Geometric algebra and covariant methods in physics and cosmology

Abstract

In this thesis we use a variety of mathematical tools to tackle problems in quantum theory, relativity and cosmology. Our choice of mathematical tool is governed by the desire to derive results in as physical a , way as possible. For example we consider objects that correspond directly with physical observables wherever possible. This allows the derivations of solutions to carry some physical meaning, often leading to a better physical insight into what is happening . . Many physical quantities have a geometric nature, and it is therefore natural to describe the physics in terms of the relevant geometric quantities. Geometric Algebra (GA) provides a framework for manipulating geometric quantities in a transparent and co-ordinate independent way. After introducing GA and establishing notation we apply, it to study the scattering of particles with spin . After explaining the technique we apply it to a variety of scattering problems, showing how to perform complicated spin-dependent calculations with a minimum of mathematical obscurity. Gravity can be derived as the result of gauge symmetries and some equations determining the dynamics of the various fields. We discuss how to construct a gravitational gauge theory using GA, and consider the possible consequences and extensions. The dynamics of the theory are not very constrained by the symmetries and there are therefore a wide variety of possibilities, some of which we discuss. We also exploit the construction of gravity as a gauge theory to consider analogues of the topological structures encountered in Yang-Mills gauge theory. In gravitational theory one of the gauge symmetries is a local displacement symmetry. Gauge invariant equations will be made up of covariant quantities, and it is these variables that we need to construct observables. Covariant quantities are therefore our preferred variables as they have some direct physical meaning. We study the evolution of covariant perturbations in cosmology, avoiding all ambiguities that can arise in other methods using non-covariant gauge-dependent quantities. We review the covariant formalism and give a covariant analysis of perturbations in single-field inflation. We give a new covariant analysis of massive neutrino perturbations, and show how it can be used in an efficient 'numerical implementation. We implement numerically the mode expanded covariant perturbation equations in order to compute predictions for Cosmic Microwave Background anisotropies. Our code handles closed , flat and op Em models efficiently and has been made publicly available. It has already proved a useful tool for extracting cosmological parameters from observational data. We conclude that Geometric Algebra and covariant methods have proved useful tools for studying a variety of problem; in physics and cosmology.