Topics in Contemporary Mathematical Physics This page is intentionally left blank Topics in Contemporary Mathematical Physics Kai S Lam California State Polytechnic University, USA V^h world Scientific wl NNeeww J Jeersrseeyy • •L Loonnddoonn • •S Si ingapore • Hong Kong Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: Suite 202,1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. TOPICS IN CONTEMPORARY MATHEMATICAL PHYSICS Copyright © 2003 by World Scientific Publishing Co. Pte. 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 written 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 981-238-404-9 Printed in Singapore by Mainland Press to Shiing-Shen Chern This page is intentionally left blank Preface Physics and mathematics have undergone an intensely symbiotic period over the last two decades, and are poised for an even more exciting and productive relationship into the twenty-first century. The present text is the result of this physicist's conviction (which I believe is shared by a growing number of physi­ cists) that it is beneficial to prepare physics students, especially those with a theoretical inclination, with the necessary background to appreciate and take advantage of this development, starting as early as practically feasible in their physics education. Various parts of the first two-thirds or so of the text, deal­ ing mainly with things algebraic, such as the theory of linear transformations, group theory, and Lie algebra theory, have been used as instructional material in an advanced undergraduate level Mathematical Physics three-quarter sequence that I have taught several times at the California State Polytechnic University, Pomona. The last third, mainly on differential geometry, is probably more suit­ able at the beginning graduate level; but the ambitious undergraduate should by no means be deterred. Mathematics students who are interested in seeing how some fundamental mathematical ideas and techniques can be applied to broad swaths of physics may also find the book useful. Because of its wide cov­ erage, the book may, in addition, serve as a reference volume, useful hopefully to student and researcher alike. The choice of material is dictated by the desire to communicate to a mainly advanced undergraduate and beginning graduate audience those topics that 1) play a significant role in contemporary applications of mathematics to physics, and 2) are not usually given prominence in conventional texts at the same level. Thus a major part of the text is focused on group representation theory, Lie groups and Lie algebras, exterior algebra, and finally, differential geometry, at the expense of more traditional (but no less important) topics such as differential equations, complex function theory, special functions, and functional analysis. This choice is limited, necessarily, by the author's knowledge or lack thereof, and also, needless to say, by space. Still, it is hoped that the reader will find between the two covers a more or less coherent body of information that is also reasonably complete, and that, above all, possesses a certain degree of thematic unity. Many excellent texts already exist which deal with the applications of either group theory or differential geometry to physics, but rarely simultaneously. In the present book we bring these two vital strands of contemporary mathematical physics together, not only for convenience, but also to demonstrate some of the Vlll Topics in Contemporary Mathematical Physics deep connections between them. The organization of the book may be described as functionally (but not log­ ically) modular - with each chapter serving as a distinct module whose contents can be clearly discerned from the title, and which is topically (but not logi­ cally) independent of the others. Yet if the book is read from beginning to end in the order presented (although this is by no means obligatory), an unbroken thread may be seen to run through all the chapters, in a loosely thematic sense. This thread weaves together linear spaces and linear operators, representations of groups and algebras, algebraic structures built on differentiable manifolds, vector and principal bundles, and finally, the algebraic objects (characteristic classes) constructed from analytical data (curvatures) that unify the local and global properties of fiber bundles. At various points, detours are taken to show: carefully how these notions have relevance in physics. In principle, the book is self-contained, the only prerequisites being sophomore- level calculus, differential equations, (three-dimensional) vector analysis, and some linear algebra. In its entirety, there is probably enough material for a four-quarter or three-semester sequence of courses. However, students engaged in self-study and instructors may select different subsets of the book as mean­ ingful units, according to their individual predilections and needs, although they should be forewarned that, for the most part, later chapters depend logically on earlier ones. In order to make the book maximally useful, a copious amount of cross-references (backward and forward) have been incorporated. This feature, together with a rather detailed index, will hopefully eliminate most sources of ambiguity. Whenever