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Twistors in mathematics and physics. (English) Zbl 0702.53003

London Mathematical Society Lecture Note Series, 156. Cambridge etc.: Cambridge University Press. 384 p. £20.00/pbk; $ 34.50/pbk (1990).
This monograph is intended to provide an overview of the diversity of ideas and techniques which constitute modern twistor theory. It consists of eighteen review articles by leading authorities in their respective areas.
These include: “Twistor theory after 25 years - its physical status and progress,” by R. Penrose; “Between integral geometry and twistors”, and “Generalized conformal structures”, by S. G. Gindikin; “Riemannian twistor spaces and holonomy groups”, by F. E. Burstall; “Twistors, ambitwistors, and conformal gravity”, by C. R. LeBrun; “The Penrose transform”, by M. G. Eastwood; “Notation for the Penrose transform”, by E. G. Dunne; “The twistor transform”, by E. G. Dunne and M. G. Eastwood; “Invariant operators”, by B. J. Baston and M. G. Eastwood; “Penrose’s quasi-local mass”, by K. P. Tod; “The Sparling 3- form, Ashtekar variables and quasi-local mass”, by L. J. Mason and J. Frauendiener, “Twistors and strings”, by W. T. Shaw and L. P. Hughston; “Integrable systems in twistor theory”, by R. S. Ward; “Twistor characterizations of stationary axisymmetric solutions of Einstein’s equations”, by J. Fletcher and N. M. J. Woodhouse; “A two-surface encoding of radiative space-times”, by C. N. Kozameh, C. J. Cutler, and E. T. Newman; “Twistors, massless fields and the Penrose transform”, by T. N. Bailey and M. A. Singer; “Twistor diagrams and Feynman diagrams”, by A. P. Hodges; and “Cohomology and twistor diagrams”, by S. A. Hoggett.
The breadth and depth of material covered makes it impractical to adequately do justice to these articles in the brief space allotted to a review. However, the initial article by Penrose deserves special notice. In it he surveys not only the goals and hopes of the ‘twistor programme’, but also where it may go in future developments. His topics include: the infinities of quantum field theory, symmetry and asymmetry in particle interactions, coupling constants and particle masses, quantum gravity, and quantum measurement. The monograph is a most welcome addition to the growing twistor literature, and must reading for anyone having any interest in this area.
Reviewer: J.D.Zund

MSC:

53-06 Proceedings, conferences, collections, etc. pertaining to differential geometry
53C80 Applications of global differential geometry to the sciences
81-06 Proceedings, conferences, collections, etc. pertaining to quantum theory
32-06 Proceedings, conferences, collections, etc. pertaining to several complex variables and analytic spaces
32L25 Twistor theory, double fibrations (complex-analytic aspects)
81R25 Spinor and twistor methods applied to problems in quantum theory
83C60 Spinor and twistor methods in general relativity and gravitational theory; Newman-Penrose formalism
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