Table of contents

Explicit travelling wave solutions with specified wave speeds are worked out for a model biochemical reaction proposed by Prigogine (1977).

A method of numerical integration is presented which leads to a Bohr-Sommerfeld type formula yielding exact eigenvalues of the Schrodinger equation. Tests on simple potentials give much better results than those of the usual JWKB methods.

Presents novel deduction of the inverse-square law of attraction between two masses in linearised gravity, utilising the Curzon and Levi-Givita solutions of Einstein's vacuum field equations.

The replica method, designed for the statistical mechanics of amorphous materials, involves evaluating the simultaneous averaging of n systems and studying the limit n to 0. It is shown that one can employ the method, for a fixed integral n, to study the phase change of normal systems such as a ferromagnet. The method throws an interesting light on the self consistency of methods of studying such systems.

Self-avoiding walks of 12 steps and fewer have been enumerated for the square lattice, for the case in which steps can occur between both nearest-neighbour and next-nearest-neighbour lattice sites. The appropriate two-variable generating function diverges along a critical line with a constant exponent gamma approximately ⁴/₃, in agreement with general universality expectations.

The authors study scalar field theories for which the interaction term of the Hamiltonian is cubic in the fields. They indicate the circumstances for which field theory models of this type represent continuous phase transitions. The renormalisation group functions for these models are presented up to, and including, three-loop contributions, giving critical exponents to order epsilon ³ in 6- epsilon dimensions. The exponent sigma which characterises the Yang-Lee edge singularity is given explicitly to this order.

The authors 'halve' an Einstein universe in two ways. For the spin-0 field they equatorially bound S³ by S², while for the spin-¹/₂ field we factor S³ to give the lens space S³/Z₂. In both cases the vacuum expectation value of the stress tensor is identical to that in the complete Einstein universe.

A rescaling argument is applied to systems with a highly degenerate ground state. It is suggested that these systems can exhibit a distinctive low-temperature phase in which correlations decay algebraically with distance. This behaviour seens to be permitted when the spatial dimensionality is sufficiently high, d>or=d_c. In addition, we find at d_c an infinite-order phase transition ( nu = infinity , alpha =- infinity ) occurring at non-zero temperature, and only paramagnetic behaviour at d<d_c. Since our results arise from a one-parameter renormalisation-group viewpoint, they should be seen as suggestive rather than definitive.

Obtains an infinite class of exact time-dependent solutions of the classical two- and four-dimensional sigma -models. In Euclidean space these solutions interpolate continuously between the corresponding meron and instanton solutions.

The invariant symbolic method initiated by Kramers in SU(2) is extended to the physically interesting group SU(4) by a projection technique which is described and applied to a reduction of direct products giving closed formulae for the multiplicities of some classes of representations.

A procedure has been outlined for generating configuration space basis states of many-particle systems in which the maximal occupancy of any single-particle orbital is arbitrary. An efficient computer program has been developed for determining the matrix elements of the generators of the unitary group U(n) over configuration space basis states. Computer times for generating these matrix elements have been presented for specific examples.

The validity of the Racah lemma concerning the relation between the Clebsch-Gordan coefficients of the representations of the groups and their subgroups is shown for corepresentations of anti-unitary groups and their subgroups. A method for calculating Clebsch-Gordan coefficients for all magnetic groups, based on this lemma, is presented. Starting from the Wigner coefficients and using this method, the Clebsch-Gordan coefficients for the single-valued and the double-valued corepresentations of the 90 anti-unitary magnetic (Shubnikov) point groups have been calculated. An example for the calculation of the coefficients for the point groups 41'=C₄(X) Theta , 41'=S₄(X) Theta , 4/m'=C_4h(C₄), 4'/m=C_4h(S₄) (as subgroups of O(3)(X) Theta ) is given. A comparison with other possible methods is discussed.

Transformation to a body fixed reference frame and direct projection of angular momentum quantum numbers are alternative ways to account for rotational invariance of a many-body Hamiltonian. Both procedures are illustrated by comparing the Born-Huang and the generator coordinate treatment of a diatomic system.

A formula due to Okubo (1975) for the eigenvalues of generalised Casimir operators of semi-simple Lie groups is used to drive an explicit expression for these eigenvalues. Full use is made of the Weyl symmetry group and is shown that this expression may be cast in the form of a symmetric power sum expansion. Expansions are derived for operators of order p constructed using the defining representation of each simple Lie group for all p<or=8. The results are in accord with the known facts regarding a complete set of algebraically independent operators and yield algebraic relations amongst those which are not independent. The expansions for the orthogonal and symplectic groups are a distinct improvement upon those obtained earlier, whilst those for the exceptional groups are the first of their kind.

