Fortschritte der Physik / Progress of Physics: Band 29, Heft 4 [Reprint 2022 ed.] 9783112655900

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FORTSCHRITTE DER PHYSIK HERAUSGEGEBEN IM AUFTRAGE DER PHYSIKALISCHEN GESELLSCHAFT DER DEUTSCHEN DEMOKRATISCHEN R E P U B L I K VON F. KASCHLUHN, A. LÖSCHE, R. RITSCHL UND R. ROMPE

H E F T 4 • 1981 • B A N D 29

A K A D E M I E

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ISSN 0015-8208

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Zeitschrift „Fortschritte der Physik'* Herausgeber: Prof. Dr. Frank Kaschluhn, Prof. Dr. Artur Lösche, Prof. Dr. Rudolf Ritsehl, Prof. D r . Robert Rompe, im Auftrag der Physikalischen Gesellschaft der Deutschen Demokratischen Republik. Verlag: Akademie-Verlag, D D R - 1080 Berlin, Leipziger Straße 3 - 4 ; Fernruf: 2236221 und 2236229; Telex-Nr. 114420; B a n k : Staatsbank der D D R , Berlin, Konto-Nr. 6836-26-20712. Chefredakteur: Dr. Lutz Rothkirch. Anschrift der Redaktion: Sektion Physik der Humboldt-Universität zu Berlin, D D R - 1040 Berlin, Hessische Straße 2. Veröffentlicht unter der Lizenznummer 1324 des Presseamtes beim Vorsitzenden de9 Ministerrates der Deutschen Demokratischen Republik. Gesamtherstellung: VEB Druckhaus „Maxim Corki", DDR - 7400 Altenburg, Carl-von-Ossietzky-Straße 30/31. Erscheinungsweise: Die Zeitschrift „Fortschritte der Physik" erscheint monatlich. Die 12 Hefte eines Jahres bilden einen Band. Bezugspreis je Band 180,— M zuzüglich Versandspesen (Preis für die D D R : 120,— M). Preis je H e f t 15,— M (Preis f ü r die D D R : 10,-M). Bestellnummer dieses Heftes: 1027/29/4. © 1981 by Akademie-Verlag Berlin. Printed in the German Democratic Republic. AN (EDV) 57618

Fortschritte der Physik 29, 1 3 5 - 1 8 5 (1981)

Physical Principles, Geometrical Aspects, and Locality Properties of Gauge Field Theories G . MACK

Max-Planck-Institut

für Physik und Astrophysik,

München,

Federal Republic of Germany

and II. Institut für Theoretische

Physik der Universität Hamburg, of Germany

Hamburg,

Federal

Republic

Contents 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

Introduction 136 Naheinformationsprinzip 136 General covariance 138 Principle of equivalence 142 Active gauge transformations 144 Global transformations 146 Other groups 147 General relativity 149 Geometry and locality 157 Gauges 163 Local gauges 166 Singular gauge transformations 169 Recovery of the geometry from the vector potential in a local gauge 171 Topological invariants 173 Topological stability of local gauge singularities 176 Unobservability of matter fields which transform nontrivially under the center of the gauge group 177 17. Infinitely many conserved currents 177

Appendix A: Field Theory on a principal fibre bundle

178

Appendix B : Some topology

181

References

182

Index

184

1

Zeitschrift „Fortschritte der Physik", Bd. 29, Heft 4

136

G . MACK

1. Introduction In these notes we shall discuss gauge field theories, particularly Yang-Mills theories, at a classical level from a geometrical point of view. The introductory chapters are partly pedagogical. We concentrate on physical principles, the necessary mathematical tools (of differential geometry) are introduced along the way. The physical principles are not only as old as YANG and MILLS' pioneering paper (1954). When reduced to their essentials they are the same general principles on which Einsteins' theory of gra vitation is built. They are a principle of equivalence, and a principle forbidding direct exchange of information at a distance (Naheinformationsprinzip). Since it appears now that all the fundamental interactions in nature are described by gauge field theories, this provides a unified geometrical point of view — contrary to expectations ten years ago 1 ). The second part (Sects. 9 — 16) is devoted to locality problems in gauge field theories. They come from the fact that only equivalence classes of connections (or vector potentials) can be observed. Examples show tjhat locality problems originate from two sources in pure Yang Mills theories (without matter fields). One is topological and the other is related to the existence of degenerate field configurations (they admit a nontrivial centralizer 2 ) of the infinitesimal holonomy groups on some extended region of space or space time). Nondegenerate field configurations in theories with semisimple gauge groups can be analyzed with the help of the concept of a local gauge. Such local gauges will play a central role in our discussion. Roughly speaking, a complete local gauge translates a gauge field theory into a theory of observable fields without ruining locality properties. Complete gauges can, however, only exist when the center of the gauge group is trivial. After the translation, topologically stable singularities in the gauge invariant potentials will, in general, appear. These local gauge singularities are interesting characteristica of a smooth connection.

2. Naheinformationsprinzip One of the most important advances in physics in the last century was the formulation of the NahewirJcungsprinzip = principle of no action at a distance. I t says that any apparent force between two particles (e.g. electrically charged ones) is not due to direct action of one of them on the other. I t must be mediated by a field which carries dynamical degrees of freedom itself. Moreover, there is in fact only a local interaction of a particle at x with the field there. As a result, the force acting on a particle at x is determined not only by the position and velocities of the other particles at the same time, but by the history of the system which determines the (electric) fields. A fundamental feature of gauge field theories and general relativity is the validity of a more general principle. We call it the Naheinformationsprinzip = principle of no information at a distance. The idea of such a principle appears natural when one considers that information has to be transferred from one point in space to another by way of signals which propagate in space time according to the laws of physics and which may be influenced by the medium trough which they pass. ) In the preface to his book on gravitation S. WEINBERG writes (1971): " B u t now the passage of time has taught us not to expect that the strong, weak and electromagnetic interactions can be understood in geometrical terms, and too great an emphasis on geometry can only obscure the deep connections between gravitation and the rest of physics". 2 ) The centralizer of H £ G in G consists of those elements of G which commute with all elements of H. G — gauge group here. x

Principles, Aspects, and Properties of Gauge Field Theories

137

More specifically, let us imagine a space time manifold M such as Minkowskispace, or a Euclidean space, or even a discrete lattice of space time points, with points x, and classical matter fields or Schrodinger wave functions W which live on it. The field XF is assumed to take its values W(x) at a given point x of M in a real or complex vector space Vx. The Naheinformationsprinzip asserts that there is no a priori possibility of comparing directions in vector spaces Vx and Vy attached to different space time points x + y.

To compare, one must exchange a signal, i.e. somehow transport (information embodied in) v £ Vx to y. We will imagine that this transport occurs step by step from one point to a neighbouring one or, in other words, along a path G from x to y. The result of the parallel transport along C is given by a map which takes