Loebl Group Theory and Its Applications

Volume III
1. Auflage 2014
ISBN: 978-1-4832-6377-9
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 496 Seiten, Web PDF

Group Theory and its Applications, Volume III covers the two broad areas of applications of group theory, namely, all atomic and molecular phenomena, as well as all aspects of nuclear structure and elementary particle theory. This volume contains five chapters and begins with an introduction to Wedderburn's theory to establish the structure of semisimple algebras, algebras of quantum mechanical interest, and group algebras. The succeeding chapter deals with Dynkin's theory for the embedding of semisimple complex Lie algebras in semisimple complex Lie algebras. These topics are followed by a review of the Frobenius algebra theory, its centrum, its irreducible, invariant subalgebras, and its matric basis. The discussion then shifts to the concepts and application of the Heisenberg-Weyl ring to quantum mechanics. Other chapters explore some well-known results about canonical transformations and their unitary representations; the Bargmann Hilbert spaces; the concept of complex phase space; and the concept of quantization as an eigenvalue problem. The final chapter looks into a theoretical approach to elementary particle interactions based on two-variable expansions of reaction amplitudes. This chapter also demonstrates the use of invariance properties of space-time and momentum space to write down and exploit expansions provided by the representation theory of the Lorentz group for relativistic particles, or the Galilei group for nonrelativistic ones. This book will prove useful to mathematicians, engineers, physicists, and advance students.

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;6
5;List of Contributors;10
6;Preface;12
7;Contents of Other Volumes;16
8;Chapter 1. Finite Groups and Semisimple Algebras in Quantum Mechanics;18
8.1;I. Introduction;19
8.2;II. Linear Associative Algebras;20
8.3;III. Semisimple Algebras;24
8.4;IV. Semisimple Algebras in Quantum Mechanics;31
8.5;V. Group Algebras;34
8.6;VI. Fundamental Representation Theory;36
8.7;VII. Sequence Adaptation;42
8.8;VIII. Induced and Subduced Representations;45
8.9;IX. Approximate Symmetries in Quantum Mechanics;50
8.10;X. Weakly Interacting Sites;55
8.11;XI. Double Sequence Adaptation and Recoupling Coefficients;60
8.12;XII. Recoupling Coefficients in Quantum Mechanics;67
8.13;XIII. Point Group Symmetry Adaptation;72
8.14;XIV. Branching Rules;77
8.15;XV. Double Cosets;94
8.16;XVI. Effective Hamiltonians for Weakly Interacting Sites;99
8.17;XVII. Conclusion;106
8.18;REFERENCES;107
9;Chapter 2. Semisimple Subalgebras of Semisimple Lie Algebras: The Algebra a5 (SU(6)) as a Physically Significant Example;112
9.1;I. Introduction;112
9.2;II. Definitions;115
9.3;III. Embedding of Subalgebras;124
9.4;IV. Regular Subalgebras;130
9.5;V. S-Subalgebras;133
9.6;VI. Classification of Subalgebras of the Algebra 5;140
9.7;VII. Inclusion Relations;153
9.8;VIII. Physically Significant Chains of Subalgebras of 5;154
9.9;REFERENCES;158
10;Chapter 3. Frobenius Algebras and the Symmetric Group;160
10.1;I. Introduction;161
10.2;II. The Frobenius Algebra and Its Centrum;162
10.3;III. The Matric Basis and Symmetry Adaptation;169
10.4;IV. The Algebra of the Symmetric Group;176
10.5;V. Isospin-Free Nuclear Theory;187
10.6;VI. Spin-Free (Supermultiplet) Nuclear Theory;200
10.7;VII. Spin-Free Atomic Theory;203
10.8;VIII. Summary;205
10.9;REFERENCES;205
11;Chapter 4. The Heisenberg–Weyl Ring in Quantum Mechanics;206
11.1;I. Introduction;207
11.2;II. The Heisenberg–Weyl Group;209
11.3;III. The Heisenberg–Weyl Ring B;220
11.4;IV. The Quantization Process;227
11.5;V. Canonical Transformations;233
11.6;VI. Quantum Mechanics on a Compact Space;244
11.7;REFERENCES;256
12;Chapter 5. Complex Extensions of Canonical Transformations and Quantum Mechanics;266
12.1;I. Introduction and Summary;267
12.2;II. Groups of Classical Canonical Transformations;270
12.3;III. Unitary Representations of Canonical Transformations in Quantum Mechanics;273
12.4;IV. Complex Phase Space and Bargmann Hilbert Space;278
12.5;V. Complex Extensions of Canonical Transformations;289
12.6;VI. Barut Hilbert Space and Angular Momentum Projection in Bargmann Hilbert Space;305
12.7;VII. Applications to Problems of Accidental Degeneracy in Quantum Mechanics;316
12.8;VIII. The Three-Body Problem;328
12.9;IX. Applications to the Clustering Theory of Nuclei;335
12.10;X. Conclusion;347
12.11;REFERENCES;347
13;Chapter 6. Quantization as an Eigenvalue Problem;350
13.1;I. Quantization;350
13.2;II. Operators on Hilbert Space;354
13.3;III. Differential Equation Theory;358
13.4;IV. Symplectic Boundary Form;363
13.5;V. Spectral Density;367
13.6;VI. Continuation in the Complex Eigenvalue Plane;371
13.7;VII. One-Dimensional Relativistic Harmonic Oscillator;376
13.8;VIII. Survey;379
13.9;REFERENCES;384
14;Chapter 7. Elementary Particle Reactions and the Lorentz and Galilei Groups;386
14.1;I. Introduction;387
14.2;II. Single-Variable Expansions for Four-Body Scattering;393
14.3;III. Lorentz Group Two-Variable Expansions for Spinless Particles and the Lorentz Amplitudes;400
14.4;IV. Two-Variable Expansions Based on the O(4) Group for Three-Body Decays;427
14.5;V. O(3, 1) and O(4) Expansions for Particles with Arbitrary Spins;434
14.6;VI. Explicitly Crossing Symmetric Expansions Based on the O(2, 1) Group;445
14.7;VII. Two-Variable Expansions of Nonrelativistic Scattering Amplitudes Based on the E(3) Group;455
14.8;VIII. Two-Variable Expansions Based on the Group SU(3) and Their Generalizations;468
14.9;IX. Conclusions;476
14.10;REFERENCES;478
15;Author Index;482
16;Subject Index;490