Table of contents

The author shows that the Potts and Temperley-Lieb representations of the projection generators of a von Neumann algebra are, in general, reducible. The author writes down a new representation with non-vanishing product of odd generators R and gives evidence that it is a common element in these reductions. The operators in this representation may be interpreted as giving the bond transfer matrices of the square lattice Whitney polynomial. The author shows that the reduction of the Temperley-Lieb representation also contains irreducible representations with R=0 which are responsible for eigenvalues in the ice-model spectrum independent of those of the Potts model.

Dimensional shadowing is introduced as a formal method to reduce extra dimensions in configuration space by considering a fractal subset. The Hausdorff dimension of the fractal is then perceived as the physical dimension of configuration space.

It can be shown that the N-soliton solution of the Korteweg-deVries equation can be decomposed into N separate solitons (cf. Gardner et al. (1974), Calogero and Degasperis and Yoneyama (1982)). However, it is not immediately clear from the form of their solutions how the separate solitons relate directly to the single soliton solution. Here the two-soliton case is considered and a decomposition is sought which can be clearly related to the single soliton solution. Although it appears that there is a family of such decompositions it is shown that only one of these is correct. Although this decomposition is equivalent to the decomposition given previously by Gardner et al., Calogero and Degasperis and Yoneyama, the form given here is different. It is suggested that the form of solution produced here is a more appropriate representation of the solution since it is clear how it relates directly to the single soliton solution and it is easy, through this form, to analyse the interaction of the two solutions.

The tunnelling propagator is determined by means of an explicit evaluation of the small energy level splittings for ground state as well as excited state doublets in a symmetric double-well potential. Squeezing the initial state is shown to generate considerable chaotic tunnelling features, which are conjectured to involve a fractal dimension.

A sequence of measured values of a property of a quantum mechanical system is required to calculate the time average of that property. The implications of using the 'projection postulate' in elementary quantum mechanics during the process is discussed in relation to a simple example.

The authors first develop a method to obtain rigorous bounds for the Lyapounov exponent of products of a random matrix. When applied to a class of 1D problems. including localisation, it reproduces the correct scaling behaviour at the band edge and gives very good approximations of the prefactors. They then study analytically the successive moments of the distribution law for the trace of the random matrix product within the whole energy band. The band centre anomaly is found to affect the whole statistics of the problem and the exact anomalous value of the Lyapounov exponent is recovered through the replica trick.

A simple general method is presented for solving mean-field spin-glass models where the bond randomness is expressible in terms of an underlying site randomness. Both separable and non-separable models can be solved.

Originally, studies of the growth of fractal objects such as percolation clusters assumed that the growth sites have an infinite lifetime. Recently Bunde, Miyazima and Stanley (1986) have studied the effect of a fixed finite lifetime and they have found that the long-time growth evolves towards the kinetic growth walk with self-avoiding walk critical exponents. Here the authors consider for two dimensions the general case in which each growth site is randomly assigned infinite lifetime (with probability q) or a finite lifetime (with probability q) or a finite lifetime (with probability 1-q). The phase diagram is similar to that of site-bond percolation, a model used to describe solvent effects in gelation.

A Monte Carlo renormalisation group method is presented to study the cluster structure of an infinite cluster, at the percolation threshold. Applying this technique to bond percolation on the square lattice, fractal dimensions of an infinite cluster, its backbone and cutting bonds are calculated. This method converges very rapidly with an increase in the size of the cell. The author estimates that the fractal dimension of an infinite cluster D=1.88+or-0.02, the fractal dimension of its backbone D_b=1.62+or-0.02 and the fractal dimension of its cutting bonds D_c=0.75+or-0.01, up to the scale factor b=10, in excellent agreement with the large-cell Monte Carlo simulation result.

The frontier in gradient percolation is generated directly by a type of self-avoiding random walk. The existence of the gradient permits one to generate an infinite walk on a computer of finite memory. From this walk, the percolation threshold p_c for a two-dimensional lattice can be determined with apparently maximum efficiency for a naive Monte Carlo calculation (+or-N^-12/). For a square lattice, the value p_c=0.592745+or-0.000002 is found for a simulation of N=2.6*10¹¹ total steps (occupied and blocked perimeter sites). The power of the method is verified on the Kagome site percolation case.

