Mukhi S., Mukunda N. Lectures on Advanced Mathematical Methods for Physicists

Lectures

on

Advanced

Mathematical

Methods

for

Physicists

This page intentionally left blankThis page intentionally left blank

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

Published by

World Scientific Publishing Co. Pte. Ltd.

5 Toh Tuck Link, Singapore 596224

USA office:

27 Warren Street, Suite 401-402, Hackensack, NJ 07601

UK office:

57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data

A catalogue record for this book is available from the British Library.

LECTURES

ON

ADVANCED

MATHEMATICAL METHODS FOR PHYSICISTS

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).

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

6

9

11

12

13

15

17

18

21

25

28

34

41

47

48

50

55

59

62

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

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

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

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

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

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

Coordinat

es

in

a Lie

Group.

9.2 Analysis

of

Associ

at

ivity . . . . . .

Vll

147

148

9.3

On

e

-p

a

rameter

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 Alge

bra

158

9.7

Comm

e

nts

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 Alge

bra.

. . . . . . . . . . . . . 171

11.2 Solvability, Levi's

Theorem,

and

Cartan's

Analysis of Complex

(Semi) Simple

Lie Algebras . . . . . . . 173

11.3

Th

e R

ea

l

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

Positiv

e

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

Th

e USp(2l) Family -

G

l

of

Cartan

. . 203

14.4

The

SU(l

+

1)

Family -

Al

of

Cartan

. 207

14.5 Coincidences for low Dimensions a

nd

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

Th

e

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

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.

Mukhi S., Mukunda N. Lectures on Advanced Mathematical Methods for Physicists

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