Methods of Mathematical Physics II

260 Pages·2012·1.62 MB·English

Methods of Mathematical Physics II
A set of lecture notes by
Michael Stone
PIMANDER-CASAUBON
Alexandria Florence London
(cid:15) (cid:15)
ii
Copyright c 2001,2002,2003 M. Stone.
(cid:13)
All rights reserved. No part of this material can be reproduced, stored or
transmitted without the written permission of the author. For information
contact: Michael Stone, Loomis Laboratory of Physics, University of Illinois,
1110 West Green Street, Urbana, IL 61801, USA.
Preface
These notes cover the material from the second half of a two-semester se-
quence of mathematical methods courses given to (cid:12)rst year physics graduate
students at the University of Illinois. They consist of three loosely connected
parts: i) an introduction to modern \calculus on manifolds", the exterior
di(cid:11)erential calculus, and algebraic topology; ii) an introduction to group rep-
resentation theory and its physical applications; iii) a fairly standard course
on complex variables.
iii
iv PREFACE
Contents
Preface iii
1 Vectors and Tensors 1
1.1 Vector Space . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Bases and Components . . . . . . . . . . . . . . . . . . 2
1.1.3 Dual Space . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.4 Inner Product and the Metric Tensor . . . . . . . . . . 5
1.2 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.1 Transformation Rules . . . . . . . . . . . . . . . . . . . 6
1.2.2 Tensor Product Spaces . . . . . . . . . . . . . . . . . . 8
1.2.3 Symmetric and Skew-symmetric Tensors . . . . . . . . 12
1.2.4 Tensor Character of Linear Maps and Quadratic Forms 15
1.2.5 Numerically Invariant Tensors . . . . . . . . . . . . . . 17
1.3 Cartesian Tensors . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.3.1 Stress and Strain . . . . . . . . . . . . . . . . . . . . . 19
1.3.2 The Maxwell Stress Tensor . . . . . . . . . . . . . . . . 25
2 Tensor Calculus on Manifolds 27
2.1 Vector Fields and Covector Fields . . . . . . . . . . . . . . . . 27
2.2 Di(cid:11)erentiating Tensors . . . . . . . . . . . . . . . . . . . . . . 32
2.2.1 Lie Bracket . . . . . . . . . . . . . . . . . . . . . . . . 32
2.2.2 Lie Derivative . . . . . . . . . . . . . . . . . . . . . . . 36
2.3 Exterior Calculus . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.3.1 Di(cid:11)erential Forms . . . . . . . . . . . . . . . . . . . . . 38
2.3.2 The Exterior Derivative . . . . . . . . . . . . . . . . . 40
2.4 Physical Applications . . . . . . . . . . . . . . . . . . . . . . . 43
2.4.1 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . 43
v
vi CONTENTS
2.4.2 Hamilton’s Equations . . . . . . . . . . . . . . . . . . . 47
2.5 * Covariant Derivatives . . . . . . . . . . . . . . . . . . . . . . 51
2.5.1 Connections . . . . . . . . . . . . . . . . . . . . . . . . 51
2.5.2 Cartan’s Viewpoint: Local Frames . . . . . . . . . . . 52
3 Integration on Manifolds 53
3.1 Basic Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.1.1 Line Integrals . . . . . . . . . . . . . . . . . . . . . . . 53
3.1.2 Skew-symmetry and Orientations . . . . . . . . . . . . 54
3.2 Integrating p-Forms . . . . . . . . . . . . . . . . . . . . . . . . 56
3.2.1 Counting Boxes . . . . . . . . . . . . . . . . . . . . . . 56
3.2.2 General Case . . . . . . . . . . . . . . . . . . . . . . . 57
3.3 Stokes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.4 Pull-backs and Push-forwards . . . . . . . . . . . . . . . . . . 62
4 Topology I 67
4.1 A Topological Miscellany. . . . . . . . . . . . . . . . . . . . . 67
4.2 Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.2.1 Retractable Spaces: Converse of Poincar(cid:19)e Lemma . . . 69
4.2.2 De Rham Cohomology . . . . . . . . . . . . . . . . . . 71
4.3 Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.3.1 Chains, Cycles and Boundaries . . . . . . . . . . . . . 72
4.3.2 De Rham’s Theorem . . . . . . . . . . . . . . . . . . . 76
4.4 Hodge Theory and the Morse Index . . . . . . . . . . . . . . . 80
4.4.1 The Laplacian on p-forms . . . . . . . . . . . . . . . . 81
4.4.2 Morse Theory . . . . . . . . . . . . . . . . . . . . . . . 85
5 Groups and Representation Theory 87
