Table of contents

It is found that the nonlinear ordinary differential equation rho + omega ²(t) rho =1/ rho ³ has a three-parameter Lie group of symmetries which are also Noether symmetries. The invariants associated with the group are calculated. The authors discuss a new way of generating n-parameter Lie symmetry groups which are associated with n-parameter nonlinear differential equations.

Considering the infinite class of relativistic wave equations wherein the transformation property of the wavefunction involves up to four (unspecified) inequivalent irreducible representations (IIRs) of the Lorentz group occurring with arbitrary multiplicities, the authors have investigated the general question as to what combinations of IIRs would be admissible and with what multiplicities, if it were required that the cautions have solutions only for a single (unspecified) spin and mass and be inequivalent to any simpler equation. It is found that the possibilities are quite limited. The main results are presented.

Shows that the generalised lattice animal model of Family and Coniglio (1980) naturally leads to a unified scaling picture for percolation and lattice animals in which the fugacity for occupied elements plays the dual role of a temperature-like and a field-like variable. Within this single-scaling-field description of percolation, there is only one independent exponent from which all others can be obtained. The author defines a new set of exponents alpha , beta and gamma for percolation and finds that they are all related to the cluster number exponent theta through the relation alpha = gamma =1- beta =3- theta , in analogy with lattice animals. To relate the cluster radius exponent nu to the other exponents he uses the generalised Ginzburg criteria to obtain a modified hyperscaling relation for isotropic and directed, percolation and lattice animals. Using this relation he finds that theta -1= nu /sub ///+_{perpendicular to} (d-1) for directed percolation and theta = nu _{perpendicular to} (d-1) for directed lattice animals where nu /sub /// and nu _{perpendicular to} are exponents characterising the parallel and perpendicular cluster radii respectively. Using the same approach he obtains the Stauffer relation theta -1=d nu and the Parisi-Sourlas relation theta -1=(d-2) nu for isotropic percolation and lattice animals respectively. The above relations give the following expressions for theta within the Flory theory: theta (percolation)=(3d+2)/(d+2), theta (directed percolation)=(6d+5)/(2d+4), theta (animals)=(7d-6)/(2d+4) and theta (directed animals)=9(d-1)/(4d+8).

The exponents nu _{perpendicular to} , gamma of directed animals in d dimensions are shown to equal those for the Lee-Yang problem in d-1 dimensions. For d=2 the author obtains the exact results nu _{perpendicular to} = theta =¹/₂, and for d<or=7 the scaling relation theta =(d-1) nu _{perpendicular to} .

It is shown that the theory of directed lattice animals is a good candidate for determining the exponents which describe river networks.

The pair connectedness for directed site percolation on the honeycomb and diamond lattices is related to that of the square and simple cubic lattices respectively. In the case of bond percolation the same correspondence leads to site-bond percolation on the latter pair of lattices.

The authors study the conductivity in directed percolation by an analogue experiment and by computer simulations. In the experiment, the dependence of the conductivity on concentration of conducting diodes, rho , shows very large fluctuations. This reflects the topological constraints of directed percolation where conduction is very sensitive to singly connected or cutting bonds. They have checked this hypothesis by studying the underlying directed backbone of the experiment using computer simulations. From the dependence of the conductivity on the backbone fraction, and the dependence of the backbone fraction on rho - rho _c. They infer a directed-conductivity exponent considerably less than unity, consistent with earlier theoretical work.

Presents a new method based on linear programming to obtain the exact ground states of three-dimensional Ising spin glasses. Numerical simulations were carried out with the +or-J model and the Gaussian model with mean mu and variance sigma ². Ferromagnetism disappears in the +or-J model at concentrations of negative bonds near 27%. This occurs, in the Gaussian model, when mu is near 0.6 sigma .

A method, akin to phenomenological renormalisation, for analysing Monte Carlo data in the critical region is proposed. The method is illustrated by an analysis of the structure factor of the two-dimensional axial next-nearest neighbour Ising model.

A generalised version of the interface method of Muller-Hartmann and Zittartz (1977) is applied to the three-state square lattice Potts antiferromagnet in a negative field. The phase boundary is found to go to zero temperature in the limit of zero magnetic field.

The zero-temperature triangular Ising antiferromagnet is mapped onto a solid-on-solid (SOS) model. The system undergoes a roughening transition characterised by a critical exponent alpha =¹/₂, by the absence of excitations in the smooth phase, and by domain wall excitations (stripes) in the rough phase. At infinite SOS temperature the height-height correlation function is explicitly calculated with the aid of known four-point Ising correlations. The authors point out that a certain six-vertex model with a comparable SOS interpretation has an identical critical temperature, critical exponent and critical amplitude. This is in support of existing ideas on university in systems with striped phases.

