Lectures on Advanced Mathematical Methods

for Physicists

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Lectures on Advanced Mathematical Methods

for Physicists

Sunil Mukhi Tata Institute of Fundamental Research, India

N Mukunda formerly of Indian Institute of Science, India

~HINDUSTAN U LQJ U BOOK AGENCY

,~World Scientific NEW JERSEY. LONDON· SINGAPORE· BEIJING· SHANGHAI· HONG KONG· TAIPEI· CHENNAI

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LECTURES ON ADVANCED MATHEMATICAL METHODS FOR PHYSICISTS

Copyright © 2010 Hindustan Book Agency (HBA)

Authorized edition by World Scientific Publishing Co. Pte. Ltd. for exclusive distribution worldwide except India.

The distribution rights for print copies of the book for India remain with Hindustan Book Agency (HBA).

All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

ISBN-13 978-981-4299-73-2 ISBN-1O 981-4299-73-1

Printed in India, bookbinding made in Singapore.

Contents

Part I: Topology and Differential Geometry

Introduction to Part I

1 Topology 1.1 Preliminaries 1.2 Topological Spaces 1.3 Metric spaces . . . 1.4 Basis for a topology 1.5 Closure ...... . 1.6 Connected and Compact Spaces. 1.7 Continuous Functions 1.8 Homeomorphisms. 1.9 Separability

2 Homotopy 2.1 Loops and Homotopies ....... . 2.2 The Fundamental Group ...... . 2.3 Homotopy Type and Contractibility 2.4 Higher Homotopy Groups ..... .

3 Differentiable Manifolds I 3.1 The Definition of a Manifold 3.2 Differentiation of Functions 3.3 Orient ability ........ . 3.4 Calculus on Manifolds: Vector and Tensor Fields 3.5 Calculus on Manifolds: Differential Forms 3.6 Properties of Differential Forms. 3.7 More About Vectors and Forms .

4 Differentiable Manifolds II 4.1 Riemannian Geometry ..

1

3

5 5 6

9 11 12 13 15 17 18

21 21 25 28 34

41 41 47 48 50

55 59 62

65 65

VI

4.2 Frames ................ . 4.3 Connections, Curvature and Torsion 4.4 The Volume Form ....... . 4.5 Isometry....... ...... . 4.6 Integration of Differential Forms 4.7 Stokes'Theorem .... 4.8 The Laplacian on Forms . .

Contents

67 69 74 76 77 80 83

5 Homology and Cohomology 87 87

100 103

5.1 Simplicial Homology ............ . 5.2 De Rham Cohomology . . . . . . . . . . . . 5.3 Harmonic Forms and de Rham Cohomology

6 Fibre Bundles 6.1 The Concept of a Fibre Bundle 6.2 Tangent and Cotangent Bundles ... 6.3 Vector Bundles and Principal Bundles

Bibliography for Part I

105 105 111 112

117

Part II: Group Theory and Structure and Representations of Com-pact Simple Lie Groups and Algebras 119

Introduction to Part II

7 Review of Groups and Related Structures 7.1 Definition of a Group ........... . 7.2 Conjugate Elements, Equivalence Classes . 7.3 Subgroups and Cosets ........... . 7.4 Invariant (Normal) Subgroups, the Factor Group 7.5 Abelian Groups, Commutator Subgroup ..... 7.6 Solvable, Nilpotent, Semi simple and Simple Groups. 7.7 Relationships Among Groups ............ .

121

123 123 124 124 125 126 127 129

7.8 Ways to Combine Groups - Direct and Semidirect Products 131 7.9 Topological Groups, Lie Groups, Compact Lie Groups 132

8 Review of Group Representations 8.1 Definition of a Representation ........... . 8.2 Invariant Subspaces, Reducibility, Decomposability

135 135 136

8.3 Equivalence of Representations, Schur's Lemma . . 138 8.4 Unitary and Orthogonal Representations. . . . . . 139 8.5 Contragredient, Adjoint and Complex Conjugate Representations 140 8.6 Direct Products of Group Representations . . . . . . . . . . . . . 144

