Group Theory in Physics: An introduction (Techniques of Physics Series)

Front Cover......Page 1
Group Theory in Physics: An Introduction......Page 2
Copyright Page......Page 3
Contents......Page 4
Preface......Page 8
1. The concept of a group......Page 12
2. Groups of coordinate transformations......Page 15
3. The group of the Schrödinger equation......Page 21
4. The role of matrix representations......Page 26
1. Some elementary considerations......Page 30
2. Classes......Page 32
3. Invariant subgroups......Page 34
4. Cosets......Page 35
5. Factor groups......Page 37
6. Homomorphic and isomorphic mappings......Page 39
7. Direct products and semi-direct products of groups......Page 42
1. Definition of a linear Lie group......Page 46
2. The connected components of a linear Lie group......Page 51
3. The concept of compactness for linear Lie......Page 53
4. Invariant integration......Page 55
1. Definitions......Page 58
2. Equivalent representations......Page 60
3. Unitary representations......Page 63
4. Reducible and irreducible representations......Page 65
5. Schur's Lemmas and the orthogonality theorem for matrix representations......Page 68
6. Characters......Page 70
1. Projection operators......Page 76
2. Direct product representations......Page 81
3. The Wigner-Eckart Theorem for groups of coordinate transfor-mations in IR3......Page 84
4. The Wigner-Eckart Theorem generalized......Page 90
5. Representations of direct product groups......Page 94
6. Irreducible representations of finite Abelian groups......Page 96
7. Induced representations......Page 97
1. The solution of the Schrödinger equation......Page 104
2. Transition probabilities and selection rules......Page 108
3. Time-independent perturbation theory......Page 111
1. The Bravais lattices......Page 114
2. The cyclic boundary conditions......Page 118
3. Irreducible representations of the group T of pure primitive translations and Bloch's Theorem......Page 120
4. Brillouin zones......Page 122
5. Electronic energy bands......Page 126
6. Survey of the crystallographic space groups......Page 129
7. Irreducible representations of symmorphic space groups......Page 132
8. Consequences of the fundamental theorems......Page 140
1. "Local" and "global" aspects of Lie groups......Page 146
2. The matrix exponential function......Page 147
3. One-parameter subgroups......Page 150
4. Lie algebras......Page 151
5. The real Lie algebras that correspond to general linear Lie groups......Page 156
2. Subalgebras of Lie algebras......Page 164
3. Homomorphic and isomorphic mappings of Lie algebras......Page 165
4. Representations of Lie algebras......Page 171
5. The adjoint representations of Lie algebras and linear Lie groups......Page 179
6. Direct sum of Lie algebras......Page 182
1. Some properties reviewed......Page 186
2. The class structures of SU(2) and SO(3)......Page 187
3. Irreducible representations of the Lie algebras su(2) and so(3)......Page 188
4. Representations of the Lie groups SU(2), SO(3) and O(3)......Page 194
5. Direct products of irreducible representations and the Clebsch-Gordan coefficients......Page 197
6. Applications to atomic physics......Page 200
2. The Killing form and Cartan's criterion......Page 204
3. Complexification......Page 209
4. The Cartan subalgebras and roots of semi-simple complex Lie algebras......Page 211
5. Properties of roots of semi-simple complex Lie algebras......Page 218
6. The remaining commutation relations......Page 224
7. The simple roots......Page 229
8. The Weyl canonical form of L......Page 234
9. The Weyl group of L......Page 235
10. Semi-simple real Lie algebras......Page 239
1. Some basic ideas......Page 246
2. The weights of a representation......Page 247
3. The highest weight of a representation......Page 252
4. The irreducible representations of L = A2, the complexification of L = su(3)......Page 256
5. Casimir operators......Page 262
1. Leptons and hadrons......Page 266
2. The global internal symmetry group SU(2) and isotopic spin......Page 267
3. The global internal symmetry group SU(3) and strangeness......Page 270
APPENDICES......Page 280
1. Definitions......Page 282
2. Eigenvalues and eigenvectors......Page 286
1. The concept of a vector space......Page 290
2. Inner product spaces......Page 293
3. Hilbert spaces......Page 297
4. Linear operators......Page 299
5. Bilinear forms......Page 303
6. Linear functionals......Page 305
7. Direct product spaces......Page 306
Appendix C. Character Tables for the Crystallographic Point Groups......Page 310
1. The simple complex lie algebra Al, l ≥ 1......Page 330
2. The simple complex Lie algebra Bl, l ≥ 1......Page 331
3. The simple complex Lie algebra Cl, 1 ≥ 1......Page 333
4. The simple complex Lie algebra D1, 1 ≥ 3 (and the semi-simple complex Lie algebra D2)......Page 335
References......Page 338
Index......Page 346