Table of contents

The Onsager-Machlup function for general Markov processes with small parameters, inclusive of the so-called system size expansion model is obtained by a conditional minimisation of the Kullback-Leibler entropy.

Two component spinor techniques, similar to those used by Witten (1981) are used to express the Bondi momentum of an asymptotically flat space-time in the form of an integral over an asymptotically null, space-like hypersurface Sigma . It is then shown that the Bondi mass is positive, given the existence of a Green function for the 'Witten equation' on Sigma .

Introduces (10) and (11) interface models for percolation on the square lattice and on the triangular lattice. In the case of p=1 (p probability) only the (10) models show surface roughening. An analytic calculation gives a square-root divergence for the interface width in the thermodynamic limit.

A q-component Potts model with ferromagnetic and antiferromagnetic interactions in the two respective directions of a square lattice is considered. An argument is given showing that an order phase can exist in this model, even though the ground state is disordered and infinitely degenerate. The authors use the Migdal-Kadanoff transformation to obtain a closed-form expression for its critical point. They also carry out a Monte Carlo simulation of the model for q=3. The specific heat exhibits a broad maximum which does not sharpen appreciably as the lattice size is increased. This suggests that the phase transition if it exists, is of an unconventional type.

Shows that the perpendicular correlation length xi _{perpendicular to} in directed percolation does not behave as a length under renormalisation but is a length times an angle. This has consequences for the renormalisation group analysis and hyperscaling which are discussed.

For Ising-correlated site-bond percolation, Delyon, Souillard and Stauffer (1981) predicted a phase transition in the number of finite droplets, without the appearance of an infinite network. The authors interpret their result as due to percolation of active bonds within very large but finite Ising clusters.

The finite size scaling method is used to study the critical properties of the spin-1 Baxter-Wu model as function of the fugacity, z, of the vacant (spin zero) sites. For z=0, the thermal exponent converges very quickly to the (exact) value y_t=3/2. For z>0, y_t monotonically increases beyond the value 2. This increase is interpreted as indicating a first-order transition. Out of several possible renormalisation group flows, the results seem to favour the one in which the critical Baxter-Wu Hamiltonian flows to the fixed point of the 4-state Potts model, with the amplitude of the marginal operator equal to zero.

The transition from periodic to chaotic behaviour via a period doubling route is studied for dissipative systems. The bifurcation ratio which characterises such a transition is calculated for the area-non-preserving Henon map by using a simple renormalisation group method. It is seen that there exists a smooth transition from the conservative case to the dissipative case. This approximate renormalisation group calculation agrees well with the numerical result obtained recently by Zisook (1981).

The results given permit the unambiguous evaluation of all possible Kronecker products of the irreducible representations (tensor and spinor) of O_n and SO_n for n=2 nu and n=2 nu +1. A complete resolution of the second and third powers of the basic spinor representations of SO_{2 nu} and SO_{2 nu +1} is given, together with a prescription for analysing the fourth power of these representations. Detailed application is made to the enumeration of properties of SO₁₀ relevant to grand unified theories, and sufficient information given to resolve the fourth power of any representation of SO₁₀.

Some representations of the Dirac delta function are considered including a new representations. A new theory of Fourier transforms is developed which is better suited to use in physics than the standard theory. The work is of general interest as well as of relevance to subsequent articles. The authors then give a brief outline of the construction of field theory from quantum mechanics as facilitated by non-standard analysis and a theorem which enables the calculation of a cross section from plane wave states.

Derives accurate compact expressions for the high-temperature specific heats of classical (S= infinity , 3-vector) spin systems on FCC, BCC and SC lattices for pure Heisenberg, XY and Ising-like couplings, respectively. The analysis of the appropriate series expansions demonstrates the utility of inhomogeneous differential approximants and supports the estimate alpha _H=-0.21+or-0.04.

The conditions for a system of second-order differential equations to be derivable from a Lagrangian-the conditions of self-adjointness, in the terminology of Santilli (1978) and others-are related, in the time-independent case, to the differential geometry of the tangent bundle of configuration space. These conditions are simply expressed in terms of the horizontal distribution which is associated with any vector field representing a system of second-order differential equations. Necessary and sufficient conditions for such a vector field to be derivable from a Lagrangian may be stated as the existence of a two-form with certain properties: it is interesting that it is a deduction, not an assumption, that this two-form is closed and thus defines a symplectic structure. Some other differential geometric properties of Euler-Lagrange second-order differential equations are described.

