Lectures on Advanced Mathematical Methods for Physicists

Cover......Page 1
Title page......Page 2
Contents......Page 4
Part I: Topology and Differential Geometry......Page 8
Introduction to Part I......Page 10
1.1 Preliminaries......Page 12
1.2 Topological Spaces......Page 13
1.3 Metric spaces......Page 16
1.4 Basis for a topology......Page 18
1.5 Closure......Page 19
1.6 Connected and Compact Spaces......Page 20
1.7 Continuous Functions......Page 22
1.8 Homeomorphisms......Page 24
1.9 Separability......Page 25
2.1 Loops and Homotopies......Page 28
2.2 The Fundamental Group......Page 32
2.3 Homotopy Type and Contractibility......Page 35
2.4 Higher Homotopy Groups......Page 41
3.1 The Definition of a Manifold......Page 48
3.2 Differentiation of Functions......Page 54
3.3 Orientability......Page 55
3.4 Calculus on Manifolds: Vector and Tensor Fields......Page 57
3.5 Calculus on Manifolds: Differential Forms......Page 62
3.6 Properties of Differential Forms......Page 66
3.7 More About Vectors and Forms......Page 69
4.1 Riemannian Geometry......Page 72
4.2 Frames......Page 74
4.3 Connections, Curvature and Torsion......Page 76
4.4 The Volume Form......Page 81
4.5 Isometry......Page 83
4.6 Integration of Differential Forms......Page 84
4.7 Stokes'Theorem......Page 87
4.8 The Laplacian on Forms......Page 90
5.1 Simplicial Homology......Page 94
5.2 De Rham Cohomology......Page 107
5.3 Harmonic Forms and de Rham Cohomology......Page 110
6.1 The Concept of a Fibre Bundle......Page 112
6.2 Tangent and Cotangent Bundles......Page 118
6.3 Vector Bundles and Principal Bundles......Page 119
Bibliography for Part I......Page 124
Part II: Group Theory and Structure and Representations of Compact Simple Lie Groups and Algebras......Page 126
Introduction to Part II......Page 128
7.1 Definition of a Group......Page 130
7.3 Subgroups and Cosets......Page 131
7.4 Invariant (Normal) Subgroups, the Factor Group......Page 132
7.5 Abelian Groups, Commutator Subgroup......Page 133
7.6 Solvable, Nilpotent, Semisimple and Simple Groups......Page 134
7.7 Relationships Among Groups......Page 136
7.8 Ways to Combine Groups - Direct and Semidirect Products......Page 138
7.9 Topological Groups, Lie Groups, Compact Lie Groups......Page 139
8.1 Definition of a Representation......Page 142
8.2 Invariant Subspaces, Reducibility, Decomposability......Page 143
8.3 Equivalence of Representations, Schur's Lemma......Page 145
8.4 Unitary and Orthogonal Representations......Page 146
8.5 Contragredient, Adjoint and Complex Conjugate Representations......Page 147
8.6 Direct Products of Group Representations......Page 151
9.1 Local Coordinates in a Lie Group......Page 154
9.2 Analysis of Associativity......Page 155
9.3 One-parameter Subgroups and Canonical Coordinates......Page 158
9.4 Integrability Conditions and Structure Constants......Page 162
9.5 Definition of a (real) Lie Algebra: Lie Algebra of a given Lie Group......Page 164
9.6 Local Reconstruction of Lie Group from Lie Algebra......Page 165
9.7 Comments on the G->G_ Relationship......Page 167
9.8 Various Kinds of and Operations with Lie Algebras......Page 168
10 Linear Representations of Lie Algebras......Page 172
11.1 Complexification of a Real Lie Algebra......Page 178
11.2 Solvability, Levi's Theorem, and Cartan's Analysis of Complex (Semi) Simple Lie Algebras......Page 180
11.3 The Real Compact Simple Lie Algebras......Page 187
12 Geometry of Roots for Compact Simple Lie Algebras......Page 190
13.2 Simple Roots and their properties......Page 196
13.3 Dynkin Diagrams......Page 201
14.1 The SO(2l) Family - D_l of Cartan......Page 204
14.2 The SO(2l+1) Family - B_l of Cartan......Page 208
14.3 The USp(2l) Family - C_l of Cartan......Page 210
14.4 The SU(l+1) Family - A_l of Cartan......Page 214
14.5 Coincidences for low Dimensions and Connectedness......Page 219
15 Complete Classification of All CSLA Simple Root Systems......Page 222
15.1 Series of Lemmas......Page 223
15.2 The allowed Graphs Gamma......Page 227
15.3 The Exceptional Groups......Page 231
16.1 Weights and Multiplicitics......Page 234
16.2 Actions of E_a and SU(2)^(a) - the Weyl Group......Page 235
16.3 Dominant Weights, Highest Weight of a UIR......Page 237
16.4 Fundamental UIR's, Survey of all UIR's......Page 240
16.5 Fundamental UIR's for A_l,_Bl,C_l,D_l......Page 241
16.6 The Elementary UIR's......Page 247
16.7 Structure of States within a UIR......Page 248
17 Spinor Representations for Real Orthogonal Groups......Page 252
17.1 The Dirac Algebra in Even Dimensions......Page 253
17.2 Generators, Weights and Reducibility of U(S) - the spinor UIR's of D_l......Page 255
17.3 Conjugation Properties of Spinor UlR's of D_l......Page 257
17.4 Remarks on Antisymmetric Tensors Under D_l = SO(2l)......Page 259
17.5 The Spinor UIR's of B_l = SO(2l+1)......Page 264
17.6 Antisymmetric Tensors under B_l = SO(2l+1)......Page 267
18.1 Definition of SO(q,p) and Notational Matters......Page 268
18.2 Spinor Representations S(A) of SO(p, q) for p+q=2l......Page 269
18.3 Representations Related to S(A)......Page 271
18.4 Behaviour of the Irreducible Spinor Representations S+-(A)......Page 272
18.5 Spinor Representations of SO(p,q) for p+q=2l+1......Page 273
18.6 Dirac, Weyl and MajorRna Spinors for SO(p,q)......Page 274
Bibliography for Part II......Page 280
Index......Page 282