Mathematical Physics

Sadri Hassani

MathematicalPhysics

A Modern Introduction toIts Foundations

Second Edition

Sadri HassaniDepartment of PhysicsIllinois State UniversityNormal, Illinois, USA

ISBN 978-3-319-01194-3 ISBN 978-3-319-01195-0 (eBook)DOI 10.1007/978-3-319-01195-0Springer Cham Heidelberg New York Dordrecht London

Library of Congress Control Number: 2013945405

© Springer International Publishing Switzerland 1999, 2013This work is subject to copyright. All rights are reserved by the Publisher, whether the wholeor part of the material is concerned, specifically the rights of translation, reprinting, reuse ofillustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way,and transmission or information storage and retrieval, electronic adaptation, computer software,or by similar or dissimilar methodology now known or hereafter developed. Exempted from thislegal reservation are brief excerpts in connection with reviews or scholarly analysis or materialsupplied specifically for the purpose of being entered and executed on a computer system, forexclusive use by the purchaser of the work. Duplication of this publication or parts thereof ispermitted only under the provisions of the Copyright Law of the Publisher’s location, in itscurrent version, and permission for use must always be obtained from Springer. Permissionsfor use may be obtained through RightsLink at the Copyright Clearance Center. Violations areliable to prosecution under the respective Copyright Law.The use of general descriptive names, registered names, trademarks, service marks, etc. in thispublication does not imply, even in the absence of a specific statement, that such names areexempt from the relevant protective laws and regulations and therefore free for general use.While the advice and information in this book are believed to be true and accurate at the date ofpublication, neither the authors nor the editors nor the publisher can accept any legal responsi-bility for any errors or omissions that may be made. The publisher makes no warranty, expressor implied, with respect to the material contained herein.

Printed on acid-free paper

Springer is part of Springer Science+Business Media (www.springer.com)

To my wife, Sarah,and to my children,Dane Arash and Daisy Bita

Preface to Second Edition

Based on my own experience of teaching from the first edition, and more im-portantly based on the comments of the adopters and readers, I have madesome significant changes to the new edition of the book: Part I is substan-tially rewritten, Part VIII has been changed to incorporate Clifford algebras,Part IX now includes the representation of Clifford algebras, and the newPart X discusses the important topic of fiber bundles.

I felt that a short section on algebra did not do justice to such an im-portant topic. Therefore, I expanded it into a comprehensive chapter dealingwith the basic properties of algebras and their classification. This required arewriting of the chapter on operator algebras, including the introduction of asection on the representation of algebras in general. The chapter on spectraldecomposition underwent a complete overhaul, as a result of which the topicis now more cohesive and the proofs more rigorous and illuminating. Thisentailed separate treatments of the spectral decomposition theorem for realand complex vector spaces.

The inner product of relativity is non-Euclidean. Therefore, in the discus-sion of tensors, I have explicitly expanded on the indefinite inner productsand introduced a brief discussion of the subspaces of a non-Euclidean (theso-called semi-Riemannian or pseudo-Riemannian) vector space. This innerproduct, combined with the notion of algebra, leads naturally to Clifford al-gebras, the topic of the second chapter of Part VIII. Motivating the subjectby introducing the Dirac equation, the chapter discusses the general prop-erties of Clifford algebras in some detail and completely classifies the Clif-ford algebras Cν

μ(R), the generalization of the algebra C13(R), the Clifford

algebra of the Minkowski space. The representation of Clifford algebras,including a treatment of spinors, is taken up in Part IX, after a discussion ofthe representation of Lie Groups and Lie algebras.

Fiber bundles have become a significant part of the lore of fundamen-tal theoretical physics. The natural setting of gauge theories, essential indescribing electroweak and strong interactions, is fiber bundles. Moreover,differential geometry, indispensable in the treatment of gravity, is most ele-gantly treated in terms of fiber bundles. Chapter 34 introduces fiber bundlesand their complementary notion of connection, and the curvature form aris-ing from the latter. Chapter 35 on gauge theories makes contact with physicsand shows how connection is related to potentials and curvature to fields. Italso constructs the most general gauge-invariant Lagrangian, including itslocal expression (the expression involving coordinate charts introduced onthe underlying manifold), which is the form used by physicists. In Chap. 36,

vii

viii Preface to Second Edition

by introducing vector bundles and linear connections, the stage becomesready for the introduction of curvature tensor and torsion, two major play-ers in differential geometry. This approach to differential geometry via fiberbundles is, in my opinion, the most elegant and intuitive approach, whichavoids the ad hoc introduction of covariant derivative. Continuing with dif-ferential geometry, Chap. 37 incorporates the notion of inner product andmetric into it, coming up with the metric connection, so essential in the gen-eral theory of relativity.

All these changes and additions required certain omissions. I was carefulnot to break the continuity and rigor of the book when omitting topics. Sincenone of the discussions of numerical analysis was used anywhere else in thebook, these were the first casualties. A few mathematical treatments thatwere too dry, technical, and not inspiring were also removed from the newedition. However, I provided references in which the reader can find thesemissing details. The only casualty of this kind of omission was the discus-sion leading to the spectral decomposition theorem for compact operators inChap. 17.

Aside from the above changes, I have also altered the style of the bookconsiderably. Now all mathematical statements—theorems, propositions,corollaries, definitions, remarks, etc.—and examples are numbered consec-utively without regard to their types. This makes finding those statementsor examples considerably easier. I have also placed important mathemat-ical statements in boxes which are more visible as they have dark back-grounds. Additionally, I have increased the number of marginal notes, andadded many more entries to the index.

Many readers and adopters provided invaluable feedback, both in spot-ting typos and in clarifying vague and even erroneous statements of thebook. I would like to acknowledge the contribution of the following peo-ple to the correction of errors and the clarification of concepts: Sylvio An-drade, Salar Baher, Rafael Benguria, Jim Bogan, Jorun Bomert, John Chaf-fer, Demetris Charalambous, Robert Gooding, Paul Haines, Carl Helrich,Ray Jensen, Jin-Wook Jung, David Kastor, Fred Keil, Mike Lieber, Art Lind,Gary Miller, John Morgan, Thomas Schaefer, Hossein Shojaie, ShreenivasSomayaji, Werner Timmermann, Johan Wild, Bradley Wogsland, and FangWu. As much as I tried to keep a record of individuals who gave me feed-back on the first edition, fourteen years is a long time, and I may have omit-ted some names from the list above. To those people, I sincerely apologize.Needless to say, any remaining errors in this new edition is solely my re-sponsibility, and as always, I’ll greatly appreciate it if the readers continuepointing them out to me.

I consulted the following three excellent books to a great extent for theaddition and/or changes in the second edition:

Greub, W., Linear Algebra, 4th ed., Springer-Verlag, Berlin, 1975.Greub, W., Multilinear Algebra, 2nd ed., Springer-Verlag, Berlin, 1978.Kobayashi, S., and K. Nomizu, Foundations of Differential Geometry,vol. 1, Wiley, New York, 1963.

