Table of contents

Bridge nodal diagrams built upon the screened Debye interaction are considered in 2+ epsilon dimensions for the classical one-component plasma (OCP) model. Their asymptotic behaviour is shown to be nearly epsilon -independent. They are evaluated analytically in one dimension. The corresponding asymptotic decay approximately exp(-K_kr), with K_k a positive integer >or=2, is thus extrapolated to any epsilon .

For any lattice in any dimension the mean square fluctuation (N_s²)-(N_s)² in the numbers N_s of percolation clusters with s sites each is found to equal (N_s) for large clusters.

Introduces a new polychromatic correlated site percolation problem, which has the novel feature that the partitioning of the sites into different species arises from a purely random process-that of random bond occupancy. A particular case of this percolation problem is shown to be of possible relevance in providing a physical mechanism which may contribute to the unusual properties displayed by liquid H₂O and D₂O under conditions of supercooling below the melting temperature.

The Monte Carlo technique is used to calculate the order parameter, Phi , of the four-dimensional spin-¹/₂ Ising ferromagnet in a simple hypercubic lattice. The order-parameter data are described by Phi (t)=B(-t)^1/2 mod ln(-t) mod ^1/3(1+Q ln mod ln(-t) mod / mod ln(-t) mod ) in the range 0.01<or approximately=-t identical to (T_c-T)/T_c<0.56 with parameters B=1.543+or-0.002 and Q=-0.030+or-0.02. The form of Phi (t) agrees with that predicted by renormalisation group theory.

It is shown that zero-fermion modes exist in the external field with non-zero topological charge in Weinberg-Salam type models. An explicit expression for the zero modes is found in a particular case. Some physical consequences of the existence of the zero modes are discussed.

The problem of existence of a family of quark stars has been investigated using the ground-state equation of state for the quark gas in second-order perturbation theory in quantum chromodynamics.

The authors have investigated the influence of the shape of pair potentials on the existence and properties of soliton-like (quasisoliton) solutions in periodic one-dimensional lattices. The classical equations of motion were solved numerically for chains of 'atoms' with nearest-neighbour interactions. The class of pair potentials studied has the form V_sigma (r) varies as exp(-2br)-2 sigma exp(-br/ sigma ), sigma >0. For all sigma 's tested ( sigma =1, 5, 10, 15), quasisoliton solutions were observed to propagate with essentially constant velocity and survived many collisions. The most interesting conclusion is that long-lived quasisoliton solutions apparently exist for most systems with realistic anharmonic potentials. The conditions these potentials have to satisfy (a sufficiently steep, short-range repulsive part and an asymmetric (V(r+r₀) not=V(r-r₀), for all r₀) overall shape) are weak. The nature of the long-range part is unimportant. The initial conditions are more decisive; they determine the nature and behaviour of the quasisolitons created. Integrability of the Hamiltonian does not seem to be necessary for the existence of quasisolitons.

The reduced determinantal forms for the character of all the irreducible representations of the classical Lie groups are derived from Weyl's character formulae. The derivations use a lemma due to Frobenius and the results are expressed in terms of characters of representations specified by hook diagrams. The results are appropriate to all irreducible representations whether specified by ordinary or composite Young diagrams, and whether tensor or spinor in nature.

Formulae are derived expressing the dimensions of each irreducible representation of each of the classical Lie groups of transformations in an N-dimensional space as a factored polynomial in N divided by a product of hook length factors.

The quantum mechanical addition of an arbitrary (very large) number of identical angular momenta is considered. The resulting states jm usually occur many times. The corresponding multiplicities P_j and Q_m are investigated. Several explicit formulae for Q_m are given. The asymptotic approximations for multiplicities P_j and Q_m, which are the Wigner and Gauss distributions respectively, are derived. All parameters of the distributions are explicitly obtained. A comparison of theoretical multiplicities with the exact ones is made when the number of angular moments is relatively small.

It is shown that the analysis of Morse (1953) of a non-Hermitian eigenvalue problem, connected with a string with non-rigid supports, leads to an erroneous expansion of the Green function into a set of related eigenfunctions. The correct expansion is derived.

