Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann 905 Differential Geometric Methods in Mathematical Physics Clausthal 1980 Proceedings of an International Conference Held at the Technical University of Clausthal, FRG, July 23-25,1980 Edited by H.-D. Doebner, S.1. Andersson, and H.R. Petry Springer-Verlag Berlin Heidelberg New York 1982
Editors H e i n z - ~ Ooebner i e t r i c h Stig I. Andersson Institut fur Theoretische Physik, Technische Universitat Clausthal 0-3392 Clausthal-Zellerfeld, FRG Herbert Rainer Petry Institut fur Theoretische Kernphysik der Universitat Bonn NuBallee 14-16,0-5300 Bonn, FRG AMS Subject Classifications (1980): 53-06, 53G05, 55R05, 58-06, 58G40, 81 EXX, 81 G30, 81 G35, 83-06 ISBN 3-540-11197-2 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-11197-2 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 Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
PREFACE The 1980 conference on "Differential Geometric Methods in Mathematical Physics" at the Technical University of Clausthal, FRG, was part of the by now fairly long series of conferences on similar themes. Initiated by K. Bleuler in 1973 (Bonn) and continued in Bonn (1975, 1977), Aix-en-Provence (1974, 1979), Warsaw (1976), Clausthal (1978) and Salamanca (1979), these conferences have gathered a large number of prominent researchers in this special branch of mathematics/mathematical physics. No doubt, these conferences have become something of an institution. As a tribute to one of the initiators of this series of conferences, one session was dedicated to K. Bleuler on the occasion of his retirement. Additionally, the 1980 conference also pursued a more local, Clausthal tradition of summer schools and meetings on special problems in mathematical physics. The topics covered in this year's conference and in the attached workshop which are included in this volume could, roughly speaking, be described by the following keywords: symplectic category, differential operators on manifolds and vector bundles, mathematics of (non-abelian) gauge fields, geometric quantization and asymptotic expansions, all of which are, of course, central issues in the contemporary differential geometric-biased approach to a variety of mathematical questions in classical and quantum physics. Notable achievements were, more specifically, reports on the a s y m ~ t for o t spherical i c s functions, bifurcation theory, mathematical structure of gauge theories, spacetime geometry and representation theory. The editors rearet that due to a general editorial requirement of homogeneity in a lecture notes volume, which applies also for these proceedings, it was not possible to include contributions (invited talks as well as contributed papers) with a very strong bias towards physics or having definitely the form of a pure review paper or of a research announcement. In some other
IV cases manucripts were not received in time. Acknowledgments We wish to express our gratitude to the following persons and organizations for financial support and for other assistance rendering the publication of this proceedings volume possible: - Der NiedersKchische Minster fur Wissenschaft und Kunst, - Alexander von Humboldt-Stiftung, Bonn. We thank especially, for generous grants - The Office for Foreign Studies and Activities at the Techn.Univ. of Clausthal (Prof.Dr. H.Quade and Dr. R.Pestel) and - The Volkswagen Foundation. We also want to thank Springer Verlag, Heidelberg, for their kind assistance in matters of publication. Last but not least, we whish to thank Mrs. J. Gardiner, Institut fur Theoretische Physik der TU Clausthal, for an exellent and speedy complete preparation of this volume, as well as the other members of the institute whose help made the organization smooth and efficient. Clausthal, January 1982 The Editors.
TABLE OF CONTENTS Preface Table of Contents I. Session in Honour of Konrad Bleuler. I.E. Segal; S. Deser; B. Kostant; J.E. Marsden; In<troduction.. Massless Limits and Dual Field Theories Poisson Commutativity and the Generalized Periodic Toda Lattice Spaces of Solutions of Relativistic Field Theories with Constraints.... 2 3 12 29 II. Symplectic Geometry A. Weinstein; The Symplectic "Category"...... V. Guillemin and S.st'ernberg; Homents and Reductions T. Ungar; A.O. Barut; Elementary Systems for Lie Algebra Bundle Actions 66 What Kind of a Dynamical System is the Radiating Electron?...... 90 45 52 III. Differential Operators on Manifolds. J.J. Duistermaat;Asymptotics of Elementary Spherical Functions.. 100 S.M. Paneitz; Hermitian Structures on Solution Varieties of Non-Linear Relativistic Wave Equations... 108 S.I. Andersson; Vector Bundle Connections and Liftings of Partial Differential Operators... 119 P.L. Garcia; Phase Space of the Coupled Vectorial Klein-Gordon-Maxwell Equations.... 133 IV. Space-Time Geometry and General Relativity I.E. Segal; Particle Theory and Global Geometry... 151
VI H.P. Jakobsen; Group Theoretical Aspects of the Chronometric Theory........ W.H. Greub and H.R. Petry; Spinor Structures W.H. Greub; The Complex Cayley Algebra and the Lorentz Group 165 170 186 V. Quant±zation Methods P.A. Horvathy; S.T. Ali; J. Tarski; Prequantisation from Path Integral Viewpoint A Geometrical Property of POV-measures and Systems of Covariance.... Path Integrals over Manifolds 197 207 229 VI. Quantum Field Theory Y. Ne'eman; Gauge-Theory Ghosts and Ghosts-Gauge Theories. P. Houston and L. O'Raifeartaigh; On Monopole Systems with B.S. Kay; P.J.M. Bongaarts; Weak Axial Symmetry Quantum Fields in Curved Space-Times and Scattering Theory The Quantized Haxwell Field and its Gauges; a Generalization of Wightman Theory...... 241 260 272 296