Table of contents

Radial moments of folding integrals for some non-spherical distributions that appear in problems relating to inelastic scattering off nuclei are evaluated in simple closed form in terms of the radial moments of the folded functions.

Numerical studies of the regularized long-wave (RLW) equation u_t+u_x+(6u²-u_xt)_x=0 suggests it has a two-soliton solution although an analytic form for this has not yet been found. The authors show that a modified form of the RLW equation u_t+u_x+(4u²+2w_xv_t-u_xt)_x=0, with u=w_t=v_x, has an exact two-soliton solution. The modified equation has the same solitary-wave solution as the original equation and its analytic two-soliton solution agrees closely with the numerical solution of the RLW equation.

Derives a new linear Dirac-like wave equation for spin-³/₂ employing four of the seven irreducible eight-dimensional matrices obeying the Clifford algebra C₇ with the wavefunction having the needed eight components only. Though this wave equation is not manifestly covariant and the wavefunction employed is not locally covariant, it is relativistically invariant and by its very derivation is connected to the Weaver, Hammer and Good (WHG) formalism (1964) for spin-³/₂ by a chain of transformations which can be arbitrarily chosen by us to be either unitary or non-unitary.

Uses a transformation suggested recently by Zwanzig and Ramshaw (1976) on the series expansion for the mean number of clusters in the square bond and triangular site percolation problems. The new series are smooth and well behaved, and the authors estimate critical exponents and amplitudes. A value of -0.668+or-0.004 is found for the exponent alpha _P, which is close to the rational fraction -2/3.

A brief analysis is undertaken of the statistics of lattice animals (connected clusters) of n cells, and the results are applied to the site percolation problem. Recent proposals by Stauffer (1975) and Leath (1976) are examined, and an alternative interpretation is offered of the relation between percolation critical exponents and cluster statistics.

The three-spin Ising model is related to a special case of the eight-vertex model by a spin transformation. It follows that the models have the same free energy magnetization and polarization.

A recent proof by Simons (see ibid., vol.9, p.413 (1976)) that the H function is convex for a system of mono-energetic particles scattering off fixed centres when the scattering matrix is isotropic is extended to show that the derivatives of H alternate in sign.

Using an exact, recently obtained solution of the non-linear Boltzmann equation for a particular type of molecular interaction, the authors show that the corresponding second time derivative of the entropy remains negative during the passage to equilibrium. The signs of higher derivatives are considered.

Properties of the seven parameter group O(3)_Lambda (T₂*T₂) are investigated using shift operator techniques. A complete classification and analysis of unitary irreducible representations is given and it is shown that a single arbitrary irreducible representation contains all those irreducible representations of the group O(4) which are realized by the hydrogen atom. Further, it is shown that the seven parameter group O(3)_Lambda (T₂*T₂) is equally as good a choice for the dynamical group of the hydrogen atom as the more usual fifteen parameter group O(4,2).

An analysis of representations of SU(3) in an SU(2) basis is given using SU(2) shift operators. It is found that, part from the hypercharge operator, there are two further SU(2) scalar Hermitian operators whose diagonalization leads to new bases for the representation space. These are modified to provide a basis which appears more suited to a description of the pseudoscalar meson octet than does the usual basis with diagonal hypercharge.

A new and straightforward method is presented for calculating the eigenvalue spectrum of a large symmetric square matrix each of whose upper triangular elements is described by a Gaussian probability density function with the same mean and variance. Using the n to 0 method, the authors derive the semicircular eigenvalue spectrum when the mean of each element is zero and show that there is a critical finite mean value above which a single eigenvalue splits off from the semicircular continuum of eigenvalues.

Bounds to overlap integrals are obtained by an operator inequality technique. The formulation involves a set of linear inequalities with several overlap integrals as unknowns. The 'solution' of this set of inequalities leads to bound expressions which complement those of Weinberger (1960) and of Cohen and Feldmann (1976).

For pt.I see ibid., vol.4, p.761 (1971). Bounds to overlap integrals between approximate and exact wavefunctions have been derived for arbitrary excited states of quantum mechanical system, using the Lagrange multipliers technique. A new lower bound to the ground state energy which is an improvement over the classical result of Temple has also been derived.

Time-dependent solutions of the Klein-Gordon equation for a massive scalar meson field in a Schwarzschild background space are obtained by the use of asymptotic methods. The solutions are found for all angular momentum states and are valid over all space exterior in a Schwarzschild radius provided the black hole is large, and the energy is very much less than the rest-mass energy of the scalar pi meson.

A general, conformally invariant theory of the general relativistic, gravitational field coupled with a scalar field is considered. The field equations of the theory are derived from a variational action principle and also from the assumption of the geodesic motion of test particles. These are shown to be identical. The scalar field is assumed to be a long-range field with the view to establishing a cosmological theory. An upper bound for the background scalar field is estimated.

Some cosmological solutions of the scalar-tensor theory of gravitation are derived. The results differ from Friedmann and Brans-Dicke cosmologies, though not sufficiently to enable an immediate decision between these theories to be made. In the case of the solar system the new theory is a special case of theories discussed by Bergmann and Wagoner.

A quasi-classical formalism is developed for spin algebras characteristic of Ising models, that is, an algebra in which all elements commute. Spin operators and their products are mapped onto continuous variables while traces are re-expressed as integrals over these variables.

