Table of contents

A critical linking number, m_c, is defined, and an expression is obtained for the mean square size, R², of an entangled polymer loop. For linking numbers m_c, the existence of a configurational phase transition is demonstrated by the occurrence of a collapsed state in the polymer molecule.

Develops an approach for describing an isolated polymer chain in solution, without appeal to the n to 0 limit of the n-vector model. The generating function for self-avoiding walks is treated as a grand partition function. In this description, the fugacity per monomer plays a fundamental role as the unique scaling field in the problem. A physical picture of a polymer chain in both an ordered and disordered phase emerges naturally from the resulting thermodynamic relations. This picture turns out to be similar to percolation, and leads one to classify the phases of a single polymer chain according to their fractal dimensionality. The authors identify the monomer density as the order parameter. Since there is only a single scaling field the fugacity per monomer, all the critical exponents can be related to a single exponent. The authors find alpha = gamma , beta =1- alpha =d nu -1. Although this theory is not equivalent to the n to 0 limit of the n-vector model the underlying generating function, and hence the correlation functions, are the same.

The influence of pinning forces on domain-wall fluctuations is studied in a continuous planar solid-on-solid model with a one-dimensional interface. The system is simple enough so that exact results can be obtained for a variety of pinning forces. The pinning of the interface is formally equivalent to the binding of a quantum mechanical particle in a temperature-dependent effective potential. In the case of a short-range pinning force applied a finite distance from the edge of the system, there is a localisation-delocalisation transition at a finite temperature. The transition is qualitatively similar to that studied in the d=2 Ising model by O.B. Abraham (see Phys. Rev. Lett., vol.44, p.1165-8, 1980).

The decay to a constant value of the correlation function (S₀^xS_r^x) in the ground state of the spin-¹/₂ XY ferromagnet in two dimensions is found to be exponential.

Introduces the concept of 'graded permutation group' in the analysis of tensor operators in the classical superalgebras. For U(m/n) and SU(m/n), irreducible tensor representations correspond to classes of Young tableaux with definite graded symmetry type. Diagram techniques are given for Kronecker products, dimensions, and branching rules. The tensor techniques are complemented by the introduction of a superfluid formalism, in which U(m/n) and SU(m/n) act on (polynomial) functions over the appropriate superspace. Such superfluids may admit constraints. A general superfield interpolates between the classes of Young tableaux which correspond to particular types of constraint. The tensor and superfield techniques are illustrated with case studies of SU(2/1) and SU(m/1).

A simple procedure is outlined for adapting the basis spanning the irreducible representations of the permutation group S(N) to those spanning the product representations of the subgroup S(N₁)*S(N₂) where N₁+N₂=N. An algorithm based on the procedure is also discussed.

It is shown that numerical differentiation can be removed from the conventional solution of Abel's integral equation. Using the Gaussian quasipotential as an example, the authors show that the new inversion method is less sensitive to random errors in the input data.

A method for evaluating the integral over pn × n Hermitian matrices is given in the limit of large n. This is an adaptation of an earlier article and should be read in conjunction with it. Explicit equations are written only for the case of three matrices.

The standard version of Noether's theorem, when applied to the classical Kepler problem, leads to the constants of energy and angular momentum but does not give the 'hidden symmetry' known as the Runge-Lenz vector. Lie's theory of differential equations is used to obtain all three constants of motion. The transformations of solutions under the point transformations to which these constants correspond are studied. The results are generalised to n dimensions.

It is argued that the field of a charge in a stationary orbit co-rotates with the charge about the centre of mass. Such a field has no radiation term and is obtained by transformation from the co-rotating frame of reference. For a rotating reference system J.F. Corum's tetrad field (1980) is employed, the object of anholonomity affecting the derivation of fields from potentials. The author finds the force between the charge +or-e of positronium. Stationary orbits are selected by Bohr quantisation of canonical angular momentum. One solution gives the well-known 'atom-like' states. The other gives 'particle-like' states for which orbital motion is ultrarelativistic. In such a state positronium is termed a 'positronium unit'. The unit has a very long lifetime for radiative transitions, and should possess a weak magnetic moment. A system, 'trionium', in which two charges of one sign orbit about a central stationary charge of the opposite sign is also explored.

A general solution of the Bhabha-Corben equations (1941) for a classical spinning particle with radiative damping is obtained. The solution only contains 'slow runaway terms' which are compatible with the principle of undetectability of small charges. Some conserved quantities are obtained which could be used to represent the total linear and angular momentum of the particle-field system. It is shown that the whole radiative effect may be renormalised in terms of 'effective variables' which describe a particle with constant mass in the frame of the Bhabha-Corben formulation without self-interaction radiative terms.