A lattice model with full rotational invariance is proposed for describing nematic liquid crystals. Orientational long-range order at low temperature is proved using the infrared bound method.

Periodic soliton structures are shown to be energetically favourable in a certain range of parameters of the Frenkel-Kontorova model. These structures form cascades which can be obtained from each other by a scaling transformation.

It is shown how to obtain an exact solution class of the nonlinear reaction diffusion equation in Schlogl's second model (1972). For large time these solutions evolve to bounded travelling kinks. When the phase coexistence conditions are satisfied, the solutions describe the collision of two travelling kinks.

Earlier work on unique labelling schemes in O_h symmetry is extended to include 48-atom shells and the symmetry coordinates of shells subject to periodicity conditions. A new approach to labelling is introduced based on the concept of cyclic regions in a crystal. These are described by finite factor groups of the crystallographic space group which contain the isogonal group as a subgroup. Some examples are worked out in detail.

It is shown that, in addition to the number of distinct equivalence classes, the representation characters of a 'labelling' can also be derived using standard combinatorial methods. These characters provide additional information on the structure of the equivalence classes which is relevant in physical applications.

An infinite number of local conservation laws are derived for the two-dimensional periodic SU(n+1) Toda lattice equations following the well known method for the sine-Gordon equation.

A geometrical description of the composition law for pure inertial transformations in relativity is developed, in close analogy with the description of rotations by means of turns. This affords a very intuitive and visual image of all problems involving a composition of velocities, and clearly shows the appearance of hyperbolic geometry in relativistic kinematics. The pertinent geometrical constructions as well as some applications are discussed. For the case of light the preceding constructions are surprisingly simple, and from them emerges a connection between the geometrical idea of parallelism angle and the physical Doppler effect. A slide rule which can be used to carry out all needed computations is also described.

The general form of nonlinear evolution equations in 1+2 dimensions integrable by the matrix non-stationary Schrodinger spectral problem is found. The infinite-dimensional group of Backlund transformations for these equations is constructed and the nonlinear superposition principle is obtained.

Product decompositions of theta -functions of multiple argument are well known in the literature of the subject. Here, additive decompositions are presented.

The authors interpret the vortex solution of Nielsen-Olesen (1973) as a complex vector bundle associated to the second Hopf sphere bundle (analogously to consider the kink of the first Hopf bundle); the peculiarity of the soliton behaviour of the two-dimensional vortex stems from the non-trivial character of the fibration; the electromagnetic and scalar (Higgs) field are the connection and the section in this bundle respectively. Properties of this mathematical construction have their natural physical translation; for example the complex structure of the sphere S² leads to a closed (Kahler) two-form, which has physical implications, and the fact that the vortex can be considered as the square root of the tangent bundle to the sphere implies a spinor nature for the vortex. A Morse theory of critical points suggests some Atiyah-Singer type of theorems, which have bearing on the stability of the multi-vortex solutions. They finish by a geometrical interpretation of the fractional charges found recently.

Introduces a general notion of non-commutative stochastic processes over complete locally convex *-algebras and proves a theorem concerning the existence and uniqueness of such processes. The author shows that quantum fields are stochastic processes in the sense of this paper. Moreover, the author applies the theory to infer the existence and uniqueness of quantum fields.

An algorithm for parametrisation of arbitrary n*n unitary matrices is presented. It is given with a minimum number of real parameters out of which a part take values in the positive unit cube, the others being arbitrary phases. The case n=4 is worked out in detail.

Proves that the necessary and sufficient condition for the existence of (non-trivial) Lagrangians s(olution)-equivalent to L=¹/₂(x²+y²)-V(x,y,t) is that L can be written as L=¹/₂(u²+v²)-V₁(u)-V₂(v) with u= alpha x+ beta y, v= gamma x+ delta y where alpha , beta , gamma , delta are constants.

Studies perturbation theory at large order for the one-dimensional Schrodinger equation with the potential U(x)=x²(R²-x²)²/8R⁴, where R is the large parameter. It is shown that for the states which 'originate' from the middle well, in addition to the Rayleigh-Schrodinger expansion of the energy eigenvalues in power series of 1/R² there is an exponentially small shift in R² due to the tunnelling into the side wells. The states which 'belong' to the side walls exhibit asymptotic eigenvalue degeneracy. Non-modal solutions for these states are also found, and by using them the authors determine the large-k behaviour of the general kth Rayleigh-Schrodinger coefficient. The comparison of asymptotic formulae with the results of numerical calculations shows an excellent agreement where it is expected.

