1. INTRODUCTION

It is an especial pleasure for me to have this opportunity to pay my respects to my friend and colleague Ivor Robinson. I have chosen to hold forth on the origins of twistor theory for two reasons. The first is that Ivor's 60th birthday very nearly coincides with what is (for me) the 20th birthday of the theory, so this seemed to be a reasonable point at which to examine how it stands today, in relation to its original aims and aspirations. But also Ivor himself played a direct and important part in those origins. In fact, I can think of several essentially independent influences that Ivor had, one of which was quite crucial. I hope the reader will bear with me and forgive me for presenting an account which, to a large extent, consists of personal or technical reminiscences. But I hope, also, that there will actually be some scientific value in these ramblings.

I should first make clear what I mean, here, by the "origins" of twistor theory. I am referring to the origins of my own rather specific approach to a physical theory. I appreciate that many of the ideas go back very much farther than twenty years. Most particularly, Felix Klein put forth his correspondence between the lines in complex projective 3-space and a general quadric in projective 5-space as long ago as 1870 (Klein 1870, 1926), this correspondence being based on the coordinates of Julius Plücker (1865, 1868/9) and Arthur Cayley (1860, 1869), (or of Hermann Grassmann, even earlier). Sophus Lie had noted essentially the key "twistor" geometric fact that oriented spheres in complex Euclidean 3-space (including various degenerate cases) could be represented as lines in complex projective 3-space (contact between spheres represented as meeting of lines) already in 1869 (cf. Lie & Scheffers 1896) as was pointed out to me by Helmuth Urbantke some years ago. The spheres may be thought of as the t = 0 representation of the light cones of events in Minkowski space, so the Lie correspondence in effect represents the points of (complexified compactified) Minkowski space by lines in complex projective 3-space, where meeting lines describe null-separated Minkowski points - the twistor correspondence! The local isomorphism between the "twistor group" SU(2,2) and the connected component of the group 0(2,4) was explicitly part of the Cartan's (1914) general study and classification of Lie groups. The physical relevance of 0(2,4) in relation to the conformal motions of (compactified) Minkowski space-time had been exploited by Paul Dirac (1936 b) and the objects which I call twistors (namely the spinors for 0(2,4)) had been explicitly studied by Murai (1953, 1954, 1958) and by Hepner (1962). (See also Gindikin 1983 for a discussion of these matters.)

Moreover, as Ivor Robinson pointed out to me some ten or so years ago, a certain line-integral expression for representing the general (analytic solution of the wave equation in terms of holomorphic functions of three complex variables was known to Bateman in 1904 (see Bateman 1904 and 1944, p. 96), this having arisen from a similar expression due to Whittaker (1903) for solving the three-dimensional Laplace equation, and Bateman also gave a similar line-integral expression for solving the free Maxwell equations (Bateman 1944, p. 100). By a simple transformation of variables, these become the helicities zero and one cases of the basic contour integral formula (Penrose 1968, 1969a) giving the linear field case of the so-called "Penrose transform" of twistor theory. The Radon transform (Radon 1917, Gel'fand Graev & Vilenkin 1966) and its generalizations may also, from a different angle, be regarded as providing models for (and generalizations of) this twistor expression. In addition, the classic Weierstrass (1866) construction (cf. Darboux 1914) (which was known to me!) had provided a paradigm for the explicit solution, in terms of free holomorphic data, of an important non-linear problem (Plateau's problem). This may be regarded as a direct antecedent of the later non-linear twistor constructions for (anti-) self dual gravitational (Penrose 1976; cf. also Hitchin 1979) and Yang-Mills fields (Ward 1977, Atiyah & Ward 19771 Aityah, Hitchin, Drinfeld & Manin 1978).

Much of this previously existing material was not known to me twenty years ago. But the Klein correspondence was something I had been well acquainted with since my undergraduate days. So some might argue that there was not a great deal left to be original about in the basic twistor scheme. Nevertheless I do feel that I have a good claim to some sort of originality! This - if we discount a fair number of (non-trivial) later mathematical developments - lies primarily in the essential "physical idea" that the actual space-time we inhabit might be significantly regarded as a secondary structure arising from a deeper twistor-holomorphic reality. The basic idea, it could be argued, pro vides little more than a shift in viewpoint, but it is this shift that provides crucial motivation and it also gives, in a sense, whatever physic al content the theory has had, so far. This viewpoint has guided us in certain unexpected and often fruitful directions, providing some surprising mathematical insights and descriptions of basic physical fields and concepts. It has enabled us to achieve results that had not seemed possible to achieve by more conventional procedures. (For accounts of some of these, see Hodges et al. 1980). Nevertheless, twistors do not, as yet, provide a new physical theory in the usual sense that predictions -different from those given by conventional procedures are yet forthcoming. However, in order that the above "physical idea" should have genuine physical content, it must at some stage lead to a successful physical theory in this sense - or else be consigned to the dustbin (1) of scientific history!