Table of contents

Canonical transformations analogous to the Backlund transformation are discussed. Explicit formulae are given for many-body systems of particles interacting in one dimension.

The authors study a family of Hilbert spaces with positive definite invariant scalar product for the quantum mechanical description of relativistic spin-¹/₂ particles. The Hermitian and anti-Hermitian parts of the Dirac operator gamma ^mu p_mu are related to energy and helicity. Replacing p^mu by p^mu -eA^mu , the difference between the squares of these operators provides a minimal coupling evolution operator, similar to the second-order Dirac operator, for which F^{mu nu} is coupled to a covariant form of the Pauli spin matrices which generate SU(2) in a space-like surface. No Dirac sea is required for the consistency of the theory.

Generalises the formalism of Ogievetsky and Sokatchev (1980) to describe the geometry of N=1 supersymmetric Yang-Mills theory (coupled in general to supergravity). This generalisation seems to be analogous to the Kaluza theory. The author's presentation uses recent developments in geometry of supergravity and exhibits some similarity with the construction of Atiyah and Ward (1977).

Starting from a Fokker-Planck equation and using a projection operator formalism and a mode-coupling approximation, the self-diffusion coefficient is calculated for a strongly interacting system of charged spherical Brownian particles.

A method of obtaining the spontaneous magnetisation from finite-lattice matrix elements of the magnetic field operator, due to Yang (1952) and Uzelac (1980), is discussed. The method is demonstrated for the case of the Ising model in (1+1) dimensions, and is shown to provide smooth and rapidly convergent finite-lattice sequences. Applied to the case of the three-state Potts (Z₃) model in (1+1) dimensions, the method yields an estimate beta =0.111 09+or-0.000 05 for the critical exponent. This confirms Alexander's conjecture of universality with the hard hexagonal model.

Investigates the percolation properties of a random network of non-directed bonds (resistors) and arbitrarily oriented directed bonds (diodes). For the square lattice, there is a multicritical line which connects the isotropic percolation threshold with a network fully occupied by randomly oriented diodes ('random Manhattan'). Along this line, symmetry and invariance properties are used to demonstrate that the correlation length exponents are constant. Furthermore, a lattice independent relation is proved which shows that at the isotropic percolation threshold, the correlation length diverges with the same exponent when the transition is approached by varying either the resistor concentration or the concentration of randomly oriented diodes.

The concept of fractal dimensionality is used to study the problem of diffusion on percolation clusters. The authors find from Monte Carlo simulations that the fractal dimensionality of a random walk on a critical percolation cluster in three-dimensional space is D=3.3+or-0.1 where the size of the cluster is restricted to be larger than the span of the walk, and is D'=3.9+or-0.1 for a walk on clusters not subject to this restriction. For two-dimensional space they find D approximately=D' approximately=2.7+or-P0.1. The exponent D (and D') is related to the scaling of the average length R of N steps via R^D varies as N. The fracton dimensionality which is related to the density of states was found to be 1.26+or-0.1. These results are in good agreement with the predictions of Alexander and Orbach (1982).

The authors study the droplet size distribution of the correlated site-bond percolation model introduced by Coniglio and Klein (1980), and also the usual clusters of two-dimensional Ising models near the critical point. Equilibrium configurations of the Ising model with nearest-neighbour interaction and also one with nearest- and next-nearest-neighbour interactions are generated through a Monte Carlo simulation, and then a cluster analysis is performed. The exponents beta and gamma for the Coniglio and Klein droplet distribution are found to agree, for both the nearest-neighbour and the next-nearest-neighbour model, with the corresponding exponents of the Ising model. The usual Ising clusters diverge only at T_c in the Ising model with nearest-neighbour interaction but not for the model with next-nearest-neighbour interaction. The Potts model formulation is used to predict the behaviour of the droplet for general further-neighbour interactions.

A universal amplitude ratio was determined from cluster numbers in the Ising model at twice the critical temperature. The authors' Monte Carlo data agree with those for random percolation (infinite temperature), in agreement with expectations from renormalisation group arguments, but in disagreement with earlier Monte Carlo simulations of Stoll and Domb (1979) on much smaller systems.

The q-state quasi Potts model is shown to be related in the q=1 limit to the statistics of alternate clusters on non-alternate lattices.

