Table of contents

The non-Stieltjes renormalised series given by the hypervirial approach are empirically found to be summable by Pade methods (using the plateau criterion). The resulting energy estimates are exact, as shown by using an alternative power series method also based on the renormalisation approach.

The authors obtain an expansion for the product of two Jacobi functions of different kinds with different arguments in terms of the Jacobi function of the first kind.

The authors examine the derivation properties of a deformed Poisson algebra of classical observables. Their considerations point to possibly new realisations concerning the quantisation mapping.

The authors develop an inverse scattering method for generating static, axially symmetric solutions for SU(N) principal sigma models.

Analytic expressions, involving gamma, incomplete gamma and degenerate hypergeometric functions, are obtained for the Laplace transforms of Airy functions, which arise in the theory of electron tunnelling from solids.

The notions of 'independent constants of motion' are clarified emphasising the role of precise definition of the background space. Using a well defined algorithm that relates symmetries of equations of motion of a classical system (Newtonian, Lagrangian or Hamiltonian) to a conserved object in one-to-one fashion, the symmetries of the Coulomb and linear force problems are determined.

It is shown that the Bondi mass of an asymptotically flat space-time which satisfies the dominant energy condition and which contains a number of trapped surfaces is positive.

The authors present for the first time an exact treatment of the production of radiation by a mirror which accelerates along a smooth, asymptotically static trajectory in two-dimensional Minkowski space. The treatment confirms the general expectations of earlier work which was based on questionable approximations and unrealistic asymptotic behaviour.

The behaviour of a domain wall is examined in a three-dimensional Ising model which has weaker exchange interactions near the surface than in the bulk. It is proved, within the Onsager-Temperley approximation, that the domain wall has a phase transition, being bound to the surface at low temperatures but not at high temperatures.

Introduces the basic features of a proposed transport theoretic approximation of the one-speed neutron transport equation. The method utilises the exact results of Case et al (1967) and retains the use of the eigenfunctions phi _0+or-( mu ) together with a properly constructed rational approximation phi _nu ( mu ) of the singular eigenfunction phi _nu ( mu ) as the complete set of basis functions. A major contribution of this work is the adaption of Case's orthogonality relations to this basis, ( phi _0+or-( mu ), phi _nu ( mu )), for obtaining the desired solution.

The author shows that a very large class of two-dimensional Z(N) spin models with multi-spin interactions is self-dual. By using Griffiths-like inequalities on correlation functions he expects that all these theories, for N sufficiently large, will show an intermediate disordered massless phase (infinite correlation length) between massive high- and low-temperature phases.

Numerical results have been obtained by several methods for a hard core two-dimensional lattice gas on a plane-square lattice, involving exclusion of first, second and third neighbours. They strongly suggest the existence of a first-order phase transition.

The transition temperature for the square-lattice Ising ferromagnet with first-, second- and third-neighbour interactions is calculated by the interface method. The results are apparently fairly accurate, except when the third-neighbour coupling is completely dominating. The author checks in particular, a series result for the equivalent-neighbour model to within a few per cent.

An m-vector classical spin glass in a field is studied at the mean-field level in terms of a Parisi replica symmetry breaking scheme. Solutions for the mean-field equations are presented which have important consequences for the irreversibility of the spin glass phase and clearly indicate the relevance of reduced temperatures scaling as H², H, H²3/.

Field theories with a trilinear interaction are used to describe both the random bond percolation problem and the Yang-Lee edge singularity problems in three dimensions at criticality. Renormalisation group functions are calculated to order four for the exponents eta , omega and the beta function, and to order five for the remaining critical exponents. Estimations of the critical coupling constant and the exponents for percolation theory are found by summing conformal transformations of the Borel transformed series.

For pt.III see ibid., vol.14, p.1825-34, (1981). The irreducible representations of the line groups (constructed in the preceding papers of this series) provide a useful labelling scheme for electron energy bands of stereo-regular polymers and quasi one-dimensional solids. To implant automatic symmetry assignation into a band-structure computing routine one needs a method of dealing with representations of line groups, which are of infinite order. The author demonstrates that the set K of coset representatives of the translational subgroup T of the line group L-which has a finite (and usually quite small) number of elements-contains all the relevant information. Dealing only with K, one can complete the symmetry assignation, decompose a reducible representation and derive selection rules.

