Clifford analysis and the Penrose transform

Abstract

The Penrose transform has its origin in a contour integral formula established by R. Penrose; a holomorphic function with suitable singularities on the twistor space was used in the formula to produce a solution of the massless field equations (i.e. the Laplace and Dirac equations and similar equations for higher spins) on Minkowski space (see [61]). This procedure quickly developed into a full mathematical theory, where the Penrose transform is presented as a 1-1 map between holomorphic solutions of the massless field equation on a domain in the complexified Minkowski space and certain cohomology groups on the corresponding region in the twistor space (see [27]). A systematic description of the Penrose transform in this setting can be found in the book by Ward and Wells ([88]).