mathematical physics - a modern introduction to its foundations

Sadri
Hassani
Mathematical Physics
A Modem
Introduction
to
Its
Foundations
With 152 Figures
, Springer
ODTlJ
KU1"UPHANESt
M. E. T. U.
liBRARY
METULIBRARY
2
~themltlcoJ
physicS:I
modem
mllllllllllllllllllllllllllllll~11111111111111111111111111IIII
002m7S69
QC20 H394
SadriHassani
Department
of
Physics
IllinoisStateUniversity
Normal, IL 61790
USA

To my wife Sarah

and to my children
DaneArash
and
Daisy Rita
336417
LibraryofCongressCataloging-in-Publication Data
Hassani,Sadri.
Mathematical physics: a modem introductionits foundations /
SadriHassani.
p. em.
Includesbibliographical referencesand index.
ISBN0-387-98579-4 (alk. paper)
1.Mathematical physics.
I. Title.
QC20.H394 1998
530.15 <1c21
98-24738
Printedon acid-freepaper.
QC20
14394
c,2.
© 1999Springer-Verlag New York,Inc.
All rightsreserved.Thiswork maynot be translatedor copiedin wholeor in part withoutthe written
permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY
10010, USA), except for brief excerpts in connection with reviews or scholarly analysis.
Use-in
connectionwith any form of informationstorageand retrieval,electronicadaptation,computersctt-
ware,or by similaror dissimilarmethodology nowknownor hereafterdevelopedis forbidden.
The use of generaldescriptivenames,trade names,trademarks,etc., in this publication,even
if

the
formerare notespeciallyidentified, isnot tobe takenas a sign that suchnames, as understoodby the
TradeMarksandMerchandise MarksAct, mayaccordinglybeused freelyby anyone.
Productionmanagedby KarinaMikhli;manufacturing supervisedby ThomasKing.
Photocomposed copypreparedfromthe author's TeXfiles.
Printedand boundby HamiltonPrintingCo.,Rensselaer, NY.
Printedinthe UnitedStatesof
America.
9 8 7 6 5 4 3 (Correctedthirdprinting,2002)
ISBN0-387-98579-4
SPIN
10854281
Springer-Verlag New York
Berlin:
Heidelberg
A
member.
of
BertelsmannSpringer Science+Business Media GmbH
Preface
"Ich
kann
es
nun
einmal
nicht
lassen,
in diesem
Drama
von

Mathematik und
Physik <lie
sich im Dunkeln befrnchten,
aber von Angesicht zu Angesicht so geme einander verkennen
und verleugnen-die Rolle des (wie ich gentigsam
erfuhr,
oft
unerwiinschten)
Boten zu
spielen."
Hermann
Weyl
It
is said that mathematics is the language
of
Nature.
If
so, then physics is its
poetry. Nature started to whisper into our ears when Egyptians and Babylonians
were compelled to invent and use mathematics in their day-to-day activities. The
faint geomettic and arithmetical pidgin
of
over four thousand years ago, snitable
for rudimentary conversations with nature as applied to simple landscaping, has
turned into a sophisticated language in which the heart of matter is articulated.
The interplay between mathematics and physics needs no emphasis. What
may need to be emphasizedis that mathematicsis not merely a tool with which the
presentation of physics is facilitated,
butthe only medium in which physics can
survive. Just as languageis the means by whichhumans can express their thoughts

and withoutwhich they lose their uniqueidentity, mathematicsis the only language
through which physics can express itself and without which it loses its identity.
And
just
as language is perfected due to its constantusage, mathematics develops
in the most dramatic way because
of
its usage in physics. The quotation by Weyl
above, an approximation to whose translation is
"In this drama
of
mathematics
and
physics-which
fertilize each other in the dark, but which prefer to deny and
misconstrue each other face to
face-I
cannot, however, resist playing the role
of
a messenger, albeit, as I have abundantly learned, often an unwelcome one:'
vi
PREFACE
is a perfect description of the natnral intimacy between what mathematicians and
physicists do, and the nnnatnralestrangementbetweenthe two camps. Some
of
the
most beantifnl mathematics has been motivated by physics (differential eqnations
by Newtonian mechanics, differential geometryby generalrelativity, and operator
theoryby qnantnmmechanics),and some of the most fundamentalphysics has been
expressed in the most beantiful poetry of mathematics (mechanics in symplectic

geometry, and fundamental forces in Lie group theory).
I do uot want to give the impression that mathematics and physics cannot
develop independently. On the contrary, it is precisely the independence
of
each
discipline that reinforcesnot only itself, but the otherdiscipline as
well-just
as the
stndy of the grammar
of
a language improves its usage and vice versa. However,
the most effective means by which the two camps can accomplish great success
is throngh an inteuse dialogue. Fortnnately, with the advent of gauge and string
theories
of
particlephysics, such a dialogue has beenreestablishedbetweenphysics
and mathematics after a relatively long lull.
Level and Philosophy
of
Presentation
This is a book for physics stndeuts interested in the mathematics they use.
It
is also a book
fur
mathematics stndeuts who wish to see some
of
the abstract
ideas with which they are fantiliar come alive in an applied setting. The level of
preseutationis that of an advancedundergraduate or beginning graduate course (or
sequence