Simple non-adiabatic model Hamiltonians are treated using the realisation of the vibrational modes on Bargmann's Hilbert space of analytic functions. In this formulation the Schrodinger equation is a system of linear first-order differential equations. The energy eigenvalues are selected by the requirement that the solutions belong to the space of entire functions. The solutions are given in terms of Neumann series; the recurrence relations for the expansion coefficients have a simple structure. Under particular conditions for the interaction constant they allow for terminating Neumann series (isolated exact solutions). In the general case the conditions for the eigenvalues are transcendental equations involving a continued fraction. The continued fraction can be approximated to any desired degree of accuracy in a rapidly convergent process based on Worpitzky's theorem and its relation to conformal mapping. The eigenvalues are calculated; a physical interpretation of the solutions which makes use of intuitive arguments is also given.

Sufficient conditions are given for the possibility to construct quantum mechanical wavefunctions by the sole knowledge of an appropriate sequence of classical canonical transformations which map a given Hamiltonian onto the new position variable. The transformation matrix element for each individual step of this sequence is given by the semiclassical limit expression of these matrix elements; it is a function of the generator of this transformation step only. The wavefunction, i.e. the transformation matrix element for the total transformation, is obtained as a multiple integral over the transformation matrix elements of the various intermediate steps. The practicability of this procedure is demonstrated by several examples. The authors consider time-independent systems with one degree of freedom.

The authors have found and investigated the important class of Riemannian space-times. In these and only in these space-times one can impose the complete set of gauge conditions on massless fields of higher spins. This set reduces the number of independent field variables to two. By this means one can simplify the field equations and interpret the solutions to the equations properly. In this paper they consider curvature of these spacetimes, their Petrov classification and the uniqueness of the gauge vector field which is used for imposing the complete set of gauge conditions.

The programme POLYNOM is used to calculate a new solution of Ernst's equation. The author then shows that the solution is an extreme limit of the so-called Kinnersley-Chitre solution, (1978).

Trautman's (1981) accelerated coloured particle solution of the Yang-Mills field equations are shown also to exist in general relativity.

A unique class of solutions of the classical spin fluid with zero pressure is obtained. The spin density vanishes at the boundary of the fluid sphere and the g_{mu v} and their derivatives satisfy continuity conditions with the exterior metric.

The Hamiltonian quantum theory of the damped harmonic oscillator is investigated by using an explicitly time-dependent Hamiltonian including a commutative stochastic force. The propagator is computed and shown to describe an inhomogeneous quantum dynamical semigroup evolution on the Banach cone of density matrices. The moments tend to the thermal equilibrium values independently of the initial conditions, when time tends to infinity and some rescaling procedure is performed. With respect to the rescaling neither the Heisenberg nor the Schrodinger picture exists. Furthermore the energy dissipates below the zero-point energy in violation of the Heisenberg uncertainty relation.

The authors generalise the arguments given in a preceding letter (unpublished, 1982) by the present authors to give a sufficient condition for the existence of the spontaneous magnetisation in the random-bond Ising model on the square lattice. They study in detail the system with a Gaussian distribution of exchange interaction, the system with the exchange interactions J>0, 0 and -J, and the system with the exchange interactions J_A>0 and -J_B<0. It is shown that in each of these systems the ferromagnetic phase certainly exists in a region in the space of the parameters describing the probability distribution of the exchange interaction at sufficiently low temperatures. The general condition for the existence of the ferromagnetic phase is given for the system without specifying the form of the probability distribution of the exchange interaction.

The authors demonstrate that all Ising antiferromagnets with arbitrary many-neighboured interactions and in the maximum critical field, have highly degenerate ground states accompanied with non-zero residual entropies. The residual entropies vanish when the range of interaction tends to infinity. The proof is realised by an explicit calculation in the case of a one-dimensional many-neighboured Ising antiferromagnet, and by establishing bounds for the residual entropy in the case of an Ising system situated on a lattice with arbitrary number of dimensions. They also show that the established bounds may serve as estimates of frequently unknown actual values of the residual entropy.

