Table of contents

A transformation is applied to low temperature series for the FCC Ising model, which enables the ratio method to be used in their extrapolation, and apparently more precise estimates to be made of the critical exponents than hitherto. These estimates are consistent with the scaling value of alpha '=1/8, and strongly favour the value of gamma '=5/4 over the previous estimates of around 1.28; they thus counter previous evidence for a possible violation of scaling in this model.

A formulation of the dilute bond Ising model is derived in terms of an equivalent Hamiltonian. A temperature concentration phase diagram is proposed. It is shown that, although no spin-glass is expected, the confluence at T=0 of the respective conditions for ferromagnetic and spin-glass transitions to occur induces the low temperature crossover to percolation behaviour.

It is suggested that in obtaining the complex refractive index from a transverse dielectric function epsilon _T(k, omega ) it is always better to treat spatial dispersion by expanding epsilon _T(k, omega ) about the free-space value of k, namely omega /c, rather than about k=O, as in the customary procedure due to Ginzburg, since this mistreats Doppler-like spatial dispersion. The proposed modification recovers correct results in zero-order and is: (i) computationally advantageous as it needs only first derivatives where second derivatives are needed in the usual method, (ii) equivalent to an infinite-order expansion about k=0, (iii) it is intuitively the obvious procedure to follow if k is approximately identified as a photon momentum, and (iv) it avoids any formal difficulties arising from the high symmetry of the k=0 neighbourhood.

A theoretical prediction concerning the density correction term to the speed of sound in a monatomic gas is presented. This correction term disappears at frequencies in excess of the frequency of three-particle collisions between gas atoms; i.e., the speed of sound reverts to the ideal gas value at these higher frequencies. Since the rate of ternary collisions is slow and can be varied by varying the density of the gas, experimental verification of this theoretical prediction appears eminently possible, at frequencies in the normal acoustic range. Estimates are given of the magnitude of the expected effect.

The influence of the strong wave E₀ sin omega ₀t on the stimulated bremsstrahlung at some frequency omega is considered. The asymptotics E₀ to infinity is investigated in two cases: when the electron velocity nu is parallel to E₀ and then the electron distribution is isotropic. The absorption coefficient alpha ( omega ) is found to be proportional to E₀^-1. Resonances in the stimulated bremsstrahlung, which occur near the points omega approximately=n omega ₀, n=1,2,..., are investigated. The conditions for negative absorption are discussed.

Some properties of Dirichlet L-series, of which the Riemann zetz function is just the simplest example, are given. The L-series are useful in expressing two-dimensional lattice sums as products of simple sums.

A condition proposed by Glasser for the double sum S= Sigma _m,n(am²+bmn+cn²)^-s:( mod m mod , mod n mod >0, to be decomposable into sums of products of simple sums is examined. The condition is that the reduced forms of the binary quadratic form am²+bmn+cn² be divisible so that there is just one class per genus. Although no general proof has yet been provided, procedures are described here for decomposing and evaluating any S satisfying the suggested condition, thus demonstrating that the condition is sufficient. Experience with double sums not satisfying the condition suggests that it is also necessary.

An integral representation is obtained for the Coulomb Green function belonging to a definite value of L_z.

A general form of two earlier relations involving 3-j symbols was found through further investigation of the helium atom problem. These relations are proved, and a very general summation is evaluated which should be most useful in summing any similar series found in connection with helium atom problem.

By formulating the generator coordinate method for bound states as a super-eigenvalue problem and using a Weyl theorem, a mathematical background is provided for the commonly used approximation schemes to solve the Hill-Wheeler integral equation.

Introducing generalized Mehler and Fourier transforms, the Funk-Hecke theorem is extended to the case of non-compact O(1,f) groups, and the exact expressions for the energy spectrum and the wavefunctions of the Schrodinger equation with non-local O(1,f), symmetric potentials in the scattering region are given.

A paper by Ben-Abraham and Lonke (see J.Math. Phys., vol.14, p.1935, (1973)) claims to resolve the ambiguity between the Schrodinger equation found by canonical quantization and that found from the Feynman propagator. It is shown that this claim is ill founded.

For pt.I see ibid., vol.9, no.1, p.35 (1976). It is shown that the radius measured by triangulation from a chord of a rotating system is identical to that obtained by using radar.

The radial equation of a static massive scalar meson field in a Schwarzschild background space is solved by asymptotic methods leading to solutions over the whole range. This approach is used to obtain the form of the Yukawa potential in the presence of a large Schwarzschild black hole, and to show that the meson field fades away to zero as the event horizon is reached, in agreement with the 'no hair' conjecture.

A static, bounded, spherically symmetric distribution of dust is considered; the constituents of this dust are simultaneously sources of gravitational, electrostatic and long-range scalar fields. Two kinds of scalar fields are considered-attractive and repulsive. Two classes of exact external solutions corresponding to these two types of fields are obtained; these solutions are asymptotically flat and satisfy the continuity conditions at the surface of the distribution. The internal solutions depend on two somewhat arbitrary functions of the radial coordinate. The solutions tend straightforwardly to all less general solutions found in the literature.

