Twistor integral representations of fundamental solutions of massless field equations
Classical twistor theory is concerned with the projective space \({\mathbb{C}\mathbb{P}}_3\) and the Grassmannian \({\text{Gr}}_2({\mathbb{C}}^4)\). The natural incidence relation between these two spaces induces a complex integral transform known as the Penrose transform. It takes cohomology classes on suitable open subsets of \({\mathbb{C}\mathbb{P}}_3\) and produces fields on \({\text{Gr}}_2({\mathbb{C}}^4)\). A link with physics is established by identifying \({\text{Gr}}_2({\mathbb{C}}^4)\) as a natural compactification of complexified Minkowski space. The fields produced by the Penrose transform satisfy well-known equations from physics. In particular, the Penrose transform of a cohomology class homogeneous of degree \(-2\) is a solution of the complex Laplace equation. If \([u^1,u^2,u^3,u^4]\) are homogeneous coordinates on \({\mathbb{C}\mathbb{P}}_3\) then the cohomology class with Čech representative \(1/u^1u^2\) is known as an elementary state and gives rise under the Penrose transform to the fundamental solution of Laplace's equation. In this article, the authors generalize this statement to higher dimensional complex Euclidean spaces. The twistor theory of these spaces is well-known but a little more complicated than the classical case. Nevertheless, the twistor spaces are homogeneous for the complex orthogonal groups and there is a Penrose transform. Now there is some difficulty is defining what is meant by an elementary state. The authors do this by summing some finite collection of Čech representatives labelled by certain graphs. The classical case is simple in that there is only one such graph. Again, the Penrose transform of an elementary state is the fundamental solution of the appropriate second order differential equation.
scientific article; zbMATH DE number 1396339
scientific article