Clifford algebra, symmetries, and vectors

Abstract

The realization of a Clifford algebra in laboratory space is considered and it is demonstrated that the elements of the algebra cannot, as often assumed, be directly identified with vectors in this space, but, rather, that they form the parametric space of the symmetry operations of the Euclidean group as performed in the laboratory space. Details of this parametrization are established and expressions are given that determine the action of the Euclidean-group operations (screws included) on laboratory-space vectors in terms of the elements of the Clifford algebra. A discussion of Clifford vectors, bivectors, and pseudoscalars and their relation to the Gibbs vectors is provided. The correct definition of axial and polar vectors within the Clifford algebra is carefully discussed. It is shown how simple it is to generate finite point groups in 4-dimensional space by means of the Clifford algebra. © 1996 John Wiley & Sons, Inc.