Non-linear Partial Differential Operators and Quantization Procedures
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Non-linear Partial Differential Operators and Quantization Procedures Proceedings of a workshop held at Clausthal Federal Republic of Germany, 1981
Edited by S.1. Andersson and H.-D. Doebner
S pri nger-Verlag Berlin Heidelberg New York Tokyo 1983
Editors
Stig I. Andersson Heinz-Dietrich Doebner Institut fUr Theoretische Physik, Technische Universitat Clausthal 3392 Clausthal-Zellerfeld, Federal Republic of Germany
AMS SUbject Classifications (1980): 53-06, 53G05, 55 R05, 58-06, 58G40, 81 EXX, 81 G30, 81 G35, 83-06 ISBN 3-540-12710-0 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-12710-0 Springer-Verlag New York Heidelberg Berlin Tokyo
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 1983 Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
PREFACE
Non-linear physical systems and their mathematical structure form one of
the most active fields in present mathematics and mathematical
physics. This volume covers parts of that topic. It reports on differential geometrical and topological properties of those non-linear systems, which can be viewed physically as models for quantized nonrelativistic particles constrained, i.e. localized, on a
(smooth) ma-
nifold or as classical or quantized fields with non-linear field equations. The contributions of this volume show how to deal with these different types of non-linearities. There are various physically motivated approaches to both of them. For systems constrained on a manifold generically geometric methods are used with promising mathematical and physical results. Now
that the feeling has dissipated, that global
solutions of non-linear field equations are"extra - terrestrial beasts" (see the contribution of I.E. SEGAL), also here a more global and geometrical approach is applied with extreme success, we refer e.g. to the application of twistor geometry or to the analysis of solution
mani-
folds of non-linear equations. The structures of both types of non-linearities are deeply related. A summer workshop in connection with the above programme was held in July 1981 at the Technical University in Clausthal, Institute for Theoretical Physics and an international conference on mathematical physics was organized parallel to the workshop. The lectures at the workshop and some of the contributions to the conference are collected and edited in an updated version in this volume.
Quantization Procedures Quantizations of non-relativistic (mechanical) systems constrained on a smooth manifold are discussed. The method of geometrical quantization is justified on more physical grounds and presented in a new cont
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