Monte Carlo methods are used to compute the renormalised coupling constant of the three-dimensional Ising model for different values of the correlation length. For an appropriate choice of parameters the correlation length scales like N for a lattice with N³ sites. By increasing N from N=3 to N-60, the authors observed a systematic downward trend by more than twice the statistical error for a quantity which should be constant if hyperscaling is valid.

The authors establish explicit relations between the residual entropy of the one-dimensional Ising chain with a nearest-neighbour ferromagnetic and kth-neighbour antiferromagnetic interaction, and the residual entropies of the one-dimensional Ising chain with many-neighboured antiferromagnetic interactions in the corresponding maximum critical fields. The obtained results are, in particular, relevant to the chains that appear in the axial next-nearest-neighbour ising model.

The authors study one-dimensional continuous-time random walks for which the pairs (W_n⁺, W_n+1^-) of nearest-neighbour transition rates are assumed to be independent, equally distributed random variables. The long-time asymptotic behaviour of the mean displacement, (x(t)), is determined exactly for a specific model system in which 'diodes' (u, 0) and 'two-way bonds' ( lambda v, v) occur with probabilities p and 1-p, respectively. For lambda <1-p, we find that (x(t)) approximately T^nu F( beta ^-1ln t), where nu =ln(1-p)/ln lambda and beta =ln lambda , and where F is a periodic function with period 1. The mean displacement thus not only increases slower than linearly in time, but exhibits superimposed, non-decaying oscillations.

A new method for simulating real polymer chains is developed and applied to self-avoiding walks (SAWs) of length 49-599 on a three-choice square lattice. Very good results for the entropy are obtained which deviate from the series expansion estimates by 0.1-2%. The author also discusses how to extend the method to models of polymer chains with both excluded volume and finite interactions (attractive or repulsive). The author's method is expected to be more efficient than other simulation methods for treating self-interacting SAWs and chains which are subject to various lattice constraints.

Smouluchowski's equation for rapid coagulation is used to describe the kinetics of gelation, in which the coagulation kernel K_ij models the bonding mechanism. For different classes of kernels the authors derive criteria for the occurrence of a gelation transition, and obtain critical exponents in the pre- and post-gelation stages in terms of the model parameters.

In the above models there exists a weak singularity of the magnetisation, in addition to its jump, on the first-order transition line. It corresponds to the Griffiths (1969) singularity above the temperature of the phase transition. Analytic continuation from positive to negative magnetic fields, yielding a metastable state is impossible.

The S-matrix for magnons in the continuous Heisenberg chain is obtained with the quantum inverse method. Using the S-matrix for the strings of magnons, the position changes and phase shifts of classical solitons in scattering are derived.

Three types of Hermite-Pade approximants are considered, known respectively as quadratic, integral and differential Pade approximants. The singularity structure of each type of approximant is described. It is more complicated than that of the standard Pade approximant, and this property may often be used to estimate the types of singularity of a function from its power series expansion, as well as evaluate it on its branch cuts. Two applications in different fields are described which illustrate these properties of the above types of Hermite-Pade approximants. The first concerns the characteristic values of Mathieu's equation which are related to the energy eigenvalues of the harmonic oscillator on a lattice. The second concerns the investigation of the singularity structure and values of various physical quantities associated with periodic and solitary water waves.

The bound states energies and eigenfunctions of the Schrodinger equation with a radial Gaussian potential are obtained using a perturbational and also a variational treatment on a conveniently chosen basis of transformed Jacobi functions. The accuracy of the results is fairly good.

Using the results of an approximation method for the Yang-Mills fields, the authors have calculated a conserved quantity corresponding to the charge at space-like infinity using a geometrical construction. The result obtained is what one would expect on physical grounds.

The coefficient r for reflection above a barrier V(x) is computed semiclassically (i.e. as h(cross) to 0) employing an exact multiple-reflection series whose mth term is a (2m+1)-fold integral. If V(x) is analytic, all terms have the same semiclassical order (exp(-h(cross)^-1)); the multiple integrals are evaluated exactly and the series summed. If V(x) has a discontinuous Nth derivative, the term m=1 dominates semiclassically and gives r approximately h(cross)^N. If V(x) has all derivatives continuous but possesses an essential singularity on the real axis, the term m=1 again dominates semiclassically, and for V approximately exp(- mod x mod ^-n) gives r approximately exp(-h(cross)^-n(n+1)/) with an oscillatory factor corresponding to transmission resonances. The formulae are illustrated by computations of mod r mod ² for four potentials with different continuity properties and show the limiting asymptotics emerging only when the de Broglie wavelength is less than 1% of the barrier width and mod r mod ² approximately 10^-1000.