The realisations of all classes of unitary irreducible representations of the generalised Poincare group P(1,4) have been found in a basis in which the Casimir operators of its important subgroup, i.e. the Galilei group, are of diagonal form. The exact form of the unitary operator which connects the canonical basis of the P(1,4) group and the Galilei basis has been established.

The kth derivative of any Meijer G-function whose argument is proportional to any rational power of the variable is obtained in terms of another G-function. Many known formulae for derivatives of G-functions are easily derived from our result. The authors also obtain, in passing, both an apparently new relationship for rearranging parameters of a special G-function, and the Fourier sine transform of a G-function whose argument is a rational power of the variable. Applications are made to several examples of physical and mathematical interest, such as the Holtsmark function used in stellar dynamics and plasma spectroscopy.

Defines a general class of problem that has been termed the inverse eigenvalue problem. Basically similar problems have already been studied as isolated and specific examples in the analysis of time eigenvalues appearing in neutron transport theory. In this work, however, the authors present a general unified method for their treatment using functional analytic methods. Specifically, the critical slab problem has been analysed as an example of such an inverse eigenvalue problem of a Fredholm integral equation using the theory of perturbation of a class of positive, analytic operator-valued functions in Banach space. Numerical calculations of the critical thickness are given. These results are encouraging, considering the simplicity of the method, which does not involve an explicit solution of the Fredholm equation.

A class of normal ordering representations of quantum operators is introduced, that generalises the Glauber-Sudarshan P-representation by using nondiagonal coherent state projection operators. These are shown to have practical application to the solution of quantum mechanical master equations. Different representations have different domains of integration, on a complex extension of the usual canonical phase-space. The 'complex P-representation' is the case in which analytic P-functions are defined and normalised on contours in the complex plane. In this case, exact steady-state solutions can often be obtained, even when this is not possible using the Glauber-Sudarshan P-representation. The 'positive P-representation' is the case in which the domain is the whole complex phase-space. In this case the P-function may always be chosen positive, and any Fokker-Planck equation arising can be chosen to have a positive-semidefinite diffusion array. Thus the 'positive P-representation' is a genuine probability distribution. The new representations are especially useful in cases of nonclassical statistics.

A simple method for calculating the extreme eigensolution of a real symmetric matrix of high order, alternative to Davidson's method, is investigated and compared with other similar methods.

Presents an exact formula for matrix elements of the free many-body propagator in the oscillator basis. It has the structure (a+bz)/(c+dz) where a, b, c and d are polynomials in the energy variable omega and the auxiliary function z( omega ) is defined by the continued fraction.

The interaction of an electromagnetic field and a set of point charges is governed by the Maxwell-Lorentz and Minkowski equations. These relate variables-the electromagnetic field tensor and the charge-current density vector-that have a direct physical significance and are uniquely defined (once a convention for their behavior under improper Lorentz transformations has been adopted). The Lagrangian formalism, however, requires the interaction to be expressed in terms of one variable that is uniquely defined together with one that is capable of being altered by a gauge transformation. It is demonstrated in an explicitly Lorentz covariant manner that there exists a class of path-dependent Lagrangians with the property that the so-called 'minimal-coupling' and 'multipolar' interactions are equal. In such Lagrangians, whether the path dependence (which is a manifestation of non-uniqueness or gauge dependence) is to be carried by the electromagnetic potential or by the polarisation-magnetisation tensor of the system of particles is entirely a question of convenience. In the analysis given, the integration paths for the potential need not lie in flat hypersurfaces, and those for the polarisation-magnetisation tensor need not move with timelike velocities, as was assumed in a previous non-relativistic treatment of the subject.

An analytic treatment of the time behaviour of spatially periodic solutions of the nonlinear Schrodinger equation is given. This predicts periodic time evolution in qualitative agreement with recent numerical results.

The repulsive phenomena which a particle experiences in the vicinity of a naked singularity are investigated in the Kerr-Newmann space-time. The aim is to extend the knowledge of this fact to charged solutions and to have a direct indication of how, in these situations, the gravitational and electrostatic interactions are competing.

The authors present a treatment of classical electromagnetic theory in the presence of a weak gravitational field, both in a vacuum and in a material medium, in terms of an effective dielectric and magnetic permeability tensor. They show that the gravitational red shift can be interpreted as the work done by the electric field of the light ray against the gravitationally induced polarisation current. They derive the dispersion relation for an electromagnetic wave in a medium and show that it depends upon the polarisation state of light.

