An Introduction to Pseudotwistors: Spinor Solutions vs. Harmonic Forms and Cohomology Groups

Abstract

Penrose observed in 1976 [27] that the points of the Minkowski space-time can be represented by two-dimensional subspaces of a complex four-dimensional vector space on which an hermitian form of signature (+ +−−) is defined. He called this flat twistor space, and the method of investigating deformation of complex structures, yielded from there the twistor program. This initiated a series of papers and monographs by various authors. In the present paper, we deal with dynamical systems generated by the hermitian Hurwitz pairs of the signature (σ,s), σ + s = 5 + 4μ, | σ + 1 − s| = 2 + 4m;μ,m = 0,1,… In particular for the signature (3,2) and its dual (1,4), the role of entropy was indicated as well as the relationship between Hurwitz and Penrose twistors, Hurwitz twistors being objects introduced by us. The signatures (1,8) and (7,6) give rise for introducing pseudotwistors and bitwistors, respectively. For pseudotwistors, we can prove a counterpart of the original fundamental Penrose theorem in the local version (on real analytic solutions of the spinor equations vs. harmonic forms, related to the original relativistic wave equations) and in the semi-global version (on holomor-phic solutions of those equations vs. Dolbeault cohomology groups). This had to be preceded [24] by basic constructions, a study of the related pseudotwistors and spinor equations as well as complex structures on spinors. In particular, we proved a theorem (which we call the atomization theorem) saying that there exist complex structures on isometric embeddings for the hermitian Hurwitz pairs concerned so that the embeddings are real parts of holomorphic mappings. The atomization theorem enables us to introduce an analysis with respect to an embedding l: (2(2,4) → M, M being a C^∞-manifold, which we call the quaternic analysis, and develop it from the point of view of quaternal spinors and quaternal harmonic forms, and finally to prove rigorously the above mentioned theorems on spinor solutions vs. harmonic forms and the cohomology group. Finally, a physical motivation is given for the double Cartan-like triality of the pairs concerned as well as an introduction to five-dimensional stochastical electrodynamics.