The branching rule for the reduction of symmetric irreducible unitary representations (IUR) of the simple Lie group SO(2n+1) into IUR of its maximal subgroup SO(2n-3)(X)SU(2)(X)SU(2) is established for all n>or=3. After the particular case n=3 is analysed in detail, a general proof is presented which is valid for all n>or=3. All branching rules (n=3,4,...) can be summed up in one formula. Also, a dimension verification is carried out. The generators of SO(2n+1) not belonging to the semi-simple subgroup can be combined into a mixed tensor-spinor representation with respect to the simple groups which occur in the direct product. The precise nature of that representation is indicated and discussed.

For pt.I see ibid., vol.15, p.2665-76 (1982). The SO(7) down arrow (SU(2))³ reduction is analysed in detail. Explicit forms for the different SU(2) generators and the remaining ones are derived. Using the results of the previous paper, basis states describing the symmetric irreducible unitary representations are introduced. Expressions for the reduced matrix elements of the occurring generators with respect to these basis states are determined.

The determination of the rank dependence of the weight multiplicities for the groups O_2k+1, Sp_2k, and O_2k is simplified using the properties of Schur functions together with a simple weight space mapping. The weight multiplicities are factorised into products of much simpler quantities whose rank dependence is readily computed. The spinor weight multiplicities for SO_2k are shown to be simply related to those of O_2k. The weight multiplicities for the k-part tensor representation of SO_2k are examined in terms of difference characters. Some specific results are tabulated for SO_2k.

Gives a coordinate-free characterisation of R separation for the Laplace equation on a pseudo-Riemannian manifold in terms of commuting conformal symmetry operators. The coordinates can be computed from a knowledge of the operators.

It is shown that the use of the finite-element method with Lagrange multipliers in scattering theory is equivalent to the Kohn variational principle. For simple test problems, the convergence of the reactance matrix is found to agree with the predicted behaviour, that is, the error in K_ij is proportional to the square of the error of the Lagrange multiplier.

A numerical procedure for the solution of the single total energy group transfer equation is given. The acceleration of the iterative method which is employed is discussed and examples are given. For simplicity isotropic scattering is used.

The problem of decoupling a system of linear coupled differential equations without changing the order of the equations will be considered. Starting from a theorem on the separation of the coupled equations, the author continues the investigations in the general case, where the conditions set up by this theorem are not satisfied, in deriving a method of separation of the equations in the strong coupling case. An example on a system of two coupled Schrodinger equations will be taken up as an illustration of this method.

Integrals of the type I(t) identical to integral integral dxdy delta (t-H(x,y)) are considered, where H is a function from an ensemble labelled by parameters. I can represent circulation times of fluid particles in the plane, orbital periods or semiclassical densities of states for one-dimensional Hamiltonian systems, spectral densities for two-dimensional crystals, or the strength of a wave pulse produced by propagation of a deformed step discontinuity. As H varies over the ensemble, I develops strong fluctuations associated with Legendre singularities. For unrestricted functions H, the probability distribution P(I) describing the fluctuations of I decays according to a universal I^-9 law, obtained by a scaling argument involving Arnold's classification of catastrophes. If H is only quadratic in y, P varies as I^-10. If the integral defining I is one-dimensional, P varies as I^-3; if it is three-dimensional, I decays no faster than I^-48. The probability distribution of I' identical to mod delta I/ delta t mod decays as (I')^-2.

It is shown that a Lagrangian of the form L=¹/₁( rho ²- omega ²(t) rho ²)+G(t)F(k(t) rho ), said to be in factored form, yields an equation of motion that is equivalent to the most general equation derivable via Noether's theorem from the unfactored Lagrangian L=¹/₂ rho ²-P( rho ,t). In view of this equivalence, the theory of extended Lie groups is applied to the factored nonlinear equation of motion p+ omega ²(t) rho =G(t)F(k(t) rho ) to obtain its Lie symmetries. The latter are obtained when G(t) and k(t), initially arbitrary, are determined in terms of a function x(t) which satisfies the auxiliary equation x+ omega ²(t)x=K/x³. It is then possible with the auxiliary equation and the equation of motion to form a coupled pair of nonlinear equations, an Ermakov system, whose first integral is not invariant under the action of the symmetry group, in contrast to previous Ermakov systems.

