TEXTS AND READINGS IN PHYSICAL SCIENCES - 6

Linear Algebra and Group Theory for Physicists

Second Edition

Texts and Readings in Physical Sciences

Managing Editors H. S. Mani, Institute of Mathematical Sciences, Chennai. h_s_mani@hotmail.com

Ram Ramaswamy, Jawaharlal Nehru University, New Delhi. r.ramaswamy@mail.jnu.ac.in

Editors V Balakrishnan, Indian Institute of Technology, Madras, Chennai. vbalki@chaos.iitm.ernet.in

Jayanta Bhattacharjee, Indian Assoc. for the Cultivation of Science, Kolkata. tpjkb@mahendra.iacs.res.in

Deepak Dhar, Tata Institute of Fundamental Research, Mumbai. ddhar@theory.tifr.res.in

Rohini Godbole, Indian Institute of Science, Bangalore. rohini@cts.iisc.ernet.in

Avinash Khare, Institute of Physics, Bhubaneswar. khare@iopb.res.in.

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Linear Algebra and Group Theory for Physicists

Second Edition

K. N. Srinivasa Rao

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ISBN 978-81-85931-64-7 ISBN 978-93-86279-32-3 (eBook) DOI 10.1007/978-93-86279-32-3

Texts and Readings in the Physical Sciences

As subjects evolve, and the teaching and study of a subject evolves, new texts are needed to provide material and to define areas of research. The TRiPS series of books is an effort to document these frontiers in the Physical Sciences.

One of the principal aims of the series is to make expOSitions of current topics accessible to the graduate student and researcher through Textbooks and Monographs. In addition, we also feel that publication of lecture notes emanating from a thematic School or Workshop, and topical volumes of contributed articles can go a long way in providing an insight into a rapidly developing field, or an introduction to a new area.

The pedagogical value of all these forms of exposition is inestimable. We thus hope, in this series of books, to both provide a forum for the physical scientist to give a personal account and definition of his field, and a source of valuable learning material for the student wishing to gain insight and knowledge

H.S. Mani Chennai

R.Ramaswamy New Delhi

Preface to the se co nd edition This introduction to Linear Algebra and Group Theory will, it is

hoped, serve as a stepping stone to students of Physics, in particular of Theoretical Physics and possibly also of Mathematics, who wish to pur­sue advanced study and research. The first few chapters on elementary Group Theory and Linear Vector Spaces are included to make the book self-contained and mayaIso be used as instructional material for grad­uate and under-graduate classes. Since there are several good books on application of Group Theory to Physical Problems, the emphasis here is almost entirely on the theory which is presented with enough care and detail to enable the student to acquire a reasonably sound grasp of the fundament als.

Of the topics discussed in this book special mention may perhaps be made of the representation of Theory of Linear Associative Algebras. With a fuIl analysis of the ideal resolution and the determination of the irreducible representations of the Dirac and Kemmer algebras, the repre­sentations of the symmetrie group via Young tableaux with application to the construction of the symmetry classes of tensors useful in the study of assemblies of identical particles, a systematic derivation of the 32 crys­taIlographic point-groups, and exhaustive discussion of the structure and representations of the Lorentz group and abrief introduction to Dynkin diagrams in the classification of Lie groups. Wigner's derivation of the Clebsch-Gordan coefficients, with a minor simplification has also been included in an Appendix to Chapter 8 on the Rotation group and its representations.

Acknowledgement I am very happy that I have a second opportunity to express my deep

sense of gratitude to my teacher Professor K. Venkatachaliyengar whose course of lectures given exclusively to me has provided the basic material from which this book has evolved. It is also a pleasure to acknowledge the contributions of my students D. Saroja, A.V. Gopala Rao and B.S. Narahari in our joint work on the structure and representations of the Lorentz group which forms a significant part of Chapter 10. I am par­ticularly thankful to D. Saroja for making available to me, her work on the Kemmer Algebra which has been chosen as an excellent non-trivial example illustrating all the aspects of the representation theory of Linear Associative Algebras discussed in Chapter 6.