5.1 Basic Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.1.1 Group Axioms . . . . . . . . . . . . . . . . . . . . . . . 87
5.1.2 Elementary Properties . . . . . . . . . . . . . . . . . . 89
5.1.3 Group Actions on Sets . . . . . . . . . . . . . . . . . . 93
5.2 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.2.1 Reducibility and Irreducibility . . . . . . . . . . . . . . 96
5.2.2 Characters and Orthogonality . . . . . . . . . . . . . . 98
5.2.3 The Group Algebra . . . . . . . . . . . . . . . . . . . . 101
5.3 Physics Applications . . . . . . . . . . . . . . . . . . . . . . . 104
5.3.1 Vibrational spectrum of H O . . . . . . . . . . . . . . 104
2
CONTENTS vii
5.3.2 Crystal Field Splittings . . . . . . . . . . . . . . . . . . 108
6 Lie Groups 111
6.1 Matrix Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.1.1 Unitary Groups and Orthogonal Groups . . . . . . . . 112
6.1.2 Symplectic Groups . . . . . . . . . . . . . . . . . . . . 113
6.2 Geometry of SU(2) . . . . . . . . . . . . . . . . . . . . . . . . 116
6.2.1 Invariant vector (cid:12)elds . . . . . . . . . . . . . . . . . . . 118
6.2.2 Maurer-Cartan Forms . . . . . . . . . . . . . . . . . . 120
6.2.3 Euler Angles . . . . . . . . . . . . . . . . . . . . . . . . 122
6.2.4 Volume and Metric . . . . . . . . . . . . . . . . . . . . 123
6.2.5 SO(3) SU(2)=Z . . . . . . . . . . . . . . . . . . . . 125
2
’
6.2.6 Peter-Weyl Theorem . . . . . . . . . . . . . . . . . . . 129
6.2.7 Lie Brackets vs. Commutators . . . . . . . . . . . . . . 131
6.3 Abstract Lie Algebras . . . . . . . . . . . . . . . . . . . . . . 132
6.3.1 Adjoint Representation . . . . . . . . . . . . . . . . . . 134
6.3.2 The Killing form . . . . . . . . . . . . . . . . . . . . . 135
6.3.3 Roots and Weights . . . . . . . . . . . . . . . . . . . . 136
6.3.4 Product Representations . . . . . . . . . . . . . . . . . 143
7 Complex Analysis I 145
7.1 Cauchy-Riemann equations . . . . . . . . . . . . . . . . . . . . 145
7.1.1 Conjugate pairs . . . . . . . . . . . . . . . . . . . . . . 147
7.1.2 Riemann Mapping Theorem . . . . . . . . . . . . . . . 151
7.2 Complex Integration: Cauchy and Stokes . . . . . . . . . . . . 155
7.2.1 The Complex Integral . . . . . . . . . . . . . . . . . . 155
7.2.2 Cauchy’s theorem . . . . . . . . . . . . . . . . . . . . . 157
7.2.3 The residue theorem . . . . . . . . . . . . . . . . . . . 159
7.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
7.3.1 Two-dimensional vector calculus . . . . . . . . . . . . . 163
7.3.2 Milne-Thomson Circle Theorem . . . . . . . . . . . . . 164
7.3.3 Blasius and Kutta-Joukowski Theorems . . . . . . . . . 165
7.4 Applications of Cauchy’s Theorem . . . . . . . . . . . . . . . . 169
7.4.1 Cauchy’s Integral Formula . . . . . . . . . . . . . . . . 169
7.4.2 Taylor and Laurent Series . . . . . . . . . . . . . . . . 171
7.4.3 Zeros and Singularities . . . . . . . . . . . . . . . . . . 176
7.4.4 Removable Singularities and the Weierstrass-Casorati
Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 179
viii CONTENTS
7.5 Meromorphic functions and the Winding-Number . . . . . . . 180
7.5.1 Principle of the Argument . . . . . . . . . . . . . . . . 181
7.5.2 Rouch(cid:19)e’s theorem . . . . . . . . . . . . . . . . . . . . . 182
7.6 Analytic Functions and Topology . . . . . . . . . . . . . . . . 184
7.6.1 The Point at In(cid:12)nity . . . . . . . . . . . . . . . . . . . 184
7.6.2 Logarithms and Branch Cuts . . . . . . . . . . . . . . 187
7.6.3 Conformal Coordinates . . . . . . . . . . . . . . . . . . 192
8 Complex Analysis II 195
8.1 Contour Integration Technology . . . . . . . . . . . . . . . . . 195
8.1.1 Tricks of the Trade . . . . . . . . . . . . . . . . . . . . 195
8.1.2 Branch-cut integrals . . . . . . . . . . . . . . . . . . . 197
8.1.3 Jordan’s Lemma . . . . . . . . . . . . . . . . . . . . . 199
8.2 The Schwarz Re(cid:13)ection Principle . . . . . . . . . . . . . . . . 203
8.2.1 Kramers-Kronig Relations . . . . . . . . . . . . . . . . 206
8.2.2 Hilbert transforms . . . . . . . . . . . . . . . . . . . . 209
8.3 Partial-Fraction and Product Expansions . . . . . . . . . . . . 211
8.3.1 Mittag-Le(cid:15)er Partial-Fraction Expansion . . . . . . . . 211
8.3.2 In(cid:12)nite Product Expansions . . . . . . . . . . . . . . . 212
8.4 Wiener-Hopf Equations Made Easy . . . . . . . . . . . . . . . 214
