"Topics in Contemporary Mathematical Physics" by Kai S. Lam

This textbook, pitched at the advanced-undergraduate to beginning-graduate level, focuses on mathematical topics of relevance in contemporary physics that are not usually covered in texts at the same level. The main purpose is to help students appreciate and take advantage of the modern trend of very productive symbiosis between physics and mathematics.

Three major areas are covered: (1) linear operators; (2) group representations and Lie algebra representations; (3) topology and differential geometry.

The following are noteworthy features of this book:
• the style of exposition is a fusion of those common in the standard physics and mathematics literatures;
• the level of exposition varies from quite elementary to moderately advanced, so that the book is of interest to a wide audience;
• despite the diversity of the topics covered, there is a strong degree of thematic unity;
• much care is devoted to detailed cross-referencing so that, from any part of the book, the reader can trace easily where specific concepts or techniques are introduced.

Contents
Preface
1 Vectors and Linear Transformations
2 Tensors
3 Symmetry and Conservation: the Angular Momentum
4 The Angular Momentum as Generators of Rotations: Lie Groups and Lie Algebras
5 Algebraic Structures
6 Basic Group Concepts
7 Basic Lie Algebra Concepts
8 Inner Products, Metrics, and Dual Spaces
9 SO (4) and the Hydrogen Atom
10 Adjoints and Unitary Transformations
11 The Lorentz Group and SL(2, C)
12 The Dirac Bracket Notation in Quantum Theory
13 The Quantum Mechanical Simple Harmonic Oscillator
14 Fourier Series and Fourier Transforms, the Dirac Delta Function, Green's Functions
15 The Continuous Spectrum and Non-normalizable States
16 Skew-Symmetric Tensors and Determinants
17 Eigenvalue Problems
18 Group Representation Theory
19 The Dihedral Group D6 and the Benzene Molecule
20 Representations of the Symmetric Groups and the General Linear Groups, Young Diagrams
21 Irreducible Representations of U(n), SL{n), SU(n) and 0(n)
22 Irreducible Representations of SU(2) and SO(3)
23 The Spherical Harmonics
24 The Structure of Semisimple Lie Algebras
25 The Representations of Semisimple Lie Algebras
26 SU(3) and the Strong Interaction
27 Clifford Algebras
28 Exterior Products
29 The Hodge-Star Operator
30 Differential Forms and Exterior Differentiation
31 Moving Frames and Curvilinear Coordinates in R3
32 Integrals of Differential Forms and the Stokes Theorem
33 Homology and De Rham Cohomology
34 The Geometry of Lie Groups
35 Connections and Curvatures on a Vector Bundle
36 Yang-Mills Equations
37 Connections on a Principal Bundle
38 Magnetic Monopoles and Molecular Dynamics
39 Riemannian Geometry
40 Complex Manifolds
41 Characteristic Classes
42 Chern-Simons Forms
43 The Atiyah-Singer Index Theorem
44 Symplectic Structures and Hamiltonian Mechanics
References
Index

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