In the general theory of relativity electromagnetic waves interact nonlinearly through the field equations. A region of space-time containing two null electromagnetic waves is considered, and a new class of exact solutions of the Einstein-Maxwell equations is given. The solutions describe two waves following shear-free null geodesics 'twisting' through each other with zero contraction, the twist of one wave being proportional to the field strength of the other.

An elementary proof is given that the statistical mechanical transfer matrix, when symmetric, has a maximum eigenvalue which is non-degenerate and larger than the absolute value of any other eigenvalue. Moreover, the corresponding eigenvector can be chosen so that all its entire entries are strictly positive.

The trail problem on the square lattice is studied by the method of exact enumeration and its relation to the self-avoiding walk problem is pointed out. The number of N-stepped trails and their mean-square sizes are enumerated on a computer up to N=17. An asymptotic analysis of the numerical data suggests that certain critical exponents obey the same values for both the trail and the self-avoiding walk problem on the square lattice.

For a completely anisotropic n-vector model with Ising-like (pure single component coupling and non-negative single-spin and pair-spin interaction the longitudinal correlation function inequalities are established.

The rate and manner of a monatomic gas relaxing to equilibrium in a confined enclosure is studied. The nature of the eigenvalue spectrum associated with the linearized Boltzmann equation and the effect on this of the gas-surface scattering interaction are considered. In particular, the existence of a long-lived sinusoidally-damped behaviour is noted which depends on the viscosity and thermal conductivity of the gas, the gas-surface interaction and the size of the container. General results are obtained using transport theory, and more explicit expressions in the hydrodynamic limit.

A study is made of distributed feedback lasers operating above threshold. A set of non-linear differential equations governing the field distribution is obtained. Some properties of these equations are discussed analytically. Then the results of some numerical calculations are presented. It is shown that the field distribution inside the laser deviates notably from the linear case. Some explanations of what the analysis means in physical terms and how the different operating regimes are characterized are provided.

The most general covariant spinor derivative compatible with the Minkowski line element leads in a unique way to an antisymmetric torsion tensor. It turns out that this can be expressed by means of a pseudovector potential, counterpart of the usual vector potential of electrodynamics. In this way some of the models for a symmetrized theory of the electromagnetic interaction with spin-¹/₂ particles are rederived.

Exact solutions of the classical non-linear field equations for the chiral-invariant model of pion dynamics are presented. The solutions depend on the space-time variable x^mu through ,_mu x^mu , where k_mu is an arbitrary constant 4-vector.

A particular multi-matrix superpropagator, occurring in exponentially parametrized quantum gravity, is calculated in detail. It is shown that the method used can be applied to calculate a whole set of multi-matrix superpropagators. Furthermore it is demonstrated that gauges exist in which the superpropagators belonging to the set behave well asymptotically.

In the semi-classical approximation to the SU(3) sigma model it is shown how the breakdown of symmetry can be viewed as an unfolding of an unstable singularity in the potential. The examples arising include the Riemann-Hugoniot and parabolic umbilic catastrophes of Thom (1972) as well as catastrophes of much larger codimension.

Communications relations between fields and their conjugate momenta cannot be imposed on the light front x+x³=constant directly because, as a rule, canonical momenta are functions of the fields themselves rather than of the velocities leading to constraints between fields and momenta. To perform light front quantization. Dirac's method for the quantization of constrained systems is applied. Quantization is carried out through the Dirac bracket evaluated by means of constraints. An interesting relation between first-class constraints and boundary conditions is pointed out. Electrodynamics is quantized in a similar manner. If the gauge A⁺=0 is imposed, constraints become second class, subsidiary conditions do not arise but constraints modify the result of naive quantization in a definite way.

The zero mass limit of massive Yang-Mills theory is investigated and it is shown that there is a conflict between Lorentz invariance and the internal symmetry group in the theory. A necessary but not sufficient condition for the resolution of this conflict is the introduction of zero mass scalar fields.

For pt.I see ibid., vol.9, no.8, p.1375, 1976. The zero mass limit of renormalizable theories involving massive Yang-Mills fields is investigated. These theories can be classified in this limit in terms of the charge of the zero mass scalar fields introduced in part I.

The possible domain structures which can arise in the universe in a spontaneously broken gauge theory are studied. It is shown that the formation of domain wall, strings or monopoles depends on the homotopy groups of the manifold of degenerate vacua. The subsequent evolution of these structures is investigated. It is argued that while theories generating domain walls can probably be eliminated (because of their unacceptable gravitational effects), a cosmic network of strings may well have been formed and may have had important cosmological effects.

Recent results have indicated that there is a significant anisotropy in the arrival directions of cosmic ray particles having median energy in the range 200-300 GeV. The assumption is made that the majority of these particles are of Galactic origin and conclusions are drawn about their propagation in the local region of the Galaxy. Observations on cosmic gamma rays indicate that there is either a long-term gradient of cosmic ray intensity along the local spiral arm or that the Vela supernova is producing a temporary one; in either case the measured anisotropy indicates that the mean free path for particle scattering is about 7 pc along the local magnetic field direction and probably about half this value in the perpendicular direction.