Maury Solomon, my editor at Springer, was immeasurably patient andcooperative on a project that has been long overdue. Aldo Rampioni has

Preface to Second Edition ix

been extremely helpful and cooperative as he took over the editorship ofthe project. My sincere thanks go to both of them. Finally, I would like tothank my wife Sarah for her unwavering forbearance and encouragementthroughout the long-drawn-out writing of the new edition.

Sadri HassaniNormal, IL, USANovember, 2012

Preface to First Edition

“Ich kann es nun einmal nicht lassen, in diesem Drama von Mathematik undPhysik—die sich im Dunkeln befruchten, aber von Angesicht zu Angesicht sogerne einander verkennen und verleugnen—die Rolle des (wie ich genügsam er-fuhr, oft unerwünschten) Boten zu spielen.”

Hermann Weyl

It is said that mathematics is the language of Nature. If so, then physicsis its poetry. Nature started to whisper into our ears when Egyptians andBabylonians were compelled to invent and use mathematics in their day-to-day activities. The faint geometric and arithmetical pidgin of over fourthousand years ago, suitable for rudimentary conversations with nature asapplied to simple landscaping, has turned into a sophisticated language inwhich the heart of matter is articulated.

The interplay between mathematics and physics needs no emphasis.What may need to be emphasized is that mathematics is not merely a toolwith which the presentation of physics is facilitated, but the only mediumin which physics can survive. Just as language is the means by which hu-mans can express their thoughts and without which they lose their uniqueidentity, mathematics is the only language through which physics can ex-press itself and without which it loses its identity. And just as language isperfected due to its constant usage, mathematics develops in the most dra-matic way because of its usage in physics. The quotation by Weyl above,an approximation to whose translation is “In this drama of mathematics andphysics—which fertilize each other in the dark, but which prefer to deny andmisconstrue each other face to face—I cannot, however, resist playing therole of a messenger, albeit, as I have abundantly learned, often an unwel-come one,” is a perfect description of the natural intimacy between whatmathematicians and physicists do, and the unnatural estrangement betweenthe two camps. Some of the most beautiful mathematics has been motivatedby physics (differential equations by Newtonian mechanics, differential ge-ometry by general relativity, and operator theory by quantum mechanics),and some of the most fundamental physics has been expressed in the mostbeautiful poetry of mathematics (mechanics in symplectic geometry, andfundamental forces in Lie group theory).

I do not want to give the impression that mathematics and physics cannotdevelop independently. On the contrary, it is precisely the independence ofeach discipline that reinforces not only itself, but the other discipline aswell—just as the study of the grammar of a language improves its usage andvice versa. However, the most effective means by which the two camps can

xi

xii Preface to First Edition

accomplish great success is through an intense dialogue. Fortunately, withthe advent of gauge and string theories of particle physics, such a dialoguehas been reestablished between physics and mathematics after a relativelylong lull.

Level and Philosophy of Presentation

This is a book for physics students interested in the mathematics they use.It is also a book for mathematics students who wish to see some of the ab-stract ideas with which they are familiar come alive in an applied setting.The level of presentation is that of an advanced undergraduate or beginninggraduate course (or sequence of courses) traditionally called “MathematicalMethods of Physics” or some variation thereof. Unlike most existing math-ematical physics books intended for the same audience, which are usuallylexicographic collections of facts about the diagonalization of matrices, ten-sor analysis, Legendre polynomials, contour integration, etc., with little em-phasis on formal and systematic development of topics, this book attemptsto strike a balance between formalism and application, between the abstractand the concrete.

I have tried to include as much of the essential formalism as is neces-sary to render the book optimally coherent and self-contained. This entailsstating and proving a large number of theorems, propositions, lemmas, andcorollaries. The benefit of such an approach is that the student will recog-nize clearly both the power and the limitation of a mathematical idea usedin physics. There is a tendency on the part of the novice to universalize themathematical methods and ideas encountered in physics courses because thelimitations of these methods and ideas are not clearly pointed out.

There is a great deal of freedom in the topics and the level of presentationthat instructors can choose from this book. My experience has shown thatParts I, II, III, Chap. 12, selected sections of Chap. 13, and selected sectionsor examples of Chap. 19 (or a large subset of all this) will be a reasonablecourse content for advanced undergraduates. If one adds Chaps. 14 and 20,as well as selected topics from Chaps. 21 and 22, one can design a coursesuitable for first-year graduate students. By judicious choice of topics fromParts VII and VIII, the instructor can bring the content of the course to amore modern setting. Depending on the sophistication of the students, thiscan be done either in the first year or the second year of graduate school.

Features

To better understand theorems, propositions, and so forth, students need tosee them in action. There are over 350 worked-out examples and over 850problems (many with detailed hints) in this book, providing a vast arena inwhich students can watch the formalism unfold. The philosophy underly-ing this abundance can be summarized as “An example is worth a thousandwords of explanation.” Thus, whenever a statement is intrinsically vague or

Preface to First Edition xiii

hard to grasp, worked-out examples and/or problems with hints are providedto clarify it. The inclusion of such a large number of examples is the meansby which the balance between formalism and application has been achieved.However, although applications are essential in understanding mathemati-cal physics, they are only one side of the coin. The theorems, propositions,lemmas, and corollaries, being highly condensed versions of knowledge, areequally important.

A conspicuous feature of the book, which is not emphasized in othercomparable books, is the attempt to exhibit—as much as it is useful andapplicable—interrelationships among various topics covered. Thus, the un-derlying theme of a vector space (which, in my opinion, is the most primitiveconcept at this level of presentation) recurs throughout the book and alertsthe reader to the connection between various seemingly unrelated topics.

Another useful feature is the presentation of the historical setting inwhich men and women of mathematics and physics worked. I have goneagainst the trend of the “ahistoricism” of mathematicians and physicists bysummarizing the life stories of the people behind the ideas. Many a time,the anecdotes and the historical circumstances in which a mathematical orphysical idea takes form can go a long way toward helping us understandand appreciate the idea, especially if the interaction among—and the contri-butions of—all those having a share in the creation of the idea is pointed out,and the historical continuity of the development of the idea is emphasized.

To facilitate reference to them, all mathematical statements (definitions,theorems, propositions, lemmas, corollaries, and examples) have been num-bered consecutively within each section and are preceded by the sectionnumber. For example, 4.2.9 Definition indicates the ninth mathematicalstatement (which happens to be a definition) in Sect. 4.2. The end of a proofis marked by an empty square �, and that of an example by a filled square �,placed at the right margin of each.

Finally, a comprehensive index, a large number of marginal notes, andmany explanatory underbraced and overbraced comments in equations fa-cilitate the use and comprehension of the book. In this respect, the book isalso useful as a reference.

