##### Abstract

This thesis deals with the interaction of quantum mechanical models and cosmologies
based on brane universes, an area of active theoretical speculation over the last five years.

For convenience, the material has been split into two parts. Part 1 deals with a selection
of background topics which are necessary and relevant to the original research. This
research is presented in Part 2. In addition, some auxiliary topics, both more elementary
and more advanced, are described in the appendices. The selection of background topics has
been influenced by the various techniques, physical theories and mathematical technologies
which play a major role in the work presented in Part 2. Although the exposition is ad
hoc, an attempt has been made to systematically develop portions where the technique (or
use of it) may be unfamiliar.

A fairly complete treatment of the necessary mathematical scaffolding is supplied. Although important, this material is familiar or strongly mathematical, and is deferred to
the appendices. This includes an elementary survey of functional analysis in Appendix A,
sufficient to support a discussion of the path integral. The path integral formalism is used
extensively throughout this thesis, and, where available, constitutes our preferred representation of quantum mechanics. The discussion is limited to the relevant portions of the
theory: functions in Banacli spaces, and the Sturm-Liouville basis (technology which appears many times in Part 2); direct evaluation of Gaussian functional integrals, ubiquitous
in field theory calculations; and ((-function regularization of the operator determinants to
which such Gaussian integrals give rise, which has a direct application in Chapter 9. In
Appendix B we describe the necessary framework of differential geometry which supports
general relativity, and low-energy discussions of string theory. All calculations in metric
gravity are based on differential geometry, together with a good proportion of the technology which buttresses quantum field theory on curved space time, string theory, and some
more advanced representations of quantum mechanics (see below). All of this is used extensively throughout both parts of the thesis. We include some more advanced topological
technology which supports the discussion of string compactification. General results from
compactification theory, when appropriately interpreted in the brane context, contribute
important stability results for zero-modes of the Kaluza—Klein fields, and provide a natural
home for the spectral KK technology used (in one form or another) throughout Part 2,
but most especially in Chapter 7 and Chapter 8. Einstein gravity and Yang-Mills theory
are set in context as examples of connexions on fibre bundles.