Comparison of Clifford and Grassmann Algebras in Applications to Electromagnetics

Abstract

This paper presents some points of comparison between Grassmann and Clifford algebras in their applications to electromagnetics. The Grassmann algebra applies directly to (exterior) differential forms leading to Cartan’s calculus. Forms of various degrees correspond most naturally to electromagnetic quantities and their relations are expressed by means of the exterior differential operator d.These relations are conveniently represented either in space or space-time by flow diagrams. They reveal the existence of potentials (Poincaré’s Lemma), the independence from coordinate systems, and the role of the metric. All of these properties are briefly reviewed. A calculus for functions having their values in a Clifford algebra can also be developed based on the Dirac operator D, whose square is a generalized Laplacian. Besides its role in Dirac’s theory of the electron, this Clifford calculus can also be adapted to electromagnetics, although less directly than with Cartan’s calculus. The point of departure is the role of a metric in a Clifford algebra. This can be remedied by introducing the Hodge star operator in the Grassmann algebra, an operation that is needed to express the property of matter usually represented by means of ε and μ.