Contents

9 Lie Groups and Lie Algebras 9.1 Local Coordinates in a Lie Group. 9.2 Analysis of Associativity . . . . . .

Vll

147 147 148

9.3 One-parameter Subgroups and Canonical Coordinates 151 9.4 Integrability Conditions and Structure Constants . . . 155 9.5 Definition of a (real) Lie Algebra: Lie Algebra of a given Lie Group157 9.6 Local Reconstruction of Lie Group from Lie Algebra 158 9.7 Comments on the G --) G Relationship . . . . . . . 160 9.8 Various Kinds of and Operations with Lie Algebras. 161

10 Linear Representations of Lie Algebras 165

11 Complexification and Classification of Lie Algebras 171 11.1 Complexification of a Real Lie Algebra. . . . . . . . . . . . . . 171 11.2 Solvability, Levi's Theorem, and Cartan's Analysis of Complex

(Semi) Simple Lie Algebras . . . . . . . 173 11.3 The Real Compact Simple Lie Algebras . . . . . . . . . 180

12 Geometry of Roots for Compact Simple Lie Algebras 183

13 Positive Roots, Simple Roots, Dynkin Diagrams 189 13.1 Positive Roots ........ . .. 189 13.2 Simple Roots and their Properties 189 13.3 Dynkin Diagrams. . . . . . . . . . 194

14 Lie Algebras and Dynkin Diagrams for SO(2l), SO(2l+1), USp(2l), SU(l + 1) 197 14.1 The SO(2l) Family - Dl of Cartan. . . 197 14.2 The SO(2l + 1) Family - Bl of Cartan 201 14.3 The USp(2l) Family - Gl of Cartan . . 203 14.4 The SU(l + 1) Family - Al of Cartan . 207 14.5 Coincidences for low Dimensions and Connectedness 212

15 Complete Classification of All CSLA Simple Root Systems 215 15.1 Series of Lemmas . . . . 216 15.2 The allowed Graphs r . 220 15.3 The Exceptional Groups 224

16 Representations of Compact Simple Lie Algebras 227 16.1 Weights and Multiplicities . . . . . . . . . . . . 227 16.2 Actions of En and SU(2)(a) - the Weyl Group 228 16.3 Dominant Weights, Highest Weight of a UlR 230 16.4 Fundamental UIR's, Survey of all UIR's 233 16.5 Fundamental UIR's for AI, B l , Gl, Dl . . . . . 234

VIll

16.6 The Elementary UIR's .... . . 16.7 Structure of States within a UIR

Contents

240 241

17 Spinor Representations for Real Orthogonal Groups 245 17.1 The Dirac Algebra in Even Dimensions. . . . . . . . . . . . .. 246 17.2 Generators, Weights and Reducibility of U(S) - the spinor UIR's

of Dl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 17.3 Conjugation Properties of Spinor UIR's of Dl . . . . . . 250 17.4 Remarks on Antisymmetric Tensors Under Dl = SO(2l) 252 17.5 The Spinor UIR's of Bl = SO(2l + 1) . . . . . . 257 17.6 Antisymmetric Tensors under Bl = SO(2l + 1) ..... 260

18 Spinor Representations for Real Pseudo Orthogonal Groups 261 18.1 Definition of SO(q,p) and Notational Matters. . . . . 261 18.2 Spinor Representations S(A) of SO(p, q) for p + q == 2l . . . 262 18.3 Representations Related to S(A) . . . . . . . . . . . . . . . 264 18.4 Behaviour of the Irreducible Spinor Representations S±(A) 265 18.5 Spinor Representations of SO(p, q) for p + q = 2l + 1 266 18.6 Dirac, Weyl and Majorana Spinors for SO(p, q) 267

Bibliography for Part II 273

Index 275

Part I: Topology and

Differential Geometry

Sunil Mukhi Department of Theoretical Physics

Tata Institute of Fundamental Research Mumbai 400 005, India

+ J.e. Bose Fellow.