of
courses) traditionally called "Mathematical Methods
of
Physics" or
some variation thereof. Unlike mostexisting mathematical physics books intended
for the same audience, which are usually lexicographic collections
of
facts about
the diagonalization of matrices, tensor analysis, Legendre polynomials, contour
integration, etc., with little emphasis on formal and systematic development of
topics, this book attempts to strike a balance between formalism and application,
between the abstract and the concrete.
I have tried to include as mnch of the essential formalism as is necessary to
render the book optimally coherent and self-contained. This entails stating and
proving a large nnmber of theorems, propositions, lemmas, and corollaries. The
benefit of such an approachis that the stndentwill recognizeclearlyboththe power
and the limitationof amathematicalidea usedinphysics. Thereis atendencyon the
part
of
the uovice to universalize the mathematicalmethods and ideas eucountered
in physics courses because the limitations
of
these methods and ideas are not
clearly pointed out.
There is a great deal
of
freedom in the topics and the level of presentation that
instructors can choose from this book. My experience has showu that Parts I,
TI,
Ill, Chapter 12, selected sections

of
Chapter 13, and selected sections or examples
of Chapter 19 (or a large snbset
of
all this) will be a reasonable course content for
advancedundergraduates.
If
one adds Chapters 14and 20, as well as selectedtopics
from Chapters 21 and 22, one can design a course snitable for first-year graduate
PREFACE
vii
students. By judicious choice
of
topics from Parts VII and VIII, the instructor
can bring the content of the course to a more modern setting. Depending on the
sophistication
of
the students, this can be done either in the first year or the second
year
of
graduate school.
Features
To betler understand theorems, propositions, and so forth, students need to see
them in action. There are over 350 worked-out examples and over 850 problems
(many with detailed hints) in this book, providing a vast arena in which students
can watch the formalism unfold. The philosophy underlying this abundance can
be summarized as
''An example is worth a thousand words
of
explanation." Thus,

whenever a statement is intrinsically vague or hard to grasp, worked-out examples
and/or problems with hints are provided to clarify it. The inclusion of such a
large number of examples is the means by which the balance between formalism
and application has been achieved. However, although applications are essential
in understanding mathematical physics, they are only one side
of
the coin. The
theorems, propositions, lemmas, and corollaries, being highly condensedversions
of
knowledge, are equally important.
A conspicuous feature
of
the book, which is not emphasized in other compa-
rable books, is the attempt to
exhibit-as
much as.it is useful and applicable-«
interrelationships among various topics covered. Thus, the underlying theme of a
vector space (which, in my opinion, is the most primitive concept at this level
of
presentation) recurs throughout the book and alerts the reader to the connection
between various seemingly unrelated topics.
Another useful feature is the presentation
of
the historical setting in which
men and women of mathematics and physics worked. I have gone against the
trend of the "ahistoricism"
of
mathematicians and physicists by summarizing the
life stories
of

the people behind the ideas. Many a time, the anecdotes and the
historical circumstances in which a mathematical or physical idea takes form can
go a long way toward helping us understand and appreciate the idea, especially
if
the interaction
among-and
the contributions
of-all
those having a share in the
creation of the idea is pointedout, and the historical continuity of the development
of
the idea is emphasized.
To facilitate reference to them, all mathematical statements (definitions, theo-
rems, propositions, lemmas, corollaries, and examples) have been nnmbered con-
secutively within each section and are precededby the section number. For exam-
ple,
4.2.9 Definition indicates the ninth mathematical statement (which happens
to be a definition) in Section4.2. The end
of
a proofis marked by an empty square
D, and that of an example by a filled square
III,
placed at the right margin of each.
Finally, a comprehensive index, a large number
of
marginal notes, and many
explanatory underbraced and overbraced comments in equations facilitate the use
viii
PREFACE
and comprehension

of
the book. In this respect, the bookis also nsefnl as a refer-
ence.
Organization and Topical Coverage
Aside from Chapter 0, which is a collection of pnrely mathematical concepts,
the book is divided into eight parts. Part I, consisting of the first fonr chapters, is
devotedto athoroughstudy of finite-dimensional vectorspaces and linearoperators
defined on them. As the unifying theme
of
the book, vector spaces demandcareful
analysis, andPart Iprovides this in the more accessiblesetting of finite dimensionin
alanguagethatisconvenientlygeneralizedto the more relevant infinite dimensions,'
the subject of the next part.
Following a brief discussion of the technical difficulties associated with in-
finity, Part
IT
is devoted to the two main infinite-dimensional vector spaces
of
mathematical physics: the classical orthogonal polynomials, and Foutier series
and transform.
Complex variables appear in Part
ill. Chapter 9 deals with basic properties
of
complex functions, complex series, and their convergence. Chapter 10 discusses
the calculus of residues and its application to the evaluation of definite integrals.
Chapter
II
deals with more advanced topics such asmultivaluedfunctions, analytic
continuation, and the method of steepest descent.
Part IV treats mainly ordinary differential equations. Chapter 12 shows how

