Topics in Contemporary Mathematical Physics

Cover......Page 1
Topics in Contemporary Mathematical Physics......Page 4
©......Page 5
Preface......Page 8
Contents......Page 12
1 Vectors and Linear Transformations......Page 14
2 Tensors......Page 23
3 Symmetry and Conservation: the Angular Momentum......Page 30
4 The Angular Momentum as Generators of Rotations: Lie Groups and Lie Algebras......Page 35
5 Algebraic Structures......Page 47
6 Basic Group Concepts......Page 51
7 Basic Lie Algebra Concepts......Page 62
8 Inner Products, Metrics, and Dual Spaces......Page 69
9 SO (4) and the Hydrogen Atom......Page 79
10 Adjoints and Unitary Transformations......Page 86
11 The Lorentz Group and SL(2, C)......Page 91
12 The Dirac Bracket Notation in Quantum Theory......Page 112
13 The Quantum Mechanical Simple Harmonic Oscillator......Page 117
14 Fourier Series and Fourier Transforms, the Dirac Delta Function, Green's Functions......Page 125
15 The Continuous Spectrum and Non-normalizable States......Page 134
16 Skew-Symmetric Tensors and Determinants......Page 140
17 Eigenvalue Problems......Page 153
18 Group Representation Theory......Page 172
19 The Dihedral Group D6 and the Benzene Molecule......Page 191
20 Representations of the Symmetric Groups and the General Linear Groups, Young Diagrams......Page 201
21 Irreducible Representations of U(n), SL{n), SU(n) and O(n)......Page 219
22 Irreducible Representations of SU(2) and SO(3)......Page 232
23 The Spherical Harmonics......Page 246
24 The Structure of Semisimple Lie Algebras......Page 255
25 The Representations of Semisimple Lie Algebras......Page 274
26 SU(3) and the Strong Interaction......Page 291
27 Clifford Algebras......Page 302
28 Exterior Products......Page 307
29 The Hodge-Star Operator......Page 317
30 Differential Forms and Exterior Differentiation......Page 323
31 Moving Frames and Curvilinear Coordinates in R3......Page 339
32 Integrals of Differential Forms and the Stokes Theorem......Page 349
33 Homology and De Rham Cohomology......Page 364
34 The Geometry of Lie Groups......Page 372
35 Connections and Curvatures on a Vector Bundle......Page 389
36 Yang-Mills Equations......Page 403
37 Connections on a Principal Bundle......Page 409
38 Magnetic Monopoles and Molecular Dynamics......Page 438
39 Riemannian Geometry......Page 454
40 Complex Manifolds......Page 482
41 Characteristic Classes......Page 495
42 Chern-Simons Forms......Page 523
43 The Atiyah-Singer Index Theorem......Page 542
44 Symplectic Structures and Hamiltonian Mechanics......Page 564
References......Page 573
Index......Page 577