Symmetry Groups and Their Applications (eBook)

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1973 | 1. Auflage
431 Seiten
Elsevier Science (Verlag)
978-0-08-087365-7 (ISBN)

Infos

Symmetry groups and their applications

Inhalt

Front Cover 1
SYMMETRY GROUPS AND THEIR APPLICATIONS 4
Copyright Page 5
Contents 6
Preface 10
Chapter 1. Elementary Group Theory 12
1.1 Abstract Groups 12
1.2 Subgroups and Cosets 15
1.3 Homomorphisms, Isomorphisms, and Automorphisms 17
1.4 Transformation Groups 19
1.5 New Groups from Old Ones 24
Problems 26
Chapter 2. The Crystallographic Groups 27
2.1 The Orthogonal Group in Three-Space 27
2.2 The Euclidean Group 31
2.3 Symmetry and the Discrete Subgroups of E(3) 34
2.4 Point Groups of the First Kind 38
2.5 Point Groups of the Second Kind 43
2.6 Lattice Groups 45
2.7 Crystallographic Point Groups 49
2.8 The Bravais Lattices 53
2.9 Crystal Structure 63
2.10 Space Groups 66
Problems 70
Chapter 3. Group Representation Theory 72
3.1 A Group Representation 72
3.2 Reducible Representations 78
3.3 Irreducible Representations 80
3.4 Group Characters 85
3.5 New Representations from Old Ones 89
3.6 Character Tables 96
3.7 The Method of Projection Operators 103
3.8 Applications 111
Problems 125
Chapter 4. Representations of the Symmetric Groups 127
4.1 Conjugacy Classes in Sn 127
4.2 Young Tableaux 130
4.3 Symmetry Classes of Tensors 139
4. 4 The Simple Characters of Sa 158
Problems 162
Chapter 5. Lie Groups and Lie Algebras 163
5.1 The Exponential of a Matrix 163
5.2 Local Lie Groups 173
5.3 Lie Algebras 177
5.4 The Classical Groups 182
5.5 The Exponential Map of a Lie Algebra 186
5.6 Local Homomorphisms and Isomorphisms 189
5.7 Subgroups and Subalgebras 194
5.8 Representations of Lie Groups 197
5.9 Local Transformation Groups 199
5.10 Examples of Transformation Groups 210
Problems 216
Chapter 6. Compact Lie Groups 217
6.1 Invariant Measures on Lie Groups 217
6.2 Compact Linear Lie Groups 221
6.3 Group Characters and Representations 228
Problems 232
Chapter 7. The Rotation Group and Its Representations 233
7.1 The Groups SO(3) and SU(2) 233
7.2 Irreducible Representations of SU(2) 239
7.3 Irreducible Representations of sl ( 2 ) 244
7.4 Expansion Theorems for Functions on SU(2) 247
7.5 New Realizations of the Irreducible Representations 250
7.6 Applications to Physics 258
7.7 The Clebsch–Gordan Coefficients 267
7.8 Applications of the Clebsch-Gordan Series 274
7.9 Double-Valued Representations of the Crystallographic Groups 282
7.10 The Wigner–Eckart Theorem and Its Applications 284
7.11 Spinor Fields and Invariant Equations 289
Problems 295
Chapter 8. The Lorentz Group and Its Representations 296
8.1 The Homogeneous Lorentz Group 296
8.2 The Physical Significance of Lorentz Invariance 304
8.3 Representations of the Lorentz Group 308
8.4 Models of the Representations 318
8.5 Lorentz-Invariant Equations 324
Problems 331
Chapter 9. Representations of the Classical Groups 332
9.1 Representations of the General Linear Groups 332
9.2 Character Formulas 345
9.3 The Irreducible Representations of GL(m, R), SL(m, E), and SU(m) 351
9.4 The Symplectic Groups and Their Representations 355
9.5 The Orthogonal Groups and Their Representations 362
9.6 Dirac Matrices and the Spin Representations of the Orthogonal Groups 373
9.7 Examples and Applications 379
9.8 The Pauli Exclusion Principle and the Periodic Table 388
9.9 The Group Ring Revisited 398
9.10 Semisimple Lie Algebras 402
Problems 406
Chapter 10. The Harmonic Oscillator Group 408
10.1 The Harmonic Oscillator 408
10.2 Representations of the Harmonic Oscillator Group 413
Problems 416
Appendix. Hilbert Space 418
References 426
Symbol Index 432
Index 436

Produkt-Details

• Erscheint lt. Verlag: 2.3.1973
• Mitarbeit: Herausgeber (Serie): Willard Miller
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie
• Themenwelt: Technik
• ISBN-10: 0-08-087365-0 / 0080873650
• ISBN-13: 978-0-08-087365-7 / 9780080873657