Organization and Topical Coverage

Aside from Chap. 0, which is a collection of purely mathematical concepts,the book is divided into eight parts. Part I, consisting of the first four chap-ters, is devoted to a thorough study of finite-dimensional vector spaces andlinear operators defined on them. As the unifying theme of the book, vectorspaces demand careful analysis, and Part I provides this in the more accessi-ble setting of finite dimension in a language that is conveniently generalizedto the more relevant infinite dimensions, the subject of the next part.

Following a brief discussion of the technical difficulties associated withinfinity, Part II is devoted to the two main infinite-dimensional vector spacesof mathematical physics: the classical orthogonal polynomials, and Fourierseries and transform.

xiv Preface to First Edition

Complex variables appear in Part III. Chapter 9 deals with basic proper-ties of complex functions, complex series, and their convergence. Chapter 10discusses the calculus of residues and its application to the evaluation of def-inite integrals. Chapter 11 deals with more advanced topics such as multi-valued functions, analytic continuation, and the method of steepest descent.

Part IV treats mainly ordinary differential equations. Chapter 12 showshow ordinary differential equations of second order arise in physical prob-lems, and Chap. 13 consists of a formal discussion of these differential equa-tions as well as methods of solving them numerically. Chapter 14 brings inthe power of complex analysis to a treatment of the hypergeometric dif-ferential equation. The last chapter of this part deals with the solution ofdifferential equations using integral transforms.

Part V starts with a formal chapter on the theory of operator and theirspectral decomposition in Chap. 16. Chapter 17 focuses on a specific typeof operator, namely the integral operators and their corresponding integralequations. The formalism and applications of Sturm-Liouville theory appearin Chaps. 18 and 19, respectively.

The entire Part VI is devoted to a discussion of Green’s functions. Chap-ter 20 introduces these functions for ordinary differential equations, whileChaps. 21 and 22 discuss the Green’s functions in an m-dimensional Eu-clidean space. Some of the derivations in these last two chapters are newand, as far as I know, unavailable anywhere else.

Parts VII and VIII contain a thorough discussion of Lie groups and theirapplications. The concept of group is introduced in Chap. 23. The theory ofgroup representation, with an eye on its application in quantum mechanics,is discussed in the next chapter. Chapters 25 and 26 concentrate on tensoralgebra and tensor analysis on manifolds. In Part VIII, the concepts of groupand manifold are brought together in the context of Lie groups. Chapter 27discusses Lie groups and their algebras as well as their representations, withspecial emphasis on their application in physics. Chapter 28 is on differentialgeometry including a brief introduction to general relativity. Lie’s originalmotivation for constructing the groups that bear his name is discussed inChap. 29 in the context of a systematic treatment of differential equationsusing their symmetry groups. The book ends in a chapter that blends many ofthe ideas developed throughout the previous parts in order to treat variationalproblems and their symmetries. It also provides a most fitting example of theclaim made at the beginning of this preface and one of the most beautifulresults of mathematical physics: Noether’s theorem on the relation betweensymmetries and conservation laws.

Acknowledgments

It gives me great pleasure to thank all those who contributed to the mak-ing of this book. George Rutherford was kind enough to volunteer for thedifficult task of condensing hundreds of pages of biography into tens ofextremely informative pages. Without his help this unique and valuable fea-ture of the book would have been next to impossible to achieve. I thank himwholeheartedly. Rainer Grobe and Qichang Su helped me with my rusty

Preface to First Edition xv

computational skills. (R.G. also helped me with my rusty German!) Manycolleagues outside my department gave valuable comments and stimulatingwords of encouragement on the earlier version of the book. I would like torecord my appreciation to Neil Rasband for reading part of the manuscriptand commenting on it. Special thanks go to Tom von Foerster, senior editorof physics and mathematics at Springer-Verlag, not only for his patienceand support, but also for the extreme care he took in reading the entiremanuscript and giving me invaluable advice as a result. Needless to say,the ultimate responsibility for the content of the book rests on me. Last butnot least, I thank my wife, Sarah, my son, Dane, and my daughter, Daisy, forthe time taken away from them while I was writing the book, and for theirsupport during the long and arduous writing process.

Many excellent textbooks, too numerous to cite individually here, haveinfluenced the writing of this book. The following, however, are noteworthyfor both their excellence and the amount of their influence:

Birkhoff, G., and G.-C. Rota, Ordinary Differential Equations, 3rd ed.,New York, Wiley, 1978.Bishop, R., and S. Goldberg, Tensor Analysis on Manifolds, New York,Dover, 1980.Dennery, P., and A. Krzywicki, Mathematics for Physicists, New York,Harper & Row, 1967.Halmos, P., Finite-Dimensional Vector Spaces, 2nd ed., Princeton, VanNostrand, 1958.Hamermesh, M., Group Theory and its Application to Physical Prob-lems, Dover, New York, 1989.Olver, P., Application of Lie Groups to Differential Equations, NewYork, Springer-Verlag, 1986.

Unless otherwise indicated, all biographical sketches have been takenfrom the following three sources:

Gillispie, C., ed., Dictionary of Scientific Biography, Charles Scribner’s,New York, 1970.Simmons, G., Calculus Gems, New York, McGraw-Hill, 1992.History of Mathematics archive at www-groups.dcs.st-and.ac.uk:80.

I would greatly appreciate any comments and suggestions for improve-ments. Although extreme care was taken to correct all the misprints, themere volume of the book makes it very likely that I have missed some (per-haps many) of them. I shall be most grateful to those readers kind enough tobring to my attention any remaining mistakes, typographical or otherwise.Please feel free to contact me.

Sadri HassaniCampus Box 4560Department of PhysicsIllinois State UniversityNormal, IL 61790-4560, USAe-mail: hassani@entropy.phy.ilstu.edu

xvi Preface to First Edition

It is my pleasure to thank all those readers who pointed out typograph-ical mistakes and suggested a few clarifying changes. With the exceptionof a couple that required substantial revision, I have incorporated all thecorrections and suggestions in this second printing.

Note to the Reader

Mathematics and physics are like the game of chess (or, for that matter, likeany game)—you will learn only by “playing” them. No amount of readingabout the game will make you a master. In this book you will find a largenumber of examples and problems. Go through as many examples as pos-sible, and try to reproduce them. Pay particular attention to sentences like“The reader may check . . . ” or “It is straightforward to show . . . ”. Theseare red flags warning you that for a good understanding of the materialat hand, you need to provide the missing steps. The problems often fill inmissing steps as well; and in this respect they are essential for a thoroughunderstanding of the book. Do not get discouraged if you cannot get to thesolution of a problem at your first attempt. If you start from the beginningand think about each problem hard enough, you will get to the solution, andyou will see that the subsequent problems will not be as difficult.

The extensive index makes the specific topics about which you may beinterested to learn easily accessible. Often the marginal notes will help youeasily locate the index entry you are after.

