Conformal transformations and Clifford algebras

Abstract

A spinor representation for the conformal group of the real orthogonal space R p , q {R^{p,q}} is given. First, the real orthogonal space R p , q {R^{p,q}} is compactified by adjoining a (closed) isotropic cone at infinity. Then the nonlinear conformal transformations are linearized by regarding the conformal group as a factor group of a larger orthogonal group. Finally, the spin covering group of this larger orthogonal group is realized in the Clifford algebra R 1 + p , q {R_{1 + p,q}} containing the Clifford algebra R p , q {R_{p,q}} on the orthogonal space R p , q {R^{p,q}} . Explicit formulas for orthogonal transformations, translations, dilatations and special conformal transformations are given in Clifford language.