Part Ⅰ Linear Algebra and Tensors

1 A Quick Introduction to Tensors

2 Vector Spaces

2.1 Definition and Examples

2.2 Span,Linear Independence,and Bases

2.3 Components

2.4 Linear Operators

2.5 Dual Spaces

2.6 Non-degenerate Hermitian Forms

2.7 Non-degenerate Hermitian Forms and Dual Spaces

2.8 Problems

3 Tensors

3.1 Definition and Examples

3.2 Change of Basis

3.3 Active and Passive Transformations

3.4 The Tensor Product—Definition and Properties

3.5 Tensor Products of V and V＊

3.6 Applications of the Tensor Product in Classical Physics

3.7 Applications of the Tensor Product in Quantum Physics

3.8 Symmetric Tensors

3.9 Antisymmetric Tensors

3.10 Problems

Part Ⅱ Group Theory

4 Groups,Lie Groups,and Lie Algebras

4.1 Groups—Definition and Examples

4.2 The Groups of Classical and Quantum Physics

4.3 Homomorphism and Isomorphism

4.4 From Lie Groups to Lie Algebras

4.5 Lie Algebras—Definition,Properties,and Examples

4.6 The Lie Algebras of Classical and Quantum Physics

4.7 Abstract Lie Algebras

4.8 Homomorphism and Isomorphism Revisited

4.9 Problems

5 Basic Representation Theory

5.1 Representations：Definitions and Basic Examples

5.2 Further Examples

5.3 Tensor Product Representations

5.4 Symmetric and Antisymmetric Tensor Product Representations

5.5 Equivalence of Representations

5.6 Direct Sums and Irreducibility

5.7 More on Irreducibility

5.8 The Irreducible Representations of su(2),SU(2)and SO(3)

5.9 Real Representations and Complexifications

5.10 The Irreducible Representations of sl(2,C)R,SL(2,C)and SO(3,1)o

5.11 Irreducibility and the Representations of O(3,1)and Its Double Covers

5.12 Problems

6 The Wigner-Eckart Theorem andO ther Applications

6.1 Tensor Operators,Spherical Tensors and Representation Operators

6.2 Selection Rules and the Wigner-Eckart Theorem

6.3 Gamma Matrices and Dirac Bilinears

6.4 Problems

Appendix Complexifications of Real Lie Algebras and the Tensor Product Decomposition of sl(2,C)R Representations

A.1 Direct Sums and Complexifications of Lie Algebras

A.2 Representations of Complexified Lie Algebras and the Tensor Product Decomposition of sl(2,C)R Representations

References

Index