Table of contents

The single-fold statistics of rays emanating from an infinite, corrugated Gaussian surface with fractal slope are investigated. Low moments of the ray-density fluctuation distribution are evaluated as a function of fractal dimension, D. It is shown that in the Brownian case, D=1.5, the distribution is exactly negative exponential, corresponding to K-distributed intensity fluctuations in a coherent scattering configuration.

The eight-dimensional Riemannian manifold which describes, via the theory of harmonic maps the self-dual SU(3) Yang-Mills equations is completely integrable. Sixteen independent Killing fields are determined, two groups of transformations which map solutions into solutions are constructed, and a conjecture on the complete integrability of systems by the inverse scattering method is formulated.

Using a nonsingular hypersurface that is not only asymptotically null, the authors prove the existence of spinor fields that yield a positive expression for the Bondi mass. Standard existence theorems based on compact hypersurfaces are applied.

The statistical mechanics of a dense fluid of charged rods is considered, and the corresponding phase diagram is calculated, using a coordinate transformation. A first-order isotropic-nematic phase transition is found, as suggested by an experiment of A.A. Brian et al. (1981).

Extend Griffiths' (1972) arguments to show that there exists the thermodynamic limit in a quenched random-bond Ising model with competing interactions on the square lattice. Peierls' (1936) arguments are extended to give 0.97394 for an upper bound of the critical concentration of the ferromagnetic phase for the system in which the exchange integrals take on J>O and -J with respective probabilities p and 1-p.

The exact surface tension for all angles and temperatures is given for the two-dimensional square Ising system with anisotropic nearest-neighbour interactions. Using this in the Wulff construction, droplet shapes are computed and illustrated. Letting temperature approach zero allows explicit study of the roughening transition in this model. Results are compared with those of the solid-on-solid approximation.

The group factorisation and correlation theorem methods used previously to provide a unique labelling of atomic shell symmetry coordinates are shown to give also a simple means for deriving explicit expressions for these coordinates. Results are given for the 12-atom shell and the two types of 24-atom shells in O_h symmetry.

The locally operating realisations of the Galilei group are studied. Levy-Leblond's equation for spin-one-half Galilei relativistic massive systems is rederived by making use of group theoretical methods and a generalisation to spin one is proposed. Finally, the relation to the corresponding Poincare relativistic equations is analysed.

Various results known for one-dimensional periodic Toda lattice equations are generalised to two dimensions. In particular, a generalisation of the Kac-van Moerbeke equations is derived from a set of first-order differential equations of which the zero gauge field strength is an integrability condition. The generalised equations are shown to be the unification of two different periodic Toda lattice equations and they naturally produce the Backlund transformation. The two lattice equations are simultaneously derived from a pair of 2(n+1)*2(n+1) potentials satisfying the zero field strength condition.

A method is given for solving a class of linear operator non-polynomial differential equations, i.e. equations with linear differential operators representing non-polynomial functions of the operator of differentiation with respect to the argument of the unknown function. Exact solutions of some particular differential equations of such type are obtained. As in a previous paper of the author it is also obtained that not all linear operator non-polynomial differential equations are of infinite order. Some of them have only a finite number of linearly independent solutions.

The need to define pointwise products and compositions with distributions is pointed out in the context of the problems of renormalisation, junction conditions and curved shock waves. Earlier definitions are briefly reviewed, and new definitions are proposed using non-standard analysis. Basic properties are established, and some products and compositions with the delta distribution are explicitly evaluated. With these definitions, the domain of validity of the nonlinear differential equations of classical field theory can be extended to include Rankine-Hugoniot equations are derived from the Euler equations. An immediate application to quantum field theory is pointed out.

Using the Backlund transformations the author has obtained infinitesimal transformations about an n-soliton solution of the KdV equation which form a complete set. They are shown to be the squares of eigenfunctions of the Schrodinger equation with the n-soliton as the potential. It is also shown that the conserved densities for an n-soliton are n in number.

An experiment is described in which an electromagnetic wave is retarded in a slow-wave structure and finally brought to rest relative to the laboratory by revolving the structure, in the opposite sense, at the same angular frequency as that of the wave. The static field system forced from the wave may be considered as electric charge having its origin in the distributed wave field.

Using Dirac's theory of canonical multipliers (as modified by Shanmugadhasan, 1973), the author investigates and derives the complete canonical formalism for a well-known relativistic model of a classical spinning particle (generalised to include asymmetry). While the procedure is well known from existing dynamical theory, its application to a relativistic asymmetric particle or top has not before been attempted. In the past, ad hoc means have been found for the formulation of a Hamiltonian method, and these have led to incomplete pictures for the model considered. The formalism restates many well known results, using the calculus of Dirac brackets, but some new results are also given. Paper I deals with the problem in general and explains the calculation of the complete set of constraints, the multipliers and the Hamiltonian equations for the model.

An approximate, variational method for the study of the gx²ⁿ anharmonic oscillator is presented. The idea of the method is to introduce into the unperturbed oscillator states the correlations originated by the presence of the anharmonic term via a unitary operator exp(iF) which is expanded up to second order in F. The variational principle determines F and one is led to a linear system of equations. The whole unperturbed basis is employed.