I am particularly grateful to Professors M.V.N. Murthy, V. Ravis­hankar and A.R. Usha Devi for bringing out this new issue, wading

viii Linear Algebra and Group Theory

through the text with a fine-toothed comb eliminating all the typograph­ical errors and to Smt. Ambika Vanchinathan for generating its excellent typescript . I gratefully acknowledge the generosity of the Infosys Foun­dation in supporting the publication of the second edition of the book.

13/2 (New no.39), 11th Main, 13th cross, Malleswaram, Bangalore-560003

K.N. Srinivasa Rao

Contents

Preface

1 Elements of Group Theory 1.1 Set-theoretic Preliminaries . 1.2 Groups ....... . ... . 1.3 Aigebraic Operations in a Group 1.4 Some Subgroups of a Given Group G . 1.5 Co sets ...... . .......... . 1.6 The Class of Conjugates of a Complex K 1. 7 The Direct Product of Two Groups . 1.8 Homomorphism and Isomorphism .

2 Some Related Aigebraic Structures 2.1 Ring ..... 2.2 Division Ring .. . . 2.3 Field ........ . 2.4 2.5

2.6

Linear Vector Space Linear Associative Algebra: Hyper Complex System . . . . Lie-ring and Lie-algebra

3 Linear Vector Space 3.1 Definition .... 3.2 3.3 3.4 3.5 3.6 3.7

Linear Dependence and Independence of Vectors Change of Basis . . . . . . . . Subspace . . ... . ..... . Isomorphism of Vector Spaces On the Matrix Product Rule The Rank of a Matrix . . . .

vii

1 1 2

14 15 17 19 24 25

31 31 34 34 35

35 37

39 39 41 47 50 54 55 57

x Linear Algebra and Group Theory

3.8 3.9 3.10 3.11 3.12

3.13 3.14

3.15 3.16 3.17

3.18 3.19

3.20

Linear Transformation . . . Sum and Produet of Operators Effeet of Change of Basis. . . . Active and Passive Points of View The Range and Kernel of a Linear Transformation . . . . . . . . . . . Linear Transformation of Rn to sm Invariant Subspaee-Eigenvalues and Eigenveetors. . . . . . . . . Euelidean Spaee. . . . . . . The Schur Canonieal Form * Thc Direct Produet of Two Veetor Spaees - The Kronecker Product Spaee ......... . The Matrix Exponential* Some Properties of Hermitian and U nitary Matriees . . . . . . The Dirae Bra-ket Notation ....

4 Elements of Representation Theory 4.1 Definition of a Representation ... 4.2 Schur Lemma . . . . . . . . . . . . 4.3 4.4

4.5

Representations of the Dirae Algebras C 2 and C 4

Elements of Representation of Linear Groups* ......... . Generalised Schur Lemma . . .

5 Representations of Finite Groups 5.1 Unitarity of a Representation 5.2 Orthogonality Relations ... 5.3 Irreps of Some Finite Groups

6 Representations of Linear Associative Algebras 6.1 Simple and Semi-Simple Algebras 6.2 Operator Homomorphism . . . 6.3 The Fundamental Theorem of

6.4 6.5

Semi-Simple Algebras* . . . . . . . . . . . Deeomposition of n into Twosided Ideals Ideal Resolution and Irreps of the Dirae and Kemmer Algebras . . . . . . . . . . .

64 71 72 74

75 76

78 87 93

98 105

111 118

123 123 129 133

139 147

151 151 155 171

179 179 182

183 191

202

Elements oE Group' Theory xi

7 Representations of the Symmetrie Group 7.1 The Characteristic of aPermutation 7.2 The Number of Elements in a Class . 7.3 The Young Tableaux ... 7.4 Lemmas for the Tableaux ..... . 7.5 Young's Theorem .......... . 7.6 The Irreducible Representations: The

Standard Tableaux . . . . . . . . 7.7 Reciprocity between the Irreps of

GL(n, c) and S f ........ .