Inequalities are applied to the lattice gas model, Ising model and higher order spin systems as well as to continuum analogies of spin-1 lattice systems. As a result, new inequalities for the case of non-ferromagnetic interactions are proved and a direct proof of a generalization of known magnetization properties is presented.

Inequalities are derived for a class of lattice systems including classical anisotropic X-Y and Heisenberg ferromagnets. Inequalities relating different models are also established and points out their connections with the existence problem of phase transitions.

The authors have studied the staggered ice-rule vertex model on a square lattice which allows different vertex weights for the four sublattices of the square lattice. The most general Pfaffian solution is given. It is shown that the model may exhibit up to three phase transitions. The specific heat diverges with an exponent ¹/₂ either above or below each transition temperature. The exact isotherm of an antiferroelectric model in the presence of both direct and staggered fields at a particular temperature is obtained. As the fields are varied, the system undergoes transitions among states of zero, partial and complete direct polarization.

The derivation of low-density series expansions for the mean cluster size in random site and bond mixtures on a three-dimensional lattice is described briefly. New data are given for the face-centred cubic, body-centred cubic, simple cubic and diamond lattices. The critical concentrations for the site problem is estimated as p_c=0.198+or-0.003 (FCC), p_c=0.245+or-0.004 (BCC), p_c=0.310+or-0.004 (SC), p_c=0.428+or-0.004 (D); for the bond problem as p_c=0.119+or-0.001 (FCC), p_c=0.1785+or-0.002 (BCC), p_c=0.247+or-0.003 (SC), p_c=0.388+or-0.005 (D). It is concluded that the data are reasonably consistent with the hypothesis that the mean cluster size S(p) approximately=C(p_c-p)^{- gamma} as p to p_c-with gamma a dimensional invariant, gamma =1.66+or-0.07 in three dimensions. Estimates of the critical amplitude C are also given.

The Niemeijer-van Leeuwen (1973) cumulant expansion for calculation of renormalized interactions is carried to third order for the Ising model on the triangular lattice. Kadanoff cells containing three spins are used. Five types of pair interactions and three types of four-spin interaction enter the calculations. The renormalization equations have one non-trivial fixed point. The accuracy of the result for the corresponding relevant eigenvalues is, however, less than in second order. The idea that the cumulant expansion is an asymptotic expansion has thus been given numerical support.

The modulation of frequency in resonant propagation can be characterized by the distance-dependent average carrier frequency of the pulse. The author shows that the propagation of this average frequency is governed by a conservation law specific for chirped pulses. He finds a relation between the average frequency, the corresponding average wavevector and the energy of the pulse.

The authors report a comprehensive ab initio all-mode second-quantized theory of super-radiance from extended systems. Spatially-dependent chirps during the cooperative motion are not consistent with the assumption of a single or symmetrical two-mode solution. Numerical integration of an associated set of c-number equations shows that super-radiance is characterized in general by a train of pulses in agreement with the two experiments. Propagational effects have an important effect on the cooperative radiative emission.

The general Lagrangian approach of Takahashi and Umezawa (1953) is used for the construction of free massive spin-1 and spin-2 Euclidean Markov fields. It is shown that these fields can be put in the framework of Markov field theory. An example of a Euclidean field which is Markovian but non-reflexive is also given. This field does not lead to a Wightman theory in the Minkowski region.

The Euclidean spin-1 massless vector field and the Euclidean spin-2 massless tensor field in general convariant gauges are shown to be Markovian. However, the reflection property is not satisfied. This indicates a close connection between the requirement of indefinite metric Hilbert space in Wightman theory and the violation of the reflection property in the Euclidean theory of electromagnetic and 'gravitational' potentials.

The problem of calculating the energy spectra of velocity and temperature fluctuations in a turbulence which is maintained by a permanent, large scale temperature gradient is studied, using a repeated cascade theory. A pair of coupled spectral equations is derived containing a turbulent momentum transport coefficient, which is identified as the eddy viscosity. An equation is formed which, when solved, leads immediately to an expression for the eddy viscosity. Solutions for the velocity and temperature spectra are found in the production-transfer, inertial and dissipation subranges. The Kolmogorov spectrum is recovered in the inertial subrange. The spectra have the form of power laws with exponents ranging from -1 to -7. Some observed spectra illustrate the predicted behaviour.

A kinetic equation formulation is used to analyse the density and current correlation functions for a classical liquid. The matrix elements of the collision integral which contain information about the behaviour of the system in the short-time and hydrodynamic limits are eliminated formally from the kinetic equation and the correlation functions derived in terms of both these matrix elements and reduced correlation functions which are solutions of a kinetic equation with the remainder of the collision integral. This procedure facilitates the comparison of most of the existing theories and leads to a flexible scheme for developing interpolation formulae for the correlation functions which are valid in both the short-time and hydrodynamic limits.

The validity of the simple Green function used in wave propagation problems is established by making use of the mathematically exact one.

The class of external fields for which the causal Green functions of the Klein-Gordon and Dirac equations can be calculated exactly by means of the methods of integrals of motion and coherent states is shown. Several important examples are considered.

For pt.II(a) see ibid., vol.9, p.463 (1976). The summation of the expressions obtained previously for the value of the absorption coefficient is performed analytically, and their asymptotic expansion derived. These results are shown to be in agreement with similar expressions obtained by other methods. It is concluded that this form represents a generalization of the Kramers-Gaunt low-field absorption coefficient.