Table of contents

Low-temperature expansions for the second moment of the spin density correlation function have been developed for a generalized ising model which includes three- and four-body interactions. The evidence for a downward shift in the value of gamma minutes +2nu minutes for a three-dimensional pure triplet ising model, which was previously predicted by the thermodynamic functions and scaling theory is well confirmed.

The charge ratio of single muons at ground level taken together with knowledge of the characteristics of high-energy nucleon-nucleus interactions in the atmosphere enables information to be gained about the mass composition of the primary cosmic radiation. In particular, the ratio of the number of neutrons (bound in nuclei) to protons and neutrons together can be determined. It is shown that the latest measurements from the Utah group (1975) can be interpreted in terms of a composition at primary energies of the order of 10¹³ eV/nucleon not very different from that below about 10¹¹ eV/nucleon where direct measurements have been made.

A variational method is presented which is based on the minimization of the norm of the function H-E mod psi ) and therefore can be considered as a generalization of the recently developed methods by Harris Ladanyi and Schmid (see abstr. A18170 of 1972). It seems more laborious than the above methods but it can be used with confidence when there does not exist, a priori, any information about the expected results. It is free of singularities, the results are obtained in terms of absolutely converging series and it provides criteria for choosing the components of the trial function. On the other hand it leads in a straightforward way to a simple formalism for upper and lower bounds of the phase shifts. This formalism needs a reasonable amount of calculation; it is easily applicable to one- or many-channel scattering as well as to the cases where exchange effects are taken into account.

A general formalism is presented which furnishes upper and lower variational bounds on squared overlaps mod ( theta , chi ) mod ², where chi and theta are vectors in a Hilbert space. The bounding functionals involve a self-adjoint operator A and a second operator B, such that A theta =0 and B chi =b chi . Most of the non-variational bounds derived by other workers on quantum-mechanical overlaps emerge from the formalism, together with optimal variational improvements in every case. In addition many new bounds are obtained.

Several aqueous dispersions of charged polystyrene spheres (mean radius =250 aa) have been studied at very low ionic strength by conventional light scattering and photon correlation spectroscopy. Plotted as a function of scattering angle, the mean scattered intensity exhibited maxima similar to those found in the structure factor of simple liquids determined by X-ray or neutron scattering. Fourier transformation of the corrected intensity data yielded radial distribution functions whose structure indicated considerable short-range ordering of the particles due to repulsive coulombic interactions. The reciprocal of the effective particle diffusion coefficient (determined from the initial decay rate of the non-exponential temporal autocorrelation function of the scattered light field) showed angular dependence similar to the mean intensity, in agreement with recent theoretical predictions.

Using a definition of percolation suitable for finite lattices the authors closely examine the percolation of such lattices. By extrapolation they calculate accurate values for the critical density p_c and discuss the size dependence of percolation. With the aid of a computer they enumerate exactly a certain class of site percolation configurations of finite square lattices with up to 81 sites. Rapidly convergent extrapolation procedures allow accurate determination of the critical density for percolation of the corresponding infinite lattice. We find p_c=0.5898+or-0.0008 for the site problem on a square lattice.

Self-avoiding paths and unrestricted or free paths are investigated on a lattice obtained from a cayley tree by replacing vertices by clusters of vertices and edges by bundles of edges. Closed analytic expressions are obtained for the generating functions, and the statistics of paths up to length 25 are explored.

Using renormalization group arguments the author expands n-point correlation functions (for non-exceptional wavevectors) in expectation values of translational invariant short-range operators O_i. He uses the fact that the Fourier components of our operators become negligible for wavevectors q large in comparison to the momentum cut-off. The correlation functions show the same non-analyticities at the critical point as the expectation values (O_i). The expansion coefficients are regular in the thermodynamic variables for q not=0. They can be expressed in terms of (a) functions which become singular at q=0 and yield the scaling behaviour, and (b) functions which are regular at q=0. The expansion coefficients of the two-point correlation function are sums of both types of functions.

High-field expansions are obtained for general three-state models on the face-centred cubic lattice. It is then shown that the method of partial generating functions can be extended to give series expansions for general multi-state models, such as the Potts models and the general spin Ising models.

