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Affine vector fields in 4-dimensional Lorentz manifolds have recently been investigated in some detail by Hall and da Costa. The picture is completed in this thesis by studying the zeros of affine vector fields. Hall and da Costa show that the problem of finding affine vector fields in non-degenerately reducible 4-dimensional Lorentz manifolds can be reduced, with one exceptional case, to the problem of finding homothetic vector fields in lower dimensional manifolds. This means that the study of affine vector fields with zeros in 4-dimensional Lorentz manifolds is aided by investigating proper homothetic and Killing vector fields with zeros in 2- or 3-dimensional manifolds. To this end proper homothetic vector fields with zeros are investigated in 2- and 3-dimensional Manifolds using techniques similar to those used by Hall. It is shown that in the 2-dimensional case the zero is necessarily isolated, whereas in the 3-dimensional case the zero set may either be isolated or 1-dimensional. In the latter case the manifold is shown to be a 3-dimensional plane wave space-time, and all of the affine and conformal vector fields that it admits are found. These results are then used to determine the nature of the zero sets of affine vector fields in 4-dimensional Lorentz manifolds. The algebraic structure of the Riemann, Ricci and Weyl tensors at such zeros is also described. This work is extended by studying affine vector fields, and their zero sets, in 3-dimensional Lorentz manifolds. An investigation of the zero sets of affine vector fields in 3- and 4-dimensional positive-definite manifolds is included for comparison.

The presence of a moderately strong magnetic field, uniform and static, in the irradiated medium modifies the spatial distribution of events. The imposition of a magnetic field produces elecron helices, characterised by their radii and pitches. The differential and integral distribution of track lengths, corresponding to electrons slowing down in water in a magnetic field, have been computed as function of radii and pitches for 200 kVp X-rays and for ⁶⁰ Co gamma-rays. Theoretical work has shown that the probability of energy deposition in a smaller volume of the absorbing medium has been significantly increased as a result of the presence of a magnetic field during photon irradiation. The distributions of track lengths as function of electron radii and pitches have been studied in strong magnetic fields (1 - 20 Tesla). The trajectories of an electron moving in water for different emission angles (up to π/2) and for magnetic fields of 5 and 10 Tesla, have been computed. The data for stopping powers used in this study, cover electron energies of 30 eV to 1200 keV (initial energy). In the presence of a magnetic field, each electron spiral has enclosed a conical volume. As the magnetic field increases, so the volume enclosed by the spiral decrease resulting in a substantial increase in the number of hits (events) compared with events in the same volume in the absence of a magnetic field. The experimental work started with the study of the characteristics of a spherical walled proportional counter. The frequency density, y.f(y), energy probability density, y^2f(y), distributions and their averages overline Y_F and overline Y_D respectively, have been computed on the basis of the pulse height distribution of low-LET radiation. Gamma rays from ¹³⁷Cs and ⁶⁰Co have been used with and without a magnetic field. Transverse magnetic fields of 0.0304, 0.13, 0.24 and 0.34 Tesla as well as a longitudinal magnetic field of 0.0304 Tesla have been used in microdosimetric measurements. An average sphere diameter of 2 μm has been simulated for the purpose. In the presence of the transverse magnetic fields, an increase of up to ~ 45% and ~ 78% has been obtained in the values of overline Y_F and overline Y_D respectively for ^137Cs gamma rays. For ^60Co gamma rays the values of both overline Y_F and overline Y_D increase by up to about 97%. For the longitudinal magnetic field when compared with the corresponding transversal magnetic field, a substantial increase in the value of overline Y_F has been found for ^137Cs gamma rays and a less significant increase for ^60Co gamma rays. Also, a significant increase in overline Y_D has been obtained for both indirectly ionizing radiations.

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Graduate / 0607

Graduate / 0607

This work is concerned with the large class of nonlinear scalar field theories known as σ-models, and in particular with their classical solutions. It is shown how the σ-models can admit solitons in (2+1) dimensions; and how, in many cases, these solitons can be classified topologically. For the Kähler c-models, the instanton (i.e. static soliton) solutions are derived explicitly via the Bogomolny equations. The main part of the thesis looks at the behaviour of solitons under the influence of small perturbations, and at their (classical) interactions. Attention is confined to the O(3) a-model and its close relatives. A recurring theme is the ability of solitons to change in size as they evolve, a feature which is attributed to the conformal invariance of the theory. There seem to be three possible approaches. In some special cases, the theory is integrable, in the sense that one can write down explicit time-dependent solutions. More often, one must resort to a numerical simulation, or else some sort of approximation. For theories that possess a topological lower bound on the energy, there is a useful approximation in which the kinetic energy is assumed to remain small. All three of these approaches are used at various stages of the thesis. Chapter IIIdeals with the properties of wave-like solitons in an integrable theory, and reveals some hitherto unseen behaviour. Chapters IV and V develop a numerical simulation based on topological arguments, which is then used in a study of soliton stability in the pure O(3) model. The conclusion is that the solitons are unstable to small perturbations, in the sense that their size is subject to large changes, even though their energy remains roughly constant. Chapter VI uses the slow-motion approximation to investigate soliton interactions in the O(3) model, and uncovers a plethora of possibilities. Finally, some suggestions are made regarding possible directions for future research. In particular, attention is focussed on ways of modifying the O(3) model in an attemptto stabilize its solitons against changes in size

This thesis is concerned with the possible sums of perturbation series in mass- less, renormalizable field theories. It shows that, given a free choice of scheme, the limit of the sequence of approximants is arbitrary. Restricting the choice to finite schemes, in particular "zero schemes", yields a perturbatively unique limit to the sequence of approximants. An operational method for calculating perturbative expansions in the class of zero schemes is discussed. A comparison of various optimization schemes is given for a few phenomenological examples in QCD and QED.

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