A cohomological description of massive fields

Abstract

In this paper we present a generalization of the theory of the P -transform that encompasses the n -twistor description of massive fields. Attention is devoted to the two-twistor description, for which it is shown that the cohomology group H 2 ( P + 3 x P + 3 , O m, s ( — ξ — 2, — ŋ — 2)) is naturally isomorphic to the space of positive-frequency free fields of mass m and spin s , provided s — 1/2 | ξ — ŋ | is a non-negative integer, and vanishes otherwise. The sheaf O m, s ( — ξ — 2, — ŋ — 2) is a subsheaf of the standard sheaf of twisted holomorphic functions O ( — ξ — 2, — ŋ — 2) on P + 3 x P + 3 and satisfies a pair of differential equations determining the mass and the spin. In establishing these results extensive use is made of a certain class of two-point fields on space-time, required to be of positive frequency and of zero rest mass in each variable separately, and also subject to a condition of definite total mass and total spin. Such fields are of considerable interest in their own right, for example in connection with the theory of twistor diagrams, and in this paper we formulate a number of their basic properties.