Misconceptions still exist regarding the nature of the approximation behind the so-called WKBJ solutions to wave propagation problems, particularly when the second-order differential equation is not in normal form, and when the order of the equation is higher than two. The authors examine (i) necessary conditions via the differential equation for the validity of the additional phase memory terms, and their possible transformation into local factors, showing how these terms arise and when they are meaningful, and (ii) sufficient conditions briefly via the corresponding integral equations. Investigation (i) is based on two sets of inequalities, the first between the moduli of the approximate solutions of the corresponding first-order equations, and the second between the moduli of the exponentials of the integrals of the effective refractive indices.

The quantal problem of a particle moving in a one-dimensional periodic potential with one barrier per period is treated rigorously by means of a phase-integral method. A general dispersion formula, relating real energies to real or complex wavenumbers, valid for any conveniently chosen order of the phase-integral approximations used, is derived. The formula allows for the use of modified as well as unmodified phase-integral approximations, and different possibilities are discussed. For instance, a modification introduced by Floyd in the usual first-order JWKB approximation and applicable to the interior of high-energy bands can be utilised in our scheme for the generation of higher-order phase-integral approximations. This approach opens the possibility of using this phase-integral method in situations where the parameters of the problem are such that with unmodified approximations it would not work at all. The accuracy obtainable is illustrated by calculations on the Mathieu potential and comparison with available accurate numerical results.

Dynamical systems with three degrees of freedom and a Hamiltonian quadratic in momenta and invariant under space rotations are considered. A class of such systems which can be interpreted as a non-linear realisation of dynamical SO(4) or SO(3,1) symmetry groups with a 'Runge-Lenze vector' linear in momenta is found. The corresponding equations of motion are solved explicitly.

The author calculates the translational friction coefficient and the translational diffusion coefficient of a permeable cylinder moving in a sheet of fluid which is embedded on both sides in a fluid of much lower viscosity. The result, which is an asymptotic expression valid in the limit of small ratios of the viscosities, is derived by the method of matched asymptotic expansions.

It is shown that -d²/dx²+V(x)+ epsilon x, with V(x) periodic and Im epsilon not=0, has a spectrum in the form of a ladder for mod epsilon mod small.

It is shown that O(p,q) and Sp(2,R) are complementary groups in the space of pseudo-oscillator H_pq eigenfunctions. The structure and irreducible representation of both invariancy algebra and generating-spectrum algebra are discussed in detail. It is proved that the transformation brackets between the basis diagonalising the compact generator and the basis diagonalising the non-compact generator in the case of discrete series of irreducible representations of the SU(1,1) group coincide with the Clebsch-Gordan coefficients for the Kronecker product D¹4+/(X)D¹4-/ of two irreducible representations of the SU(1,1) group belonging to positive and negative discrete series respectively.

The Dirac equation in the presence of the classical field of a standing wave is investigated for an electron moving parallel to the beam direction. The main differences from the spinless case are: if the two plane waves which produce the standing wave have opposite polarisation, the energy bands for the two spin components are different; for the spin-averaged transmission coefficient this is of relevance only for very low energies; if the waves have equal polarisation, the momentum bands as found in the spinless case are completely absent.

The problem of the vacuum state definition in a part of the Minkowski space bounded by a single or two spherically symmetric mirrors which expand or contract with uniform acceleration is considered. The Euclidean approach is used to obtain the explicit expressions for the corresponding causal Green functions. The vacuum stress-energy tensor is obtained and its properties are investigated.

It is shown that spin-¹/₂-particles with non-zero rest-mass have certain anomalous properties in a curved space-time. The sign of their energy density may be observer-dependent. In addition, the complete energy-momentum tensor may vanish in some space-times.

An effective Lagrangian for the conformally coupled Klein-Gordon field in an expanding 3-flat Robertson-Walker universe is deduced, and from this the number density of the particle pairs created is calculated exactly. An approximation formula for the imaginary part of the effective action density is given. It is applied to the already solved case and to an as yet not exactly solved one.

A more detailed consideration of a trial wavefunction antisymmetry makes it possible to raise an energy lower bound of a many-fermion system. The method is particularly effective for a small number of particles.

The authors have generated series expansions for the layer susceptibility chi ₁ of the Ising model with a free surface, on both the square and simple cubic lattices. In both cases they have extended known series expansions by four terms. Analysis of these series yields eta =1.37+or-0.01 (square) and eta =0.78+or-0.02 (simple cubic). These results are consistent with both surface scaling and with the recent RG scaling result obtained by Bray and Moore.

