A CP 5 calculus for space-time fields

Abstract

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 ijX iX 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. We provide here a systematic review of the basic framework: the underlying projective geometry; the calculus of tensor fields; the characterization of spinors as twistor-valued fields ψ α( X) which satisfy a geometrical condition ( ψ α X αβ = 0 on Ω ); and the introduction of the conformally invariant Laplacian operator ∇ 2 = Ω ij ∂ 2/∂ X i ∂ X j . In addition a number of subsidiary topics are discussed which illustrate the general scheme, including: the breaking of conformal symmetry to Poincaré symmetry; a derivation of the zero rest mass equations for all helicities; and a new and manifestly conformally covariant form of the twistor contour integral formulae for massless fields.