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

318 Pages·1982·7.38 MB·English

by  I. E. Segal (auth.)

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
Heinz-~ietrich Ooebner
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
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© by Springer-Verlag Berlin Heidelberg 1982
Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr.
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PREFACE
The 1980 conference on "Differential Geometric Methods in Mathema
tical 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 gathe
red 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 retire
ment. 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 work
shop 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 differen
tial geometric-biased approach to a variety of mathematical questions
in classical and quantum physics. Notable achievements were, more
specifically, reports on the asym~totics for spherical functions,
bifurcation theory, mathematical structure of gauge theories, space
time 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 organi
zations 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; In<troduction. . 2
S. Deser; Massless Limits and Dual Field Theories 3
B. Kostant; Poisson Commutativity and the Generalized
Periodic Toda Lattice 12
J.E. Marsden; Spaces of Solutions of Relativistic Field
Theories with Constraints • . . . . 29
II. Symplectic Geometry
A. Weinstein; The Symplectic "Category" . . . . . . 45
V. Guillemin and S.st'ernberg; Homents and Reductions 52
T. Ungar; Elementary Systems for Lie Algebra Bundle Actions 66
A.O. Barut; What Kind of a Dynamical System is the
Radiating Electron? . . . . . . 90
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. . . . . . . . 165
W.H. Greub and H.R. Petry; Spinor Structures 170
W.H. Greub; The Complex Cayley Algebra and the
Lorentz Group 186
V. Quant±zation Methods
P.A. Horvathy; Prequantisation from Path Integral Viewpoint 197
S.T. Ali; A Geometrical Property of POV-measures and
Systems of Covariance . . . . 207
J. Tarski; Path Integrals over Manifolds 229
VI. Quantum Field Theory
Y. Ne'eman; Gauge-Theory Ghosts and Ghosts-Gauge
Theories· . 241
P. Houston and L. O'Raifeartaigh; On Monopole Systems with
Weak Axial Symmetry 260
B.S. Kay; Quantum Fields in Curved Space-Times
and Scattering Theory 272
P.J.M. Bongaarts; The Quantized Haxwell Field and its
Gauges; a Generalization of Wightman
Theory· .........•.... 296
I. SESSION IN HONOUR OF KONRAD BLEULER
SEGAL, I.E. INTRODUCTION ........................................... 2
DESER, S. MASSLESS LIMITS AND DUAL FIELD THEORIES ... .... ..... .... 3
KOSTANT, B. POISSON COMMUTATIVITY AND THE GENERALIZED PERIODIC
TODA LATTICE ........................................... 12
MARSDEN, J.E. SPACES OF SOLUTIONS OF RELATIVISTIC FIELD THEORIES
WITH CONSTRAI NTS ....................................... 29
2
INTRODUCTION
This session is dedicated to Professor Konrad Bleuler, who is technically scheduled
for retirement this year, but who we all know will continue as always to be in the
forefront of the seekers for a true understanding of the physical world. As a man of
broad scientific experience and deep vision, he knew that fundamental progress required
coordination of a variety of scientific disciplines, and could see which strands needed
pulling together. One of the means he chose towards this end was the sponsorship of
the present series of conferences on Differential Geometry and Mathematical Physics.
His scientific culture, communal dedication, and not least, personal interest and
warmth have contributed enormously to the effectiveness of these conferences.
Today, theoretical physicists speak of bundles, group actions, operator algebras, and
topology almost as easily as they did a decade or two ago of diagrams, form factors,
Green's functions, and the like. From gauge theories of elementary particles to
general relativity and cosmology, these notions are vital to the discussions of the
theoretical community dedicated to their explication. At the same time, a new genera
tion of mathematicians has found inspiration and motivation in central problems and
ideas of theoretical physics. Konrad Bleuler has been one of the primary catalyzers
of the interaction between disciplines that these changes represent. As a fundamental
innovator and broad authority in his own right, and as one who has worked so effec
tively to keep the torch burning in other times and places, we salute him.
But we do so as he would want, by hearing now about new insights into challenging
and fundamental problems.
I.E. Segal
Massachusetts Institute of Technology
MASSLESS LIMITS AND DUAL FIELD THEORIES
*
S. Oeser
Brandeis University,
Waltham, MA 02254, USA
We discuss some aspects of mass zero limits of massive gauge theories and gauge in
variant representations of massless spin 0 and} systems constructed from higher rank
fields, which are not reachable from massive ones. A number of open problems of
interest are mentioned.
1. Introduction
Two different but related topics will be discussed in this lecture. Neither is new,
but both have found recent applications in areas of current interest such as super
gravity, and the second especially is under active current investigation.
The first topic is the mass zero limit of massive gauge theories, which is well
understood in the abelian vector case. I will give an amusing application of this
limit in a model involving supergravity, where just the opposite of the usual situa
tion occurs, and then, in the spirit of this conference. state some open mathematical
problems from a physicist's point of view in the non-abelian case. The second and
main topic concerns alternative representations of lower spin by higher rank fields
in terms of gauge invariant actions. which are sometimes called dual field theories.
A considerable literature is accumulating on this, from a variety of motivations. and
this subject too should have some interest for mathematicians. Rather than give
details or extensive bibliography, I refer to the works on which this talk is based
and where these may be found.
*
Supported in part by NSF grant PHY 78-08644 AOI
4
II. Massless Limit of Massive Gauge Theories
We begin with the abelian case. It is an old story (see e.g. [IJ for some earlier
literature) that the massless limit of Proca theory (massive photons) exists if and
only if the current to which it couples is such that m-1 at' fl'- stays finite in this
limit. The latter can be analyzed in terms of the decomposition
(1 )
applied to the action
I (m) j dx [- "14 F,..."2 (A) - 21 m2 A 2 + (2 )
}<-
The point is that the kinetic term depends only on B/, i.e. Fr<-v (A) "'" F"....v (B),
and orthogonality of the transverse and longitudinal parts of (1) in the mass term,
+kae -2
SA2 =JB2 )2 m , decouples them there. Consequently, the action reads
J [
I (m) dx -"41F ,..v2 (B)-21 m2
f t ( (3)
+ dx [- dfJ- e )2 - m- Ie <3,.. f'" ]
The first part of the action is no longer constrained by the Bianchi identity
& ~v F }J-V == 0, since both m2 Br<-T and {~ are automatically divergenceless, and the
m ~ 0 limit there simply consists in dropping the B2 term. This is precisely the
usual Maxwell theory coupled to a conserved current 1~T , and although it appears to
be expressed in terms of a vector potential in a specific (Lorentz) gauge ( a· B = 0),
e
it is in fact gauge invariant. The second part represents a scalar *f.i eld related
to the original longitudinal photon, coupled to the quantity m-1C). Normal cur
II'-
rents, such as = e 1} 'g'P4-, remain strictly conserved and the longitudinal field
decouples from the charges. Consequently, walls become transparent to these photons,
which removes the old equipartition paradox about the jump from three to two degrees
of freedom. There are always three degrees of freedom, even at m = 0, but one is
decoupled from charges (although it still couples to gravity, for example, where its
energy can be measured in principle).
However, not all sources need be of this type. The first dynamical example, to my
knowledge, occurs when a massive vector supermultiplet is coupled to supergravity.
In this system, the current consists of an identically conserved part from a non
minimal coupl ing '" M1 '-" Ff ov together with a nonconserved part expl icitly propor
tional to m, i.e. a term of the form m II< A,... This leads to just the opposite