Formulates a generalisation of the droplet theory of Ising systems to droplet systems containing more than two phases, valid near one dimension and based on the configurational energy of surface tension. Thermal and magnetic exponents for these multi-phase systems are compared with the results for a q-state Potts model in the external field. The Potts exponents are obtained using a Migdal recursion formula and are shown to be of very similar structure.

It is argued that the scaling transformation which is currently being used to obtain numerical estimates of critical exponents cannot safely be employed without prior knowledge that the points in question are second-order transition points. The transformation when defined on finite or semi-infinite systems will not distinguish between order-disorder critical points and any other point on the coexistence surface. This in turn means that the full phase diagram of the equilibrium surface can be found using the scaling transformation. The three-state Potts model is used to illustrate this.

A phase equilibrium surface for the 2D (square lattice) antiferromagnetic Potts model in the presence of its ordering fields has been constructed using the phenomenological scaling transformation.

Making use of the Bogoliubov inequality, a necessary condition for spontaneous local magnetisation is derived for a d-dimensional classical planar model with exchange interactions of arbitrary distribution and range except for systems with infinite energies of elementary excitations. For ferromagnetic systems with long-range interaction Upsilon ^{-d- sigma} , it gives the new result that the ferromagnetic phase is ruled out for d<or= sigma with 2> sigma >0. For spin glass systems of the planar model the condition is y>1 for the density of spin wave states with N(E) approximately E^y for small E.

The magnetic behaviour of a Dirac electron gas in the presence of the inhomogeneous magnetic field, H sech²(ay), is studied. Using the single-particle energy eigenvalue of a Dirac electron, the magnetic interaction energy density is written down, from which an explicit expression for the magnetic moment is derived. The magnetic moment density is evaluated numerically in the degeneracy limit for several values of the magnetic field strength and the chemical potential. The transition from para- to diamagnetism exhibited by the electron gas in the homogeneous magnetic field is found to persist in the inhomogeneous field also. Further, the distinct possibility of spontaneous magnetisation (i.e. ferromagnetic behaviour) of the electron gas in this inhomogeneous field is discussed.

The magnetic flux sensitivity in electrical circuits is discussed in the time domain. It is shown that the quantum electrodynamic uncertainty principle implies that measurements of the magnetic flux at two different times interfere. Thus, an initial measurement of flux, at time t₁, to within Delta phi (t₁) will induce a random back EMF which will affect the uncertainty Delta phi (t₂) of the flux measurement at a later time t₂. The authors derive the 'interference' rule that Delta phi (t₁) Delta phi (t₂)>or=(h(cross)/2) mod G(t₁,t₂) mod , where G(t₁,t₂) describes the transient circuit response in flux to a weak current source impulse. As an example, this rule is applied to the case of a current-fed, singly connected Josephson weak link to give an expression for the minimum measurement uncertainty in the weak link critical current.

It is shown that in gauge theories of the electroweak interaction, the symmetry-breaking process always defines a genuine connection on the electromagnetic sub-bundle.

A variational theory is presented for the isotropic triplet pair-condensed state of liquid ³He. Using a correlated pairing wavefunction the authors express the expectation values of the required quantities in terms of the diagonal and off-diagonal matrix elements of the Hamiltonian and unit operators with respect to a correlated basis of normal wavefunctions. The correlated basis consists of Slater determinants modified by a Jastrow correlation factor which incorporates the strong short-range correlations induced by the interparticle potential. The formalism has been developed within the weak-coupling approximation. The study of the instability of the normal state with respect to the triplet pair-condensed state is achieved via an exact criterion. This criterion has been expressed in terms of normal-state quantities which have been evaluated within the correlated basis functions scheme with the accuracy of the Fermi hypernetted chain approximation. The numerical calculations indicate an instability of the normal ground state with respect to the ³P₀ superfluid state but at too low a density.

Proposes to apply the method of moments not to the spectral density itself but rather to one of its derivatives. The spectral density obtained by integration is therefore continuous or smooth depending on the order of the derivative which is used. The problems arising from the fact that the derivatives of the spectral function are generally not positive definite have been resolved. An additional advantage of the method is its ability to reproduce singular spectral densities (e.g. van Hove singularities) as is shown for three illustrative examples; a one dimensional crystal, a two-dimensional lattice and a three dimensional simple cubic lattice.

Comments upon a remarkable cancellation which occurs when the logarithm of the configurational partition function is taken. This little-known fact has interesting consequences for the form of the partition function. Indeed, it may explain why there are no logarithmic terms in the perturbation series for the expansion factor of the chain.