## Geometric algebra and covariant methods in physics and cosmology.

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##### Authors

Lewis, Antony Martin

##### Date

2001-05-08##### Awarding Institution

University of Cambridge

##### Author Affiliation

Department of Physics

##### Qualification

Doctor of Philosophy (PhD)

##### Type

Thesis

##### Metadata

Show full item record##### Citation

Lewis, A. M. (2001). Geometric algebra and covariant methods in physics and cosmology. (Doctoral thesis). https://doi.org/10.17863/CAM.16632

##### Description

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.