Table of contents

A simple derivation is given of the explicit value of the class operator for the rotation group SO(3).

Using non-local (non-Lie) symmetry of the linear Dirac equation the authors have constructed a number of new ansatze reducing it to systems of ordinary differential equations.

The Thomas equation has the Lie-Backlund algebra recurrence operator of the first order, the infinite set of the many-parameter Backlund autotransformations and no reduction to the Painleve transcendents. The Burgers equation has the same properties which are probably the general indicators of the hidden linearity.

The authors present a 'correction' method to match electrostatic boundary conditions for the Poisson equation in the presence of unscreened interior point charges. This method is applicable to any general electrostatic boundary conditions, is independent of the geometry of the boundaries, and is well suited to numerical computation. It is based on correcting the changes in boundary values caused by the presence of the point charges. Thus, the corrected boundary conditions include not only the effects of all exterior charges, but also the effects of the interior point charges.

A unique family G_n-1 of conformal invariants, polynomial in the extrinsic curvature of a hypersurface embedded in an n-dimensional Riemannian manifold M, is constructed. In the quantum theory of a conformally coupled scalar field on a manifold M with boundary delta M, the counterterm that regularises the one-loop effective action contains members of G_n-1 integrated over delta M. The counterterms through n=4 are given explicitly and a derivation of the n=4 boundary contribution is given based on a flat-space result.

The existence of many equilibrium states of the spin glass on a Bethe lattice at low temperatures is confirmed numerically, directly on the lattice. Following a method due to Nemoto and Takayama (1985), approximate solutions of the exact equations for the Bethe lattice site magnetisations (m_i) are found by minimising the norm mod Del F mod identical to ( Sigma _i( delta F/ delta m_i)²)^1/2 where F((m_i)) is the free energy. In order to examine the stability of solutions on the Bethe lattice using the Hessian delta ^2/F delta m_i delta m_j, it is necessary to connect up boundary sites. Evidence is presented for the bifurcation of equilibrium states with decreasing temperature.

As a step toward proving the generally held belief that the two-dimensional O(n) model exhibits no phase transition for n>or=3, the authors consider a special O(n) model on the honeycomb lattice and establish a rigorous lower bound on n such that the free energy of the O(n) model is analytic for all n above this bound.

The overlaps of spin-spin correlations inside one phase space valley and between different valleys are calculated in the Gaussian approximation around Parisi's mean-field solution for the Ising spin glass. The large mass of the theory is shown to be related to the coherence length of the longitudinal fluctuations of the order parameter inside a single pure state. The long-wavelength asymptotics of correlation functions is worked out in detail and it is shown that an exact inequality between intra- and intervalley overlaps breaks down below dimension d=6, signalling an internal inconsistency of the theory for d(6.

The authors extend the known series expansion coefficients for the mean cluster size in two-dimensional bond percolation and use them to accurately approximate the size function for all probabilities less than the critical probability.

The author derives the possible values of the conformal anomaly number c for the integrable Heisenberg antiferromagnet with planar anisotropy gamma = pi / upsilon (0( gamma <or= pi /2) in the case where upsilon is a rational number. For given spin S, with upsilon )2S, the author finds c=3S/(S+1), suggesting a renormalisation onto the k=2S Wess-Zumino-Witten fixed point. In contrast, for upsilon <or=2S a new renormalisation behaviour is revealed, with 2S-1 distinct universality classes indexed by the integer part of upsilon .

It is shown that there exist polynomial (or non-trivial) zeros for the 9-j coefficient. A simple closed form expression for the polynomial zeros of degree one of the 9-j coefficient is derived from the triple-sum series due to Jucys and Bandzaitis (1977). Polynomial zeros of degree one of the 9-j coefficient are generated from either this closed form expression or from a set of parametric solutions of the multiplicative Diophantine equations: xyz=uvw.