I have included a large collection of biographical sketches of mathemat-ical physicists of the past. These are truly inspiring stories, and I encourageyou to read them. They let you see that even under excruciating circum-stances, the human mind can work miracles. You will discover how theseremarkable individuals overcame the political, social, and economic condi-tions of their time to let us get a faint glimpse of the truth. They are our trueheroes.

xvii

Contents

1 Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . 11.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Equivalence Relations . . . . . . . . . . . . . . . . 31.2 Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . 81.4 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . 101.5 Mathematical Induction . . . . . . . . . . . . . . . . . . . 121.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Part I Finite-Dimensional Vector Spaces

2 Vectors and Linear Maps . . . . . . . . . . . . . . . . . . . . 192.1 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . 19

2.1.1 Subspaces . . . . . . . . . . . . . . . . . . . . . . 222.1.2 Factor Space . . . . . . . . . . . . . . . . . . . . . 242.1.3 Direct Sums . . . . . . . . . . . . . . . . . . . . . 252.1.4 Tensor Product of Vector Spaces . . . . . . . . . . 28

2.2 Inner Product . . . . . . . . . . . . . . . . . . . . . . . . . 292.2.1 Orthogonality . . . . . . . . . . . . . . . . . . . . 322.2.2 The Gram-Schmidt Process . . . . . . . . . . . . . 332.2.3 The Schwarz Inequality . . . . . . . . . . . . . . . 352.2.4 Length of a Vector . . . . . . . . . . . . . . . . . . 36

2.3 Linear Maps . . . . . . . . . . . . . . . . . . . . . . . . . 382.3.1 Kernel of a Linear Map . . . . . . . . . . . . . . . 412.3.2 Linear Isomorphism . . . . . . . . . . . . . . . . . 43

2.4 Complex Structures . . . . . . . . . . . . . . . . . . . . . 452.5 Linear Functionals . . . . . . . . . . . . . . . . . . . . . . 482.6 Multilinear Maps . . . . . . . . . . . . . . . . . . . . . . . 53

2.6.1 Determinant of a Linear Operator . . . . . . . . . . 552.6.2 Classical Adjoint . . . . . . . . . . . . . . . . . . . 56

2.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3 Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.1 From Vector Space to Algebra . . . . . . . . . . . . . . . . 63

3.1.1 General Properties . . . . . . . . . . . . . . . . . . 643.1.2 Homomorphisms . . . . . . . . . . . . . . . . . . . 70

3.2 Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733.2.1 Factor Algebras . . . . . . . . . . . . . . . . . . . 77

xix

xx Contents

3.3 Total Matrix Algebra . . . . . . . . . . . . . . . . . . . . . 783.4 Derivation of an Algebra . . . . . . . . . . . . . . . . . . . 803.5 Decomposition of Algebras . . . . . . . . . . . . . . . . . 83

3.5.1 The Radical . . . . . . . . . . . . . . . . . . . . . 843.5.2 Semi-simple Algebras . . . . . . . . . . . . . . . . 883.5.3 Classification of Simple Algebras . . . . . . . . . . 92

3.6 Polynomial Algebra . . . . . . . . . . . . . . . . . . . . . 953.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4 Operator Algebra . . . . . . . . . . . . . . . . . . . . . . . . . 1014.1 Algebra of End(V) . . . . . . . . . . . . . . . . . . . . . . 101

4.1.1 Polynomials of Operators . . . . . . . . . . . . . . 1024.1.2 Functions of Operators . . . . . . . . . . . . . . . 1044.1.3 Commutators . . . . . . . . . . . . . . . . . . . . . 106

4.2 Derivatives of Operators . . . . . . . . . . . . . . . . . . . 1074.3 Conjugation of Operators . . . . . . . . . . . . . . . . . . 113

4.3.1 Hermitian Operators . . . . . . . . . . . . . . . . . 1144.3.2 Unitary Operators . . . . . . . . . . . . . . . . . . 118

4.4 Idempotents . . . . . . . . . . . . . . . . . . . . . . . . . 1194.4.1 Projection Operators . . . . . . . . . . . . . . . . . 120

4.5 Representation of Algebras . . . . . . . . . . . . . . . . . 1254.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

5 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1375.1 Representing Vectors and Operators . . . . . . . . . . . . . 1375.2 Operations on Matrices . . . . . . . . . . . . . . . . . . . 1425.3 Orthonormal Bases . . . . . . . . . . . . . . . . . . . . . . 1465.4 Change of Basis . . . . . . . . . . . . . . . . . . . . . . . 1485.5 Determinant of a Matrix . . . . . . . . . . . . . . . . . . . 151

5.5.1 Matrix of the Classical Adjoint . . . . . . . . . . . 1525.5.2 Inverse of a Matrix . . . . . . . . . . . . . . . . . . 1555.5.3 Dual Determinant Function . . . . . . . . . . . . . 158

5.6 The Trace . . . . . . . . . . . . . . . . . . . . . . . . . . 1605.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

6 Spectral Decomposition . . . . . . . . . . . . . . . . . . . . . 1696.1 Invariant Subspaces . . . . . . . . . . . . . . . . . . . . . 1696.2 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . 1726.3 Upper-Triangular Representations . . . . . . . . . . . . . . 1756.4 Complex Spectral Decomposition . . . . . . . . . . . . . . 177

6.4.1 Simultaneous Diagonalization . . . . . . . . . . . . 1856.5 Functions of Operators . . . . . . . . . . . . . . . . . . . . 1886.6 Real Spectral Decomposition . . . . . . . . . . . . . . . . 191

6.6.1 The Case of Symmetric Operators . . . . . . . . . . 1936.6.2 The Case of Real Normal Operators . . . . . . . . . 198

6.7 Polar Decomposition . . . . . . . . . . . . . . . . . . . . . 2056.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

Contents xxi

Part II Infinite-Dimensional Vector Spaces

7 Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 2157.1 The Question of Convergence . . . . . . . . . . . . . . . . 2157.2 The Space of Square-Integrable Functions . . . . . . . . . 221

7.2.1 Orthogonal Polynomials . . . . . . . . . . . . . . . 2227.2.2 Orthogonal Polynomials and Least Squares . . . . . 225

7.3 Continuous Index . . . . . . . . . . . . . . . . . . . . . . 2277.4 Generalized Functions . . . . . . . . . . . . . . . . . . . . 2337.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

8 Classical Orthogonal Polynomials . . . . . . . . . . . . . . . . 2418.1 General Properties . . . . . . . . . . . . . . . . . . . . . . 2418.2 Classification . . . . . . . . . . . . . . . . . . . . . . . . . 2448.3 Recurrence Relations . . . . . . . . . . . . . . . . . . . . 2458.4 Details of Specific Examples . . . . . . . . . . . . . . . . 248

8.4.1 Hermite Polynomials . . . . . . . . . . . . . . . . 2488.4.2 Laguerre Polynomials . . . . . . . . . . . . . . . . 2498.4.3 Legendre Polynomials . . . . . . . . . . . . . . . . 2508.4.4 Other Classical Orthogonal Polynomials . . . . . . 252