Starting from the time-independent Schrodinger equation for elastic scattering a theory of wave-particle duality is developed which takes into account effects due to the quantum potential by means of a generalised ray equation. The duality between waves and particles is expressed in terms of two basic equations. Wave effects are described by a nonlinear amplitude equation which is coupled to a generalized ray equation that describes particle effects. The coupling between these two aspects of duality occurs by means of a function called the quantum expansion coefficient which describes the way in which a bundle of ray paths expand as the propagation distance increases. For systems in which the phase of the one-body wavefunction is a separable function of its coordinates, the authors derive simplified forms for the ray equations. As an example the authors show how the theory may be used to calculate tunnelling ray paths in planar electron channelling. The reflected rays in this case originate from beyond the classical turning points, thereby demonstrating the existence of a Goos-Hanchen effect in electron channeling.

The article deals with non-commutative (boson) probability theory. The phase space considered is a linear symplectic space built up from a Hilbert space. Generating functionals for representations of the canonical commutation relations are related to characteristic functionals for random distributions on the underlying Hilbert space. An attempt is made to characterise those Levy-Khinchin characteristic functionals on phase space which are generating functionals by means of a generalised Heisenberg inequality.

The angular momentum projection technique is worked out for systems with an intrinsic asymmetric deformation. These include asymmetric top molecules and triaxial nuclei. A parameterisation of the rotation group is introduced which allows an approximate but analytical evaluation of angular momentum projected matrix elements. Via symmetric orthonormalisation a quantal definition of the moments of inertia is obtained.

The analytical solution of the Schrodinger equation is presented in the case m=0 for the angular equation of the dipolar potential.

The author presents an exact Kantowski-Sachs solution with torsion. The solution represents a magnetic model with stiff matter and is shown to be geodesically incomplete both to the past and the future.

The Kelin-Kaluza version of the combined Einstein-Yang-Mills system is based on a larger-dimension manifold, admitting an O(4+N) symmetry, which undergoes compactification. The theory provides a unified treatment of all gauge fields and its action consists of a single invariant Lagrangian and, separately, a single gauge-fixing term. The authors show that both Lagrangians can be combined within a larger framework, admitting an OSp(4+N/2) supersymmetry, which undergoes appropriate dimensional reduction; here the extra graded dimensions are associated with the fictitious fields needed for consistent quantisation and unitarity.

The authors show how to redefine differentially constrained and unconstrained fields so that they become either auxiliary or propagating fields when the equations of motion are used.

A general and unifying theory of nth-order phase transitions is developed by combining catastrophe theory with self-similar fractal lattices. The scaling laws governing the latter imply the relation D=(c+2)(2- eta )/c between the dimension D of the fractal lattice and the codimension c=2n-2 of the free energy topological normal forms characterising isolated nth-order transitions. This together with the Euclidean dimension of the real system yields numerical values for bicritical, tricritical and tetracritical exponents which are in very good agreement with experiment and those of solvable models. The results are independent of the details of the interaction because nth-order phase transitions are essentially topological phenomena.

The dynamics of strongly interacting colloidal particles is studied in the range of medium to high particle concentrations. Both the short-range ordered liquid phase and the long-term ordered crystal phase are investigated by means of photon correlation spectroscopy. The results are discussed in terms of a memory function formalism which is shown to be applicable for both the liquids and the crystals. The viscoelastic approximation for the memory function describes the experimental results in the liquid phase very well. For the first time results on single crystals are reported which exhibit a purely elastic behaviour. The theory enables the authors to extract information on the elastic moduli and sound velocities of the systems from the data, which in principle yield information on the particle interaction potential. The 'phonon' dispersion of colloidal crystals is also presented.