The authors generalise a previously introduced group manifold approach to quantisation in order to apply it to the quantisation of free fields. The procedure is based on the consideration of infinite-dimensional groups, for which the appropriate generalisations of certain concepts of ordinary Lie group theory are introduced. The cases of the Klein-Gordon and the Proca fields are treated in detail, the latter to illustrate the treatment of constraints. The zero-mass limit of the vector fields is also briefly discussed in connection with the Stuckelberg formalism.

The authors describe the linear T ⊗ τ₂ Jahn-Teller interaction as a pseudoscalar operator in electron-phonon space. In this form it is found to mix states within an invariant minimal subspace for which they give basis vectors in an analytic form for arbitrary phonon numbers. These vibronic states are labelled by spherical symmetry quantum numbers and possess definite cubic symmetry at the same time. It is shown how to extend the method to the case when the Jahn-Teller interaction competes with spin-orbit coupling.

For pt.III, see ibid., vol.20, p.3577 (1987). An algebra of operators commuting with a given compact internal symmetry group action on a Fock space is introduced as a means for constructing unitary invariant operators on the Fock space. The algebra is generated by invariant polynomials and, in general, is an infinite-dimensional Lie algebra with a Cartan-Weyl structure. As an example, the algebra A^SO(3)₂ generated by the l=2 representation of SO(3) is analysed. Irreducible representations of A^SO(3)₂ are given by raising operators acting on lowest-weight states; the coefficients which connect states generated by raising operators are computed. Some of the multiplicity appearing in an irreducible representation is dealt with by introducing an SL(2,R) subalgebra of A^SO(3)₂.

For pt.IV, see ibid., vol.21, p.4305 (1988). The pseudoscalar mesons associated with SU(3)_flavour are used to generate a many-particle Fock space. For the eight pseudoscalar mesons the algebra of operators commuting with SU(3)_flavour, A^SU(3)₈, is shown to be isomorphic to an infinite-dimensional Lie algebra with a Cartan-Weyl structure discussed previously. When the Fock space is generated by quark-antiquark pairs, the algebra of operators commuting with SU(3)_flavour becomes a direct sum of A^SU(3)₈ and a finite-dimensional Lie algebra. The relationship between elements of the algebra of commuting operators and multiparticle amplitudes is discussed and a model phase operator is chosen to compute some perturbative amplitudes.

The procedure for getting an effective Lagrangian and Hamiltonian for the low-lying states of an interacting string, by integrating over some ultraviolet degrees of freedom, is extended to fermionic variables. The perturbation on the fermionic levels is determined and a crossed term which describes the mutual effect of the bosonic and fermionic excitations is also obtained.

In a book on Bessel functions by Watson (1980), there appear to be minor errors in one particular integral of Bessel functions. This integral is given a series expansion which may be useful in certain physical problems. The asymptotic nature of this series is very briefly discussed.

Using Fourier transforms within two-dimensional spaces is one of the usual methods employed to obtain a convenient image reconstruction from a sample of transaxial tomography data. However, this task can also be achieved via a thorough study of a Fredholm equation of a first kind with finite boundaries, i.e. admitting a disc as integration domain. With appropriate polar coordinates and utilising furthermore a power series representing 1/r_MM, the Fredholm equation can be split into a denumerable set of new one-dimensional equations. Afterwards, suitable methods of solution are offered, taking into account the fact that all the equations studied belong to a class of ill-posed problems. The transformations pointed out are also helpful for handling as correctly as possible some related and more difficult non-linear problems, e.g. image reconstruction from time-of-flight data measured on ultrasonic pulses.

A Green function procedure is presented by means of which exact decay integrals can be derived without recourse to the eigenvalue problem. General solutions are given for Fano and Lifshitz type decay setups. Specific versions of these models are discussed. In particular the Bixon-Jortner model and generalisations thereof, and the excitonic transfer problem, are discussed in more detail. Novel decay features are found, such as multiple decay channels, some of which are of non-golden-rule character, and a 'crossover' phenomenon from complete to incomplete decay, which is accompanied by alterations in the damping features.