8.5 Expansion in Terms of Orthogonal Polynomials . . . . . . 2548.6 Generating Functions . . . . . . . . . . . . . . . . . . . . 2578.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

9 Fourier Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 2659.1 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . 265

9.1.1 The Gibbs Phenomenon . . . . . . . . . . . . . . . 2739.1.2 Fourier Series in Higher Dimensions . . . . . . . . 275

9.2 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . 2769.2.1 Fourier Transforms and Derivatives . . . . . . . . . 2849.2.2 The Discrete Fourier Transform . . . . . . . . . . . 2869.2.3 Fourier Transform of a Distribution . . . . . . . . . 287

9.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 288

Part III Complex Analysis

10 Complex Calculus . . . . . . . . . . . . . . . . . . . . . . . . 29510.1 Complex Functions . . . . . . . . . . . . . . . . . . . . . 29510.2 Analytic Functions . . . . . . . . . . . . . . . . . . . . . . 29710.3 Conformal Maps . . . . . . . . . . . . . . . . . . . . . . . 30410.4 Integration of Complex Functions . . . . . . . . . . . . . . 30910.5 Derivatives as Integrals . . . . . . . . . . . . . . . . . . . 31510.6 Infinite Complex Series . . . . . . . . . . . . . . . . . . . 319

10.6.1 Properties of Series . . . . . . . . . . . . . . . . . 31910.6.2 Taylor and Laurent Series . . . . . . . . . . . . . . 321

10.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 330

11 Calculus of Residues . . . . . . . . . . . . . . . . . . . . . . . 33911.1 Residues . . . . . . . . . . . . . . . . . . . . . . . . . . . 33911.2 Classification of Isolated Singularities . . . . . . . . . . . 34211.3 Evaluation of Definite Integrals . . . . . . . . . . . . . . . 344

xxii Contents

11.3.1 Integrals of Rational Functions . . . . . . . . . . . 34511.3.2 Products of Rational and Trigonometric Functions . 34811.3.3 Functions of Trigonometric Functions . . . . . . . 35011.3.4 Some Other Integrals . . . . . . . . . . . . . . . . 35211.3.5 Principal Value of an Integral . . . . . . . . . . . . 354

11.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 359

12 Advanced Topics . . . . . . . . . . . . . . . . . . . . . . . . . 36312.1 Meromorphic Functions . . . . . . . . . . . . . . . . . . . 36312.2 Multivalued Functions . . . . . . . . . . . . . . . . . . . . 365

12.2.1 Riemann Surfaces . . . . . . . . . . . . . . . . . . 36612.3 Analytic Continuation . . . . . . . . . . . . . . . . . . . . 372

12.3.1 The Schwarz Reflection Principle . . . . . . . . . . 37412.3.2 Dispersion Relations . . . . . . . . . . . . . . . . . 376

12.4 The Gamma and Beta Functions . . . . . . . . . . . . . . . 37812.5 Method of Steepest Descent . . . . . . . . . . . . . . . . . 38212.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 388

Part IV Differential Equations

13 Separation of Variables in Spherical Coordinates . . . . . . . 39513.1 PDEs of Mathematical Physics . . . . . . . . . . . . . . . 39513.2 Separation of the Angular Part . . . . . . . . . . . . . . . . 39813.3 Construction of Eigenvalues of L2 . . . . . . . . . . . . . . 40113.4 Eigenvectors of L2: Spherical Harmonics . . . . . . . . . . 406

13.4.1 Expansion of Angular Functions . . . . . . . . . . 41113.4.2 Addition Theorem for Spherical Harmonics . . . . 412

13.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 413

14 Second-Order Linear Differential Equations . . . . . . . . . . 41714.1 General Properties of ODEs . . . . . . . . . . . . . . . . . 41714.2 Existence/Uniqueness for First-Order DEs . . . . . . . . . 41914.3 General Properties of SOLDEs . . . . . . . . . . . . . . . 42114.4 The Wronskian . . . . . . . . . . . . . . . . . . . . . . . . 425

14.4.1 A Second Solution to the HSOLDE . . . . . . . . . 42614.4.2 The General Solution to an ISOLDE . . . . . . . . 42814.4.3 Separation and Comparison Theorems . . . . . . . 430

14.5 Adjoint Differential Operators . . . . . . . . . . . . . . . . 43314.6 Power-Series Solutions of SOLDEs . . . . . . . . . . . . . 436

14.6.1 Frobenius Method of Undetermined Coefficients . . 43914.6.2 Quantum Harmonic Oscillator . . . . . . . . . . . . 444

14.7 SOLDEs with Constant Coefficients . . . . . . . . . . . . 44614.8 The WKB Method . . . . . . . . . . . . . . . . . . . . . . 450

14.8.1 Classical Limit of the Schrödinger Equation . . . . 45214.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 453

15 Complex Analysis of SOLDEs . . . . . . . . . . . . . . . . . . 45915.1 Analytic Properties of Complex DEs . . . . . . . . . . . . 460

15.1.1 Complex FOLDEs . . . . . . . . . . . . . . . . . . 46015.1.2 The Circuit Matrix . . . . . . . . . . . . . . . . . . 462

Contents xxiii

15.2 Complex SOLDEs . . . . . . . . . . . . . . . . . . . . . . 46315.3 Fuchsian Differential Equations . . . . . . . . . . . . . . . 46915.4 The Hypergeometric Function . . . . . . . . . . . . . . . . 47315.5 Confluent Hypergeometric Functions . . . . . . . . . . . . 478

15.5.1 Hydrogen-Like Atoms . . . . . . . . . . . . . . . . 48015.5.2 Bessel Functions . . . . . . . . . . . . . . . . . . . 482

15.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 485

16 Integral Transforms and Differential Equations . . . . . . . . 49316.1 Integral Representation of the Hypergeometric Function . . 494

16.1.1 Integral Representation of the ConfluentHypergeometric Function . . . . . . . . . . . . . . 497

16.2 Integral Representation of Bessel Functions . . . . . . . . . 49816.2.1 Asymptotic Behavior of Bessel Functions . . . . . 502

16.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 505

Part V Operators on Hilbert Spaces

17 Introductory Operator Theory . . . . . . . . . . . . . . . . . 51117.1 From Abstract to Integral and Differential Operators . . . . 51117.2 Bounded Operators in Hilbert Spaces . . . . . . . . . . . . 513

17.2.1 Adjoints of Bounded Operators . . . . . . . . . . . 51717.3 Spectra of Linear Operators . . . . . . . . . . . . . . . . . 51717.4 Compact Sets . . . . . . . . . . . . . . . . . . . . . . . . 519

17.4.1 Compactness and Infinite Sequences . . . . . . . . 52117.5 Compact Operators . . . . . . . . . . . . . . . . . . . . . 523

17.5.1 Spectrum of Compact Operators . . . . . . . . . . 52717.6 Spectral Theorem for Compact Operators . . . . . . . . . . 527