The master equation approach is used to relate the calculation of correlation functions to the calculation of single-time expectation values. The quantum regression theorem is shown to result on neglecting a certain term. The properties of this neglected term are briefly discussed.

A family of complete integrable three-dimensional N-body quantum systems is introduced and completely solved by the dynamical algebras O(3N+1,2) and their representations. In a particular realisation of these algebras the particles interact by N-body 'Coulomb-type' potentials. A complete set of commuting integrals of motions, their spectra, the energy levels for both discrete and continuous spectra and their degeneracy have been explicitly determined. Relativistic generalisations and applications are briefly discussed.

For pt.I see ibid., vol.13, no.8, p.2673-88 (1980). Concepts of classical and quantum global measurability introduced in an earlier paper are discussed in greater detail and in the context of an n-dimensional Riemannian manifold. The ideas are illustrated with diagrams and examples. A notion of exact measurability is introduced and is shown to imply quantum global measurability. The physically important Killing momenta are shown to be exactly, and hence quantum globally, measurable.

A mechanism of the algebraical confinement of colour based on the quaternionic structure of the space of states is investigated. A new dynamical interpretation of the colour group SU(3)_c is given.

A method suggested by Holevo (1979) is used to obtain a lower bound for the information capacity of a quantum narrow-band free-space link without extraneous noise. At high photon rates the bound is better than those previously proposed and comes close to a fundamental upper bound. It is not so good at low rates, where it is beaten by photon-detecting systems. The normal modes are not treated as independent channels. A system is described where the normal modes are used in pairs.

The classical nonlinear Schrodinger equation may be solved using the inverse scattering transform, but there are difficulties in carrying this over to the case of quantum fields. These difficulties are overcome by explicitly constructing a Fock space representation of the states, together with quantum fields properly defined over this space.

Suppose the function epsilon _k( nu ) which represents the kth bound-state eigenvalue of the Hamiltonian h=(- Delta + nu phi (r)+U(r)) is known exactly for all allowed values of nu >or=0. The authors present the corresponding eigenvalue E_k( nu ) of the Hamiltonian H=(- Delta + nu f( phi (r))+U(r)), where f( phi ) is a smooth, increasing, and either convex or concave transformation of the potential phi (r). An application of the method of potential envelopes yields a simple formula for an upper or lower bound to E_k( nu ) according to whether the transformation f( phi ) is concave or convex. The example phi (r)=(-r^-1+ omega r), U(r)= omega ²r², and f( phi )= lambda ^-1(e^{lambda phi} -1) for lambda >0 is discussed in detail.

The Bianchi identities are analysed to first order. There are two cases to consider. First, the author examines the case in which the sources for the gravitational field are ignored, and obtain the solutions of the resulting homogeneous equations for the frame components of the Weyl spinor. Second, he includes the sources and obtain solutions of the wave-like equation which occurs. The two cases are compared, especially with reference to the multipole moments which arise from these analyses. The moments arising from the solutions of the homogeneous case are due to contributions from terms with different dimensions. The moments of the inhomogeneous case are associated with a single dimension only, and are used together with certain integral expressions involving the Bondi-Sachs news function to determine the main contributions to the energy momentum losses of gravitationally radiating isolated sources.

Determines the frequency shift between the emitter and the receiver due to nonlinear effects of gravitational and electromagnetic radiation from a bounded source.

Develops a cell position-space renormalisation group PSRG with which the authors study the scaling properties of isolated polymer chains. They model a chain by a self-avoiding walk constrained to a lattice. For rescaling factors b<or=6, they calculate recursion relations analytically on the square lattice with several different choices for the PSRG weight function. They also calculate implicit cell-to-cell transformations in which a cell of size b is rescaled to a cell of size b'. They also develop a constant-fugacity Monte Carlo method which enables them to simulate-in an unbiased way within the grand canonical ensemble-chains of up to 10³ bonds. With this method they extend the PSRG to larger cells (b<or=150) on the square lattice. Their numerical method provides high statistical accuracy for all cell sizes.

The present work evaluates two ways of approximating the spectral function, S(k, omega ) in the kinetic regime (Kn to 1), by calculating the predictions of these methods for a simple model of the Boltzmann equation, and comparing those predictions with exact results. The perturbed eigenvalue approximation, used in this context for the first time, is shown to be clearly superior to the other method considered (a generalised hydrodynamic approximation), both in rapidity of convergence at fixed Kn, and in range of applicability in Kn.

