A CP5 calculus for space-time fields

Compactified Minkowski space can be embedded in projective five-space CP⁵ (homogeneous coordinates Xⁱ, i = 0, …, 5) as a four dimensional quadric hypersurface given by $\text{Ω}{}_{\mathrm{ij}}\text{X}{}^{\mathrm{i}}\text{X}{}^{\mathrm{j}}\text{= 0.}$ 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 Xⁱ?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 ($\text{ψ}{}^{\mathrm{\alpha }}\text{X}{}_{\mathrm{\alpha \beta }}\text{= 0 on Ω}$ ); and the introduction of the conformally invariant Laplacian operator $\text{?}{}^2\text{= Ω}{}^{\mathrm{ij}}\text{?}{}^2\text{/?}\text{X}{}^{\mathrm{i}}\text{?}\text{X}{}^{\mathrm{j}}$. 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.