ordinary differential equations of second order arise in physical problems, and
Chapter 13 consists
of
a formal discussion
of
these differential equations as well
as methods of solving them numerically. Chapter 14 brings in the power
of
com-
plex analysis to a treatment
of
the hypergeometric differential equation. The last
chapter of this part deals with the solution of differential equations using integral
transforms.
Part V starts with a formal chapter on the theory
of
operator and their spectral
decomposition in Chapter 16. Chapter 17 focuses on a specific type
of
operator,
namely the integral operators and their corresponding integralequations. The for-
malism and applications of Stnrm-Liouville theory appear in Chapters 18 and 19,
respectively.
The
entire Part VI is devoted to a discussion
of
Green's functions. Chapter
20 introduces these functions for ordinary differential equations, while Chapters
21 and 22 discuss the Green's functions in an m-dimensional Euclidean space.
Some

of
the derivations in these last two chapters are new and, as far as I know,
unavailable anywhere else.
Parts VII and
vrncontain a thorough discussion
of
Lie groups and their ap-
plications. The concept of group is introduced in Chapter 23. The theory
of
group
representation, with an eye on its application in quantom mechanics, is discussed
in the next chapter. Chapters 25 and 26 concentrate on tensor algebra and ten-,
sor analysis on manifolds.
In Part
vrn,
the concepts of group and manifold are
PREFACE
ix
brought together in the coutext
of
Lie groups. Chapter 27 discusses Lie groups
and their algebras as well as their represeutations, with special emphasis on their
application in physics. Chapter 28 is on differential geometry including a brief
introduction to general relativity. Lie's original motivation for constructing the
groups that bear his name is discussedin Chapter 29 in the context of a systematic
treatment of differential equations using their symmetry groups. The
book
ends in
a chapter that blends many of the ideas developed throughout the previous parts
in order to treat variational problems and their symmetries.

It
also provides a most
fitting example of the claim made at the beginning
of
this preface and one of the
most beautiful results of mathematical physics: Noether's theorem ou the relation
between symmetries and conservationlaws.
Acknowledgments
It
gives me great pleasure to thank all those who contributed to the making
of
this book. George Rutherford was kind enough to voluuteer for the difficult task
of condensing hundreds of pages
of
biography into tens
of
extremely informative
pages. Without his help this unique and valuable feature
of
the book would have
been next to impossible to achieve. I thank him wholeheartedly. Rainer Grobe and
Qichang Su helped me with my rusty computational skills. (R. G. also helped me
with my rusty German!) Many colleagues outside my department gave valuable
comments and stimulating words
of
encouragement on the earlier version of the
book. I would
like to recordmy appreciationto Neil Rasbandfor readingpart of the
manuscript and commenting on it. Special thanks go to Tom von Foerster, senior
editor

of
physics and mathematics at Springer-Verlag, not ouly for his patience and
support, but also for the,extreme care he took in reading the entire manuscript and
giving me invaluable advice as a result. Needless to say, the ultimateresponsibility
for the content of the book rests on me. Last but not least, I thank my wife, Sarah,
my son, Dane, and my daughter, Daisy, for the time taken away from them while
I was writing the book, and for their support during the long and arduous writing
process.
Many excellent textbooks, too numerous to cite individually here, have influ-
enced the writing
of
this book. The following, however, are noteworthy for 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, Van Nostrand,
1958.
x
PREFACE
Hamennesh, M. Group Theory
and

its Application to Physical Problems, Dover,
New York, 1989.
Olver, P.Application
of
Lie Groups to DifferentialEquations, New York, Springer-
Verlag, 1986.
Unless otherwise indicated, all biographical sketches have beentakenfrom the
following three sources:
Gillispie, C., ed., Dictionary
of
ScientificBiography,CharlesScribner'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 wonld greatly appreciate any comments and suggestions for improvements.
Although extreme care was taken to correct
all the misprints, the mere volume of
the
book
makes it very likely that I have missed some (perhaps many) of them. I
shall be most grateful to those readers kind enough to bring to my attention any
remaining mistakes, typographical or otherwise. Please feel free to contact me.
Sadri Hassani
Campus
Box
4560
Department
of

Physics
Illinois State University
Normal, IL 61790-4560,
USA
e-mail:
It
is my pleasureto thankall thosereaders who pointed out typographical mistakes
and suggestedafew clarifyingchanges.With the exception
ofa couplethat required
substantial revisiou, I have incorporated all the corrections and suggestions in this
second printing.
Note to the Reader
Mathematics and physics are like the game
of
chess (or,
for
that matter, like any
gamej-i-you
willleam
only by ''playing'' them. No amount of reading about the
game will make you a master.
In
this bookyou will find alarge number
of
examples
and problems.Go throughas many examples as possible,and try to reproducethem.
Pay particular attention to sentences like "The reader may check .

"or
"It

is
straightforward to show .

