Twistor quantisation and curved space-time

Abstract

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(Z ^α) (homogeneous of degree −2s −2). The four complex variablesZ ^α 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(Z ^α) form a disconnected pair of regions in the upper half of twistor space, fields of positive frequency are generated.

The twistor formalism is adapted so as to be applicable in curved space-times. The effect of conformai curvature in theM-picture is studied by consideration of plane (-fronted) gravitational ‘sandwich’ waves. TheC-picture still exists, but its complex structure ‘shifts’ as it is ‘viewed’ from different regions of the space-time. A weaker symplectic structure remains. The shifting of complex structure is naturally described in terms of Hamiltonian equations and Poisson brackets, in the twistor variablesZ ^α,$\bar Z_\alpha$. This suggests the correspondence$\bar Z_\alpha = \partial /\partial Z^\alpha$ as a basis for quantization. The correspondence is then shown to be, in fact, valid for the Hubert space of functionsf(Z ^α), which give the above twistor description of zero rest-mass fields. For this purpose, the Hubert space scalar product is described in (conformally invariant) twistor terms. The twistor expressions for the charge and the mass, momentum and angular momentum (both in ‘inertial’ and ‘active’ versions, in linearised theory) are also given.

It is suggested that twistors may supply a link between quantum theory and space-time curvature. On this view, curvature arises whenever a ‘shift’ occurs in the interpretation of the twistor variablesZ ^α,$\bar Z_\alpha$ as the twistor ‘position’ and ‘momentum’ operators, respectively.