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Introduction to Part I

These notes describe the basic notions of topology and differentiable geometry in a style that is hopefully accessible to students of physics. While in mathematics the proof of a theorem is central to its discussion, physicists tend to think of mathematical formalism mainly as a tool and are often willing to take a theorem on faith. While due care must be taken to ensure that a given theorem is actually a theorem (i.e. that it has been proved), and also that it is applicable to the physics problem at hand, it is not necessary that physics students be able to construct or reproduce the proofs on their own. Of course, proofs not only provide rigour but also frequently serve to highlight the spirit of the result. I have tried here to compensate for this loss by describing the motivation and spirit of each theorem in simple language, and by providing some examples.

The examples provided in these notes are not usually taken from physics, however. I have deliberately tried to avoid the tone of other mathematics-for­physicists textbooks where several physics problems are set up specifically to illustrate each mathematical result. Instead, the attempt has been to highlight the beauty and logical flow of the mathematics itself, starting with an abstract set of points and adding "qualities" like a topology, a differentiable structure, a metric and so on until one has reached all the way to fibre bundles.

Physical applications of the topics discussed here are to be found primarily in the areas of general relativity and string theory. It is my hope that the enterprising student interested in researching these fields will be able to use these notes to penetrate the dense physical and mathematical formalism that envelops (and occasionally conceals!) those profound subjects.

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Chapter 1

Topology

1.1 Preliminaries

Topology is basically the study of continuity. A topological space is an abstract structure defined on a set, in such a way that the concept of continuous maps makes sense. In this chapter we will first study this abstract structure and then go on to see, in some examples, why it is a sensible approach to continuous functions.

The physicist is familiar with ideas of continuity in the context of real analysis, so here we will use the real line as a model of a topological space. For a mathematician this is only one example, and by no means a typical one, but for a physicist the real line and its direct products are sufficient to cover essentially all cases of interest.

In subsequent chapters, we will introduce additional structures on a topo­logical space. This will eventually lead us to manifolds and fibre bundles, the main "stuff" of physics.

The following terms are in common use and it will be assumed that the reader is familiar with them: set, subset, empty set, element, union, intersection, integers, rational numbers, real numbers. The relevant symbols are illustrated below:

subset: c empty set: ¢> element: E

union: U intersection: n set of integers: 7L set of rational numbers: Q set of real numbers: 1R

We need some additional terminology from basic set theory that may also be known to the reader, but we will explain it nevertheless.

Definition: If A c B, the complement of A in B, called A' is

A' = { x E B I x ~A }

6 Chapter 1. Topology

For the reader encountering such condensed mathematical statements for the very first time, let us express the above statement in words: A' is the set of all elements x that are in B such that x is not in A. The reader is encour­aged to similarly verbalise any mathematical statement that is not immediately comprehensible.

We continue with our terminology.

Definition: The Cartesian product of two sets A and B is the set

A @ B = { (a, b) I a E A, bE B }

Thus it is a collection of ordered pairs consisting of one element from each of the original sets.

Example: The Cartesian product lRx lR is called lR2 , the Euclidean plane. One can interate Cartesian products any number of times, for example lRx lRx· .. x lR is the n-dimensional Euclidean space lR n .

We continue by defining functions and their basic properties.

Definition: A function A --> B is a rule which, for each a E A, assigns a unique bE B, called the image of a. We write b = f(a).

A function f: A --> B is surjective (onto) if every b E B is the image of at least one a E A.

A function f: A --> B is injective (one-to-one) if every b E B is the image of at most one a E A.

A function can be surjective, or injective, or neither, or both. If it is both, it is called b~jective. In this case, every b E B is the image of exactly one a E A.

Armed with these basic notions, we can now introduce the concept of a topological space.

1.2 Topological Spaces

Consider the real line lR. We are familiar with two important kinds of subsets of lR:

Open interval: (a, b) = { x E lR I a < x < b }

Closed interval: [a, bj = { x E lR I a :::::: x :::::: b } (1.1 )

A closed interval contains its end-points, while an open interval does not. Let us generalize this idea slightly.

Definition: X C lR is an open set in lR if:

x EX::::} x E (a, b) c X for some (a, b)