"These
are red flags warning you that for a good
understanding of the material at hand, yon need to provide the missing steps.
The
problems often fill in missing steps as well; and in this respect they are essential
for a thorough understanding of the book. Do not get discouragedif you cannot get
to the solution of a problem at your first attempt.
If
you start from the beginning
and think about each problem hard enough, you
will get to the solution, .and you
will see that the subsequent problems will not be as difficult.
The extensive index makes the specific topics about which you may be in-
terested to leam easily accessible. Often the marginal notes will help you easily
locate the index entry you are after.
I have included a large collection of biographical sketches of mathematical
physicists
of
the past. These are truly inspiring stories, and I encourage you to read
them. They let you seethat even underexcruciatingcircumstances,the human mind
can
work
miracles. Youwill discover how these remarkable individuals overcame
the political, social, and economic conditions of their time to let us get a faint
glimpse of the truth. They are our true heroes.
1
1

1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Contents
Preface
Note to
the
Reader

Listof Symbols
o Mathematical Preliminaries
0.1 Sets .
0.2 Maps .
0.3 Metric Spaces .
0.4
Cardinality
0.5 MathematicalInduction
0.6
Problems
I Finite-Dimensional VectorSpaces
1 Vectors
and
Transformations
1.1 VectorSpaces . . . . .
1.2 Inner Product .
1.3 Linear Transformations .
1.4
Algebras.
1.5
Problems
2
Operator
Algebra
2.1 Algebra
«s:
(V)
v
xi
xxi

1
1
4
7
10
12
14
17
19
19
23
32
41
44
49
49
xiv
CONTENTS
2.2 Derivatives of Functions of Operators .
2.3 Conjugationof Operators . . . .
2.4 HermitianandUnitary Operators
2.5 ProjectionOperators . . . . . . .
2.6 Operators
in Numerical Analysis
2.7
Problems
3 Matrices: OperatorRepresentations
3.1
Matrices
3.2 Operationson Matrices

. . . .
3.3 OrthonormalBases .
3.4 Changeof Basis and SimilarityTransformation .
3.5 TheDeterminant . .

3.6 TheTrace
3.7
Problems
4 SpectralDecomposition
4.1 Direct Sums . . . . . . . . . .
4.2 InvariantSubspaces . . . . . .
4.3 EigeuvaluesandEigenvectors .
4.4 SpectralDecomposition
4.5 Functionsof Operators
4.6 PolarDecomposition
4.7 Real VectorSpaces
4.8
Problems
IT
Infinite-Dimensional Vector Spaces
5 HilbertSpaces
5.1 The Questionof Convergence .
5.2 The Space of Square-IntegrableFunctions
5.3
Problems
6 Generalized Functions
6.1 ContinuousIndex
6.2 GeneralizedFunctions .
6.3

Problems
7 Classical Orthogonal Polynomials
7.1 GeneralProperties . .
7.2
Classification
7.3 RecurrenceRelations

7.4 Examplesof ClassicalOrthogonalPolynomials.
56
61
63
67
70
76
82
82
87
89
91
93
101
103
109
109
112
114
117
125
129
130

138
143
145
145
150
157
159
159
165
169
172
172
175
176
179
7.5
7.6
7.7
Expansion in Terms of Orthogonal Polynomials

GeneratingFunctions . . . . . . . . . . . .
Problems .
CONTENTS
xv
186
190
190
8 Fourier Analysis
8.1 Fourier Series

8.2 The FourierTransform .
8.3
Problems
III Complex Analysis
196
196
208
220
225
9 Complex Calculus 227
9.1 ComplexFunctions " 227
9.2 AnalyticFunctions. . . . . . . " 228
9.3 ConformalMaps

. . . . . . . . . . . 236
9A Integration of Complex Functions . . . . . . . . . . . . . " 241
9.5 Derivativesas Integrals 248
9.6 Taylorand Laurent Series . 252
9.7
Problems
263
10 Calculus of Residues
10.1 Residues . . . . . . . . . . . . . . . .
10.2 Classificationof Isolated Singularities
10.3 Evaluation of DefiniteIntegrals . .

lOA Problems .
11 Complex Analysis: Advanced Topics
ILl
MeromorphicFunctions . . .

11.2 MultivaluedFunctions . . . . . . . .
11.3 Analytic Continuation . . . . . . . . .
1104
The Gamma and Beta Functions. . . .
11.5 Method of SteepestDescent . . . . . .
11.6 Problems. . . . . . . . . . . . . . . .
270
270
273
275
290
293
293
295
302
309
312
319
IV DifferentialEquations 325
12 Separation
of
Variables in Spherical Coordinates 327
12.1 PDEs of MathematicalPhysics

. . . . . . . . . . . . . . 327
12.2 Separation of theAngularPart of the Laplacian. . . . . .

331
12.3 Construction of Eigenvaluesof
L

2.
. . . . . . . . . . 334
1204
Eigenvectorsof L
2:
Spherical Harmonics . . 338
12.5
Problems
. . . . . . . . . . . . .
.,
. . . . . . . 346
xvi
CONTENTS
13
Second-Order
Linear
Differential
Equations
348
13.1 General Properties of
ODEs.
. . . . . . . . . . . . 349
13.2 Existence and Uniqneness for First-Order DEs . 350
13.3 General Properties
of
SOLDEs . . . 352
13.4 The Wronskian. . . . . . . . . . . . 355
13.5 Adjoint Differential Operators. . . . 364
13.6 Power-Series Solntions
of

