Clifford algebra approach to twistors

Abstract

Local particle interpretation or, equivalently, an enlargement of a structure group to the Poincaré group at each point of a Riemannian space-time manifold naturally results in a complexification of the Clifford algebra for the tangent Minkowski space. Following Crumeyrolle, twistor space is identified with an appropriate one-sided ideal of this algebra. Every antiautomorphism of the latter provides a unique projection from the complexified Clifford algebra onto the affine complex Minkowski space. This projection commutes with the action of the Poincaré group. Using the above approach, three projections (the cases of symmetric, antisymmetric, and Hermitian tensors) are derived. The projection in terms of the antisymmetric, decomposable tensors is shown to give the Penrose projection.