The conventional interpretation of the quantum mechanical probability amplitudes and probabilities is examined in detail with respect to gauge transformations and the Foldy-Wouthuysen transformations. It is shown that the conventional interpretation is gauge dependent and, when applied to the relativistic (Dirac) and the non-relativistic quantum mechanics, cannot be consistent with itself through the Foldy-Wouthuysen transformations. It is also shown that these two difficulties are not present in a gauge invariant formulation that explicitly incorporates Poynting's theorem and the conservation law of energy into the definition of probability amplitudes.

It is shown that in the interior region of a finite-range potential a continuum outgoing wave solution may be expressed as an infinite sum involving standing natural functions, which are defined as the residues at the poles, real and complex, of the corresponding standing Green function of the problem. The above representation is valid provided the elastic scattering by the potential is known.

The scattering of electromagnetic waves from underground tunnels is investigated analytically. A Green function approach is employed to formulate the problem for horizontally polarised incident waves. This approach results in an integral equation for the unknown interior field E(r) for the buried scatterer region. Expansion of the unknown E(r) field in terms of cylindrical wavefunctions is conjunction with the basic integral equation leads to an infinite set of linear equations. An analytical procedure is developed to decouple this system of equations when the excessive phaseshift inside the scattering region is small. This determines the inner field E(r) of the scatterer. In order to compute the scattered field for the far-field region a steepest descent integration technique is employed. Numerical results are obtained for several cases and are presented.

The one-graviton transition operator and, consequently, the classical energy-loss formula for gravitational radiation are derived from the Feynman graphs of helicity+or-2 theories of gravitation. The calculations are done both for the case of electromagnetic and gravitational scattering. The departure of the in and out states from plane waves owing to the long-range nature of gravitation is taken into account to improve the Born approximation calculations. This also includes a long-range modification of the graviton wavefunction which is shown to be equivalent to the classical problem of the true light cones deviating logarithmically at large distances from the flat space light cones. The transition from the S-matrix elements calculated graphically to the graviton transition operator is done by using complementarity of space-time and momentum descriptions. The energy-loss formula drove originally by Einstein is shown to be correct.

A general Lagrangian formulation of the metric-affine, metric compatible theories of gravitation is given. The applicability of the metric-teleparallel geometry to gravitation is considered. It is pointed out that a teleparallel theory with the Lagrangian usually accepted in the literature leads to a non-predictable behaviour of torsion. A new choice of the Lagrangian is proposed.

The Bethe ansatz solutions of the massive Thirring model are renormalised and used to obtain exactly the ground-state properties of the sine-Gordon model with a finite density of topological solitons. Using a theory of 1D quantum fluids, the correlation function exponents and low-energy excitation spectrum are obtained. Expansions in various limits are given, and various crossovers described.

Van Kampen's systematic expansion of the master equation is used to obtain some non-equilibrium properties of a one-dimensional model. The authors derive the phenomenological equation of motion and to zero order in the expansion parameter m/(m+M). They also calculate the velocity autocorrelation function, the fluctuation spectrum and the time-dependent diffusion coefficient for two degrees of departure from equilibrium. It is found that the correlation function does not exhibit a long-time tail pattern. For a generalisation of the model considered, for further away from equilibrium situations a memory sets into the system, giving rise to a specific structure of the hydrodynamical equation for the local concentration variable.

Discusses two models of polymer adsorption. In one model a self-avoiding walk on a D-dimensional lattice interacts with a (D-1)-dimensional hyperplane; in the other model, this walk must also lie in or on one side of this hyperplane. Both models exhibit non-analytic behaviour corresponding to a phase transition, but these phase transitions do not occur at the same point. The authors give numerical estimates of the locations of these transitions.

Reports studies of the critical behaviour of the Ising model with first and second neighbour interactions J₁, J₂ on the triangular lattice. High-temperature series have been used to determine the locations of the ferromagnetic critical line and the critical line in the region (J₁<0, J₂>0). On the latter line the ordering susceptibility appears to diverge algebraically with the exponent gamma approximately 2.4. The existence of a sinusoidal phase and an associated Lifshitz point are conjectured.

Considers the pure Z(N) spin systems (including the standard Ising and Potts models) as well as generalised gauge systems (plaquettes or more complex simplex) in d-dimensional hypercubic lattices. These models are self-dual, and the authors show how this duality can be thought of as a series-parallel transformation. The simplicity of the equations enables conjectures to be made on the approximate critical frontier of the diluted version of the above systems, including some particular asymptotic behaviours which are exact. As an illustration the d=2 diluted Z(4) spin system is discussed in some detail: for those regions where exact results are available the agreement is satisfactory.

Statistical assemblies of non-intersecting closed lines (loops) with a variable loop multiplicity are considered which may have relevance for defect models of certain phase transitions. A loop gas model on a lattice is formulated and shown to be formally equivalent to an m-anisotropic n-vector model in the limit n=0. In two dimensions, the equivalence with the eight-vertex model in a field is discussed. In a few special cases the novel model is equivalent to known ones.

