Part Ⅰ Linear Algebra and Tensors 3
1 A Quick Introduction to Tensors 3
2 Vector Spaces 9
2.1 Definition and Examples 9
2.2 Span,Linear Independence,and Bases 14
2.3 Components 17
2.4 Linear Operators 21
2.5 Dual Spaces 25
2.6 Non-degenerate Hermitian Forms 27
2.7 Non-degenerate Hermitian Forms and Dual Spaces 31
2.8 Problems 33
3 Tensors 39
3.1 Definition and Examples 39
3.2 Change of Basis 43
3.3 Active and Passive Transformations 49
3.4 The Tensor Product—Definition and Properties 53
3.5 Tensor Products of V and V＊ 55
3.6 Applications of the Tensor Product in Classical Physics 58
3.7 Applications of the Tensor Product in Quantum Physics 60
3.8 Symmetric Tensors 68
3.9 Antisymmetric Tensors 70
3.10 Problems 81
Part Ⅱ Group Theory 87
4 Groups,Lie Groups,and Lie Algebras 87
4.1 Groups—Definition and Examples 88
4.2 The Groups of Classical and Quantum Physics 96
4.3 Homomorphism and Isomorphism 103
4.4 From Lie Groups to Lie Algebras 111
4.5 Lie Algebras—Definition,Properties,and Examples 115
4.6 The Lie Algebras of Classical and Quantum Physics 121
4.7 Abstract Lie Algebras 127
4.8 Homomorphism and Isomorphism Revisited 131
4.9 Problems 138
5 Basic Representation Theory 145
5.1 Representations：Definitions and Basic Examples 145
5.2 Further Examples 150
5.3 Tensor Product Representations 159
5.4 Symmetric and Antisymmetric Tensor Product Representations 165
5.5 Equivalence of Representations 169
5.6 Direct Sums and Irreducibility 177
5.7 More on Irreducibility 184
5.8 The Irreducible Representations of su(2),SU(2)and SO(3) 188
5.9 Real Representations and Complexifications 193
5.10 The Irreducible Representations of sl(2,C)R,SL(2,C)and SO(3,1)o 196
5.11 Irreducibility and the Representations of O(3,1)and Its Double Covers 204
5.12 Problems 208
6 The Wigner-Eckart Theorem andO ther Applications 213
6.1 Tensor Operators,Spherical Tensors and Representation Operators 213
6.2 Selection Rules and the Wigner-Eckart Theorem 217
6.3 Gamma Matrices and Dirac Bilinears 222
6.4 Problems 225
Appendix Complexifications of Real Lie Algebras and the Tensor Product Decomposition of sl(2,C)R Representations 227
A.1 Direct Sums and Complexifications of Lie Algebras 227
A.2 Representations of Complexified Lie Algebras and the Tensor Product Decomposition of sl(2,C)R Representations 229
References 235
Index 237