Studies a correlation function which is given by the canonical average of a product of one or more spin variables, for the random-bond Ising model, in which the exchange integrals are +J(J>O) and -J with probabilities p and 1-p, respectively. The authors show that an upper bound to the configurational average of the correlation functions, calculated in the thermodynamic limit in the zero external field limit, is the product of the same correlation function for the corresponding ferromagnetic Ising model at the temperature under consideration and the same quantity at the temperature T₁ which is determined by the condition T₁=2J/ mod k_B1n(p/(1-p)) mod , where k_B is the Boltzmann constant. The results are given for the random-bond Ising model of an arbitrary spin S, and also for the diluted random-bond Ising models of an arbitrary spin S, with the pair interaction and with general interaction, and for the diluted and undiluted random-bond n-vector model.

In the models proposed by Longa and Oles (1980) there exist two oppositely magnetised equilibrium states, if the temperature is sufficiently low. Spins in the frustrated cells also carry a moment, the magnitude of which is 1/5^1/2 at zero temperature.

The scaling theory of finite size effects in the limiting bulk behaviour is extended to treat crossover phenomena. The method is used to study quantum critical phenomena in the XY chain in a transverse field at zero temperature, through the scaling of the longitudinal susceptibilities and of the energy gaps between the ground and two first excited states. As expected, no abrupt change in critical exponents is observed for small anisotropy. Because of the finiteness of the system, but the limiting isotropic and anisotropic regions display quite distinct critical behaviour, in good agreement with known results. Another interesting result obtained is that the two energy gaps examined vanish at the critical line with the same critical exponent.

The critical properties of a random Ising model with long-range isotropic interactions decaying as R^{-(d+ sigma )} are analysed by using renormalisation group methods in an expansion in epsilon '=2 sigma -d, where d is dimensionality. For epsilon '>O the critical behaviour is described by a stable fixed point O( epsilon ¹2/') in the physical region of parameter space. The crossover to short-range behaviour is analysed and takes place when sigma =2+ epsilon /106, epsilon =4-d>O. There is then a small region O< sigma -2<O( epsilon ) where the system is still dominated by long-range behaviour. The results for d=1 are compared with those obtained previously for the random hierarchical model and it is found that the two models do not show the same critical behaviour in the epsilon ' expansion.

Derives an appropriate generalisation of Baxter's variational method and define a sequence of variational approximations for 'antiferromagnetic' models on the triangular lattice. Expressions for the sublattice magnetisations are derived from a variational principle for the partition function per site. He applies the method to the triangular antiferromagnet and obtains approximate phase diagrams in the temperature-field and temperature-magnetisation (density) planes.

Calculates the intrachain bead-to-bead square distance of a real polymer chain, close to the critical dimensionality d=4. From this the authors obtain the mean-square radius of gyration (S²) of the polymer as a function of the molecular weight N of the chain and the excluded volume parameter mu . A proportionality relationship between (S²) and the mean square end-to-end distance (R²) of the coil, previously suggested by enumerations of self-avoiding walks on lattice, is shown to be true. An estimate of the universal ratio (S²)/(R²) is given.

The thermodynamic potential for independent electrons in a magnetic field is calculated by a method that arises from a more general Green function approach. Results are obtained in the weak field and the quantum limits.

It is shown that invariance of Lagrangian field theory under a class of the coordinate-dependent Lorentz group of transformations requires the introduction of a massless axial vector gauge field which gives rise to a super-weak long-range spin-spin force between particles in vacuum. Recent experiments demonstrating repulsion and attraction between circularly polarised laser beams are interpreted to be due to such a force enhanced by spin polarisation of sodium vapour, through which these beams pass.

The Noether-type point transformation symmetry of the one-dimensional single-particle system is investigated systematically. All four possible potentials which possess symmetry larger than time translation have been found. The connection between symmetries on the classical level and the quantum level is also established.

Upper bounds on the interband transitions for Bloch electrons in homogeneous electric fields are obtained. The bounds are powerful enough to imply the existence of 'oscillating Bloch electrons' in weak electric fields. The existence of the effective Wannier Hamiltonians of arbitrary order is also proved.

The cosmological equations for the general scalar tensor theory proposed by Nordvedt (1970) in a Bianchi type-I radiation-filled Universe are solved and the behaviour of the model is discussed. There are two distinct situations. Either the Universe will explode from a bit band type singularity and continuously increase, or the Universe may continuously contract to approach the singularity at the end.