Group Theory and Its Applications
Volume III
- 1st Edition - May 10, 2014
- Editor: Ernest M. Loebl
- Language: English
- Paperback ISBN:9 7 8 - 1 - 4 8 3 2 - 4 2 8 8 - 0
- eBook ISBN:9 7 8 - 1 - 4 8 3 2 - 6 3 7 7 - 9
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… Read more
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Request a sales quoteGroup 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.
List of Contributors
Preface
Contents of Other Volumes
Finite Groups and Semisimple Algebras in Quantum Mechanics
I. Introduction
II. Linear Associative Algebras
III. Semisimple Algebras
IV. Semisimple Algebras in Quantum Mechanics
V. Group Algebras
VI. Fundamental Representation Theory
VII. Sequence Adaptation
VIII. Induced and Subduced Representations
IX. Approximate Symmetries in Quantum Mechanics
X. Weakly Interacting Sites
XI. Double Sequence Adaptation and Recoupling Coefficients
XII. Recoupling Coefficients in Quantum Mechanics
XIII. Point Group Symmetry Adaptation
XIV. Branching Rules
XV. Double Cosets
XVI. Effective Hamiltonians for Weakly Interacting Sites
XVII. Conclusion
References
Semisimple Subalgebras of Semisimple Lie Algebras: The Algebra 𝒂5(SU(6)) as a Physically Significant Example
I. Introduction
II. Definitions
III. Embedding of Subalgebras
IV. Regular Subalgebras
V. S-Subalgebras
VI. Classification of Subalgebras of the Algebra 𝒂5
VII. Inclusion Relations
VIII. Physically Significant Chains of Subalgebras of 𝒂5
References
Frobenius Algebras and the Symmetric Group
I. Introduction
II. The Frobenius Algebra and Its Centrum
III. The Matric Basis and Symmetry Adaptation
IV. The Algebra of the Symmetric Group
V. Isospin-Free Nuclear Theory
VI. Spin-Free (Supermultiplet) Nuclear Theory
VII. Spin-Free Atomic Theory
VIII. Summary
References
The Heisenberg-Weyl Ring in Quantum Mechanics
I. Introduction
II. The Heisenberg-Weyl Group
III. The Heisenberg-Weyl Ring 𝔚
IV. The Quantization Process
V. Canonical Transformations
VI. Quantum Mechanics on a Compact Space
References
Complex Extensions of Canonical Transformations and Quantum Mechanics
I. Introduction and Summary
II. Groups of Classical Canonical Transformations
III. Unitary Representations of Canonical Transformations in Quantum Mechanics
IV. Complex Phase Space and Bargmann Hilbert Space
V. Complex Extensions of Canonical Transformations
VI. Barut Hilbert Space and Angular Momentum Projection in Bargmann Hilbert Space
VII. Applications to Problems of Accidental Degeneracy in Quantum Mechanics
VIII. The Three-Body Problem
IX. Applications to the Clustering Theory of Nuclei
X. Conclusion
References
Quantization as an Eigenvalue Problem
I. Quantization
II. Operators on Hilbert Space
III. Differential Equation Theory
IV. Symplectic Boundary Form
V. Spectral Density
VI. Continuation in the Complex Eigenvalue Plane
VII. One-Dimensional Relativistic Harmonic Oscillator
VIII. Survey
References
Elementary Particle Reactions and the Lorentz and Galilei Groups
I. Introduction
II. Single-Variable Expansions for Four-Body Scattering
III. Lorentz Group Two-Variable Expansions for Spinless Particles and the Lorentz Amplitudes
IV. Two-Variable Expansions Based on the O(4) Group for Three-Body Decays
V. O(3,1) and O(4) Expansions for Particles with Arbitrary Spins
VI. Explicitly Crossing Symmetric Expansions Based on the O(2,1) Group
VII. Two-Variable Expansions of Nonrelativistic Scattering Amplitudes Based on the E(3) Group
VIII. Two-Variable Expansions Based on the Group SU(3) and Their Generalizations
IX. Conclusions
References
Author Index
Subject Index
- No. of pages: 496
- Language: English
- Edition: 1
- Published: May 10, 2014
- Imprint: Academic Press
- Paperback ISBN: 9781483242880
- eBook ISBN: 9781483263779
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