which are far from the boundary. Finally, the ferromagnetic percolation problem on usual lattices is also considered as a boundary problem.

The real-space time-dependent renormalisation group approach is applied to the study of a square lattice. A decimation and a spin-block transformation are performed according to the potential moving trick suggested by Kadanoff (see Ann. Phys. Lpz., vol.100, p.359, 1976). A value z=2.2 is found for the dynamic index characterising the two-dimensional lattice.

The assumption of the universal atomic scale is not justified by observations. The latter indicate just the contrary, namely that the scale is a degree of freedom. The author attempts to state a concise general definition of the principle of relativity. The incorporation of the dilatational degree of freedom into the principle of relativity requires that physical events take place in the six-dimensional space V₆, the subspace of which is the Minkowski space M₄. The norms and angles of 6-vectors in V₆ are assumed to be conserved, whilst their projections into M₄ are in general not conserved under the transformations of the group ISO(4,2). The speed of light is invariant in V₆ but its projection into M₄ is invariant only under certain transformations. The theory provides a description of the muon to electron mass ratio. The 'primordial' transverse momentum of partons is proposed to originate from the dilatational momentum.

The recently detected photoelectron antibunching effect is considered to be evidence for the quantised electromagnetic field, i.e. for the existence of photons. Direct-action quantum absorber theory, on the other hand, has been developed on the basis that the quantised field is illusory, with quantisation being required only for atoms. The author shows that photoelectron antibunching is readily explicable in terms of absorber theory and in fact is directly attributable to the quantum nature of the emitting and detecting atoms alone. The physical nature of the reduction of the wavepacket associated with the detection process is briefly discussed in terms of absorber theory.

The Lagrangian of a 'supersymmetric Weinberg-Salam model' which is invariant under the action of the gauge group SU(2/1) and under rotations in a six-dimensional graded space is given. The author shows that such a Lagrangian does not, in fact, naturally give rise to the Weinberg-Salam model.

The static classical solution of the field equations, describing charged particles has been modified by a small perturbation obeying linearised field equations. Assuming time dependence of the perturbation in the form exp(-i omega t) it has been shown that omega ² is non-negative and the classical static configuration of the field describing charged particles is stable. Detailed analysis of the linearised field equations has been performed for the Born-Infeld model of the non-linear electromagnetic field. There is only a continuous spectrum of omega ² with certain distinct values of omega corresponding to the resonant states. The problem of the zero modes connected with translational invariance is also studied.

A method utilising the coherent state variables for the electromagnetic field and appropriate spin variables for two-level atomsis developed for obtaining the dynamical evolution of the field and the atoms separately. The case of a single radiation mode interacting with two two-level atoms is studied. The time evolution of the photon number, starting from n photons and both atoms at their higher levels, is given. Furthermore, when the initial state consists of n photons and one of the atoms in its excited state with the other in its ground state, the temporal development of the photon number, as well as the motion of each atom, are obtained. In the latter case it becomes apparent that energy exchange takes place among the atomic systems, while the field acts as a transfer agent.

A method is described for calculating the scattering of electromagnetic waves by a grid of equally spaced parallel cylindrical wires with finite conductivity. It is an adaptation of the 'Green function' method used in solid-state band-structure theory, as described by Ham and Segall (see Phys. Rev., vol.124, p.1786, 1961). The theory is restricted to the case when the direction of incidence is in a plane perpendicular to the wires. Curves are shown for the transmission at normal incidence in situations typical of infrared spectroscopy.

Three-wave interactions are investigated by the method of nonlinear perturbation, with special emphasis on nonlinear explosive instabilities in the presence of linear damping or growth.

Recently it has been shown that the derivative non-linear Schrodinger equation, concerned with wave propagation in plasmas, can be associated with a modified Zakharov-Shabat inverse scattering problem. The authors produce an operator formula for the most general system of equations is solvable by this method and develop a perturbation theory capable of determining the variation in the scattering data to first order. They illustrate the theory by applying it to the derivative non-linear Schrodinger equation containing an additional perturbing harmonic forcing term, and consider the effect of this perturbation on an algebraic soliton.

The permanent profile solutions of the continuous classical Heisenberg chain is reviewed and the author expounds on the application of the inverse scattering method. Extending and amplifying the work of Takhtajan (1977) he exhibits the 'diagonal' action angle representation of the model. The spectrum of the Hamiltonian is exhausted by a magnon band and a soliton band. The magnons have no internal degrees of freedom and can be characterised by the dispersion law E=p². Like the sine Gordon doublet, the solitons have internal structure, they carry a continuous angular momentum m, and are characterised by the dispersion law E=16 sin²(p/4)/ mod m mod , in accordance with Tjon and Wright (1977). The continuous Heisenberg chain is a completely integrable Hamiltonian system possessing an infinite number of constants of motion. The recursive procedure for the determination of the conserved integrated densities is established.

For pt.I see ibid., vol.12, p.2087 (1979). A mixture of the site Ising models, each constituent of which has a different value of spin, is investigated. The phase boundaries between paramagnetic and ferromagnetic phases, paramagnetic and antiferromagnetic phases, and paramagnetic and spin glass phases are calculated. The condition under which the spin glass phase is realised is obtained.

The most general spin model with nearest-neighbour interactions invariant under a global Z_p symmetry in two dimensions is considered. Dual transformations are discussed, and the subset of self-dual models is characterised. The phase diagrams for p>or=5 are particularly rich, containing first-order, second-order and infinite-order phase transitions. In particular, the existence of a massless phase similar to the low-temperature phase of the XY model is established.