100 Years of Relativity Space-Time Structure: Einstein and Beyond ...

The Twistor Approach to Space-Time Structures 477which gives a self-dual field (for integer spin), describing a positive-helicity(right-handed) massless particle, or toχ A ′ B ′ ...L ′ = 0which gives an anti-self-dual field, describing a negative-helicity (lefthanded)massless particle.We ask for the relation between these massless field equations and atwistor wavefunction of homogeneity degree −n − 2. The answer is largelyexpressed in the contour-integral expressions oϕ(x) = c 0∮f(ω, π)δZfor the case n = 0,χ A′ B ′ ...L ′(x) = c n∮π A ′π B ′ . . . π L ′f(ω, π)δZfor n > 0, andφ AB...L (x) = c n∮∂∂ω A∂∂ω B . . . ∂f(ω, π)δZ∂ωL for n < 0. (The constants c n are here left undetermined, their most appropriatevalues to be perhaps fixed at some later date.) In each case, ω is firstto be eliminated by means of the incidence relationω = ixπbefore the integration is performed, the quantity δZ being defined by eitherof the following definitionshomogeneous case ( ∮ with 1-dimensional real contour): δZ = π A ′dπ A′inhomogeneous case ( ∮ with 2-dimensional real contour): δZ= 1 2 dπ A ′∧dπA′ .In either case, the integration removes the π-dependence, and we are leftwith a function solely of x. Moreover, it is a direct matter to verify thatthe appropriate massless field equation is indeed satisfied in each case.In the homogeneous case we get a genuine contour integral — in thesense that the answer does not change under continuous deformations ofthe (closed) contour, within regions where f remains holomorphic — providedthat the entire integrand (including the 1-form δZ) has homogeneitydegree zero, that being the condition that its exterior derivative vanishes.This condition is ensured by the nature of the 1-form δZ and the balancing