The equation governing the propagation of spiral beams in a non-linear medium is analysed. It is shown that, taking into account the saturation effect, spiral beams have a tube-like structure with a periodicity along the axis of a non-linear autoguide. A critical power is found which it is necessary to exceed in order to give rise to the indicated autoguided regime of the beam propagation. Furthermore, in this work a strict self-similar solution is obtained of a non-linear Schrodinger-type equation that describes the collapse of spiral beams, and a new class of self-similar solutions is found for the case of spherical symmetry.

The authors are concerned with the critical properties of antiferromagnetic Takhtajan-Babujian models with spin S=1, 3/2 and 2. The leading eigenenergies of this Hamiltonian, in a finite chain, are calculated by investigating numerically and analytically the Bethe ansatz equations for the finite system. The critical exponents and the conformal anomaly are obtained from their relations with the eigenspectrum of the finite Hamiltonian. The appearance of logarithmic corrections produces poor estimates. However, a combination of analytical and numerical methods produces very good estimates. Their results strongly support the conjecture that the Wess-Zumino-Witten-Novikov non-linear sigma models with topological charge k=2S are the underlying field theories for these spin-S statistical mechanics models.

The author obtains the exact finite lattice partition functions for a sequence of two-dimensional Z_n symmetric spin models, interpolating between the two-phase ferromagnetic Potts model and the three-phase (n>4) vector model. The author presents these results through the distribution of zeros of the partition function in the complex coupling variable. The author finds that the phase structure for both two- and three-phase models is manifested in this presentation. In the Potts model case the zeros approaching the ferromagnetic phase transition point are restricted to a circular locus (although well away from the ferromagnetic region this restriction does not apply). As the interpolation proceeds the simple ferromagnetic locus is modified, revealing a three-phase structure.

The decay constants for diffusion through inhomogeneous media are known to be proportional to the eigenvalues of the corresponding elliptic operator. A new method of obtaining a hierarchy of upper bounds on sums and products of these eigenvalues as well as the eigenvalues themselves is presented. The first member of this hierarchy is just the usual Rayleigh-Ritz quotient. The other members of the hierarchy are generalised Rayleigh-Ritz quotients which can be derived simply using properties of integrals of the solutions of the diffusion equation. Explicit bounds are presented for the first three eigenvalues, but general methods of obtaining bounds for higher-order eigenvalues are also outlined. For fixed time t, many of the bounds reduce to results given by the classical method of moments. The hierarchy of rigorous variational bounds on the eigenvalues studied may be generated using simple recursion relations based on properties of the characteristic orthogonal polynomials. The conditions on the trial functions used to obtain bounds on eigenvalues higher than the first are much simpler than those required by the traditional Rayleigh-Ritz procedure.

A neural circuit model is presented that consists of a hierarchy of nested clusters of neuron-like elements. Each hierarchical level of cluster organisation encodes a different level of distributed memory. Associate memory properties within and between levels are investigated numerically in simple versions of the model. The results may provide insight into mechanisms of memory storage, recall and loss in real neural circuits, such as those found in the cerebral cortex.

The particle spectrum of a spontaneously broken supersymmetric gauge theory may in principle contain massive vector bosons without the scalars required for complete vector supermultiplets (supersymmetry 'shattering' or breaking 'in spins'). It is conjectured, however, that this truncation of the spectrum is incompatible with gauge anomaly freedom, and thus cannot occur in a unitary quantum field theory. A supersymmetric Higgs model is presented illustrating clearly the role of anomalies in ensuring an untruncated spectrum of supersymmetry multiplet modulo mass splittings, and providing strong evidence in favour of the conjecture.

In this sequel to Srivastava's work (1987), a number of new expansions (in series of various classes of hypergeometric polynomials) are derived for the general multivariable hypergeometric function which provides an interesting and useful unification of numerous families of hypergeometric functions of one, two or more variables encountered naturally in a wide variety of physical and quantum chemical applications. By suitably specialising some of these polynomial expansions, Clebsch-Gordan type linearisation relations are deduced for the products of several Jacobi or Laguerre polynomials. It is also shown how one of the linearisation relations involving Laguerre polynomials would yield the corrected (and modified) versions of a couple of results given by Niukkanen (1985).