The choice of a convenient self-dual cell within a real space renormalisation group framework enables a satisfactory treatment of the anisotropic square lattice q-state Potts ferromagnet criticality. The exact critical frontier and dimensionality crossover exponent Hf as well as the expected universality behaviour (renormalisation flow sense) are recovered for any linear scaling factor b and all values of q(q<or=4). The b=2 and b=3 approximate correlation length critical exponent nu is calculated for all values of q and compared with the den Nijs conjecture. The same calculation is performed, for all values of b for the exponent nu (d=1) associated with the one-dimensional limit and the exact result nu (d=1)=1 is recovered in the limit b to infinity .

A Monte Carlo study of self-avoiding walks on the diamond lattice is presented. This model incorporates some of the steric effects and short-range stiffness of real alkanes, and for a nearest-neighbour attractive interaction- epsilon is found to have a collapse transition at k_B theta / epsilon =2.25+or-0.05. The behaviour of the chains in the vicinity of this theta -temperature is analysed with the help of recent 'crossover scaling' theories. It is shown that for finite chain length N there is a rather broad theta -region where the chains' behaviour is quasi-ideal. The width of this region w_theta behaves as omega ₀ varies as N^-12/, consistent with the blob picture. The peak of the specific heat occurs at the boundary between the theta -region and the region of collapsed chains. The authors also give rough estimates for the scaling functions describing the crossovers of the end-to-end distance and structure factor of the chains.

The duality transformation between the discrete Gaussian model of a d-dimensional interface in a (d+1)-dimensional bulk system and the Coulomb gas in d dimensions is extended to include a pinning force which tends to localise the interface either in the centre or near the edge of the bulk system. Arguments are given for the probable phase diagrams of each system.

A representation of the real part of the Heisenberg-Euler Lagrangian density in quantum electrodynamics by means of special functions is obtained. It is shown that this representation is very convenient for numerical calculations of the real part of the Heisenberg-Euler Lagrangian density. It is indicated that this representation is of use for calculations of a quantum electrodynamical field energy density in the absence of real charges and for calculations of polarisation and magnetisation of the vacuum.

The authors examine some of the problems with scale-covariant quantum field theories, as developed by Klauder (1979) and summarise their progress in solving them.

For pt.I see ibid., vol.15, p.2911-27 (1982). The authors examine the possibility of developing a perturbation expansion about the pseudo-free scalar theory. The degeneracy of the subtracted scale-covariant equations requires that additional information be provided. They suggest two ways in which this can be done.

For pt.II see ibid., vol.15, p.2929-45, (1982). The authors examine the augmented formalism for a scalar pseudo-free field theory. Motivated by a diagrammatic expansion for mass renormalisation, they justify a constraint equation for the theory that gives additional information over the subtracted scale-invariant formalism.

Properties of an alternative supersymmetric extension of the Poincare algebra are investigated. Relations to other superalgebras are studied and the explicit form of Casimir operators is written down. Representations on a superspace as well as an analogue of the Wess-Zumino supersymmetric Lagrangian are found by a contraction procedure.

Fix a periodic potential, V, and let k(E,B) be the integrated density of states up to energy E in a constant magnetic field B. The author proves that k(E,B) is continuous in B at all points where it is continuous in E. He proves a similar result when V is zero and B is a multiple of a periodic magnetic field.

The S^z=0 excited states of the anisotropic antiferromagnetic Heisenberg Hamiltonian H= Sigma _j=1^N(S_j^xS_j+1^x+S_j^yS_j+1^y+ rho S_j^zS_j+1^z) are studied when 0<or= rho <or=1. The original set of secular equations is reduced to a simpler one, which contains the parameters of excitations only. The energy-momentum dispersion is also found. It is shown that the typical low-lying excitations are either two-strings or not of string form. The simplest excitations are described in more detail.

The author points out that the solution given by Moussiaux et al. (1981) and MacCullum et al. (1982) has been (in principle) independently obtained by Stewart and Ellis (1968). In addition he presents new exact Bianchi-Kantowski-Sachs solutions of type III with dust and a cosmological constant, which are generalisations of the vacuum solutions.

Ginzburg-Landau equations for the pairing of relativistic electrons are derived. It is observed that relativistic effects tend to make superconductors type I and tend to make the superconducting phase transition strongly first order.

We develop the idea presented in a previous paper by including the case of non-identical coupling terms and show that, under some conditions, separation of the equations is also possible.