The authors quantise the classical canonical scattering transformation of Hunziker (1968) with the representation method of Moshinsky and Seligmann (1977). This leads to a sheeted phase space characterised by the number of turns around the scatterer. The usual detection device projects on the trivial representation of the corresponding ambiguity group. This operation extracts the integer values of the angular momentum.

The author shows that properties like spectrum condition, analyticity of n-point functions, space-like clustering of correlation functions stronger than any inverse power, the Reeh-Schlieder property do hold in Galilei invariant quantum theory. Furthermore, the range of validity of Haag's theorem is briefly discussed. The results seems to be of relevance both in the non-relativistic regime proper and as a hint that many of the properties typically attributed to relativistic field theories are actually a common feature of every theory with a zero mean-particle density and translation-invariant Hamiltonian.

Symmetry breaking for twisted scalar fields in non-simply connected space-times is studied. A general procedure is described whereby an approximation to the ground state in the broken phase may be found when the length scales associated with the non-trivial topology are close to their critical values at which the phase transition occurs. Application to twisted fields in S¹*R³ and R¹*RP³ is given.

The radiative Green functions for the massless scalar, electromagnetic and gravitational perturbations of the Kerr space-time are constructed using the Teukolsky (1973) formalism. The reaction force acting upon a test particle, which can emit radiation of any spin s=0, 1, 2, is calculated and shown to account correctly for the energy and the angular momentum carried away by radiation to infinity and to the event horizon. The azimuthal component of the reaction force is found to remain finite for a particle at rest in the Boyer-Lindquist coordinates owing to non-zero angular momentum transfer to the rotating hole. This anomalous static force of radiation reaction emerges as the counteraction to Hawking's tidal friction.

For a static dust distribution charged in both the electric and scalar sense the authors prove that the sum of squares of these two types of charge densities must be either greater than or equal to the square of the mass density, when the scalar potential is a function of the electrostatic potential. In the absence of either the electrical or scalar field the equality sign holds. In general, if there is no singularity in the matter distribution the three-space is conformally Euclidean. They show how to generate a special class of exact solutions of empty space in the presence of electrostatic and zero mass scalar fields and some properties are discussed. Finally they present two exact spherically symmetric solutions as examples.

The kinematics of the relativistically rigid motion of a surface is investigated in a Lorentzian manifold and the influence of the rigidity conditions on the congruence of the world lines of the points of that surface is examined.

The authors derive Poincare, de Sitter and conformal supersymmetry algebras, in all dimensions allowing Majorana spinors. They consider only minimal gradings (N=1), and show that these always exist. A brief discussion of fermionic central charges is given.

The minimal N=1 supergravity with local U(1) invariance is analysed in the framework of complex superspace. The existence of a new geometric dimensionless scalar invariant is shown. It is not present in real superspace geometry. In order to write down an action this invariant must be constrained. The constraint can be implemented in the action by means of a Lagrange multiplier. A weaker constraint leads to an action with 16+16 fields.

The truncated quantum mechanical n-vector model in one dimension is studied by means of the phenomenological renormalisation group (PRG) and the block renormalisation group (BRG) for general values of n. Particular emphasis is put on the extrapolation of the finite size results using the finite size scaling hypothesis. Accurate estimates of the critical properties can be made for 0<or approximately=n<or approximately=2. The two methods used are critically compared.

The authors introduce and analyse Z_p-symmetric models on the Union Jack lattice. They show that these models have the same self-dual structure already known for other Z_p systems. By performing renormalisation group calculations as well as Monte Carlo simulations they analyse their phase diagrams showing in particular that for p>or=5 they exhibit a disordered massless phase.

