Twistor Space, Minkowski Space and the Conformal Group

The theory of twistors, developed by Penrose’,2,3) gives an alternative geometry of Minkowski space. It is therefore remarkable that the symmetry group of compactified Minkowski space, the conformal group C, is different from the symmetry group of twistor space, which is SU(2,2). The relation between these two groups is that SU(2,2) is 4: 1 homomorphic to C,, the identity-connected component of C. The aim of this paper is to present an alternative approach to the symmetry transformations of twistor space which answers the following two questions: i) Why is the homomorphism between the two symmetry groups a 4: 1 homomorphism? ii) What about the components of C which are not connected with the identity? We will take the view that the symmetry group of the twistor formalism is the group G of transformations of projective twistor space which preserve the orthogonality of projective twistors. It will be shown that G is isomorphic to C. Also it will be shown that G is isomorphic to a group G of rays of semilinear transformations of twistor space; this will reveal the relationship between C and SU(2,2). The transformation of twistor space under the conformal group is given by an explicit realization of the isomorphism between C and G. This explicit realisation will be established; in particular the transformation of twistor space under space inversion and time inversion is determined.