8.4.1 Wiener-Hopf Sum Equations and the Index Theorem . 214
9 Special Functions II 221
9.1 The Gamma Function . . . . . . . . . . . . . . . . . . . . . . 221
9.2 Linear Di(cid:11)erential Equations . . . . . . . . . . . . . . . . . . . 226
9.2.1 Monodromy . . . . . . . . . . . . . . . . . . . . . . . . 226
9.2.2 Hypergeometric Functions . . . . . . . . . . . . . . . . 227
9.3 Solving ODE’s via Contour integrals . . . . . . . . . . . . . . 231
9.3.1 Bessel Functions . . . . . . . . . . . . . . . . . . . . . 233
9.4 Asymptotic Expansions . . . . . . . . . . . . . . . . . . . . . . 237
9.4.1 Stirling’s Approximation for n! . . . . . . . . . . . . . 239
9.4.2 Airy Functions . . . . . . . . . . . . . . . . . . . . . . 240
9.5 Elliptic Functions . . . . . . . . . . . . . . . . . . . . . . . . . 246
Chapter 1
Vectors and Tensors
Linear algebra is one of the most useful branches of mathematics for a physi-
cist. We (cid:12)nd ourselves dealing with matrices and tensors far more frequently
than with di(cid:11)erential equations. Often you will have learned about vectors
and matrices by example, and by intuitive absorption of the concepts. This
mode of learning will only take you so far, however. Sometimes even a physi-
cist has to actually prove something. For this one needs a somewhat more
formal approach.
In this chapter we will attempt to describe material that is probably
familiar to you, but in slightly more \mathematical" style than is customary
in physics courses. Our aim is to provide a more structured approach to the
subject and to give you some familiarity with the notions and notations that
you will meet if you have to look something up in a mathematics book. This
background will alsoprovide you with the toolsand languageyou will require
to understand the calculus of di(cid:11)erential forms, which will occupy the second
and third chapters of these notes.
1.1 Vector Space
We begin with a rapid review of real vector spaces.
1.1.1 Axioms
A vector space V over the (cid:12)eld of the real numbers (denoted by R) is a
set with a binary operation, vector addition which assigns to each pair of
1
2 CHAPTER 1. VECTORS AND TENSORS
elements x, y V a third element denoted by x+y, and another operation
2
scalar multiplication which assigns to an element x V and (cid:21) R a new
2 2
element (cid:21)x V. There is also a distinguished element 0 V such that the
2 2
following axioms are obeyed. In this list 1, (cid:21), (cid:22); R and x, y;0 V.
2 2
1) Vector addition is commutative: x+y = y+x.
2) Vector addition is associative: (x+y)+z = x+(y+z).
3) Additive identity: 0+x = x.
4) Existence of additive inverse: x V; ( x) V, such that x +
8 2 9 (cid:0) 2
( x) = 0.
(cid:0)
5) Scalar distributive law i) (cid:21)(x+y) = (cid:21)x+(cid:21)y.
6) Scalar distributive law ii) ((cid:21)+(cid:22))x = (cid:21)x+(cid:22)x.
7) Scalar multiplicative associativity: ((cid:21)(cid:22))x = (cid:21)((cid:22)x).
8) Multiplicative identity: 1x = x.
The elements of V are called real vectors.
You have no doubt been working with vectors for years, and are saying
to yourself \I know this stu(cid:11)". Perhaps so, but try the following exercise |
not just to prove what is asked, but also to understand why these \obvious"
things are not quite obvious.
Exercise: Use the axioms to show that:
i) If x+0~ =x, then 0~ =0.
ii) We have 0x=0 for any x V. Here 0 is the additive identity in R.
2
iii) If x+y =0, then y = x. Thus the additive inverse is unique.
(cid:0)
iv) Givenx,yinV,thereisauniquezsuchthatx+z =y,towhitz = x y.
(cid:0)
v) (cid:21)0 =0 for any real (cid:21).
vi) If (cid:21)x =0, then either x =0 or (cid:21)=0.
vii) ( 1)x = x.
(cid:0) (cid:0)
1.1.2 Bases and Components
Let V be a vector space V over the (cid:12)eld of real numbers R. For the moment
this space has no additional structure beyond that of the previous section |
no inner product and so no notion of what it means for two vectors to be
orthogonal. There is still much that can be done, though. Here are the most
basic concepts and properties that we will need:
i) A set of vectors e ;e ;:::;e is linearly dependent if there exist (cid:21)(cid:22),
1 2 n
f g
not all zero, such that
(cid:21)1e +(cid:21)2e + +(cid:21)ne = 0:
1 2 n
(cid:1)(cid:1)(cid:1)