Table of contents

Linear equations which can be used for inverse scattering solutions for solitons in g phi ³+h phi ⁴ theory have been determined.

Discusses the relationship of space-time with three spatial and 2n+1 temporal dimensions, which may be of interest in the theory of electroweak interactions, to ordinary four-dimensional space-time.

A simple technique is used to extend and clarify recent results on some charmonium confining potentials. One such potential has a convergent Rayleigh-Schrodinger series which does not give the correct bound-state energy.

A certain relationship between a recent conjecture by F.Y. Wu (1979) and one by W. Klein et al. (1978) is established through considering a three-colour site percolation problem, where the three sublattices of a triangular lattice are populated with probability s₁, s₂ and s₃, respectively. Both conjectures imply the same critical condition for various special cases of this three-colour model, including the threshold probability 1/ square root 2 for the honeycomb site problem. Existing numerical estimates provide strong evidence against this threshold value, hence against both conjectures.

Introduces a field-like variable to develop a generating function for the percolation problem which, in the appropriate limits, also describes the statistics of random animals (dilute branched polymers) and compact clusters (collapsed branched polymers). The crossover from percolation to random animals and compact clusters is studied using a two-parameter position space renormalisation-group approach. The authors obtain the global flow diagram in the two-parameter space and calculate the critical properties. They find that the critical behaviour is described below the percolation threshold p_c by the random-animal fixed point, and above p_c by the compact-cluster fixed point. A Hamiltonian formulation using the Q-state Potts model is proposed. The crossover from percolation to random animals can be described by taking a specific limit of the field-like variable.

A three-state Potts model with pure three-site interactions is analysed using low-temperature series expansions. Universality predicts that the exponents alpha ' and beta should be the same as for the 'hard-hexagons' lattice gas: ¹/₃ and ¹/₉ respectively. The series estimates are consistent with these predictions, and the estimates of gamma ' are consistent with the scaling value of 1⁴/₉.

The q-state, d-dimensional Potts models exhibit a variety of phase-transition behaviour in the limit d to 1⁺, q to infinity , and l identical to (d-1) in q finite. The regions l<1, 1<l<2, and 2<l are distinguished, respectively, by no transition, second-order transitions (with a new changeover phenomenon at l=ln 4), and first-order transitions. The latter are due to the condensation of effective vacancies. Critical and tricritical exponent values are given.

Generalisations of R.P. Wigner's formulae (1971) relating to the counting problem when one considers the restriction of the irreducible representations of a group to its subgroups are obtained in the context of the symmetric group. The usefulness of these in the enumeration problem of many-body diagrams is described.

Invariant volumes are calculated for all compact simple Lie groups. The measure is defined by means of the Riemannian metric, induced in the group by the Killing form, so the total normalisation is fixed. The coset space aspects are discussed.

The backward Kolmogorov equation is established and solved for a Markovian system of coagulating droplets. The resulting probability generating function of the total number of droplets coincides with the expression recently obtained by M.M.R. Williams (1979) from the forward equation. Thus the equivalence of the two approaches is verified for a system whose evolution implies a probability balance giving rise to terms nonlinear in the state variable. It is shown that the backward equation lends itself to being solved directly in terms of moments of any order, whereas the forward question shows in this respect the problem of closure.

Results are given for computer searches for those configurations of N points on the surface of a four-dimensional hypersphere which have maximum values of the least angular distance between pairs of points. The generalised inverse was used to solve the equations giving the corrections to the hyperspherical coordinates in terms of the overlaps of the domains of neighbouring spheres. It is concluded that the use of this curved space offers an alternative to cyclic boundary values for simulations involving the packing of points in three dimensions.

A model consisting of a hydrogen atom coupled to a single quantised radiation mode in the long-wavelength approximation is studied mathematically. It is shown that the Hamiltonian H for this model is dilatation analytic. This result makes it possible to describe various resonance phenomena associated with atoms in radiation fields by means of analytic continuation techniques. Furthermore, it is shown that H possesses an infinite number of bound states with negative energy. Nevertheless the photoionisation probability tends to one with increasing field strength.

A double generating function, phi ( xi , xi ' mod Z), for the harmonic oscillator function psi _n(r) is derived. Then phi ( xi , xi ' mod Z) is applied to obtain a double generating function, P(k mod st) for the radiative matrix elements P_mn(k). To obtain explicit matrix elements the author expands P(k mod st) in powers of s and t and picks out terms proportional to sⁿt^m.

Calculates expectation values and uncertainties of the radial coordinate and momentum, and the azimuthal angle and angular momentum components for a three-dimensional harmonic oscillator in a coherent state. The operators corresponding to powers of the angular variable are defined by their Fourier expansions. For large distances of the particle from the origin, where the oscillator becomes classical, radial and angular uncertainty products approach the value h(cross)/ square root 2, except for the point of discontinuity, where the angular uncertainty becomes infinite. For small distances, where the oscillator tends to its quantum ground state, the radial uncertainty stays finite and the angular uncertainty product tends to zero. It is also shown that the concept of a polar momentum operator is meaningless.

The problem of calculating psi (0) for radial Hamiltonians is shown to be reducible to a sequence of energy calculations, which can be performed easily by using a pseudoangular momentum term in a numerical integration approach. A first-principles approach to s-state hypervirial relations is shown to produce terms not given by the usual commutator approach.