Derives exact relations that allow one to describe unambiguously and quantitatively the structure of clusters near the percolation threshold p_c. In particular, the author proves the relations p(dp_ij/dp)=( lambda _ij) where p is the bond density, p_ij is the pair connectedness function and ( lambda _ij) is the average number of cutting bonds between i and j. From this relation it follows that the average number of cutting bonds between two points separated by a distance of the order of the connectedness length xi , diverges as mod p-p_c mod ^-1. The remaining (multiply connected) bonds in the percolating backbone, which lump together in 'blobs', diverge with a dimensionality-dependent exponent. He also shows that in the cell renormalisation group of Reynolds et al. (1978, 1980) the 'thermal' eigenvalue is simply related to the average number of cutting bonds in the spanning cluster. He discusses a percolation model in which the 'blobs' can be controlled by varying a parameter, and study the influence on the critical exponents.

The effects of the quantum diffraction corrections at high temperature (k_BT>or=1 Ryd) are quantitatively investigated for the high-frequency component of the thermal microfield in a dense plasma. The authors make use of the double expansion of the static pair correlation function with respect to the classical plasma parameter Lambda and h(cross) omega _p/k_BT. Neutral-point and singly-charged-point cases are treated. Numerical data are plotted up to nu =1.4.

For pt.I see ibid., vol.15, no.4, p.1271 (1982). The free energy density and one- and two-particle distribution functions for a two-dimensional one-component plasma of particles with charge q confined to a strip bearing charge densities on its surfaces are calculated exactly at the temperature for which Gamma =q²/kT=2 (this parameter being dimensionless in two dimensions). The external dielectric constant is the same as that in the system interior so that there are no image forces. The density profiles look roughly like a sum of two independent double-layer profiles, except at extremely close separation. At large enough separation (more than the strip width) the two-particle distribution function decays with the inverse square of particle separation.

A two-dimensional lattice with harmonic interactions to nearest and next-nearest neighbours is studied in detail. The local anharmonic ion-electron interaction is described by a double-quadratic potential. The ground state is a periodic structure with different periods for variable parameters. In a simple mean-field approximation, it is found that with changing temperature, transitions between different periodic structures are possible.

In a weak-link constriction between two bulk superconductors it is well known that the condensed matter Hamiltonian exhibits periodicity in magnetic flux space. It is shown that a quantum electrodynamic treatment of the voltage across the weak link yields energy bands periodic in the space of Maxwell electric flux displacement charge.

Short-range spin systems with random interactions are considered. A simple proof is given showing that the free energy of almost every sample converges to the average free energy in the thermodynamic limit. A stronger criterion, thermodynamic convergence, is also demonstrated. This implies that the N to infinity and n to 0 limits may be interchanged in the replica method.

A thermodynamic observable is approximately additive under the decomposition of the volume. This property is of particular relevance to disordered systems. If scaled by the size of the system (N) a thermodynamic observable converges with probability one to a non-random limit as N to infinity inasmuch as one may apply the ergodic theorem. The author presents a simple argument to prove that the density of states in the Anderson model is a thermodynamic observable. Both diagonal and off-diagonal disorder are discussed, and the relation to the replica method is indicated.

Extended series expansions are derived for the high-temperature susceptibility of the classical Heisenberg model, on three-dimensional lattices. Series coefficients are presented to twelfth order for the simple cubic (SC) and face centred cubic (FCC) lattices and to eleventh order for the body centred cubic (BCC) lattice. The results are in agreement with earlier calculations apart from a small discrepancy at the tenth order on the FCC lattice. Extrapolation studies on the extended series are used to obtain revised estimates for the critical points (K_c) and the susceptibility exponent ( gamma ). On the FCC lattice, the authors also investigate the possibility of a confluent non-analytic correction to the dominant singularity. While the coefficients are consistent with the presence of such a correction term with an exponent ( Delta ₁) of 0.55, as predicted by renormalisation group (RG) calculations, the amplitude of the correction term appears to be very small compared with that of the first analytic correction term.

The high-temperature susceptibility of the four-dimensional classical Heisenberg model is studied by the method of series expansions. High-temperature series are presented to order K⁹ for the hyper face centred (HFC) lattice, to order K¹¹ for the hyper body centred cubic (HBCC) lattice and to order K¹² for the hyper simple cubic (HSC) lattice. The last three coefficients for the HSC lattice and all the coefficients for the other two lattices are new. The series are analysed for singularities of the form t^-1 mod In t mod ^p, predicted by the renormalisation group theory (t=1-K_c/K, where K is the high-temperature expansion variable J/kT). Fairly good convergence is obtained for p approximately=0.45 for all three lattices, in agreement with renormalisation group calculations.