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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. [email protected]
Ram Ramaswamy, Jawaharlal Nehru University, New Delhi. [email protected]
Editors V Balakrishnan, Indian Institute of Technology, Madras, Chennai. [email protected]
Jayanta Bhattacharjee, Indian Assoc. for the Cultivation of Science, Kolkata. [email protected]
Deepak Dhar, Tata Institute of Fundamental Research, Mumbai. [email protected]
Rohini Godbole, Indian Institute of Science, Bangalore. [email protected]
Avinash Khare, Institute of Physics, Bhubaneswar. [email protected].
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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