Clifford algebras and their application in the Dirac equation

Abstract

The aim of this thesis will be to study the Clifford algebras that appear in the derivation of the Dirac equation and investigate alternative formulations of the Dirac equation using (complex) quaternions. To this end, we will first look at the symmetries of the Dirac equation and some of the additional insights that follow from the Dirac equation. We will also give a derivation of the Dirac equation starting from the Schrödinger equation, in which we will come across the gamma matrices. These gamma matrices are a representation of a Clifford algebra. We then give a mathematical description of Clifford algebras and explore some of the properties of Clifford algebras. We eventually classify the universal Clifford algebras over regular quadratic spaces for all possible dimensions and find that the Clifford algebra mostly used as the algebra of space-time is actually isomorphic to a matrix algebra with entries from the quaternions. In the last part, we therefore look at different (complex) quaternionic formulations of the Dirac equation and some of the operators and operations in these formalisms. We end with the conclusion that even though some of these (complex) quaternionic formulations are very elegant, it is highly unlikely that it will yield any new physical results and that there is no real reason to prefer a (complex) quaternionic formulation of the Dirac equation over the standard complex formulation.