Table of contents

A fixed frequency Lagrangian variational principle is formulated for the invariant tori of conservative dynamical systems. It avoids the singularities due to small frequency divisors, and for pure rotation provides a strict bound which can be used as a basis for an effective variational method.

An attempt is made to generalise the solutions of the SU(2) Yang-Mills-Higgs system obtained by Mecklenburg and O'Brien (Phys. Rev. D. vol.18, p.1327 (1978)). A new soliton solution is obtained when the Higgs' field is zero.

A new method of generating classes of equations which possess an inverse scattering formulation in two spatial dimensions is proposed and developed in detail.

Endomorphisms of the real axis with one extremum have some universal metric properties which depend only on their analytic dependence near the extremum (bifurcation velocity, reduction parameter). It is shown how this problem is similar to the renormalisation problem, and how the bifurcation velocity may be derived from a fixed-point theory.

Present treatments of the Fresnel equations are criticised. A full treatment without the defects is presented, the amplitudes and phases of the electric fields being calculated, and curves displayed for a variety of cases.

The two potentials for which a particle moving non-relativistically in a spherical space under the action of conservative central force executes closed orbits are found. When the curvature is zero they reduce to the familiar Coulomb and isotropic oscillator potentials of Euclidean geometry. The corresponding vector (for the former) and symmetric tensor (for the latter) constants of motion are constructed. For each system in N dimensions the Poisson bracket algebra in classical mechanics, and the commutator algebra in quantum mechanics, of these constants of motion and the angular momentum components are constructed. It is proved that by an appropriate choice of independent constants of motion these Poisson bracket algebras may be transformed into Lie algebraic structures, those of the symmetry groups SO(N+1) and SU(N) respectively: the Hamiltonian of each system is expressed as a function of the Casimir operators of its symmetry group. The corresponding transformations of the quantum mechanical commutator algebras are performed only for N=2: the corresponding expressions for the Hamiltonian as functions of the Casimir operators yield the energy levels of the two systems.

A functional is given which can be used to derive lower and upper bounds for the discrete eigenvalues of self-adjoint operator. The lower bounds of the functional are deduced and the connections to other variational principles are pointed out.

The spheroidal phase shifts for high-energy electron scattering from a target consisting of two point charges are calculated by a semiclassical method. The calculations include the contribution of terms linear in h(cross)² for two different model equations, and the results obtained are compared. Some general features of the spheroidal phase shifts are pointed out. The variation of the phase shifts with the separation between the charges is also studied.

The coefficient of the R_{mu nu} R^{mu nu} -¹/₃R² part of the conformal anomaly of the stress tensor for arbitrary spin Weinberg-type fields in a curved background spacetime is calculated. The method used involves obtaining terms in the De-Witt-Schwinger expansion for the two-point function by explicit calculation of the complete two-point function in de Sitter space.

The method of time-ordered cumulants is used to investigate the behaviour of pulses in a one-dimensional medium in which the phase speed is a random function of space and time. A linear partial differential equation is obtained for the average pulse profile (E(x,t)). The dispersion relation between frequency and wavenumber is obtained and used to solve the initial-value problem for the infinite medium.

The authors compare various modifications of general relativity theory (GRT) from the viewpoint of the equivalence principle. In GRT, gravitational collapse and the classical particle problem are closely connected with Einstein's version of the strong principle of equivalence. It is argued that theories which violate that principle or start from its 'telescopic' formulation might avoid collapse. On the other hand, one should invoke Einstein's equations with fourth-order corrections in order to solve the classical particle problem.

It is argued that general relativity allows more freedom in the choice of the metric than is usually supposed. In particular, conformally equivalent classes of metrics are to be regarded as physically equivalent. The theory is thus conformally invariant-a fact which is obscured by the non-conformally covariant way in which the field equations are usually written. Furthermore, there may be situations in which this usual way of writing the field equations is liable to give misleading results in the same way that a bad choice of coordinate system can lead to non-physical results. Another representation is given which does not suffer from this disadvantage and in a companion paper this new representation is applied to the problem of black holes and obtain some rather surprising results.

For pt.I see ibid., vol.12, p.367 (1979). In general relativity there are certain singularities, like the initial singularity of the Friedmann cosmologies, which are due to the presence in the metric of a conformal factor which becomes zero. Using the version of the general relativity field equations produced in paper I, it is shown that there is such a singularity outside the horizon of a Schwarzchild black hole. This means that the horizon, and its interior, are mathematical constructs which have no physical meaning.

Einstein-Maxwell vacuum field equations of an axially symmetric stationary charged rotating source are studied. A class of asymptotically flat solutions representing the exterior field of a stationary rotating charged oblate spheroidal source is obtained. By examining eigenvectors of Einstein tensor G_ij, invariants R_ijR^ij and Petrov classification of Weyl's conformal tensor, it is proved that the metric potentials describe null electromagnetic fields.

A superspace with two commuting and two anti-commuting co-ordinates is discussed with particular emphasis on its superconformal properties. A complete expansion of the supervierbein is given and the local supersymmetry transformations of the component fields derived. Super Weyl transformations are defined and it is shown that (2+2)-dimensional superspace is superconformally flat. The spinning string is re-examined and the problems of previous approaches resolved.

A gauge-invariant and microscopically causal theory of the sourceless electromagnetic field in a topologically arbitrary globally hyperbolic asymptotically flat background metric is proposed. The topology manifests itself in an algebra of superselected quantities, one of which is the net (apparent) charge. Relative to a particular Cauchy hypersurface, the field resolves into a 'Coulombic' part generating the above algebra and a 'radiative' part expressible in terms of photon creation and destruction operators. An appendix extends 'Hodge theory' to non-compact, but asymptotically-flat three-manifolds.

The authors give a detailed general solution of the Dirichlet problem for two exterior touching spheres. From this general result they derive an explicit formula giving electrostatic capacity of two unequal exterior adhering spheres.