Within the method developed by the authors (1977, 1978, 1979) for calculation of the skeleton diagrams in conformally invariant field theory they find anomalous dimensions of fields and the coupling constants in some models of quantum field theory. They also demonstrate the efficiency of the method for finding critical indices in the theory of second-order phase transitions directly in three-dimensional space.

Differential cross sections are written in terms of a time autocorrelation function involving the transition operator. Such a formulation is useful when time-dependent approximations are considered. Eikonal theory provides a good example. Using the time autocorrelation function representation, a general eikonal differential cross section is obtained which is parametrised by a momentum. If this parameter is taken as the incoming momentum, then the resulting eikonal differential cross section is found to be complementary to the standard Glauber total cross section, that is, the angular average of the differential cross section is identical to the total cross section. Weak potential limits of these cross sections are Mott cross sections. On the other hand, if the parameter is taken as the average of the incoming and outgoing momenta, then an eikonal differential cross section is produced whose complementary total cross section is not of Glauber form. However, the weak potential limits of these cross sections are the Born cross sections.

The dynamics of a perfect fluid with pressure equal to energy density is considered in space-times with planar, pseudoplanar, cylindrical and toroidal symmetries. A scheme for obtaining the general exact solution of the field equations is described. It is shown that this scheme admits inclusion of additional massless, minimally coupled scalar fields and (for planar symmetry) electromagnetic plane waves. Two physical examples are discussed: finite perturbations of a static regular cylinder, and waves of finite magnitude in a Kasner-like universe. In the first case it is shown that all physically allowed perturbations preserving cylindrical symmetry are of standing-wave type and cannot propagate. Thus a perturbed static stiff cylinder cannot radiate the Einstein-Rosen waves. In the second case it is shown that travelling-wave perturbations obey a sort of momentum-conservation law; besides, it is argued that sound perturbations can provide a mechanism for fragmentation in a universe filled with stiff matter.

The relativistic theory of axially symmetric, rotating charged dust is compared with the classical theory. The equations of motion are obtained from the field equations as compatibility conditions. It is proved that, if comoving coordinates are used for a rigid rotation in which the Lorentz force does not vanish, sigma / rho and g₄₄ are functions of the electric potential. Two new classes of exact solutions are obtained, one being the general solution for the case of rigid rotation with vanishing Lorentz force.

It is shown that a Hermitian phase operator exists for quantum spins. Its spectrum is not continuous but has the values (2 pi /(2S+1))n, 0<or=n<or=2S for spin S. The precession of a high-S spin in a magnetic field is shown to consist essentially of a set of jumps from one value of phase to the next one, at equal time intervals. At intermediate times the value of the phase is uncertain. By going to a larger Hilbert space than the spin space, a Hermitian phase operator whose eigenvalue spectrum covers the entire real line is defined, and its relation to the formerly defined operator is clarified. Possible applications to spin and other systems are outlined.

A study has been made of the layer and local susceptibilities of the four-dimensional n=0 (self-avoiding walk) model. A renormalisation group calculation has been made for general n. Series expansions to 11th order have been generated for n=0.

Shows that in the usual treatment of the O(n) symmetric Ginzburg-Landau-Wilson field theory Goldstone modes induce a negative susceptibility for n<1 as the co-existence curve is approached. This situation affects the whole of the semi-dilute regime of polymer solutions. To examine the role of Goldstone modes within this formalism, the authors consider the generalised Heisenberg model of H.E. Stanley (1969) with self-avoiding constraint on a d=2 square lattice in the limit n to 0. By taking highly anisotropic couplings, the authors take the continuum limit in one direction for the transfer matrix and obtain a quantum mechanical Hamiltonian on a d=1 lattice. This is used to obtain perturbation series in the coupling between lattice sites for the inverse correlation length and susceptibility. The authors obtain the equation of state to second order, and show the existence of spontaneous magnetisation even though it is absent in d=1. The susceptibility is well behaved and positive. The calculations can be readily extended to systems of higher dimensionality.

For a polymer chain in a good solvent, the author calculates the probability distribution functions between an endpoint and an interior point, and between two interior points, by using exact enumeration to study a lattice self-avoiding walk model. These distribution functions are different from the usual distribution function between endpoints. At small distance scales, the probability of nearest-neighbour contacts between two interior points is smaller than the probability of contact between two endpoints. FCC and triangular lattices are considered.

Linear analytic solutions of the homogeneously broadened, unidirectional Maxwell-Bloch equations with field losses are found in circumstances where amplification can occur. Particular stress is laid upon the similarity-variable solution. A number of inequalities are derived which suffice to differentiate between the physical regimes of superfluorescence, amplified spontaneous emission and swept-gain amplification.

The motion of a non-relativistic free charged particle in a classical electromagnetic plane wave field is investigated. By introducing the ansatz of a modulated plane wave for the wavefunction, the problem is reduced to the solution of the general Mathieu equation for a circularly polarised field and to the solution of the Hill equation for a linearly polarised field. The corresponding eigenvalue equations show that physical values of the energy are separated into bands. The relationship with earlier methods is also discussed.

The relations investigated by J. Kulesza and J. Rembielinski (see ibid., vol.13, p.1189-95, 1980) are particular cases of a recoupling equation.