For the site dilution model on the hypercubic lattice Z^d, d>or=2, the authors examine the density of states for the tight-binding Hamiltonian projected onto the infinite cluster. It is shown that, with probability one, the corresponding integrated density of states is discontinuous on a set of energies which is dense in the band. This result is proved by constructing states supported on finite regions of the infinite cluster, analogous to the Kirkpatrick and Eggarter zero-energy molecular state.

The dynamic Monte Carlo renormalisation group method introduced by Jan, Moseley and Stauffer (1983) is used to determine the dynamic critical exponent, z, for the q=3 and q=4 state Potts model in two dimensions. The authors find that z=2.43+or-0.15 and 2.36+or-0.20 for the q=3 and q=4 state Potts models. These results are in disagreement with the recent conjecture of Domany (1984).

The authors present evidence for the partial suppression of self-avoidance effects in the rod-to-coil transition scaling, observed recently in numerical simulations. The scaling ansatz is examined critically, and also checked explicitly in a soluble model on a partially directed lattice.

The detailed proof presented in an earlier paper, justifying Vdovichenko's method (1965) which gives an exact expression of the free energy of the Ising model in a plane, is extended to the system on a torus. This justifies the exact expression of the free energy in the form given by Vdovichenko for the Ising model on the general two-dimensional lattice in the thermodynamic limit.

Animal counting techniques suggest that the fraction of rigid animals among all animals with s sites on a triangular lattice varied roughly as s^1/2 (0.46+or-0.01)^s.

An analytical real space renormalisation group transformation is obtained for the one-dimensional spin-1/2 antiferromagnetic dimerised XY model. Using the method of Sarker (1984) the recursion relations for the parameters are derived from the mapping between blocks of sizes L and L/b, for arbitrary size L and scale factor b. The results for the gap and for the critical exponent are shown to converge rapidly toward their exact values. The dynamical exponent has 1/L corrections (b approximately 1) instead of the usual 1/ln L ones (b=L). The differential renormalisation group equations are obtained explicitly. The authors show that the non-analyticity of the scaling function (for the density of energy) stems from a dangerous irrelevant variable, the antiferromagnetic coupling.

The authors report here a calculation of the spatial correlation function of the fully polymerised four-functional units in the kinetic gelation model in three dimensions. The results indicate that the fully polymerised four-functional units form small correlated regions in space. This is in sharp contrast to normal percolation where the spatial distribution of fully polymerised units is uniform. The size of these clusters of crosslinks depends on the extent of reaction, the concentration of initiators and the solvent diffusivity. The results suggest a picture of the microscopic structure of a gel as consisting of bundles of chains linked by clusters of crosslinks in agreement with electron microscopic studies of polyacrylamide gels.

A detailed study is presented of the finite-size scaling in systems with vanishing critical exponent, alpha , which usually have logarithmic specific heat singularity. The appropriate form of the finite-size hyperuniversality is established. Recent results on complex temperature plane zeros of the partition function are extended to the alpha =0 case.

The authors define directly a new set of symmetries for the AKNS system and prove that they constitute an infinite-dimensional Lie algebra with the 'old' symmetries. They also point out the relation between the new symmetry and the non-isospectral problem. They use the reduction technique and point out that the NLS, MKdV, SG and sinh-G hierarchies have two sets of symmetries which constitute an infinite-dimensional Lie algebra. These results are extensions of those of Chen et al. (1982).

The algebra of the restricted Lorentz group and the Weyl formulation of massless Poincare irreducible fields have been developed by primarily Lorentz-covariant Pauli matrix methods which facilitate the generation of both indexed Weyl spinor identities and Dirac identities. The results have been used to display a complex set of symmetries for the (anti)self-dual Weyl field strengths of arbitrary helicity j(>or=1). These symmetries and the requirement of Poincare irreducibility have then been used to give a direct and uniform determination of the forms of the gauge invariant (Lagrangian) free field wave equations for the Fierz-Pauli and Rarita-Schwinger potentials of spin 1, 3/2 and 2. The procedures set out indicate that it should be possible to establish the gauge invariant Lagrangian free field wave equations of arbitrary helicity in a uniform and direct manner from the corresponding much simpler equations governing the field strengths in unmixed spin representations.

