Riemann, Topology, and Physics

169 Pages·1987·4.766 MB·English

by  Michael Monastyrsky (auth.), R. O. Wells Jr. (eds.)

Riemann, Topology, and Physics
Michael Monastyrsky
Riemann, Topology,
and Physics
With a Foreword by Freeman J. Dyson
Translated by James King and Victoria King
Edited by R.O. Wells, Jr.
With 20 Illustrations and 16 Photographs
Springer Science+Business Media, LLC
Michael Monastyrsky James King (Translator)
Institute for Theoretical and Victoria King (Translator)
Experimental Physics 7326 55th Avenue N.E.
Moscow 117259 Seattle, WA 98115
U.S.S.R. U.S.A.
R.O. Wells, Jr. (Editor)
Department of Mathematics
Weiss School of Natural Sciences
Rice University
Houston, TX 77251
U.S.A.
Library of Congress Cataloging in Publication Data
Monastyrskiï, Mikhail Il'ich.
Riemann, topology, and physics.
Translated from the Russian; part 1 is translation
of Bernhard Riemann.
1. Riemann, Bernhard, 1826-1866. 2. Topology.
3. Mathematical physics. 4. Mathematics—Germany—
Biography. I. Wells, Ronny O. II. Title.
QA29.R425M6613 1987 530.1'5 86-28399
CIP-Kurztitelaufnahme der Deutschen Bibliothek
Monastyrskij, Michail:
Riemann, topology, and physics / Michael Monastyr
sky. With a foreword by Freeman J. Dyson. Transi.
by James King and Victoria King. Ed. by Ronny O.
Wells, Jr.—Boston ; Basel ; Stuttgart :
Birkhäuser, 1986.
ISBN 978-1-4899-3516-8 ISBN 978-1-4899-3514-4 (eBook)
DOI 10.1007/978-1-4899-3514-4
Part I of this book previously appeared as Bernhard Riemann, © IzdateFstvo "Znanije", Moscow,
1979.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system,
or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or
otherwise, without prior permission of the copyright owner.
© Springer Science+Business Media New York 1987
Originally published by Birkhäuser Boston in 1987
Softcover reprint of the hardcover 1st edition 1987
Typeset by Asco Trade Typesetting Ltd., Hong Kong.
9 8 7 6 5 4 3 21
Foreword
Soviet citizens can buy Monastyrsky's biography of Riemann for eleven
kopeks. This translated edition will cost considerably more, but it is still good
value for the money. And we get Monastyrsky's monograph on topological
methods in the bargain. It was a good idea of Birkhiiuser Boston to publish
the two translations in one volume. The economics of publishing in a capitalist
country make it impossible for us to produce the small cheap paperback
booklets, low in quality of paper and high in quality of scholarship, at which
the Soviet publishing industry excels. Monastyrsky's two booklets are out
standing examples of the genre. By putting them together, Birkhiiuser has
enabled them to fit into the Western book-marketing system.
The two booklets were written separately and each is complete in itself, but
they complement each other beautifully. The Riemann biography is short and
terse, like Riemann's own writings. It describes in few words and fewer
equations the revolutionary ideas which Riemann brought into mathematics
and physics a hundred and twenty years ago. The topological methods booklet
describes how some of these same ideas, after lying dormant for a century,
found new and fruitful applications in the physics of our own time. The two
parts of the story together illustrate one of the central themes of science, the
mysterious power of mathematical concepts to prepare the ground for physical
discoveries which could not have been foreseen or even imagined by the math
ematicians who gave the concepts birth. In telling this story, Monastyrsky
does not begin, as Dostoyevskii began The Brothers Karamazov, with a
quotation from the gospel of St. John: "Verily, verily, I say unto you, except a
corn of wheat fall into the ground and die, it abideth alone: but if it die, it
bringeth forth much fruit." Dostoyevskii's epigraph would be as appropriate
to the history of mathematical physics as to the history of the human soul.
vi Foreword
Riemann's grains of wheat, many of them unknown and unpublished during
his lifetime, are still bringing forth fruit abundantly today.
The quality that makes Monastyrsky unique among expositors of contem
porary physics is his depth of historical focus. He sees modern ideas in a
perspective that goes back all the way to Riemann. But this does not mean
that his account is only of interest to historians. On the contrary, his descrip
tions of recent developments in physics are thoroughly modern and may be
read with profit by Americans who agree with the dictum of Henry Ford
that history is bunk. The reading of the Riemann biography is not a pre
requisite for understanding the monograph on topological methods. The
monograph is self-contained and provides a splendidly clear explanation of
such modern inventions as liquid crystals, 't Hooft monopoles, and twisted
hedgehogs. The menagerie of strange objects populating the recent literature
of mathematical physics is made intelligible by viewing them all as special
cases of a single unifying concept. The unifying concept is the classification of
spaces and mappings into d.iscrete topological types. Each species of crystal,
monopole, or hedgehog corresponds to a particular class of mappings, and
