Clifford algebras and Möbius transformations

Abstract

It is well-known that Clifford algebras are geometrical algebras in the sense that they afford the spin representation of the orthogonal groups. By virtue of modern results, however, it is clear that Clifford algebras make a structure which is suited also to other purposes. Such another purpose we want to pay attention to is the representation of conformai groups. We shall present a most rewarding Clifford algebra approach to conformai geometry. Our employment of the modulo (1,1) periodicity of Clifford algebras brings about the full generalization of the classical Möbius transformations. Moreover, the treatment of conformai groups which we advocate, gives rise to a generalization of twistors (this subject shall not be pursued here). In this paper all proofs are omitted. For a detailed treatment of the subject one is referred to [5].