## Geometric algebra and its application to mathematical physics

##### Authors

Doran, Christopher John Leslie

##### Date

1994##### Awarding Institution

University of Cambridge

##### Author Affiliation

Department of Applied Mathematics and Theoretical Physics

##### Qualification

Doctor of Philosophy (PhD)

##### Type

Thesis

##### Metadata

Show full item record##### Citation

Doran, C. J. L. (1994). Geometric algebra and its application to mathematical physics (Doctoral thesis). https://doi.org/10.17863/CAM.16148

##### Description

Clifford algebras have been studied for many years and their algebraic properties are well
known. In particular, all Clifford algebras have been classified as matrix algebras over one
of the three division algebras. But Clifford Algebras are far more interesting than this
classification suggests; they provide the algebraic basis for a unified language for physics
and mathematics which offers many advantages over current techniques. This language is
called geometric algebra - the name originally chosen by Clifford for his algebra - and
this thesis is an investigation into the properties and applications of Clifford's geometric
algebra. The work falls into three broad categories:
- The formal development of geometric algebra has been patchy and a number of
important subjects have not yet been treated within its framework. A principle feature
of this thesis is the development of a number of new algebraic techniques which
serve to broaden the field of applicability of geometric algebra. Of particular interest
are an extension of the geometric algebra of spacetime (the spacetime algebra)
to incorporate multiparticle quantum states, and the development of a multivector
calculus for handling differentiation with respect to a linear function.
- A central contention of this thesis is that geometric algebra provides the natural
language in which to formulate a wide range of subjects from modern mathematical
physics. To support this contention, reformulations of Grassmann calculus, Lie
algebra theory, spinor algebra and Lagrangian field theory are developed. In each
case it is argued that the geometric algebra formulation is computationally more
efficient than standard approaches, and that it provides many novel insights.
- The ultimate goal of a reformulation is to point the way to new mathematics and
physics, and three promising directions are developed. The first is a new approach
to relativistic multiparticle quantum mechanics. The second deals with classical
models for quantum spin-I/2. The third details an approach to gravity based on
gauge fields acting in a fiat spacetime. The Dirac equation forms the basis of this
gauge theory, and the resultant theory is shown to differ from general relativity in
a number of its features and predictions.