SOLDEs . 367
13.7 SOLDEs with Constant Coefficients 376
13.8 The WKB Method . . . . . . . . . . . . 380
13.9 Numerical Solntions of DEs . . . . . . . 383
13.10
Problems.
. . . . . . . . . . . . . . . . . . . . . . . . . . 394
14 Complex Aualysis
of
SOLDEs 400
14.1 Analytic Properties of Complex DEs 401
14.2 Complex SOLDEs . . . . . . . . . . 404
14.3 Fuchsian Differential Equations . . . 410
14.4 The Hypergeometric Functiou . . . . 413
14.5 Confiuent Hypergeometric Functions 419
14.6
Problems.
. . . . . . . . . . . . . . . . . 426
15
Integral
Transforms
and
Differential
Equations
433
15.1 Integral Representation
of
the Hypergeometric Function . . 434
15.2 Integral Representation
of

the Confiuent Hypergeometric
Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 437
15.3 Integral Representation of Bessel Functions 438
15.4 Asymptotic Behavior of Bessel Functions. 443
15.5
Problems
. . . . . . . . . . . . . . . . . . . . . . . .

445
V Operators on Hilbert Spaces
16 An
Inlroduction
to
Operator
Theory
16.1 From Abstract to Integral and Differential Operators .
16.2 Bounded Operators in Hilbert Spaces . .
16.3 Spectra
of
Linear Operators . .
16.4 Compact Sets .
16.5 Compact Operators .
16.6 Spectrum
of
Compact Operators
16.7 Spectral Theorem for CompactOperators .
16.8 Resolvents
16.9
Problems.
. . . . . . . . . . . . . . . . .

17
Integral
Equations
17.1 Classification.
449
451
451
453
457
458
464
467
473
480
485
488
488
CONTENTS
xvii
17.2 Fredholm Integra!Equations . 494
17.3
Problems
505
18 Sturm-Liouville Systems: Formalism 507
18.1 UnboundedOperatorswith CompactResolvent 507
18.2 Sturm-Liouville Systemsand SOLDEs . . . . 513
18.3 Other Propertiesof Sturm-Liouville Systems . 517
18.4 Problems. . . . . . . . . . . . . . . . . . . . 522
19 Sturm-Lionville Systems: Examples 524
19.1 Expansionsin Termsof Eigenfunctions . 524

19.2 Separationin CartesianCoordinates. . . 526
19.3 Separationin CylindricalCoordinates. . 535
19.4 Separationin SphericalCoordinates. 540
19.5
Problems
545
VI Green's Functions
20 Green's Functions
in
One Dimension
20.1 Calculationof SomeGreen's Functions .
20.2 Formal Considerations . . . . . . . . . .
20.3 Green's Functionsfor SOLDOs . . . . .
20.4 EigenfunctionExpansion of Green's Fnnctions .
20.5
Problems
. . . . . . . . . . . . . . . . . . . . . . . . . .
21 Multidimensional Green's Functions: Formalism
21.1 Properties of Partial DifferentialEquations .
21.2 MultidimensionalGFs and DeltaFunctions .
21.3 Formal Development

21.4 IntegralEquations and GFs
21.5 PerturbationTheory
21.6 Problems. . . . . . . . . .
22 Multidimensional Green's Functions: Applications
22.1 Elliptic Equations . .
22.2 ParabolicEquations . . . . . . . . . . . . . . .
22.3 HyperbolicEquations . . . . . . . . . . . . . .
22.4 The FourierTransformTechnique . . . .

22.5 The EigenfunctionExpansionTechnique
22.6 Problems.






.

551
553
r-
554
557
565
577
580
583
584
592
596
600
603
610
613
613
621
626
628

636
641
xviii
CONTENTS
VII Groups and Manifolds
23 Group Theory
23.1
Groups.
23.2 Subgroups . . .
23.3 Group Action .
23.4 The Symmetric Group
s,
23.5 Problems. . . . . . . . .
24 Group Representation Theory
24.1 Definitionsand Examples
24.2 OrthogonalityProperties. . .
24.3 Analysis of Representations . .
24.4 GroupAlgebra . . . . . . .

. . . . .
24.5 Relationship of Charactersto Those of a Subgroup .
24.6 IrreducibleBasis Functions . . . . . . .
24.7 TensorProduct of Representations

24.8 Representationsof the Symmetric Group
24.9
Problems

.



25 Algebra of Tensors
25.1 Multilinear Mappings
25.2 Symmetries of Tensors .
25.3 Exterior Algebra . . . . .
25.4 hmer Product Revisited .
25.5 The Hodge Star Operator
25.6 Problems. . . . . . . . .
26 Analysis of Tensors
26.1 DifferentiableManifolds. .
26.2 Curves andTangentVectors .
26.3 Differentialof a Map

26.4 TensorFields on Manifolds
26.5 Exterior Calculus

26.6 SymplecticGeometry
26.7 Problems. . . . . . . .
VIII Lie Groups and Their Applications
27 Lie Groups and Lie Algebras
27.1 Lie Groups and Their Algebras . . . . . . .
27.2 An Outlineof Lie Algebra Theory