In percolation theory the critical probability P_c(G) of an infinite connected graph G is defined as the supremum of those values of the occupation probability for which only finite clusters occur. An interesting question is the following: is each number between 0 and 1 the critical probability of some graph? It is shown that the answer is positive. A remarkable intermediate result is that for an important class of graphs the following holds: for each p>or=P_c(G) there exists a subgraph of G with critical probability equal to p.

A superfield action is proposed within an OSp(4/2) framework whose component form reproduces the covariant xi -gauge Yang-Mills action, but with modified ghost-compensating terms. (The case xi =0 reduces to the usual Landau gauge.) 'Supertranslations' give rise to extended BRS transformations, and lead to constraints amongst the renormalisation constants. In addition, the system admits 'super-Lorentz' transformations, which mix vector and ghost fields. For other field representations, the ghost structure suggested by the space-time supersymmetry OSp(4/G) is also exhibited. This simplifies the results for counting ghosts and their own ghosts.

Field theories describing particles with zero spin, but utilising higher-rank Lorentz representations than usual, are considered. Of the three new first-order formulations given, two have consistent and causal minimal couplings.

If quarks and gluons are truly unobservable in a direct manner, this should be an inevitable consequence of the theory describing them. A formalism, within which 'internal fields' are automatically 'confined', is developed by considering a class of transformations of quantum mechanical states that have no explicit macroscopic manifestation. One of the necessary consequences of this formalism is standard non-Abelian gauge theory, which arises in such a way that the results presented here seem to be consistent with accepted gauge-theoretic descriptions of subnuclear phenomena.

The reduced wave equation can describe a non-dynamical system with infinite-dimensional state space and a control at the boundary. A generalised controllability property is defined and it is shown that it holds in a bounded domain. This property states that the linear subspace spanned by the trace of the system solution (i.e. the monochromatic scalar field) on a given boundary is dense in the corresponding Hilbert space as the boundary control is made to span an adequate Hilbert space. By duality, the 'controllability' property is shown to be equivalent to 'observability' of the adjoint system. The connection of this statement with the uniqueness proof for inverse diffraction problems is discussed. The adjoint system solution is a Green function. The inverse problem is ill posed. It calls for a regularisation procedure related to minimising a real-valued convex functional which depends on complex variables. At this point Lions's theory of optimal control is applied to state the existence and uniqueness of the minimising control. Lagrangian theory is used to derive the explicit form of the functional gradient. The latter allows one to design a minimisation algorithm where primal and adjoint systems must be solved sequentially.

A special class of Heisenberg spin Hamiltonians with short-range antiferromagnetically signed interactions is described. It is demonstrated via a simple group-theoretic argument that the ground states to many of these Hamiltonians are Kekule states, involving singlet spin-pairing between nearest-neighbour pairs of sites. A number of examples are given and previous related work is indicated. In addition to standard applications to exchange-coupled atomic ions, the relevance of this work for the simple valence-bond model of conjugated pi -electron networks is mentioned.

The problem of generating ground states of a quenched random Ising spin system with variable concentration of mixed-neighbour exchange couplings (J_ij()0) on a planar lattice (frustration model) is mapped into the problem of the Chinese postman which has been solved by a polynomial algorithm known as Edmond's algorithm. This algorithm is transposed and applied to the frustration problem. Not only is one particular ground state generated, but a post-optimal algorithm is established which gives the map of the rigid bonds and solidary spins (bonds in the same state for all ground states). This study of the rigidity on a square lattice reveals three distinct regimes by varying x, the concentration of negative bonds: a low-concentration regime where the ground states are rigid and ferromagnetic; an intermediate regime 0.1<or approximately=x<or approximately=0.15 where the rigid ground states are structured in an antiphase domain separated by magnetic walls; a high-concentration regime where the clustering of solidary spins is finite and separated by fracture lines. These defects characterise the phase transitions between the ferromagnetic, the random antiphase and the paramagnetic states which occur with increasing x.

Considers the problem of transition radiation in the presence of an alternating electric field. It is assumed that the particle travels at a uniform relativistic velocity and that the alternating electric field is applied at the boundary between two media in a direction parallel to the velocity of the particle. The method of Fourier transforms is used to solve the modified Maxwell equations with appropriate boundary conditions. Assuming that modulation energy is small compared with mean particle energy and using a linear approximation, the relativistic formula specifying the intensity of transition radiation in the presence of the field is presented in a compact form. It shows that in the presence of the field, the backward transition radiation increases, but forward transition radiation decreases compared with the radiation without a field. The contribution to the radiation due to the field varies linearly with the field strength, but varies inversely with the field frequency.

An elementary derivation of the expression of the Voigt functions in terms of circular functions and confluent hypergeometric functions is presented, correcting some errors in the analysis by H. Exton (see ibid., vol.14, no.4, p.L75, 1981). Voigt functions are used in neutron physics calculations.