Modern Differential Geometry for Physicists PDF
Preview Modern Differential Geometry for Physicists
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
Description:The list of books you might like
