Problems and Solutions in Group Theory for Physicists

by Ma, Zhong-Qi; Gu, Xiao-Yan

• ISBN13: 9789812388322
• ISBN10: 981238832X
• Format: Hardcover
• Copyright: 2004-08-01
• Publisher: World Scientific Pub Co Inc

Summary

This book is aimed at graduate students and young researchers in physics who are studying group theory and its application to physics. It contains a short explanation of the fundamental knowledge and method, and the fundamental exercises for the method, as well as some important conclusions in group theory. This book is also suitable for some graduate students in theoretical chemistry.

Table of Contents

Preface	v
1. REVIEW ON LINEAR ALGEBRAS	1	(26)
1.1 Eigenvalues and Eigenvectors of a Matrix	1	(3)
1.2 Some Special Matrices	4	(3)
1.3 Similarity Transformation	7	(20)
2. GROUP AND ITS SUBSETS	27	(16)
2.1 Definition of a Group	27	(2)
2.2 Subsets in a Group	29	(4)
2.3 Homomorphism of Groups	33	(10)
3. THEORY OF REPRESENTATIONS	43	(72)
3.1 Transformation Operators for a Scalar Function	43	(4)
3.2 Inequivalent and Irreducible Representations	47	(18)
3.3 Subduced and Induced Representations	65	(14)
3.4 The Clebsch-Gordan Coefficients	79	(36)
4. THREE-DIMENSIONAL ROTATION GROUP	115	(58)
4.1 SO(3) Group and Its Covering Group SU(2)	115	(8)
4.2 Inequivalent and Irreducible Representations	123	(17)
4.3 Lie Groups and Lie Theorems	140	(6)
4.4 Irreducible Tensor Operators	146	(20)
4.5 Unitary Representations with Infinite Dimensions	166	(7)
5. SYMMETRY OF CRYSTALS	173	(20)
5.1 Symmetric Operations and Space Groups	173	(4)
5.2 Symmetric Elements	177	(9)
5.3 International Notations for Space Groups	186	(7)
6. PERMUTATION GROUPS	193	(76)
6.1 Multiplication of Permutations	193	(4)
6.2 Young Patterns, Young Tableaux and Young Operators	197	(8)
6.3 Primitive Idempotents in the Group Algebra	205	(6)
6.4 Irreducible Representations and Characters	211	(26)
6.5 The Inner and Outer Products of Representations	237	(32)
7. LIE GROUPS AND LIE ALGEBRAS	269	(48)
7.1 Classification of Semisimple Lie Algebras	269	(10)
7.2 Irreducible Representations and the Chevalley Bases	279	(20)
7.3 Reduction of the Direct Product of Representations	299	(18)
8. UNITARY GROUPS	317	(58)
8.1 The SU(N) Group and Its Lie Algebra	317	(4)
8.2 Irreducible Tensor Representations of SU(N)	321	(15)
8.3 Orthonormal Bases for Irreducible Representations	336	(26)
8.4 Subduced Representations	362	(7)
8.5 Casimir Invariants of SU(N)	369	(6)
9. REAL ORTHOGONAL GROUPS	375	(58)
9.1 Tensor Representations of SO(N)	375	(28)
9.2 Spinor Representations of SO(N)	403	(12)
9.3 SO(4) Group and the Lorentz Group	415	(18)
10. THE SYMPLECTIC GROUPS	433	(24)
10.1 The Groups Sp(2l, R) and USp(2l)	433	(7)
10.2 Irreducible Representations of Sp(2l)	440	(17)
Bibliography	457	(4)
Index	461