17.6.1 Compact Hermitian Operator . . . . . . . . . . . . 52917.6.2 Compact Normal Operator . . . . . . . . . . . . . 531

17.7 Resolvents . . . . . . . . . . . . . . . . . . . . . . . . . . 53417.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 539

18 Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . 54318.1 Classification . . . . . . . . . . . . . . . . . . . . . . . . . 54318.2 Fredholm Integral Equations . . . . . . . . . . . . . . . . . 549

18.2.1 Hermitian Kernel . . . . . . . . . . . . . . . . . . 55218.2.2 Degenerate Kernels . . . . . . . . . . . . . . . . . 556

18.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 560

19 Sturm-Liouville Systems . . . . . . . . . . . . . . . . . . . . . 56319.1 Compact-Resolvent Unbounded Operators . . . . . . . . . 56319.2 Sturm-Liouville Systems and SOLDEs . . . . . . . . . . . 56919.3 Asymptotic Behavior . . . . . . . . . . . . . . . . . . . . 573

19.3.1 Large Eigenvalues . . . . . . . . . . . . . . . . . . 57319.3.2 Large Argument . . . . . . . . . . . . . . . . . . . 577

19.4 Expansions in Terms of Eigenfunctions . . . . . . . . . . . 57719.5 Separation in Cartesian Coordinates . . . . . . . . . . . . . 579

19.5.1 Rectangular Conducting Box . . . . . . . . . . . . 57919.5.2 Heat Conduction in a Rectangular Plate . . . . . . . 581

xxiv Contents

19.5.3 Quantum Particle in a Box . . . . . . . . . . . . . . 58219.5.4 Wave Guides . . . . . . . . . . . . . . . . . . . . . 584

19.6 Separation in Cylindrical Coordinates . . . . . . . . . . . . 58619.6.1 Conducting Cylindrical Can . . . . . . . . . . . . . 58619.6.2 Cylindrical Wave Guide . . . . . . . . . . . . . . . 58819.6.3 Current Distribution in a Circular Wire . . . . . . . 589

19.7 Separation in Spherical Coordinates . . . . . . . . . . . . . 59019.7.1 Radial Part of Laplace’s Equation . . . . . . . . . . 59119.7.2 Helmholtz Equation in Spherical Coordinates . . . 59319.7.3 Quantum Particle in a Hard Sphere . . . . . . . . . 59319.7.4 Plane Wave Expansion . . . . . . . . . . . . . . . . 594

19.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 595

Part VI Green’s Functions

20 Green’s Functions in One Dimension . . . . . . . . . . . . . . 60520.1 Calculation of Some Green’s Functions . . . . . . . . . . . 60620.2 Formal Considerations . . . . . . . . . . . . . . . . . . . . 610

20.2.1 Second-Order Linear DOs . . . . . . . . . . . . . . 61420.2.2 Self-adjoint SOLDOs . . . . . . . . . . . . . . . . 616

20.3 Green’s Functions for SOLDOs . . . . . . . . . . . . . . . 61720.3.1 Properties of Green’s Functions . . . . . . . . . . . 61920.3.2 Construction and Uniqueness of Green’s Functions . 62120.3.3 Inhomogeneous BCs . . . . . . . . . . . . . . . . . 626

20.4 Eigenfunction Expansion . . . . . . . . . . . . . . . . . . 63020.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 632

21 Multidimensional Green’s Functions: Formalism . . . . . . . 63521.1 Properties of Partial Differential Equations . . . . . . . . . 635

21.1.1 Characteristic Hypersurfaces . . . . . . . . . . . . 63621.1.2 Second-Order PDEs in m Dimensions . . . . . . . 640

21.2 Multidimensional GFs and Delta Functions . . . . . . . . . 64321.2.1 Spherical Coordinates in m Dimensions . . . . . . 64521.2.2 Green’s Function for the Laplacian . . . . . . . . . 647

21.3 Formal Development . . . . . . . . . . . . . . . . . . . . . 64821.3.1 General Properties . . . . . . . . . . . . . . . . . . 64821.3.2 Fundamental (Singular) Solutions . . . . . . . . . . 649

21.4 Integral Equations and GFs . . . . . . . . . . . . . . . . . 65221.5 Perturbation Theory . . . . . . . . . . . . . . . . . . . . . 655

21.5.1 The Nondegenerate Case . . . . . . . . . . . . . . 65921.5.2 The Degenerate Case . . . . . . . . . . . . . . . . 660

21.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 661

22 Multidimensional Green’s Functions: Applications . . . . . . 66522.1 Elliptic Equations . . . . . . . . . . . . . . . . . . . . . . 665

22.1.1 The Dirichlet Boundary Value Problem . . . . . . . 66522.1.2 The Neumann Boundary Value Problem . . . . . . 671

22.2 Parabolic Equations . . . . . . . . . . . . . . . . . . . . . 67322.3 Hyperbolic Equations . . . . . . . . . . . . . . . . . . . . 67822.4 The Fourier Transform Technique . . . . . . . . . . . . . . 680

Contents xxv

22.4.1 GF for the m-Dimensional Laplacian . . . . . . . . 68122.4.2 GF for the m-Dimensional Helmholtz Operator . . . 68222.4.3 GF for the m-Dimensional Diffusion Operator . . . 68422.4.4 GF for the m-Dimensional Wave Equation . . . . . 685

22.5 The Eigenfunction Expansion Technique . . . . . . . . . . 68822.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 693

Part VII Groups and Their Representations

23 Group Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 70123.1 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70223.2 Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . 705

23.2.1 Direct Products . . . . . . . . . . . . . . . . . . . 71223.3 Group Action . . . . . . . . . . . . . . . . . . . . . . . . 71323.4 The Symmetric Group Sn . . . . . . . . . . . . . . . . . . 71523.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 720

24 Representation of Groups . . . . . . . . . . . . . . . . . . . . 72524.1 Definitions and Examples . . . . . . . . . . . . . . . . . . 72524.2 Irreducible Representations . . . . . . . . . . . . . . . . . 72824.3 Orthogonality Properties . . . . . . . . . . . . . . . . . . . 73224.4 Analysis of Representations . . . . . . . . . . . . . . . . . 73724.5 Group Algebra . . . . . . . . . . . . . . . . . . . . . . . . 740

24.5.1 Group Algebra and Representations . . . . . . . . . 74024.6 Relationship of Characters to Those of a Subgroup . . . . . 74324.7 Irreducible Basis Functions . . . . . . . . . . . . . . . . . 74624.8 Tensor Product of Representations . . . . . . . . . . . . . 750

24.8.1 Clebsch-Gordan Decomposition . . . . . . . . . . . 75324.8.2 Irreducible Tensor Operators . . . . . . . . . . . . 756

