Twistor Geometry and Non-Linear Systems Review Lectures given at the
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970 Twistor Geometry and Non-Linear Systems Review Lectures given at the 4th Bulgarian Summer School on Mathematical Problems of Quantum Field Theory, Held at Primorsko, Bulgaria, September 1980
Edited by H. D. Ooebner and T. O. Palev
Springer-verlag Berlin Heidelberg New York 1982
Editors
Heinz-Dietrich Doebner Institut fur Theoretische Physik, Technische Universitat Clausthal 3392 Clausthal-Zellerfeld, Federal Republic of Germany Tchavdar D. Palev Institute of Nuclear Research and Nuclear Energy Bulgarian Academy of Sciences 1184 Sofia, Bulgaria
ISBN 3-540-11972-8 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-11972-8 Springer-Verlag New York Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich.
© by Springer-Verlag Berlin Heidelberg 1982 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
PREFACE
The mathematical structure and the physical application of twistor geometry together with special properties of solution varieties of non-linear PDG and their quantisation have been an active and fruitful field of research in mathematical physics durinq the last years and no doubt this situation will prevail in the next future. The twistor approach emerged directly from a description of physical systems in Minkowski space with non-linear dynamics in general. Examples are field theory, includinq non-abelian gauqe fields, and general relativity with the Einstein equations. The approach relates physical problems and complex manifold theory, alqebraic topology and sheaf theory thus providing one example where more theoretical parts of mathematics are applicable to fundamental and practical physical problems, yielding fruitful results presumably not obtainable otherwise. The non-linearity of the physical system in question reflects itself in the twistor geometry. Part of this relation has to be explored yet; infinite-dimensional Lie-algebras will be useful there and the singularity structure as well as the dynamical symmetries are of special interest. Furthermore the quanti sat ion of such systems will rely also on complex manifold techniques.
Review lectures covering authoritatively part of the above programme were given at the Fourth Bulgarian Summer School on Elementary Particles and High Energy Physics: "Mathematical Problems in Quantum Field Theory" held in Primorsko in September 1980. The lectures are collected and edited in an updated version in this volume. Twistor geometry and its application to certain non-linear physical systems were treated. Some reviews present a detailed account of the formalisms, others show its applicability and relevance to physical systems.
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