27.3 Representationof Compact Lie Groups . . .
649
651
652

656
663
664
669
673
-J
673
680
685
687
692
695
699
707
723
728
729
736
739
749
756
758
763
763
770
776
780
791
801
808

813
815
815
833
845
27.4 Representationof the GeneralLinear Group .
27.5 Representationof Lie Algebras
27.6 Problems .
CONTENTS
xix
856
859
876
28
DifferentialGeometry 882
28.1 VectorFields and Curvature . . . . . . 883
28.2 RiemannianManifolds. . . . 887
28.3 CovariantDerivative and Geodesics . 897
28.4 Isometriesand KillingVectorFields 908
28.5 GeodesicDeviationand Curvature . 913
28.6 GeneralTheory of Relativity 918
28.7 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . 932
29 Lie Gronpsand DifferentialEquations 936
29.1 Synunetriesof Algebraic Equations. . . . . . 936
29.2 Synunetry Groupsof DifferentialEquations 941
29.3 The CentralTheorems. . . . . . . 951
29.4 Applicationto Some KnownPDEs 956
29.5 Applicationto ODEs. 964
29.6 Problems. . . . . . . . . . . . . . 970
30 Calcnlusof Variations, Symmetries,and ConservationLaws 973

30.1 The Calculusof Variations. . . . . . . . . . . 973
30.2 SymmetryGroups of VariationalProblems . . . . . . . . 988
30.3 ConservationLaws and Noether's Theorem. . . . . . . . 992
30.4 Applicationto ClassicalField Theory . 997
30.5 Problems. . . . . . . . . . . . . . . . . . . . . . . . . .

1000
Bibliography 1003
Index 1007
I
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j

j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
J
List
of
Symbols
E,
(11')
z
R
R+
C
l\I
ill
~A
AxB
An

U,
«»
A=B
x
f(x)
V
3
[a]
gof
iff
ek(a, b)
Cn(or
JRn)
pClt]
P'[t]
P~[t]
Coo
(al
b)
lIall
"belongs to", ("does not belong to")
Set of integers
Set of real nnmbers
Set of positive real nnmbers
Set of complex nnmbers
Set of nonnegative integers
Set of rational nnmbers
Complement
of
the set A

Set of ordered pairs (a, b) with a E A and b E B
{(aI,
a2,

, an)lat E Al
Union, (Intersection)
A is eqnivalent to B
x is mapped to
f(x)
via the map f
for all (valnes of)
There exists (a valne
of)
Eqnivalence class to which a belongs
Composition
of
maps f and g
if
and only
if
Set of functions on (a, b) with continnons derivatives np to order k
Set of complex (or real) n-tnples
Set of polynomials in
t with complex coefficients
Set of polynomials in
t with real coefficients
Set of polynomials with complex coefficients of degree
n or less
Set of
all complex seqnences

(atl~1
snch that
L:~I
latl
2
<
00
Inner prodnct
of
la) and
Ibl
Norm (length)
of
the vector la)
xxii
LIST
OF
SYMBOLS
£-(V)
[8, T]
Tt
At, or A
l.lE/lV
8(x - xo)
Res[f(zolJ
DE, ODE,
PDE
SOLDE
GL(V)
GL(n,q

SL(n,q
71
®72
AJ\B
AP(V)
Set of endomorphisms (linear operators) on vector space V
Commutator of operators 8 and T
Adjoint (hermitian conjugate) of operator T
Transpose of matrix
A
Direct sum of vector spaces
1.l
and V
Dirac delta function nonvanishing only at
x = xo
Residue
of
f at point zo
Differential equation, Ordinary DE, Partial DE
Second orderlinear (ordinary) differential equation
Set
of
all invertible operators on vector space V
Set
of
all n x n complex matrices
of
nonzero determinant
Set
of

all n x n complex matrices
of
unit determinant
Tensor product
of
71
and
72
Exterior (wedge) product
of
skew-symmetric tensors A and B
Set of
all skew-symmetric tensors of type
(p,
0) on V
0 _
Mathematical
Preliminaries
This introductory chapter gathers together some of the most basic tools and notions
that are used throughout the book.
It
also introduces some common vocabulary
and notations used in modem mathematical physics literature. Readers familiar
with
suchconcepts as sets,
maps,
equivalence
relations,
and
metric

spaces
may
wish to skip this chapter.
0.1.
Sets
Modem mathematics starts with the basic (and undefinable) concept of set. We
think
of
a set as a structureless family, or collection, of objects. We speak, for
example, of the set of students in a college,
of
men in a city,
of
women working
concept
ofset for a corporation, of vectors in space,
of
points in a plane, or
of
events in the
elaborated
continuum
of
space-time. Each member a of a set A is called an element of that
sel. This relation is denoted by a E A (read "a is an element
of
A"
or "a belongs
to
A"),

and its negation by a ¢ A. Sometimes a is called a
point
of
the set A to
emphasize a geometric connotation.
A set is usually designated by enumeration
of
its elements between braces.
For example, {2,4, 6, 8}represents the set consisting
of
the first four even natural
numbers;
{O,
±I,
±2,
±3,

