The primary aim of this thesis is to investigate the utility of a Clifford-Dirac algebra Cℓ1,3 in describing relativistic electromagnetism. The motivation for choosing this algebra over other potential choices lies in the success of this algebra in describing relativistic quantum mechanics. The unit imaginary in is excluded in this algebra because a purely real description is sought and the supposition that the element is not required in order to formulate a covariant electromagnetic theory is to be tested. Square roots of the basis elements of the algebra are also investigated. An exhaustive computer search algorithm is developed to search for roots. This analysis provides general conditions for the formation of square roots where the basis elements have coefficients of equal magnitude. It is shown that such roots with two terms exist only for those elements which square to -1. No square roots are found with more than six terms. A simple algorithm is developed which enables the computational manipulation of Dirac-Clifford basis elements. A modified bubble-sort is used to perform multiplication of basis elements. This new algorithm is a reliable mechanism for performing multiplication. Maxwell's equations are developed using the Clifford-Dirac algebra. The derivation of the complete set of field equations appear in a particularly compact and elegant form. A further motivation is to explore the potential for new kinds of wave functions within the algebra. The larger number of basis elements which square to -1 presents the opportunity to study wave equation using replacements for the complex imaginary. Finally, the aim of this thesis is to examine the behaviour of the electromagnetic field equations under relativistic transformations to determine whether or not the field equations and algebra form a relativistically covariant system.
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Investigation of electromagnetism in a real Dirac algebra
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