24.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 758

25 Representations of the Symmetric Group . . . . . . . . . . . 76125.1 Analytic Construction . . . . . . . . . . . . . . . . . . . . 76125.2 Graphical Construction . . . . . . . . . . . . . . . . . . . 76425.3 Graphical Construction of Characters . . . . . . . . . . . . 76725.4 Young Operators . . . . . . . . . . . . . . . . . . . . . . . 77125.5 Products of Representations of Sn . . . . . . . . . . . . . . 77425.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 776

Part VIII Tensors and Manifolds

26 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78126.1 Tensors as Multilinear Maps . . . . . . . . . . . . . . . . . 78226.2 Symmetries of Tensors . . . . . . . . . . . . . . . . . . . . 78926.3 Exterior Algebra . . . . . . . . . . . . . . . . . . . . . . . 794

26.3.1 Orientation . . . . . . . . . . . . . . . . . . . . . . 80026.4 Symplectic Vector Spaces . . . . . . . . . . . . . . . . . . 80126.5 Inner Product Revisited . . . . . . . . . . . . . . . . . . . 804

26.5.1 Subspaces . . . . . . . . . . . . . . . . . . . . . . 80926.5.2 Orthonormal Basis . . . . . . . . . . . . . . . . . . 812

xxvi Contents

26.5.3 Inner Product on Λp(V,U) . . . . . . . . . . . . . 81926.6 The Hodge Star Operator . . . . . . . . . . . . . . . . . . 82026.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 823

27 Clifford Algebras . . . . . . . . . . . . . . . . . . . . . . . . . 82927.1 Construction of Clifford Algebras . . . . . . . . . . . . . . 830

27.1.1 The Dirac Equation . . . . . . . . . . . . . . . . . 83227.2 General Properties of the Clifford Algebra . . . . . . . . . 834

27.2.1 Homomorphism with Other Algebras . . . . . . . . 83727.2.2 The Canonical Element . . . . . . . . . . . . . . . 83827.2.3 Center and Anticenter . . . . . . . . . . . . . . . . 83927.2.4 Isomorphisms . . . . . . . . . . . . . . . . . . . . 842

27.3 General Classification of Clifford Algebras . . . . . . . . . 84327.4 The Clifford Algebras Cν

μ(R) . . . . . . . . . . . . . . . . 84627.4.1 Classification of C0

n(R) and Cn0(R) . . . . . . . . . 849

27.4.2 Classification of Cνμ(R) . . . . . . . . . . . . . . . 851

27.4.3 The Algebra C13(R) . . . . . . . . . . . . . . . . . 852

27.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 856

28 Analysis of Tensors . . . . . . . . . . . . . . . . . . . . . . . . 85928.1 Differentiable Manifolds . . . . . . . . . . . . . . . . . . . 85928.2 Curves and Tangent Vectors . . . . . . . . . . . . . . . . . 86628.3 Differential of a Map . . . . . . . . . . . . . . . . . . . . 87228.4 Tensor Fields on Manifolds . . . . . . . . . . . . . . . . . 876

28.4.1 Vector Fields . . . . . . . . . . . . . . . . . . . . . 87728.4.2 Tensor Fields . . . . . . . . . . . . . . . . . . . . . 882

28.5 Exterior Calculus . . . . . . . . . . . . . . . . . . . . . . 88828.6 Integration on Manifolds . . . . . . . . . . . . . . . . . . . 89728.7 Symplectic Geometry . . . . . . . . . . . . . . . . . . . . 90128.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 909

Part IX Lie Groups and Their Applications

29 Lie Groups and Lie Algebras . . . . . . . . . . . . . . . . . . 91529.1 Lie Groups and Their Algebras . . . . . . . . . . . . . . . 915

29.1.1 Group Action . . . . . . . . . . . . . . . . . . . . 91729.1.2 Lie Algebra of a Lie Group . . . . . . . . . . . . . 92029.1.3 Invariant Forms . . . . . . . . . . . . . . . . . . . 92729.1.4 Infinitesimal Action . . . . . . . . . . . . . . . . . 92829.1.5 Integration on Lie Groups . . . . . . . . . . . . . . 935

29.2 An Outline of Lie Algebra Theory . . . . . . . . . . . . . . 93629.2.1 The Lie Algebras o(p,n − p) and p(p,n − p) . . . 94029.2.2 Operations on Lie Algebras . . . . . . . . . . . . . 944

29.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 948

30 Representation of Lie Groups and Lie Algebras . . . . . . . . 95330.1 Representation of Compact Lie Groups . . . . . . . . . . . 95330.2 Representation of the General Linear Group . . . . . . . . 96330.3 Representation of Lie Algebras . . . . . . . . . . . . . . . 966

30.3.1 Representation of Subgroups of GL(V) . . . . . . . 967

Contents xxvii

30.3.2 Casimir Operators . . . . . . . . . . . . . . . . . . 96930.3.3 Representation of so(3) and so(3,1) . . . . . . . . 97230.3.4 Representation of the Poincaré Algebra . . . . . . . 975

30.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 983

31 Representation of Clifford Algebras . . . . . . . . . . . . . . 98731.1 The Clifford Group . . . . . . . . . . . . . . . . . . . . . 98731.2 Spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995

31.2.1 Pauli Spin Matrices and Spinors . . . . . . . . . . . 99731.2.2 Spinors for Cν

μ(R) . . . . . . . . . . . . . . . . . . 1001

31.2.3 C13(R) Revisited . . . . . . . . . . . . . . . . . . . 1004

31.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 1006

32 Lie Groups and Differential Equations . . . . . . . . . . . . . 100932.1 Symmetries of Algebraic Equations . . . . . . . . . . . . . 100932.2 Symmetry Groups of Differential Equations . . . . . . . . 1014

32.2.1 Prolongation of Functions . . . . . . . . . . . . . . 101732.2.2 Prolongation of Groups . . . . . . . . . . . . . . . 102132.2.3 Prolongation of Vector Fields . . . . . . . . . . . . 1022

32.3 The Central Theorems . . . . . . . . . . . . . . . . . . . . 102432.4 Application to Some Known PDEs . . . . . . . . . . . . . 1029

32.4.1 The Heat Equation . . . . . . . . . . . . . . . . . . 103032.4.2 The Wave Equation . . . . . . . . . . . . . . . . . 1034

32.5 Application to ODEs . . . . . . . . . . . . . . . . . . . . . 103732.5.1 First-Order ODEs . . . . . . . . . . . . . . . . . . 103732.5.2 Higher-Order ODEs . . . . . . . . . . . . . . . . . 103932.5.3 DEs with Multiparameter Symmetries . . . . . . . 1040

32.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 1043

33 Calculus of Variations, Symmetries, and Conservation Laws . 104733.1 The Calculus of Variations . . . . . . . . . . . . . . . . . . 1047

33.1.1 Derivative for Hilbert Spaces . . . . . . . . . . . . 104733.1.2 Functional Derivative . . . . . . . . . . . . . . . . 105033.1.3 Variational Problems . . . . . . . . . . . . . . . . . 105333.1.4 Divergence and Null Lagrangians . . . . . . . . . . 1060

