Twistor quantisation and curved spacetime
 Roger Penrose
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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 restmass 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 (Cpicture) it is analytic structure which takes the place of field equations in ordinary Minkowski spacetime (Mpicture). 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 spacetimes. The effect of conformai curvature in theMpicture is studied by consideration of plane (fronted) gravitational ‘sandwich’ waves. TheCpicture still exists, but its complex structure ‘shifts’ as it is ‘viewed’ from different regions of the spacetime. 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 restmass 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 spacetime 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.
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 Title
 Twistor quantisation and curved spacetime
 Journal

International Journal of Theoretical Physics
Volume 1, Issue 1 , pp 6199
 Cover Date
 19680501
 DOI
 10.1007/BF00668831
 Print ISSN
 00207748
 Online ISSN
 15729575
 Publisher
 Kluwer Academic PublishersPlenum Publishers
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 Authors

 Roger Penrose ^{(1)}
 Author Affiliations

 1. Department of Mathematics, Birkbeck College, University of London, London, England