| A Clifford Algebra is an algebra associated with a finite dimensional vector space and a symmetric form on that space. It contains a multiplicative subgroup, the group of spinors, which is related to the group of orthogonal transformations of the vector space. This group may act on the algebra via multiplication on the left or right, or by the adjoint action. In the first part of this paper we consider the problem of classifying the orbits of these actions in the algebras C(3,1) and C(3,2). For a certain subclass of orbit this problem is completely solved and the isotropy groups for elements in these orbits are determined. After writing the Dirac and Maxwell equations in terms of Clifford Algebras we show how a classification of the solutions to these equations is related to the orbit and isotropy group calculations. Finally, we show how Clifford algebras may be used to define spinor and r-vector fields on manifolds, gradients of such fields, and other more familiar concepts from differential geometry. The end result is that the calculations for C(3,1) and C(3,2) may be applied to fields on space-time and on the five dimensional space of the gauge theory of electromagnetism, respectively. In the latter case the slightly different version of gauge theory presented here gives rise to wave functions corresponding to electrons and neutrinos. But there are also free particle wave functions resulting from the earlier classification which do not have such a ready interpretation and whose corresponding particles would have some unusual properties. This gauge theory also allows us to relate Einstein's equations for free space to Maxwell's equations in a natural manner. |
