# Group Theory and Its Applications

## 1st Edition

### Volume III

**Editors:**Ernest M. Loebl

**eBook ISBN:**9781483263779

**Imprint:**Academic Press

**Published Date:**28th January 1974

**Page Count:**496

## Description

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.

## Table of Contents

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

## Details

- No. of pages:
- 496

- Language:
- English

- Copyright:
- © Academic Press 1975

- Published:
- 28th January 1974

- Imprint:
- Academic Press

- eBook ISBN:
- 9781483263779