C J. L. Doran.
Geometric Algebra and its Application to Mathematical Physics
Ph.D. thesis, University of Cambridge (1994).
Abstract: 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-1/2. The third details an approach to gravity based
on gauge fields acting in a flat 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.