Analyses the critical behaviour of a three-dimensional spherical model with nearest-neighbour ferromagnetic interactions (of strength J>0) in which there is a regular array of planes of bond defects (perpendicular to the planes) of strength J':0<or=J'<or=J spaced n lattice sites apart. For a fixed, non-zero defect strength J', the critical temperature has the form T_c(J',n)=T_c(pure)+c₁(J')n^-2+c₂(J')n^-2+o(n^-2). For J'=0 the problem reduces to that of finite size effect, while at J'=J the system becomes homogeneous. Two asymptotically distinct scaling regions correspond to these limiting cases. In both scaling regions the spherical field scales with the variable x identical to n² phi where phi is proportional to the deviation in spherical field. In terms of the variable lambda identical to 1-(J('/J)² and its conjugate lambda =(J'/J)², the impurity strength scales as y=n lambda in the region near J'=J( lambda =0) and as lambda for J' away from zero. These scaling relationships are valid for the spherical constraint equation and the thermodynamic quantities such as the specific heat and entropy.

Critical indices of several operators in the generalised Villain model are calculated to second order along a critical line emerging from the model's multicritical point. Relations between these expansions support the identification suggested between this line and a critical line of the Ashkin-Teller model.

The authors present a calculation of the bond percolation problem in a square lattice in presence of a 'magnetic field', using the position space renormalisation group and cells of dimension b*b, where b runs from 2 up to 5. Due to symmetry, the calculation splits into two parts, one determining the 'thermal' exponent nu and the other, the 'magnetic' exponent eta . For the largest cell in each case, one gets nu =1.355 (b=5) and eta =0.244 (b=4), in good agreement with series results of Dunn et al. (1975). Comments are made on the extrapolation of the results to b= infinity .

The authors introduce a new recursive method which allows one to solve exactly one-dimensional Ising problems with higher-order interactions. The new method presents some important features, for example its exactness and the computational ease of its solution. They present a new method for evaluating the partition function. Moreover an application of this to the general approach is given and an analytic solution is found.

The compressible Ising chain with the spins S=1 and S=³/₂ is studied. The lattice and the spin variables are decoupled by using a direct generalisation of the Mattis-Schultz transformation for S=¹/₂. The effective spin Hamiltonian appears to be equivalent to the rigid Ising chain Hamiltonian with an additional biquadratic interaction. The authors calculate exactly the zero-field free energy, the initial susceptibility and the correlation functions of the effective Hamiltonian. Numerical analysis of the obtained results reveals considerable effects induced by the lattice compressibility.

The Lagrange multiplier method is applied to the description of a spin-2 field. The method bypasses the problem of constraint breakdown and inconsistency of quantisation in the presence of interactions and the theory is shown to remain causal when coupled to an external electromagnetic field. Canonical quantisation of the field is carried out and a covariant propagator obtained. The massless limit of the theory is also discussed.

The infinite class of massless spin-1 actions formed from the two algebraic invariants F_{mu nu} F^{mu nu} , F_{mu nu} *F^{mu nu} which allow a supersymmetric extension is derived. It is shown that (to second nonlinear order at least) these extensions all have causal propagation, even though only one of them (Born-Infeld) was causal before supersymmetrisation.

Gluons in a fractionally charged quark model, arising within a non-Abelian unified gauge theory, are likely to be massless of colour SU(3) is to serve as a good classification symmetry. Unless gluons are selectively confined and quarks are not, the absence of massless gluons in weak decays of hadrons would seem to argue against observable fractionally charged quarks. This, however, does not preclude the existence of fractionally charged pre-quarks (preons, objects of which integer-charged quarks may be composed). The authors remark that if pre-quarks carry charge +or-e/3, the flavour symmetry group must be larger than SU(4) (possible SU(6)), within the context of the Yang-Mills type of unified gauge theories.

The technique of Brownian dynamics is used to calculate the time and scattering vector dependence of the structuring in colloidal systems. The van Hove space-time correlation functions and the corresponding intermediate scattering functions are presented; the latter are in good qualitative agreement with those obtained experimentally.

An exact formulation of the framework of the so-called 'local mean-field' theory is given in a plasma with an arbitrary number of components and it is proved that an approximate theory with a frequency-independent 'local-field' correction factor G_ij(q) has a basic difficulty. The difficulty is as follows. When G_ij(q) is chosen so as to satisfy the third frequency moment sum rule, the theory has the unphysical result: (a) 1/ epsilon (q,0) does not vanish as q to 0 in an electrically neutral system without a uniform, neutralising background charge and (b) chi _s=0 in a magnetically neutral system, epsilon (q, omega ) and chi _s being the generalised dielectric function and the spin susceptibility, respectively. An approximation procedure for removing this difficulty is discussed.

By an extension of the generalised Wronskian technique the authors reobtain the Backlund transformations and the class of nonlinear discrete evolution equations whose first member is the Toda lattice.

Earlier references to the capacity of two unequal tangent spheres are given.

The authors show that the effective Hamiltonian approximation and in a large volume of parameter space, quenched and annealed random locally anisotropic magnetic systems have the same critical temperature versus concentration behaviour.

This article is an addendum to J. Phys. A: Math. Gen., vol 11 (1978), No 7, 1347-51.