The asymptotics of the state of the n-vector model with a finite number of spins in the spherical limit is studied. Besides re-deriving the limit of the free energy, corresponding to a generalised spherical model (with 'spherical constraint' at every site), the authors also obtain the limit of the correlation functions, which allows a precise definition of the state of the latter model. Correlation inequalities are proved for ferromagnetic interactions in the asymptotic regime. In particular, it is shown that the generalised spherical model fulfils the expected Griffiths-type inequalities, differing in this respect from the spherical model with overall constraint.

A diagrammatic method of comparable status to the exact enumeration machine techniques is developed for the determination of the configurational properties of isolated self-avoiding polymer chains of freely jointed hard sphere segments. A close upper bound on the exponent in (R_1N²) approximately n^gamma is determined to be 1.26> gamma >1.0, which is a substantial improvement upon earlier diagrammatic analyses ( gamma =2.0). Moreover, the end-to-end distribution function has a non-zero contact value, indicating the possible development of closed ring configurations-again contrary to earlier diagrammatic analyses.

For pt.I see ibid., vol.12, no.12, p.2475 (1979). The internal distributions within a freely self-avoiding polymer are determined on the basis of a simple diagrammatic convolution technique in conjunction with a superposition approximation. It is found that the mean square lengths of internal subsets are expanded with respect to their isolated values to an extent depending upon the proximity of the subset to either end of the chain. For chains of length much greater than the length of subset, asymmetric screening by the tail 'crowds-in' the internal subset, and shows signs of incipient collapse-a result familiar in dense fluid systems.

For previous work see ibid., vol.12, no.12, p.2487 (1979). A screened convolution approximation is introduced for the specification of internal and end-to-end probability distributions at various packing fractions. The competing effects of inter- and intra-chain exclusion are investigated, and the conformations of the chains determined as a function of packing fractions. The equation of state for relatively short chains at intermediate to high concentrations is determined.

A renormalisation-group approach to three-dimensional site percolation systems on the diamond, simple cubic, body-centered cubic, face-centered cubic and hexagonal close-packed lattices is developed using a scaling transformation in real space.

A generalised Gibbs equation for viscous, incompressible fluids is proposed. It takes into account the dependence of the entropy on the viscous pressure tensor. When the restrictions of the objectivity principle are imposed on the phenomenological equations, the expression for the pressure tensor is seen to be equivalent to that of second-order fluids.

The solution of the Dirac equation for the electron in the inhomogeneous magnetic field H_z=H sech²(ay) is discussed and it is shown that spontaneous electron-positron pair creation is possible, provided certain relations between the strength of the field, the inhomogeneity parameter and momenta are satisfied. The realisation of this possibility in the case of a neutron star is examined.

The general structure of field theories on multiply connected spaces is presented using the universal covering space concept. Restriction to rigid gauge theories allows the use of representation theory to answer several relevant mathematical questions.

The authors calculate the vacuum stress-energy tensor for scalar fields having an explicit U(2) symmetry and for neutrinos in three flat space-times whose constant-t hypersurfaces are a 3-torus, Klein bottle and twisted 3-torus. For special values of the parameters, previously calculated values are regained.

The convergence of a certain sequence of Legendre Pade approximants defined in a previous work (1979) is illustrated numerically for the case of the Yukawa potential, and in particular when this potential is such that the partial wave series converges only slowly. The sequences converge appreciably faster than this series in the physical region, and also converge quite quickly even in regions where the series diverges. The Legendre Pade approximants are used to construct convergent sequences of approximants to the impact parameter amplitude, and numerical results are given in the Yukawa case. The unitarity of the extrapolated partial waves is also studied numerically.

For Moller scattering in a monochromatic laser field and energy conservation law is valid which differs considerably from that for free Moller scattering. The consequences of this modified conservation law for the scattering energy are examined for strong circularly polarised laser fields. The laser field acts in two ways on the electrons, which experience both the collective phenomenon of an intensity-dependent energy shift, and can also absorb or emit laser quanta. The second effect produces a discrete energy spectrum. In most cases the intensity-dependent shift is covered by this discrete spectrum. For both effects approximate formulae and numerical results for the scattering energy of non-relativistic electrons are presented and the possibility of an experimental verification of these energy changes is discussed.

A study is made of the one-dimensional Ginzburg-Landau system with a cubic term, a model of the first-order phase transition. Fluctuations are taken into account by means of the WKB method. It is pointed out that the transition is of continuous type for any value of the coefficient of the cubic term. This results from the excitation of domain walls between two coexisting phases.