the variety of species is a consequence of the inherent richness of structure
allowed by the topological classification. The exploration of possible topo
logical structures that may be relevant to physics is not yet at an end.
Monastyrsky's lucid account of the modern topological zoo can be under
stood by people who have no interest in Riemann, and his biography of
Riemann can be understood by people ignorant of modern physics. Never
theless, both classes of readers could profit enormously from studying that
half of this book which is less familiar to them. The physicist could learn some
history, and the historian could learn some physics. Both could gain a deeper
understanding of their own fields by seeing them as part of a broader vi
sion, a vision combining historical scholarship with mathematical expertise.
Monastyrsky is a living bridge between the two cultures. Riemann's life and
death, and the slow fruition of his ideas a hundred years later, constitute a
human and intellectual drama which must be seen as a whole in order to be
fully understood.
Princeton FREEMAN J. DYSON
Preface
This book grew out of a suggestion by Klaus Peters, who was at the time with
Birkhauser Boston, and who asked me to help with the editing of the transla
tion. I looked over the draft of the manuscript and found it to be a fascinating
account of Riemann's work and a lucid discussion of the contemporary
interaction of topology and physics. The manuscript had been translated by
Victoria and James King, but it still needed some additional editing due to
the translation of various German, French, and English quotations into
Russian and then back into English. I had an opportunity to visit with
Monastyrsky in his Moscow apartment in 1984 and we discussed various fine
points of the book at the time. The title Riemann, Topology, and Physics was
chosen because of the strong relation that Riemann had to both fields of
science. In fact, Riemann is credited with being the founder of the theory of
manifolds, the cornerstone of modern topology. Moreover, he made contribu
tions to electromagnetism and shock waves, to mention only two physical
phenomena he investigated. Physics and topology both evolved over the next
century, and their interaction today would have made Riemann quite proud,
I am sure.
This book represents then an historical as well as quite a contemporary
perspective on the relation between topology and physics, with the personage
of Riemann being a key historical link.
I would like to thank Klaus Peters for his initial encouragement and
support. To the editors at Birkhauser I worked with: Priscilla White, Carol
MacPherson, and Iris Kramer, I would like to express my appreciation for
their efforts to get this book out under sometimes complicated circumstances.
I'd like to thank Rena Schwarze Wells for her efforts to check all of the original
sources and for her help in so many other ways. My warm admiration goes
to M. Monastyrsky for the wonderful manner in which he captured the essence
viii Preface
of the spirit of both mathematics and physics and shows the magical way in
which they continue to interact vigorously today.
Houston R.O. WELLS, Jr.
Contents
Foreword (Freeman J. Dyson) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Preface (R.O. Wells, Jr.). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
PART I. Bernhard Riemann. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1. Beginnings...................................... 3
2. Doctoral Dissertation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3. Riemann-Lecturer at Gottingen University. . . . . . . . . 18
4. Riemann and Dirichlet. . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5. A Chair in Gottingen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
6. Last Years . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
7. The Fate of Riemann's Work. . . . . . . . . . . . . . . . . . . . . . 64
Notes to Part I. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Bibliography to Part I. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
PART II. Topological Themes in Contemporary Physics. . . . . . . . . . . 71
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2. Topological Structures. . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3. The Connectivity of a Manifold and the Quantization
of Magnetic Flux. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4. Systems with Spontaneous Symmetry-Breaking. . . . . . . 91
5. Topology and Liquid Crystals. . . . . . . . . . . . . . . . . . . . . . 95
6. The Theory of Gauge Fields. . . . . . . . . . . . . . . . . . . . . .. 108
7. Topological Particles. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 125
8. Soliton Particles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 130
9. What Next? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 133
10. A Brief Historical Survey. . . . . . . . . . . . . . . . . . . . . . . . .. 134
x Contents
Notes to Part II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 137
Bibliography to Part II. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 138
Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 141