231 231 233 234 240 243

247

254

8 The Rotation Group and its Representations 273 8.1 Rotation Matrix in Terms ofAxis

and Angle . . . . . . . . . . . . . . . . . . . . . . . . . . 273 8.2 The Angle and Axis of an Arbitrary

Proper Orthogonal Matrix . . . . . . 278 8.3 The Eigenvalues of a Rotation Matrix 279 8.4 The Canonical Form of a Rotation

8.5 8.6 8.7

8.8 8.9 8.10

8.11 8.12 8.13

8.14 8.15

8.16

8.17

8.18 8.19

Matrix .............. . The Euler Resolution of Rotation Quaternions and Rotations Stereographie Projection and the SU(2) Representation ... Invariant Integration . . . . . . . Irreps of the Algebra 80(3) ... Exponentiation of the Infinitesimal Operators ............. . The Character Formula ..... . The Dj Representation through SU(2) . Orthogonality and Completeness of the D-functions ............ . Additional Properties of the Dj Irreps Representations in Function Space: Irreducible Tensors ..... . Differential Operators for the Infinitesimal Transformations -Spherical Functions* Kronecker Product Representation: Clebsch-Gordan Theorem . . Clebsch-Gordan Coefficients . The Wigner-Eckart Theorem

280 283 291

298 301 304

313 322 322

328 330

336

340

345 350 355

xii Linear Algebra and Group Theory

8.20 Appendix . . . . . . . . . . . . . . . . . . . . . . 359 8.20.1 Wigner's Derivation of the C-G Coefficients 359

9 The Crystallographic Point Groups 9.1 Preliminaries ........... . 9.2 Finite Dimensional Subgroups of 80(3)* . 9.3 The Crystallographic Point Groups

(First Kind) ............. . 9.4 The Crystallographic Point Groups

(Second Kind)* .......... . 9.5 The Character Tables of the Point

Groups ............... .

10 The Lorentz Group and its Representations 10.1 TheLorentz Transformation. 10.2 Minkowski Space ........ . 10.3 The Lorentz Group ....... . 10.4 Eigenvalues and Eigenvectors of

an OPLT ............ . 10.5 Planar Transformations .... . 10.6 Canonical Forms of Planar OPLTs 10.7 The Canonical Form of an Arbitrary

Non-Null OPLT* ......... . 10.8 Synge's Physical Interpretation of

Null and Non-Null OPLTs ..... 10.9 OPLT as a Polynomial in the ILT . 10.10 Determination of the Blades of a

Screw-like OPLT ......... . 10.11 Planar Resolutions of an OPLT .. 10.12 Complex Lie-Cartan Parameters of

80(3,1) ............ . 10.13 Quaternions and OPLT's .... . 10.14 The 8L(2, C) Representation of

80(3,1) ............ . 10.15 Spinors .............. . 10.16 The 80(3, C) Representation of

80(3,1) ............ . 10.17 The Finite Dimensional Irreps of

80(3,1) ............ . 10.18 Irreps of (80(3,1) in General- The

Gelfand-Naimark Basis ........ .

367 367

· 369

· 384

385

396

407 · 407 · 413 · 420

· 427 · 434 · 447

· 451

· 458 · 459

· 462 · 468

· 483 · 490

502 507

514

521

544

Elements oE Group Theory xiii

11 Introduction to the Classification of Lie Groups - Dynkin Diagram 567 11.1 Preliminaries . . . . . . . . . 567 11.2 Complex Extension of aReal

Lie-Algebra . . . . . . . . . . 568 11.3 Simple and Semisimple Lie Algebras 569 11.4 Cartan's Criterion for a Lie Algebra to be Semisimple 570 11.5 The Adjoint Representation. . . . . 572 11.6 Classification of Lie Groups; Dynkin

Diagrams ............... 574

Index 587