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This paper deals with the formalism of local twistors, which has developed from the twistor algebra, and extends some of the basic twistor concepts to curved space-time. Essentially, the central ideas are to define a twistor space at each point of the spacetime, and to define a covariant derivative so that an operation of local twistor transport is possible; this leads to the definition of a conformally invariant curvature twistor. In an appendix, some conformally invariant spinors are discussed.

The connected symmetry group SU(2, 2) of twistor space (T), a four‐dimensional complex manifold with metric dsT2≡dX0 dX2¯+dX2 dX0¯+dX1 dX3¯+dX3 dX1¯, and the connected symmetry group O0(2, 4) of conformal space (C), a six‐dimensional real manifold with metric dsC2≡(dX0)2−(dX1)2−(dX2)2−(dX3)2−(dX4)2+(dX5)2, are 4:1 and 2:1 homomorphic, respectively, to the restricted conformal group in compactified Minkowski space M. We obtain explicit realizations for these homomorphisms and explore the invariant geometrical relationships they imply between T, C, and M. As an application of the twistor formalism, we show that every continuous conformal transformation has a unique decomposition as the product of a Lorentz transformation, a translation, an acceleration, a dilation, and one of four special conformal transformations.

The formalism of twistors [the ‘spinors’ for the group O(2,4)] is employed to give a concise expression for the solution of the zero rest-mass field equations, for each spin (s=0, 1/2, 1, ...), in terms of an arbitrary complex analytic functionf(Zα) (homogeneous of degree −2s −2). The four complex variablesZα are the components of a twistor. In terms of twistor space (C-picture) it is analytic structure which takes the place of field equations in ordinary Minkowski space-time (M-picture). By requiring that the singularities off(Zα) form a disconnected pair of regions in the upper half of twistor space, fields of positive frequency are generated.

Twistor theory offers a new approach, starting with conformally-invariant concepts, to the synthesis of quantum theory and relativity. Twistors for flat space-time are the SU(2,2) spinors of the twofold covering group O(2,4) of the conformal group. They describe the momentum and angular momentum structre of zero-rest-mass particles. Space-time points arise as secondary concepts corresponding to linear sets in twistor space. They, rather than the null cones, should become “smeared out” on passage to a quantised gravitational theory. Twistors are represented here in two-component spinor terms. Zero-rest-mass fields are described by holomorphic functions on twistor space, on which there is a natural canonical structure leading to a natural choice of canonical quantum operators. The generalisation to curved space can be accomplished in three ways; i) local twistors, a conformally invariant calculus, ii) global twistors, and iii) asymptotic twistors which provide the framework for an S -matrix approach in asymptotically flat space-times. A Hamiltonian scattering theory of global twistors is used to calculate scattering cross-sections. This leads to twistor analogues of Feynman graphs for the treatment of massless quantum electrodynamics. The recent development of methods for dealing with massive (conformal symmetry breaking) sources and fields is briefly reviewed.

The purpose of this paper is to describe a geometrical approach to the twistor formalism. After introducing the fundamental concepts of conformal geometry, a description of the compactification M of the Minkowski space M , compatible with its conformal structure, is given. The null geodesics in M are investigated. Then in the Section III, the concept of the twistor space, i.e. a complex, four-dimensional vector space endowed with a Hermitian form of the signature (+, +, -, -), is introduced. It is shown that notions of points and null geodesics in M can be translated into the twistor formalism. Relations between the groups of automorphisms of M and of the twistor space are given. In the last section, the structure of the Minkowski space M is investigated by means of the twistor formalism. Simultaneously, it is shown that a certain two-dimensional subspace of the twistor space should be identified with the spinor space. This fact justifies the use of spinors in the Penrose approach.