Following Flaschka and Newell (1979) the authors have formulated the inverse problem for Painleve IV, with the help of similarity variables. The Painleve IV arises as the eliminant of the two second-order differential equations originating from the non-linear Schrodinger equation. They have obtained the asymptotic expansions near the singularities at 0 and infinity of the complex eigenvalue plane. The corresponding analysis then displays Stokes phenomena. The monodromy matrices connecting the solution Y_j in the sector S_j to that in S_j+1 are fixed in structure by the imposition of certain conditions. They then show that a deformation keeping the monodromy data fixed leads to the non-linear Schrodinger equation. At this point they may mention that, while Flaschka and Newell did not make any absolute determination of the Stokes parameter, the approach yields the values of the Stokes parameter in an explicit way, which in turn can determine the matrix connecting the solutions near 0 and infinity . Such a realisation was not possible in the approach of Flashchka and Newell. Lastly they show that the integral equation originating from the analyticity and asymptotic nature leads to the similarity solution previously determined by Boiti and Pempinelli (1979).

The solution of the third-order isospectral equation of the Caudrey-Dodd-Gibbon-Sawada-Kotera equation (CDGSKE) for soliton potential is obtained recursively from the Riccati equation derived by iterating once the auto-Backlund transformation. It is then shown that the discrete eigenfunctions of the sixth-order recursion operator for this equation can be written in terms of the solutions of the isospectral equation. The behaviour of the 1-soliton solution which has certain novel features is studied. A sine-Gordon-like equation resembling the double-sine-Gordon equation is derived from the CDGSKE.

The authors study the mechanisms of enstrophy transfer in 2D turbulence. By theoretical discussion of the fragmentation process at very small scales, the authors show that enstrophy transfer must be space filling, due to the regularity of the flow at any time. Detailed analysis of a numerical decaying experiment confirms this picture. Yet 'intermittency' is acting at large scale due to the presence of coherent structures, namely vortices, which inhibit local enstrophy transfer.

The author obtains an exact integral representation of the sum describing the evolution of the inversion of the two-level atom in the Jaynes-Cummings model when the field is initially prepared in a coherent state or a state generated by classical sources at non-zero temperature. The author uses the saddle point method to estimate the integrals.

A phase integral approximation, based on the notion that large N is a semiclassical limit, is presented for spherically symmetric systems of the Coulomb type. Since the dynamical group SO(2,1) is the spectrum generating group for the radial point Coulomb problem and the Casimir operators of O(N) and SO(2,1) are coupled, the authors write a path integral expression of the Green function in terms of SO(2,1) coherent states. The resulting phase integral approximation is applied to the Yukawa potential and the Coulomb plus linear r confining potential.

The direct point-by-point application of minimal sensitivity optimisation to the first-order perturbative approximation to the ground-state wavefunction of the quartic oscillator, carried out previously, is extended to low-lying excited states. An analytic formula for the derivative with respect to the redundant parameter is developed and used to obtain the minimally sensitive value for this parameter as a function of configuration space. Generally, more than one such minimally sensitive value is found at each configuration space point-these values are categorised into disjoint continuous curves. For all the cases studied, it has proved possible to piece together sections of such curves in an unambiguous manner so as to obtain optimised wavefunction approximations which are both renormalisable and continuously differentiable. Comparison of these optimised wavefunction approximations with the exact results revealed satisfactory agreement, particularly in the large mod x mod asymptotic region where orthodox perturbation methods fail completely due to the strong coupling.

The generalised E* epsilon Jahn-Teller Hamiltonian is treated in configuration space using polar coordinates. The expansion in radial oscillator states leads to simple recurrence relations. These are equivalent to a system of two ordinary linear first-order differential equations. The isolated exact solutions are polynomials multiplied with an exponential function in this formulation. They are also calculated in configuration space. The connection of the present treatment with Reik's treatment is established. This leads to a new understanding of Reik's Neumann series expansion.

