Loebl Group Theory and Its Applications

1. Auflage 2014
ISBN: 978-1-4832-6401-1
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 724 Seiten, Web PDF

Group Theory and Its Applications focuses on the applications of group theory in physics and chemistry. The selection first offers information on the algebras of lie groups and their representations and induced and subduced representations. Discussions focus on the functions of positive type and compact groups; orthogonality relations for square-integrable representations; group, topological, Borel, and quotient structures; and classification of semisimple lie algebras in terms of their root systems. The text then takes a look at the generalization of Euler's angles and projective representation of the Poincare group in a quaternionic Hilbert space. The manuscript ponders on group theory in atomic spectroscopy, group lattices and homomorphism, and group theory in solid state physics. Topics include band theory of solids, lattice vibrations in solids, stationary states in the quantum theory of matter, coupled tensors, and shell structure. The text then examines the group theory of harmonic oscillators and nuclear structure and de Sitter space and positive energy. The selection is a dependable reference for physicists and chemists interested in group theory and its applications.

Autoren/Hrsg.

Loebl, Ernest M.

Inhaltsverzeichnis

1;Front Cover;1
2;Group Theory and its Applications;4
3;Copyright Page;5
4;Table of Contents;14
5;List of Contributors;8
6;Preface;10
7;Glossary of Symbols and Abbreviations;18
8;Chapter 1. The Algebras of Lie Groups and Their Representations;30
8.1;I. Introduction;30
8.2;II. Preliminary Survey;36
8.3;III. Lie's Theorem, the Rank
Theorem, and the First Criterion of Solvability;44
8.4;IV. The Cartan Subalgebra and Root Systems;48
8.5;V. The Classification of Semisimple Lie Algebras in Terms of
Their Root Systems;69
8.6;VI. Representations and Weights for Semisimple Lie Algebras;81
8.7;References;84
9;Chapter 2. Induced and Subdued Representations;86
9.1;I. Introduction;86
9.2;II. Group, Topological, Borel, and Quotient Structures;90
9.3;III. The Generalized Schur Lemma and Type I Representations;96
9.4;IV. Direct Integrals of Representations;102
9.5;V. Murray-von Neumann Typology;106
9.6;VI. Induced Representations of Finite Groups;109
9.7;VII. Orthogonality Relations for Square-Integrable Representations;119
9.8;VIII. Functions of Positive Type and Compact Groups;125
9.9;IX. Inducing for Locally Compact Groups;129
9.10;X. Applications;139
9.11;References;145
10;Chapter 3. On a Generalization of Euler's Angles;148
10.1;I. Origin of the Problem;148
10.2;II. Summary of Results;151
10.3;III. Proof;153
10.4;IV. Corollary;157
10.5;References;158
11;Chapter 4. Projective Representation of the Poincare Group in a Quaternionic Hilbert
Space;160
11.1;I. Introduction;160
11.2;II. The Lattice Structure of General Quantum Mechanics;164
11.3;III. The Group of Automorphisms in a Proposition System;171
11.4;IV. Projective Representation of the Poincare Group in Quaternionic Hilbert Space;181
11.5;V. Conclusion;208
11.6;References;210
12;Chapter 5. Group Theory in Atomic Spectroscopy;212
12.1;I. Introduction;212
12.2;II. Shell Structure;214
12.3;III. Coupled Tensors;222
12.4;IV. Representations;227
12.5;V. The Wigner-Eckart Theorem;235
12.6;VI. Conclusion;247
12.7;References;248
13;Chapter 6. Group Lattices and Homomorphisms;250
13.1;I. Introduction;250
13.2;II. Groups;252
13.3;III. Symmetry Adaption of Vector Spaces;258
13.4;IV. The Lattice of the Quasi-Relativistic Dirac Hamiltonian;267
13.5;V. Applications;278
13.6;References;293
14;Chapter 7. Group Theory in Solid State Physics;294
14.1;I. Introduction;295
14.2;II. Stationary States in the Quantum Theory of Matter;296
14.3;III. The Group of the Hamiltonian;300
14.4;IV. Symmetry Groups of Solids;314
14.5;V. Lattice Vibrations in Solids;333
14.6;VI. Band Theory of Solids;341
14.7;VII. Electromagnetic Fields in Solids;357
14.8;References;365
15;Chapter 8. Group Theory of Harmonic Oscillators and Nuclear Structure;368
15.1;I. Introduction and Summary;369
15.2;II. The Symmetry Group U (3n); the Subgroup U(3) X
U(n); Gelfand States;374
15.3;III. The Central Problem: Permutational Symmetry of the Orbital States;388
15.4;IV. Orbital Fractional Parentage Coefficients;416
15.5;V. Group Theory and
n-Particle States in Spin-Isospin Space;431
15.6;VI. Spin-Isospin Fractional Parentage Coefficients;440
15.7;VII. Evaluation of Matrix Elements of One-Body and Two-Body Operators;451
15.8;VIII. The Few-Nucleon Problem;463
15.9;IX. The Elliott Model in Nuclear Shell Theory;470
15.10;X. Clustering Properties and Interactions;477
15.11;References;495
16;Chapter 9. Broken Symmetry;498
16.1;I. Introduction;498
16.2;II. Wigner-Eckart Theorem;503
16.3;III. Some Relevant Group Theory;512
16.4;IV. Particle Physics SU(3) from the Point of View of the
Wigner-Eckart Theorem;520
16.5;V. Foils to SU(3) and the Eightfold Way;532
16.6;VI. Broken Symmetry in Nuclear and Atomic Physics;542
16.7;VII. General Questions concerning Broken Symmetry;546
16.8;VIII. A Note on SU(6);554
16.9;References;566
17;Chapter 10. Broken SU (3) as a Particle Symmetry;570
17.1;I. Introduction;570
17.2;II. Perturbative Approach;572
17.3;III. Algebra of SU(3);577
17.4;IV. Representations;586
17.5;V. Tensor and Wigner Operators;603
17.6;VI. Particle Classification, Masses, and Form Factors;607
17.7;VII. Some Remarks on R and
SU(3)/Z3;625
17.8;VIII. Couplings and Decay Widths;625
17.9;IX. Weak Interactions;640
17.10;X. Appendix;654
17.11;References;656
18;Chapter 11. De Sitter Space and Positive Energy;660
18.1;I. Introduction and Summary;660
18.2;II. Ambivalent Nature of the Classes of de Sitter Groups;664
18.3;III. The Infinitesimal Elements of Unitary Representations of the de Sitter Group;668
18.4;IV. Finite Elements of the Unitary Representations of Section III;672
18.5;V. Spatial and Time Reflections;676
18.6;VI. The Position Operators;684
18.7;VII. General Remarks about Contraction of Groups and Their Representations;693
18.8;VIII. Contraction of the Representations of the 2 + 1 de Sitter Group;696
18.9;References;704
19;Author Index;706
20;Subject Index;712