33.2 Symmetry Groups of Variational Problems . . . . . . . . . 106233.3 Conservation Laws and Noether’s Theorem . . . . . . . . . 106533.4 Application to Classical Field Theory . . . . . . . . . . . . 106933.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 1073

Part X Fiber Bundles

34 Fiber Bundles and Connections . . . . . . . . . . . . . . . . . 107934.1 Principal Fiber Bundles . . . . . . . . . . . . . . . . . . . 1079

34.1.1 Associated Bundles . . . . . . . . . . . . . . . . . 108434.2 Connections in a PFB . . . . . . . . . . . . . . . . . . . . 1086

34.2.1 Local Expression for a Connection . . . . . . . . . 108734.2.2 Parallelism . . . . . . . . . . . . . . . . . . . . . . 1089

34.3 Curvature Form . . . . . . . . . . . . . . . . . . . . . . . 109134.3.1 Flat Connections . . . . . . . . . . . . . . . . . . . 1095

xxviii Contents

34.3.2 Matrix Structure Group . . . . . . . . . . . . . . . 109634.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 1097

35 Gauge Theories . . . . . . . . . . . . . . . . . . . . . . . . . . 109935.1 Gauge Potentials and Fields . . . . . . . . . . . . . . . . . 1099

35.1.1 Particle Fields . . . . . . . . . . . . . . . . . . . . 110135.1.2 Gauge Transformation . . . . . . . . . . . . . . . . 1102

35.2 Gauge-Invariant Lagrangians . . . . . . . . . . . . . . . . 110535.3 Construction of Gauge-Invariant Lagrangians . . . . . . . . 110735.4 Local Equations . . . . . . . . . . . . . . . . . . . . . . . 111235.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 1115

36 Differential Geometry . . . . . . . . . . . . . . . . . . . . . . 111736.1 Connections in a Vector Bundle . . . . . . . . . . . . . . . 111736.2 Linear Connections . . . . . . . . . . . . . . . . . . . . . 1120

36.2.1 Covariant Derivative of Tensor Fields . . . . . . . . 112336.2.2 From Forms on P to Tensor Fields on M . . . . . . 112536.2.3 Component Expressions . . . . . . . . . . . . . . . 112836.2.4 General Basis . . . . . . . . . . . . . . . . . . . . 1132

36.3 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . 113736.3.1 Riemann Normal Coordinates . . . . . . . . . . . . 1138

36.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 1140

37 Riemannian Geometry . . . . . . . . . . . . . . . . . . . . . . 114337.1 The Metric Connection . . . . . . . . . . . . . . . . . . . 1143

37.1.1 Orthogonal Bases . . . . . . . . . . . . . . . . . . 114837.2 Isometries and Killing Vector Fields . . . . . . . . . . . . . 115537.3 Geodesic Deviation and Curvature . . . . . . . . . . . . . 1159

37.3.1 Newtonian Gravity . . . . . . . . . . . . . . . . . . 116137.4 General Theory of Relativity . . . . . . . . . . . . . . . . 1163

37.4.1 Einstein’s Equation . . . . . . . . . . . . . . . . . 116337.4.2 Static Spherically Symmetric Solutions . . . . . . . 116737.4.3 Schwarzschild Geodesics . . . . . . . . . . . . . . 1169

37.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 1174

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1179

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1181

List of Symbols

General∀ for all (values of)∃ there exists (a values of)iff if and only ifA ≡ B A is identical to (equivalent to, defined as) B

Set Theory∈ set theoretic membership sign: “belongs to”�∈ set theoretic exclusion sign: “does not belong to”⊂ subset sign⊆ subset with the possibility of equality emphasized∅ empty set∪ union of sets∩ intersection of sets∼ A complement of the set A

A × B set of ordered pairs (a, b) with a ∈ A and b ∈ B

� equivalence relation�a� equivalence class of a

idX identity map of the set X

N the set of natural (non-negative integer) numbersZ the set of integersQ the set of rational numbersR the set of real numbersR

+ the set of positive real numbersC the set of complex numbersH the set of quaternionsf : A → B map f from set A to set B

x �→ f (x) x is mapped to f (x) via the map f

g ◦ f composition of the two maps f and g

Vector SpacesV a generic vector space|a〉 a generic vector labeled a

Cn, Rn complex (real) n-tuples

Mm×n m × n matricesC

∞ absolutely convergent complex seriesPc[t] polynomials in t with complex coefficientsPr [t] polynomials in t with real coefficientsPc

n[t] complex polynomials of degree n or less

xxix

xxx List of Symbols

C0(a, b) continuous functions on real interval (a, b)

Cn(a, b) n-times differentiable functions on real interval (a, b)

C∞(a, b) infinitely differentiable functions on real interval (a, b)

Span{S} the span of a subset S of a vector space〈a | b〉 inner product of vectors |a〉 and |b〉‖a‖ norm of the vector |a〉⊕ direct sum of vectors or vector spaces⊕n

k=1 direct sum of n vectors or vector spacesV∗ dual of the vector space V

〈a,ααα〉 pairing of the vector |a〉 with its dual ααα

AlgebrasA a generic algebraL(V) the algebra of linear maps of the vector space V; same as

End(V)

End(V) the algebra of linear maps of the vector space V; same asL(V)

⊕ direct sum of algebras⊕V direct sum of algebras only as vector spaces⊗ tensor product of algebrasM(F) total matrix algebra over the field F

C(V) Clifford algebra of the inner-product space V

∨ Clifford product symbol

GroupsSn symmetric group; group of permutations of 1,2, . . . , n

GL(V) general linear group of the vector space V

GL(n,C) general linear group of the vector space Cn

GL(n,R) general linear group of the vector space Rn

SL(V) special linear group; subgroup of GL(V) with unitdeterminant

O(n) orthogonal group; group of orthogonal n × n matricesSO(n) special orthogonal group; subgroup of O(n) with unit

determinantU(n) unitary group; group of unitary n × n matricesSU(n) special unitary group; subgroup of U(n) with unit

determinantg Lie algebra of the Lie group G∼= isomorphism of groups, vector spaces, and algebras

TensorsTr

s (V) set of tensors of type (r, s) in vector space V

Sr (V) set of symmetric tensors of type (r,0) in vector space V

Λp(V) set of p-forms in vector space V

Λp(V∗) set of p-vectors in vector space V

∧ wedge (exterior) product symbolΛp(V,U) set of p-forms in V with values in vector space U

F∞(P ) set of infinitely differentiable functions at point P of amanifold

TP (M) set of vectors tangent to manifold M at point P ∈ M

List of Symbols xxxi

T (M) tangent bundle of the manifold M

T ∗(M) cotangent bundle of the manifold M

X(M) set of vector fields on the manifold M

exp(tX) flow of the vector field X parametrized by t

T rs (M) set of tensor fields of type (r, s) in the manifold M