The magnetic field produced in a plane conductor by a non-relativistic line charge moving parallel to the face is examined on the basis of classical electromagnetic theory. It is discovered that, at a fixed speed less than the local speed of light, skin effect is not a significant feature and the magnetic intensity decays algebraically rather than exponentially with distance from the face of the conductor. The multiplying factor does, however, depend upon the conductivity and becomes zero for a perfect conductor. For Cerenkov radiation, on the other hand, the position is reversed; skin effect is dominant and the attenuation is exponential. This result may affect the interpretation of the Aharonov-Bohm effect which is currently under discussion.

A quantum statistical analysis of the interaction of a radiation field with a system of electrons in cyclotron motion in a homogeneous field is presented. With high magnetic fields the interaction offers a possible means for detecting far infrared radiation. The redistribution of electrons in the Landau ladder of states after interaction with the field is calculated for various initial quantum states. The sensitivity of the detector is shown to depend crucially upon the initial quantum state of the electrons.

The field generalizes in a natural way the c-number theory of radiation reaction and frequency shifts due to Lorentz. By a trivial extension the operator field can be applied to the more complicated cases of a single multi-level atom and the super-radiant problem of N atoms on the same site where these are coupled either to the vacuum or the vacuum and additional imposed quantized fields. By further extension it also applies to the case of N atoms on different sites. Explicit expressions for positive and negative frequency parts of the total operator field are given in terms of arbitrary dipole operator sources and are apparently valid for times t>> omega _s^-1>> omega _max^-1 where omega _s is the atomic resonance and omega _max a cut-off used systematically throughout the theory.

In a massless fermion-gravity interaction field theory, the triangle graph with three external graviton lines is considered. The infinities arising from this and associated graphs are calculated in the dimensional regularization scheme, putting the energy-momentum of one of the external legs equal to zero. These infinities are exactly cancelled by those of the three-graviton graph originating from the graviton self-energy counter-Lagrangian density.

Evanescent interactions which disappear in four dimensions but which lead to divergences on the basis of power counting, are shown to be non-renormalizable. This result is not apparent at the one-loop level when the S-matrix elements are simple polynomials in the four-dimensional limit.

Whitham's theory (1965) of nonlinear water waves is applied to a nonlinear c-number field ( lambda phi ⁴ model) to investigate the propagation characteristics of the field in the plane-wave modes. A system of first-order partial differential equations is set up for the assumed slow variations of the wave parameters corresponding to each of the different types of Jacobian elliptic function wave modes. By the method of characteristics it is shown that the system is hyperbolic in two of the regimes, indicating that the disturbances in the wave parameters propagate with two different amplitude-dependent velocities.

A realization is given, in terms of Dirac matrices and suitable SL(2,c) generators, of Gel'fand-Yaglom matrices associated with the simplest parity-invariant wave equations for particles with half-odd-integral spin. Commutation and anti-commutation relations for these matrices are thereby derived, replacing those given previously by Lorente, Huddleston and Roman (see abstr. A8608 of 1974) which are shown to be incorrect.

It is shown that the Lie Algebra O(4,2) is an exact invariance algebra of Dirac's equation. The elements of this algebra become equivalent to the generators of conformal transformations in the zero-mass or high energy limit, provided the helicity of the electron is held at an appropriate value. A picture is suggested of the electron as 'a massless spin ¹/₂ particle in a space-time knot'.

The microwave power scattered by a circular cylinder of inhomogeneous plasma contained in a glass tube is computed with the aid of the Frobenius method. It is found that computed and experimental values are in good agreement in the range 0.2< lambda /R<3.0. Values are also computed for wavelengths of 8 mm and 3 cm for a range of collision frequencies with the electric vector parallel or perpendicular to the cylinder axis.

An obliquely impinging electromagnetic wave exerts a negative radiation pressure on an abrupt plasma-vacuum boundary. The radiation pressure on a conductor immersed in plasma consists of an electromagnetic part and a mechanical part. The equation of motion of a quasi-particle corresponding to a wave packet is derived. The time-averaged motion of a charged particle in a non-monochromatic wave packet is given. No assumption is made concerning the validity of Abraham or Minkowski momentum densities.

A series of model calculations is described and the results for average proton-initiated extensive air showers of about 10¹⁸ eV are compared with those of other model calculations and with a wide range of experimental data. Particular care has been taken to allow the model calculations to be matched with the experimental data and the conclusion of those authors that protons are present in the primary cosmic ray beam at about 10¹⁸ eV is supported. Further work is suggested to test whether the fluctuations found experimentally can be explained on other hypotheses.