} is the set
of
all integers; {I, x, x
2
,
x
3,

} is the
set of all nonnegative powers
of
x; and {I, i,
-1,

-i}
is the set of the four complex
fourth roots
of
unity.
In
many cases, a set is defined by a (mathematical) statement
that holds for all
ofits
elements. Such a
set
is generally denoted by {x IP
(x)}
and
read "the set of all
x's such that
P(x)
is true." The foregoing examples of sets can
be written alternatively as follows:
{n In is even and I < n < 9}
2
O.
MATHEMATICAL
PRELIMINARIES
{±n In is a natural number}
{y I y =
x"
and n is a uatural uumber}
{z IZ4 = I and z is a complex uumber}
singleton

(proper)
subset
empty
set
union,
intersection,
complement
universal
set
Cartesian
product
ordered
pairs
In a frequently used shorthand uotation, the last two sets
can
be abbreviated as
[x"
I n 2: aand n is an integer} and [z E
iC
I Z4 = I}. Similarly, the uuit circle
can be deuoted by {z
[z] = I}, the closed interval [a, b] as {xla ::; x ::; b}, the
open interval
(a, b) as {x I a < x < b}, and the set of all nonnegative powers
of
x as
{x"}~o'
This last notation will be used frequeutly iu this book. A set with a
single element is called a singleton.
If

a E A whenever a E B, we say that B is a
subset
of
A and write B C A or
A:J
B.
If
Be
A and A c B, then A =
B.1f
Be
A and A
i'
B,
thenB
is called
a
proper
subset of A. The set defined by {ala
i'
a}is
called the
empty
set
and
is denoted by 0. Clearly, 0 contains no elements and is a subset
of
any arbitrary
set. The collection of all subsets (including 0) of a set
A is denoted by 2

A
•
The
reason for this notation is that the number of subsets
of
a set containing n elements
is 2" (Problem
a.I).1f
A and B are sets, their
union,
denoted by A U B, is the set
containing all elements that belong to
A or B or both. The intersection of the sets
.!\ and B, denoted by A n B, is the set containing all elements belonging to both
A and B.
If
{B.}.El
is a collection of sets,
1
we denote their union
by
U.B.
and
their intersection by
n.B
a-
In
any application of set theory there is an underlying
universal
set whose

subsets are the objects
of
study. This universal set is usually clear from the context.
For
exaunple, in the study
of
the properties of integers, the set of integers, denoted
by Z, is the universal set. The set
of
reals,
JR,
is the universal set in real analysis,
and the set of complex numbers,
iC,
is the universal set in complex analysis. With
a universal set X in mind, one can write X
~
A insteadof ~ A. The
complement
of
a set A is denoted by ~ A and defined as
~
A sa {a Ia
Ii!
A}.
The complement of B in A (or their difference) is
A
~
B
==

{ala E A and a
Ii!
B}.
From two given sets A and B, it is possible to form the
Cartesian
prodnct
of A
and B, denoted by A x B, which is the set
of
ordered
pairs
(a, b), where a E A
and b E B. This is expressed in set-theoretic notatiou as
A x B ={(a, b)la E A and b e B}.
1Here I is an index
set or
a counting
set-with
its typical element denoted by ct. In most cases, I is the set
of
(nonnegative)
integers, but, in principle,
it can be any set, for example, the set
of
real numbers.
relation
and
equivalence
relation
0.1

SETS
3
We can generalize this to an arbitrary number
of
sets.
If
AI, A
z,

, An are sets,
then the Cartesian product of these sets is
Al x Az x

x An = {(ai, az,

, an)!ai E
Ad,
which is a set of ordered n-tuples.
If
Al = Az = = An = A, then we write
An instead of A x A
x···
x A, and
An = {(ai, az,

, an) Ia; E Aj.
The most familiar example of a Cartesian product occurs when A = R Then
JRz
is the set of pairs (XI,xz) with XI,xz E
JR.

This is simply the points in the
Euclidean plane. Similarly,
JR3
is the set
of
triplets (XI,xz, X3), or the points in
space, and
JRn
=
{(XI,
Xz,

,
Xn)!Xi
E
JRj
is the set
of
real n-tuples.
0.1.1 Equivalence Relations
There are many instances in which the elements
of
a set are naturally grouped
together. For example, all vector potentials that differ by the gradient of a scalar
function can be grouped together because they all give the same magnetic field.
Similarly, all quantum state functions
(of
unit "length") that differ by a multi-
plicative complex number of unit length
can

be grouped together because they all
represent the same physical state. The abstraction
of
these ideas is summarized in
the following definition.
0.1.1. Definition.
Let
A be a set. A relation on A is a comparison test between
ordered pairs
of
elements
of
A.
If
the pair (a, b) E A x A pass this test, we write
at>
b and read "a is related to
b"
An equivalence relation an A is a relation that
has the fallowing properties:
af>a
V'aEA,
a s-b
~
b s a
a.b
e A,
a
i-b.b»
c

==>-
a[>c
a.b;c
E A,
(reflexivity)
(symmetry)
(transivity)
When a t> b, we say that "a is equivalent to
b"
The set
[a]
= {b E
Alb
t> aj
of
all
equivalence
class elements that are equivalent to a is called the equivalence class
of
a.
The reader may verify the following property of equivalence relations.
0.1.2. Proposition.
If
» is an equivalence relation an A and a, b E A, then either
[a]
n
[b]
= 0 or
[a]
=

