A CP5 Calculus for Space-Time Fields

Compactified Minkowski space can be embedded in projective five-space CP 5 (homogeneous coordinates X i , i = 0, …, 5) as a four dimensional quadric hypersurface given by Ω ij X i X j = 0. Projective twistor space (homogeneous coordinates Z α , α = 0, …, 3) arises via the Klein representation as the space of two-planes lying on this quadric. These two facts of projective geometry form the basis for the construction of a global space-time calculus which makes use of the coordinates X i ↔X αβ (=-X βα ) to represent spinor and tensor fields in a manifestly conformally covariant form. This calculus can be regarded as a synthesis of work on conformal geometry by Veblen, Dirac, and others, with the theory of twistors developed by Penrose.