Table of contents

The equivalence between direct-interaction theories and non-linear field theories is discussed on a classical (non-quantum) basis. It is shown that to have such an equivalence three-body forces must be introduced, if there is not coupling between different interactions, in the action-at-a-distance framework. Finally a new type of Poincare-invariant theory containing three-body forces is proposed.

A lower-bound renormalisation transformation of the type introduced by Kadanoff (see J. Statist. Phys., vol.14, p.171, 1976) is applied to the three-state Potts model. For both d=2 and d=3 the Kadanoff transformation predicts a continuous transition. Values for the critical exponents and the critical temperature are reported. The consistency of the d=2 results with series estimates gives the authors some confidence in the predictions for d=3.

It is shown that the 8-vertex model on a Kagome lattice is equivalent to the 32-vertex model on a triangular lattice. In particular, the ice-rule vertex model on a Kagome lattice is equivalent to the 20-vertex model on a triangular lattice.

Two definitions of the scattering phase shift are given. For non-local interactions giving rise to continuum bound states, the two definitions yield different phase shifts. Accordingly two different versions of Levinson's theorem result; one of them takes the continuum bound states into account, the other one does not. The former is usually used by physicists, in spite of indications that the latter corresponds more closely to physical phenomena.

It is pointed out that reflection and transmission coefficients may easily be obtained for Alfven waves normally incident upon the solar transition region, by a suitable analytic choice of a density profile which is a good approximation to reality.

Attention is given to the problem of explaining the shape of the spectrum of very energetic cosmic rays, particularly those above 10¹⁹ eV where evidence for a flattening of the spectrum is becoming stronger. It is proposed that the particles above 10⁹ eV are mainly extragalactic and that they originate from heavy nuclei generated in clusters of galaxies which are largely trapped but from which fragmentation neutrons are able to escape and decay to protons outside the region of confining fields.

Graphical techniques are developed in the case that the labels on legs (or edges) span a representation of a compact group. This generalizes previous work, in which the labels had to be irreducible representation labels, to cover many matrices or reducible tensors arising in theoretical physics. Diagrams corresponding to tensors are defined so that basis transformations, such as group operations or parity interchange, may be performed by standard graphical operations. Bipartite diagrams play a central role in the theory. The Jucys diagrammatic reduction theorems may be generalised so as to cover reducible tensors. Two examples are considered: group-subgroup bases are discussed and a graphical approach to group-theoretic restrictions on the elements of crystal tensors is given.

In perturbation calculations using basis states defined in terms of spherically symmetric potentials it is often necessary to simplify complicated expressions involving n-j symbols. A well known graphical technique can be used to aid in this process. The graphs are represented by their incidence matrices, so that the algebraic manipulations can be carried out by matrix arithmetic. It is shown that the sequence of operations required to simplify a given graph can be determined from structural considerations based on the properties of certain polynomials in the adjacency matrix. This provides a method of performing complete perturbation calculations of this type on a computer.

The operators r², p², and r.p+p.r are a basis of a representation of the algebra O_2,1. Using harmonic oscillator wavefunctions with a fixed angular momentum as basis of the representation space, the isometrically equivalent representation in work by Barut (1967) can be identified. This allows one to write down formulae for matrix elements, relative to oscillator wavefunctions, of any operator of the corresponding group representation. These include exp(icr²), exp(iq²p²/h(cross)²), their products, and the scaling operator. One can thus obtain an algebraic derivation of the matrix elements of a momentum-dependent nuclear interaction proposed by Tabakin and Davies (1966).

A canonical transformation which removes the coherent oscillatory motion of a particle in a stochastic potential (the renormalised oscillation-centre transformation) is constructed by a new classical perturbation method using Lie operators and Green function techniques. A frequency and wavevector dependent particle-wave collision operator is calculated explicitly for stationary, homogeneous electrostatic turbulence in the short wavelength limit. The width of the resonance is proportional to the one-third power of the quasilinear diffusion coefficient, in agreement with Dupree's result (1966). However the k dependence is quite different from that expected from a simple Wiener process model. At large k spatial diffusion dominates over velocity diffusion in sharp contrast with previous theories.

The states of a quantum mechanical system form a convex set with the pure states as extremals. For a system of spin-¹/₂, the set is a 3-dimensional ball. For spin j, the convex set is stratified by rank r, 1<or=r<or=2j+1. The dimension of the stratum of rank r is r(2(2j+1)-r)-1. A geometrical description, for j=1, of how the strata with r=1, 2, 3 fit together in the 8-dimensional convex set is presented. As a simpler example, the real section of this set is given, i.e., the states of the real 3*3 matrix algebra are described.