calculations are presented, they tend to be quite explicit with step-by-step justifications and relatively few steps left out. This practice has no doubt increased the physical size of the book, but hopefully will sub­ stantially decrease the frustration level of the reader. Numerous exercises are inserted at strategic locations throughout the text. In principle, they should be completely workable once the material in the text is comprehended. They mainly serve to amplify, concretize, and reinforce things learned, but never to intimidate. I have decided to adopt a somewhat mathematical style of presentation, at odds with the usual practice in the physics literature. This by and large means frequent definition-theorem-proof sequences, with a level of rigor some­ where between the mathematical and the physical. There are primarily three reasons for doing this. The first is that the mathematical style affords a certain compactness, precision, generality, and economy of presentation that is quite in­ dispensable for a text of this size. The second is that this style, when used with moderation, will often facilitate comprehension of deep and general concepts significantly, especially those that find very diversified applications in physics. Most physicists usually learn the mathematics that they need through specific, multiple, and contextual applications. While this approach has the definite ad­ vantage of making abstract ideas concrete, and thus initially less intimidating, the many different physical guises under which a single mathematical notion may appear frequently tend to obscure the essential unity of the latter. We need only mention two related examples, one elementary (and assumed familiar Preface IX to the reader), the other relatively more advanced (but dealt with exhaustively in this book): the derivative and the covariant derivative (connection on a fiber bundle). The last, perhaps most controversial, reason is my belief that even physicists should learn to "speak", with a reasonable degree of fluency, the lan­ guage of mathematics. This belief in turn stems from the observation that the period of "acrimonious divorce" (in Freeman Dyson's words) between physicists and mathematicians seems to be drawing to an end, and the two groups will find it increasingly rewarding to communicate with each other, not just on the level of trading applications, but also on the deeper one of informing each other of their different but complementary modes of thinking. In this book, however, rigor is never pursued for rigor's sake. Proofs of theorems are only presented when they help clarify abstract concepts or illustrate special calculational tech­ niques. On the other hand, when they are omitted (usually without apologies), it can be assumed that they are either too lengthy, too technically difficult, or simply too distracting. A good many complete chapters deal exclusively with physics applications. These tend to be in close proximity to the exposition of the requisite mathemat­ ics, and one may notice a somewhat abrupt change in style from the mathemati­ cal to the physical, and vice versa. This is again done with some deliberation, in order to prepare the reader for the "culture shock" that she/he may experience on going from the standard literature in one discipline to the other. In some cases, the physics applications are presented even before the necessary mathe­ matics has been completely explained. This may disrupt the logical flow of the presentation, but I suspect that the physicist reader's anxiety level (and blood pressure!) may be considerably lowered on being reassured frequently that there are serious physics applications to rather esoteric pieces of mathematics. Indeed, if I have succeeded in this volume to convince some physics students (or even practicing physicists) that the mathematical style and contents therein are not just fancy garb and window dressing, that they are there not to obfuscate, but rather to clarify, unify, and even to lend depth and hence generality to a host of seemingly disconnected physics ideas, my purpose would have been more than well-served. Unfortunately, my lack of training and knowledge does not permit me to relate the story of the other direction of flow in this fascinating two-way traffic: that physical reasoning and techniques (for example, in quantum field theory) have recently provided significant insights and tools for the solution of long-standing problems in pure mathematics. The writing of much of this text would not have been possible without a recent unique collaborative experience which was the author's great fortune to enjoy. Over the course of about two years, Professor S. S. Chern generously and patiently guided me through the translation and expansion of his introductory text "Lectures on Differential Geometry" (Chern, Chen and Lam, 1999). This immensely valuable learning experience deeply enhanced not only my technical knowledge, but perhaps more importantly, my appreciation of the mysteriously fruitful but sometimes tortuous relationship between mathematics and physics. It also provided a degree of much-needed confidence for a physicist with rela­ tively little formal training in mathematics. The last third of the book, which