The behaviour of heat conducting viscous fluids is described through a suitable set of hidden variables whose (objective) evolution equations account also for cross-effect coupling terms. Such equations are incorporated into a thermodynamic theory which includes Muller's (1967) as a particular case and leads to Navier-Stokes' and Fourier's laws when uniform constant gradients of velocity and temperature are concerned. Meanwhile, the whole nonlinear theory turns out to be hyperbolic; this is shown via a direct analysis of the propagation modes. Finally, the authors outline an operative way of testing whether the co-rotational derivative is the required objective time derivative.

Proves lower bounds on the radius of convergence of various Born expansions associated with partial wave Schrodinger operators h_l involving Coulomb plus short-range potentials. The authors' estimates, clearly indicating the high-energy character of these expansions, also exhibit the connection between the increase of the radii of convergence with increasing energy and the behaviour of the short-range potential at the origin. In the case of a repulsive Coulomb potential, the authors prove bounds on the number of eigenvalues of h_l and discuss the relation between the absence of bound states of h_l and the convergence of Born expansions.

Whereas the standard expansion of the 'quantum field theory' belonging to the s-d exchange Kondo Hamiltonian for a spin impurity in a paramagnetic host metal describes the high-energy scattering, the alternative renormalised perturbation expansion describes the low-energy scattering. In the high-energy regime the scattering centre acts as if it was an anti-ferromagnetically coupled spin, but in the low-energy regime it acts as if it was a spin of a different magnitude that is ferromagnetically coupled. Correspondingly, it is shown that two different Hamiltonians can give rise to the same field theory. In solid state physics this equivalence is the generalisation of the Schrieffer-Wolff transformation (1960) between the Anderson (or Wolff) model and the spin-¹/₂ Kondo Hamiltonian.

Application is made of two different generalised fluctuation-dissipation theorems and their derivations to the calculation of the gravitational quadrupole radiation resistance using the radiation-reaction force given by Misner, Thorne and Wheeler (1973) and the usual tidal force on one hand and the tidal force and the free gravitational radiation field on the other hand. The quantum-mechanical version (including thermal generalisations) of the well known classical quadrupole radiation damping formula is obtained as a function of the radiation resistance.

The surface and local susceptibility exponents gamma ₁ and gamma ₁₁ for semi-infinite phi ⁴, O(n) systems are calculated to order epsilon =4-d at the ordinary bulk transition temperature. The results agree with the predictions of correlation function scaling.

Cluster statistics obtained by the Monte Carlo method for percolation processes in systems of dimensionality two to seven are analysed for the percolation analogue of the thermodynamic equation of state. In particular, the authors calculate the scaling functions for the analogues of the thermodynamic potentials and their derivatives, and investigate their dependence on dimension d. They are guided by the two exactly soluble limits of d=1 and the Bethe lattice (d= infinity ). The scaling region, where a good degree of data collapsing can be observed, is investigated in terms of the two 'thermodynamic' variables, one of which is analogous to the temperature and the other to the magnetic field. The characteristic forms of the scaling functions are closely related to the 'thermodynamic' stability conditions.

The q-state Potts model on the triangular lattice with nearest-neighbour interactions and three-site interactions in half of the triangular faces is considered. The exact duality relation is re-examined from the point of view of determining its critical point. Using the continuity and uniqueness arguments the authors determine the exact critical point in the ferromagnetic model. It is argued that a transition exists in an antiferromagnetic model only for q=3. A conjecture is then made on the phase diagram for the q=3 isotropic model. These results are used to determine the exact criticality of a dilute Potts model on the honeycomb lattice.

An integro-differential equation is constructed which describes the evolution of a system of masses evolving by completely inelastic collisions and spontaneous fragmentations. The development of a characteristic size spectrum is demonstrated for a simple example.

Recalls how the field theory formalism is very useful for computing the density of electronic levels in disordered materials, and stresses those characteristics of the associated field theory which are peculiar to this model. In particular, the author shows that the localisation transition associated with the mobility edge has rather strange properties from the point of view of pure field theory: a sound computation of the critical exponents associated with the mobility edge is rather difficult, due to this unusual behaviour.

Using for spin-³/₂ matrices a direct-product structure involving the usual Pauli spin matrices, the authors derive the Dirac-Clifford matrices in terms of certain algebraic combinations of spin-³/₂ matrices in a representation-independent way, thus achieving an extension of the Pauli spin matrices from the usual spin-¹/₂ space to the spin-³/₂ space. Basing the derivation directly on this analysis, two algebras satisfied by spin-³/₂ matrices are derived. One of these, which is also satisfied by spin-¹/₂ matrices, is directly related to the spin-³/₂ algebras of Weaver (1978) and of Bhabha and Madhava Rao (for three objects). The other algebra is new and curiously is not satisfied by spin-¹/₂ matrices.

Starting from the spectral representation of the two-point functions in the axial gauge, the author solves the gauge identities so as to express the higher-point Green functions, linearly in terms of the two-point spectral function. The four-point functions are an important input for investigations of scalar electrodynamics and vector chromodynamics based on the gauge technique.

The extended (2*2) algorithm is supplemented by a variation procedure which improves the convergence rate of the algorithm.