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.
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