The geometry of twistors was first introduced in Penrose [28]. Since that time it has played a significant role in solutions of various problems in mathemetical physics of both a linear and nonlinear nature (cf. Penrose [29], Penrose [35], Ward [48], and Atiyah-Hitchin-Drinfeld-Manin [2], see Wells [52] for a recent survey of the topic with a more extensive bibliography). The major role it has played has been in setting up a general correspondence which translates certain important physical field equations in space-time into holomorphic structures on a related complex manifold known as twίstor space. The purpose of this paper is to give a rigorous discussion of this correspondence for the case of the linear massless free-field equations, including Maxwell's source-free equations, the wave equation, the Dirac-Weyl neutrino equations, and the linearized (weakfield limit of) Einstein's vacuum equations. These equations may also be analyzed from this point of view on a background provided by the nonlinear Yang-Mills or Einstein equations in the (anti-) self-dual case. The correspondence is effected by an integral-geometric transform, which transforms complex-analytic data on twistor space to solutions of the linear massless field equations, and is, in fact, a generalization of the classical Radon transform, which is discussed further below. The motivation for finding such a correspondence in general is that it forms an essential part of the \"twistor programme\" according to which one attempts to eliminate the equations of physics by deriving them from the rigidity of complex geometry and holomorphic functions (see, e.g. Penrose [38]). It is, in fact, rather remarkable the extent to which it is possible to achieve this. Success apparently comes about because in twistor-space descriptions the information is \"stored\" nonlocally. The (local) value of a field at a point in space-time depends upon the way that the holomorphic structure in the twistor-space is fitted together in the large. So sheaf cohomology and function theory of several complex variables turn out to be the appropriate tools in the twistor framework. It is hoped that, as part of the general twistor programme, some deeper insights may eventually be gained as to the interrelation between quantum mechanics or quantum field

https://projecteuclid.org/download/pdf_1/euclid.cmp/1103908690

Cohomology and massless fields

On the pseudo-conformal geometry of hypersurfaces of the space of n complex variables

Group Theoretical Concepts and Methods in Elementary Particle Physics Edited by F. Gursey. (Lectures of the Istanbul Summer School of Theoretical Physics, July 16–August 4, 1962.) (Quantum Physics and Its Applications, Vol. 1.) Pp. vii + 425. (New York and London: Gordon and Breach, Science Publishers, 1964.) 19.50 dollars.

/pdf/elementary-particle-physics-1m2sae58w9.pdf

Elementary Particle Physics

Construction of instanton and monopole solutions and reciprocity

On etudie la mecanique lagrangienne des particules sans masse a spin, basee sur l'interpretation que le spin correspond au deplacement dans des directions complexes

Lagrangian Mechanics of Massless Particles With Spin

This lecture represents joint work with Claude LeBrun of State University of New York at Stony Brook and Yat-Sun Poon of Rice University, [LPW]. There is some overlap with similar work by Jeffrey Rabin and P.T. Topiwala in two recent reprints [RT1], [RT2], and this work is reported in [R].

/pdf/projective-embeddings-of-complex-supermanifolds-4f9zrfbvc4.pdf

Projective Embeddings of Complex Supermanifolds

In this paper it is shown how the Penrose transform maps tangential hyperfunction Dolbeault groups with coefficients in a power of the hyperplane section bundle on the hyperquadric of null twistors in projective twistor space isomorphically to all hyperfunction solutions of the massless field equations of nonnegative helicity on compactified Minkowski space. This is an extension of the Penrose transform which generated real-analytic solutions of the same field equations on the same space (cf. Eastwood, M., Penrose, R., Wells, R.O., [10]). In additions, one obtains the result that each hyperfunction solution of the massless field equations of nonnegative helicity is the sum of massless fields of positive and negative frequency, a generalization of the usual Fourier decomposition for solutions with appropriate growth conditions.

/pdf/hyperfunction-solutions-of-the-zero-rest-mass-field-58o9z3gvae.pdf

R. O. Wells

Papers

Cohomology and massless fields

Projective Embeddings of Complex Supermanifolds

Hyperfunction solutions of the zero-rest-mass field equations