The constants of the Krori-Barua metric (see ibid., vol.8, p.508, 1975) are explicitly fixed in terms of the physical constants of mass, charge and radius of the source. The conditions for physical relevance are investigated leading to a functional dependence of the ratio of mass-to-radius on the ratio of charge-to-mass and also to upper and lower limits on these ratios. The functional dependences of the physical variables of the problem on the fractional radius, with the square of the charge-to-mass ratio as a parameter, are presented. The information obtained regarding the Krori-Barua solution is illustrated graphically throughout.

The model studied consists of a two-dimensional triangular lattice of which some sites are occupied by the centres of molecules. The lattice is wrapped around a semi-infinite cylinder terminating in a row of boundary sites. Maxima of the isothermal compressibility are located and it is shown that these maxima become steeper as the width of the lattice increases. The 'phase diagram' of these maxima is compared with the corresponding phase diagram obtained from first-order calculations. The probabilities for different types of occupation of the boundary sites are determined by the interaction with the environment and the successive occupations of the sites of the lattice, proceeding in a direction away from the boundary, are given by a stationary Markov chain.

A lattice model first proposed by Ginibre (1971, unpublished) and a related model are considered in the classical or equivalent neighbour limit. It is found that there exists an infinite number of phase transitions in the appropriate field space.

The consequences of the Lee and Yang representation (see Phys. Rev., vol.87, p.404, 410 (1952)) for a ferromagnetic Ising model are reviewed. Bounds for the Lee and Yang angle are given and the critical magnetic behaviour is analysed. A main result is that identities for the Mayer-Yvon coefficients are deduced, which lead to the expansion of the magnetization with respect to the variable (1-tanh² beta H). This expansion appears to exhibit remarkable positivity properties and the corresponding coefficients are tabulated. A central role is played by the Bethe lattice which is completely analysed, since its coefficients seem to give upper bounds for the more realistic regular lattices.

For pt.II see ibid., vol.9, p.479 (1976). An earlier study of the conventional nearest-neighbour Ising model on the tetrahedron lattice is extended to include four-spin interactions about each tetrahedron. The problem of deriving high-temperature susceptibility series can be converted into a form for which previous graphical techniques can be readily applied. Analysis of the series by Pade approximants and the ratio method suggests that the exponent gamma has the constant value of 1.25 independent of the strength of the four-spin coupling, in agreement with universality.

The derivation of series expansions for the mean size of finite clusters in the Ising model is described briefly. From the analysis of low temperature series it is concluded that for a two-dimensional lattice in zero magnetic field the mean size probably diverges at the Ising critical temperature, T_c, as (T_c-T)^{- theta} , with theta =1.91+or-0.01. It appears therefore that theta < gamma '=1.75 the corresponding Ising susceptibility exponent. For a three-dimensional lattice it is tentatively concluded that the mean size diverges at some temperature T*<T_c.

An extension of Gibbs' theory of ordinary critical points is presented with the aim of describing and classifying higher-order critical phases in multicomponent fluid mixtures. A classification scheme based on two topological invariants, namely the dimensionality of the critical null space (corank) and the number of thermodynamic equations which characterise a critical phase (codimension) is proposed. A classical theorem of Marston Morse (1925) is used to discuss the phase diagram in the neighbourhood of critical phases of corank one and two.

The building up of concentration gradients in a critical mixture of aniline and cyclohexane is observed. A non-perturbative technique that allows the observation of the gradients along the entire size of the sample is used. The experiments are performed at three temperatures, i.e. (T-T_c)/T_c=1.8*10^-3, 6.0*10^-4 and 2.3*10^-4. The experimental results allow the calculation of the magnitude of the barodiffusion coefficients and of the osmotic compressibility, as well as their critical behaviour. In addition, the Onsager coefficient can also be evaluated. The results are discussed in the framework of existing theories. It is shown that the observed values are close to theoretical predictions both in magnitude and temperature dependence.

A numerical method for designing reflector surfaces that produce a specified far-field over a given solid angle when illuminated by an isotropic source is presented. The technique, which uses the geometrical optics approximation, requires the solution of a particular non-linear, elliptic partial differential equation, previously derived by the authors. The equation is solved iteratively by applying a finite difference model to a linearised form. Examples of some generated reflector surfaces are given showing the existence of two distinct solutions for each case. The method is applicable to problems in microwave antenna design optics and acoustics.

The mass operator of an electron in a plane wave field of arbitrary polarisation is calculated to lowest order of alpha (but exactly with respect to the external field). It is represented by a threefold integral over elementary functions or a double integral over exponential integral functions. For a monochromatic plane wave and circular polarisation the number of integrations reduces by one and the mass operator turns out to be almost diagonal. In this case the corrected propagator is obtained and the modifications of the quasi-levels of the electron are investigated. The property of the mass operator and the propagator of being almost diagonal is related to a special symmetry of the circularly polarised field and found to be valid to all orders.