The quantum dynamics of the Hamiltonian of a two-level system coupled to a boson mode is formulated in terms of the Wigner matrix. The trace of the Wigner matrix gives complete information for the bosonic degree of freedom. The time evolution and the long time average of the trace of the Wigner matrix is calculated numerically for small perturbations and in resonance. Some aspects of 'quantum chaos' of the two-level system are discussed.

The author discusses a simple separable two-freedom Hamiltonian H describing the motion of a point particle in a plane region (V=0) between two infinite parallel hard walls (V= infinity ) and with a rectangular finite potential well (V=-V₀). The quantal spectrum of H is discrete below the continuum limit (ionisation energy) different from the classical escape energy (E=0) by the zero point energy of the transverse motion and has infinitely many bound states embedded in the continuum. The author shows that this spectrum is fragile against perturbations of H: if the hard walls are made diverging, the bound states in the continuum disappear, while if they are converging, the continuum disappears and the spectrum is pure point.

Dirac's prescription for quantisation does not lead to a unique phase operator for the electromagnetic field. The authors consider the commonly employed phase operators due to Susskind and Glogower (1964) and their extension to unitary exponential phase operators. However, they find that phase measuring experiments respond to a different operator. They discuss the form of the measured phase operator and its properties.

A whole class of two-loop finite N=1 supersymmetric Yang-Mills theories for all groups with the exception of SU(N) is obtained.

The authors construct two-stage second-order algorithms for integrating stochastic differential equations with variable diffusivity in one and higher dimensions.

The paper studies the random bond Ising model of a spin glass of a tree of coordination number q=3. The model is studied as an example of the method of ring recurrence. A set of recursion relations between the moments of the probability distribution of the order parameter are produced. It is shown that solutions of the recursion relations exist which form an infinite bifurcating set below a critical temperature tau _c. The nature of the first such solution is investigated in detail and the critical exponent analogous to beta is found.

The authors discuss the possibility that space has to be described as being of non-integer dimension. They elaborate on some of the problems associated with this idea and investigate the chance to observe effects due to such a fractional dimension. Two experiments show rather stringent bounds for the deviation of the space dimension from the value three. These are the perihelion shift of planetary motion and the Lamb shift in hydrogen.

A new order parameter for percolative systems, the minimum gap (x*), is calculated on diluted Cayley trees. x* is self-averaging and finite for concentrations p below p_c and zero above. Numerical work (including finite-size scaling) and analytic arguments show that on approaching p_c from below, x* approximately epsilon /(ln(1/ epsilon )+1+ln( alpha -1)), where alpha +1 is the coordination number of the Cayley tree and epsilon =(p_c-p)/p_c. The behaviour of x* as a function of p (for all p<or=p_c) is calculated in terms of the solution to a transcendental equation, and as p to 0, x* approximately 1+ln alpha /lnp.

The authors prove two theorems on the simultaneous diagonalisability of a set of complex square matrices by a biunitary transformation.

The pointwise convergence of the probability densities induced from an initial density on (0,1) under repeated iteration of the map x to 4s(1-x) is established for a large class of initial densities.

Based on work of Itzykson et al., see ibid., vol.19, no.3, p.L111-16 (1986). For a d-dimensional rectangular box the function zeta (T;s) considered by Itzykson et al. is related to Epstein's zeta function and can be written in a form which exhibits the analytic structure explicitly.

Using a VAX 11/780, a random walk motion is studied in a random-field disordered system. A non-monotonic transport behaviour is observed as a function of local field intensity. In contrast to the random walk motion in random percolating fractals, the root mean square displacement here increases as a function of time faster than that of diffusion. A crossover from diffusion to disorder-induced transport is discussed and its spectral dimensionality is estimated.

Clusters grown by the Eden process have few empty sites in their interior; these sites are concentrated near the cluster surface. The number of clusters of connected empty sites is investigated by computer simulation and shown to disagree with a simple percolation approximation.