Date of Award


Degree Type


Degree Name

Doctor of Philosophy


The unusually rich topological structure accommodated by three dimensional manifolds and their consequent importance to the exploration of fundamental realistic issues, has recently generated interest in 2+1-dimensional physics. The stated dimensionality offers, as a result, a challenging perspective to the fundamentally important issues of second quantization and renormalization. The present project explores the peculiar implications which this odd dimensionality has for the technique of dimensional regularization in whose context the perturbative evaluation of the renormalization group functions characterizing a renormalizable scalar field theory with self-interacting {dollar}\phi\sp6{dollar} and {dollar}\phi\sp4{dollar} couplings, is subsequently pursued in flat Euclidean space. The analysis is naturally extended to the exploration of the dynamical effects which any background 2+1-dimensional curved geometry has on the ultra-violet behaviour of a {dollar}\phi\sp6{dollar} self-interacting scalar theory manifesting conformal invariance at the classical level. It is shown that parity conservation precludes the presence of the Chern-Simons term--peculiar to that dimensionality--in the gravitational action. In its Euclidean spherical version, a de Sitter space formulation readily yields information related to the aforestated effects, by invariably resulting in Green functions depending at most linearly on the constant radius of that manifold. This effect is exclusively restricted to that dimensionality, and in its context, it is evident at least to third order in the self-coupling, that the vacuum processes do not generate any conformally non-invariant counterterms in the bare action.



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