[bl
representative
ofan
equivalence
class
Therefore,
a' E
[a]
implies that
[a']
=
[a].
In
other words, any element
of
an equivalence class
can
be chosen to be a
representative
of
that class. Because
of
the symmetry
of
equivalence relations, sometimes we denote them by
c-o.
4
O.
MATHEMATICAL
PRELIMINARIES

0.1.3.
Example.
Let A bethe setof humanbeings.Leta
»b
beinterpretedas"a is older
than b." Then clearly, I> is a relation but not an equivalence relation.
On
the other hand,
if
we interpret a
E>
b as "a and b have the same paternal grandfather," thenl> is an equivalence
relation, as the reader
may
check. The equivalence class
of
a is the set of all grandchildren
of
a's paternal
grandfather.
Let
V be the set of vector potentials. Write A l> A'
if
A -
A'
= V f for some function
f. The reader may verify
that"
is an equivalence relation.and that
[A]

is the set of all
vector potentials giving rise to the
same
magnetic field.
Let the underlying set be Z x (Z -
{OJ).
Say "(a, b) is relatedto (c,
d)"
if ad = be.
Thenthis relation is an equivalence relation. Furthermore,
[(a,
b)]
can
be identified as the
ratio
a/b.
l1li
0.1.4. Definition.
Let
A be a set
and
{R
a}
a collection
of
subsets
of
A. Wesay that
partition
ofa set {R

a}
is a partition
of
A, or {R
a}
partitions A, if the R
a'
s are disjoint, i.e., have
noelementin
common,
and
Uo;B
a
= A.
Now consider the collection
{[a]
Ia E A}
of
all equivalence classes
of
A.
quotient
set These classes are disjoint, aod evidently their union covers all
of
A. Therefore,
the collection
of
equivalence classes
of
A is a partition

of
A. This collection is
denotedby
A/1><1
aod is calledthe
quotient
set
of
A underthe equivalencerelation
1><1.
0.1.5.
Example.
Let the underlyingsetbe
lll.3.
Definean equivalence relationon
lll.3
by
saying that PI E lR
3
and P2 E
}R3
are equivalent
if
they lie on the
same
line passing through
the origin. Then
]R3
I l><l is the set
of

all lines in space passing through the origin.
If
we
choose the unit vectorwith positive third coordinate along a given line as the representative
of
thatline, then]R3I l><lcan be identified with the upper unit hemisphere.e
]R3
I l><lis called
projective
space
the
projective
space
associated with]R3.
On
the set
IE
of
integers define a relation by writing m
e-
n for m, n E
IE
if
m - n is
divisible by
k,where
k is a fixed integer. Then e-is not only a relation, but an equivalence
relation.
In this case, we have
Z/"

= {[O],
[1],

,
[k
- I]},
as the readeris urged to verify.
For
the equivalence relation defined on
IE
x
IE
of
Example 0.1.3, the set
IE
x
lEI
l><l
can
be identified with
lQ,
the set
of
rational numbers.
II
0.2 Maps
map,
domain,
codomain,
image

To communicate between sets, one introduces the concept
of
a map. A
map
f
from a set X to a set Y, denoted by f : X
->
Y
or
X
~
Y, is a correspondence
between elements
of
X aod those
of
Y in which all the elements
of
X participate,
2Purthermore, we need to identify any two points on the edge of the hemisphere which lie on the same diameter.
0.2 MAPS 5
Figure 1 The map f maps all of the set X onto a subset of Y. The shaded areain Y is
f(K),
therangeof
f.
and each element
of
X corresponds to only one element
of
Y (see Figure 1).

If
y E Y is the element that corresponds to x E X
via
the map
f,
we write
y =
f(x)
or x f >
f(x)
or
and call
f (x) the
image
of
x onder
f.
Thus, by the definition
of
map, x E X can
have only one image.
The
set X is called the
domain,
and Y the
codomain
or
the
target
space. Two maps f : X

>
Y and g : X > Y are said to be equal if
function
f(x)
=
g(x)
for all x E X.
0.2.1. Box. A map whose codomain is the set
of
real numbers
IR
or the set
of
complex numbers
iC
is commonly called afunction.
A special map that applies to all sets A is idA : A > A, called the
identity
identity
map
map
of
A, and defined by
VaEA.
graph
ofa
map
The
graph
r f

of
a map f : A > B is a subset
of
Ax
B defined by
r f = {(a,
f(a))
Ia E A} C A x B.
This general definition reduces to the ordinary graphs encounteredin algebra and
calculus where
A = B =
IR
and A x B is the xy-plane.
If
A is a subset
of
X,
we call
f(A)
=
{f(x)lx
E A} the image
of
A. Similarly,
if
B C
f(X),
we call
preimage
f-

1(B)
= {x E
Xlf(x)
E B) the inverse image, or
preimage,
of
B. In words,
f-
1
(B) consists
of
all elements in X whose images are in B C
Y.1f
B consists
of
a single element b, then
r:'
(b) = {x E
Xlf(x)
= b) consists
of
all elements
of
X that are mapped to b. Note that it is possible for many points
of
X to have
the same image in
Y.
The
subset

f(X)
of
the codomain
of
a
map
f is called the
range
of
f (see Figure 1).