Modern Differential Geometry for Physicists

307 Pages·1999·14.99 MB·English

by  C. J. Isham

Modern Differential
Geometry for Physicists
World Scientific Lecture Notes in Physics
Published
Vol. 47: Some Elementary Gauge Theory Concepts
H M Chan and S T Tsou
Vol. 48: Electrodynamics of High Temperature Superconductors
A M Portis
Vol. 49: Field Theory, Disorder and Simulations
G Parisi
Vol. 50: The Quark Structure of Matter
M Jacob
Vol. 51 : Selected Topics on the General Properties of Quantum Field Theory
F Strocchi
Vol. 52: Field Theory: A Path Integral Approach
A Das
Vol. 53: Introduction to Nonlinear Dynamics for Physicists
H D I Abarbanel, et a/.
Vol. 54: Introduction to the Theory of Spin Glasses and Neural Networks
V Dotsenko
Vol. 55: Lectures in Particle Physics
D Green
Vol. 56: Chaos and Gauge Field Theory
T S Biro, et a/.
Vol. 57: Foundations of Quantum Chromodynamics (2nd edn.)
T Muta
Vol. 59: Lattice Gauge Theories: An Introduction (2nd edn.)
H J Rothe
Vol. 60: Massive Neutrinos in Physics and Astrophysics
R N Mohapatra and P B Pal
Vol. 61 : Modern Differential Geometry for Physicists (2nd edn.)
C J lsham
-
World Scientific Lecture Notes in Physics Vol. 61
Modern Differential
Geometry for Physicists
Second Edition
Chris J lsham
Theoretical Phvsics Grow
Imperial College of Science, fechnolog y and Medicine
UK
World Scientific
Singapore New Jersey. London Hong Kong
Published by
World Scientific Publishing Co. Re. Ltd.
P 0 Box 128, Farrer Road, Singapore 912805
USA once: Suite lB, 1060 Main Street, River Edge, NJ 07661
UKoflce: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-PublicationD ata
A catalogue record for this book is available from the British Library.
First published 1999
Reprinted 2001
-
World Scientific Lecture Notes in Physics Vol. 61
MODERN DIFFERENTIAL GEOMETRY FOR PHYSICISTS (2nd Edition)
Copyright 0 1999 by World Scientific Publishing Co. Re. Ltd.
All rights reserved. This book, or parts thereof; may not be reproduced in any form or by any means,
electronic or mechanical, including photocopying, recording or any information storage and retrieval
system now known or to be invented, without wriften permission from the Publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright
Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to
photocopy is not required from the publisher.
ISBN 981-02-3555-0
ISBN 981-02-3562-3 (pbk)
This book is printed on acid-free paper.
Printed in Singapore by UtcFPrint
Preface
This book is based on lecture notes for the introductory course on
modern, coordinate-free differential geometry which is taken by our
first-year theoretical physics PhD students, or by students attending
the one-year MSc course “Fundamental Fields and Forces” at Imperial
College.
The course is concerned entirely with the mathematics itself, al-
though the emphasis and detailed topics have been chosen with an
eye on the way in which differential geometry is applied to theoretical
physics these days. Such applications include not only the traditional
area of general relativity, but also the theory of Yang-Mills fields, non-
linear sigma models, superstring theory, and other types of non-linear
field systems that feature in modern elementary particle theory and
quantum gravity.
The course is in four parts dealing with, respectively, (i) an intro-
duction to general topology; (ii) introductory coordinate-free differ-
ential geometry; (iii) geometrical aspects of the theory of Lie groups
and Lie group actions on manifolds; and (iv) the basic ideas of fibre
bundle theory.
The first chapter contains a short introduction to general topol-
ogy with the aim of providing the necessary prerequisites for the later
chapters on differential geometry and fibre bundle theory. The treat-
ment is a little idiosyncratic in so far as I wanted to emphasise certain
algebraic aspects of topology that are not normally mentioned in in-
troductory mathematics texts but which are of potential interest and
importance in the use of topology in theoretical physics.
V
vi PREFACE
The second and third chapters contain an introduction to differ-
ential geometry proper. In preparing this part of the text, I was par-
ticularly conscious of the difficulty which physics graduate students
often experience when being exposed for the first time to the rather
abstract ideas of differential geometry. In particular, I have laid con-
siderable stress on the basic ideas of ‘tangent space structure’, which
I develop from several different points of view: some geometric, some
more algebraic. My experience in teaching this subject for a number
of years is that a firm understanding of the various ways of describing
tangent spaces is the key to attaining a grasp of differential geometry
that goes beyond just a superficial acquiescence in the jargon of the
subject. I have not included any material on Riemannian geometry
as this aspect of the subject is well covered in many existing texts on
differential geometry and/or general relativity.
Chapter four is concerned with the theory of Lie groups, and the
action of Lie groups on differentiable manifolds. I have tried here to
emphasise the geometrical foundations of the connection between Lie
groups and Lie algebras, but the latter subject is not treated in any
detail and readers not familiar with this topic should supplement the
text at this point.
The theory of fibre bundles is introduced in chapter five, with a
treatment that emphasises the theory of principle bundles and their
associated bundles. The final chapter contains an introduction to the
theory of connections and their use in Yang-Mills theory. This is
fairly brief since many excellent introductions to the subject aimed at
physicists have been published in recent years, and there is no great
point in replicating that material in detail here.
The second edition of this book differs from the first mainly by
the addition of the chapter on general topology; it has also been com-
pletely reset in LaTeX, thus allowing for a more extensive index. In
addition, I have taken the opportunity to correct misprints in the
original text, and I have included a few more worked examples. A
number of short explanatory remarks have been added in places where
readers and students have suggested that it might be helpful: I am
most grateful to all those who drew my attention to such deficien-
cies in the original text. However, I have resisted the attention to
PREFACE vii
add substantial amounts of new material-other than the chapter on
topology-since I wanted to retain the flavour of the original as bona
fide lecture notes that could reasonably be read in their entirety by a
student who sought an overall introduction to the subject.
Chris Isham
Imperial College, June 1998

Contents
1 An Introduction to Topology 1
1.1 Preliminary Remarks . . . . . . . . . . . . . . . . . . . 1
1.1.1 Remarks on differential geometry . . . . . . . . 1
1.1.2 Remarks on topology . . . . . . . . . . . . . . . 2
1.2 Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 The simple idea of convergence . . . . . . . . . 3
1.2.2 The idea of a metric space . . . . . . . . . . . . 5
1.2.3 Examples of metric spaces . . . . . . . . . . . . 8
1.2.4 Operations on metrics . . . . . . . . . . . . . . 10
1.2.5 Some topological concepts in metric spaces . . . 11
1.3 Partially Ordered Sets and Lattices . . . . . . . . . . . 14
1.3.1 Partially ordered sets . . . . . . . . . . . . . . . 14
1.3.2 Lattices . . . . . . . . . . . . . . . . . . . . . . 18
1.4 General Topology . . . . . . . . . . . . . . . . . . . . . 23
1.4.1 An example of non-metric convergence . . . . . 23
1.4.2 The idea of a neighbourhood space . . . . . . . 25
1.4.3 Topological spaces . . . . . . . . . . . . . . . . 32
1.4.4 Some examples of topologies on a finite set . . . 37
1.4.5 A topology as a lattice . . . . . . . . . . . . . . 40
1.4.6 The lattice of topologies T(X)o n a set X . . . . 42
ix