# Group Theory and Its Applications

## 1st Edition

**Editors:**Ernest M. Loebl

**eBook ISBN:**9781483264011

**Imprint:**Academic Press

**Published Date:**1st January 1968

**Page Count:**724

## Description

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.

## Table of Contents

List of Contributors

Preface

Glossary of Symbols and Abbreviations

The Algebras of Lie Groups and Their Representations

I. Introduction

II. Preliminary Survey

III. Lie's Theorem, the Rank Theorem, and the First Criterion of Solvability

IV. The Cartan Subalgebra and Root Systems

V. The Classification of Semisimple Lie Algebras in Terms of Their Root Systems

VI. Representations and Weights for Semisimple Lie Algebras

References

Induced and Subdued Representations

I. Introduction

II. Group, Topological, Borel, and Quotient Structures

III. The Generalized Schur Lemma and Type I Representations

IV. Direct Integrals of Representations

V. Murray-von Neumann Typology

VI. Induced Representations of Finite Groups

VII. Orthogonality Relations for Square-Integrable Representations

VIII. Functions of Positive Type and Compact Groups

IX. Inducing for Locally Compact Groups

X. Applications

References

On a Generalization of Euler's Angles

I. Origin of the Problem

II. Summary of Results

III. Proof

IV. Corollary

References

Projective Representation of the Poincare Group in a Quaternionic Hilbert Space

I. Introduction

II. The Lattice Structure of General Quantum Mechanics

III. The Group of Automorphisms in a Proposition System

IV. Projective Representation of the Poincare Group in Quaternionic Hilbert Space

V. Conclusion

References

Group Theory in Atomic Spectroscopy

I. Introduction

II. Shell Structure

III. Coupled Tensors

IV. Representations

V. The Wigner-Eckart Theorem

VI. Conclusion

References

Group Lattices and Homomorphisms

I. Introduction

II. Groups

III. Symmetry Adaption of Vector Spaces

IV. The Lattice of the Quasi-Relativistic Dirac Hamiltonian

V. Applications

References

Group Theory in Solid State Physics

I. Introduction

II. Stationary States in the Quantum Theory of Matter

III. The Group of the Hamiltonian

IV. Symmetry Groups of Solids

V. Lattice Vibrations in Solids

VI. Band Theory of Solids

VII. Electromagnetic Fields in Solids

References

Group Theory of Harmonic Oscillators and Nuclear Structure

I. Introduction and Summary

II. The Symmetry Group U (3n); The Subgroup U(3) X U(n); Gelfand States

III. The Central Problem: Permutational Symmetry of the Orbital States

IV. Orbital Fractional Parentage Coefficients

V. Group Theory and n-Particle States in Spin-Isospin Space

VI. Spin-Isospin Fractional Parentage Coefficients

VII. Evaluation of Matrix Elements of One-Body and Two-Body Operators

VIII. The Few-Nucleon Problem

IX. The Elliott Model in Nuclear Shell Theory

X. Clustering Properties and Interactions

References

Broken Symmetry

I. Introduction

II. Wigner-Eckart Theorem

III. Some Relevant Group Theory

IV. Particle Physics SU(3) from the Point of View of the Wigner-Eckart Theorem

V. Foils to SU(3) and the Eightfold Way

VI. Broken Symmetry in Nuclear and Atomic Physics

VII. General Questions concerning Broken Symmetry

VIII. A Note on SU(6)

References

Broken SU(3) as a Particle Symmetry

I. Introduction

II. Perturbative Approach

III. Algebra of SU(3)

IV. Representations

V. Tensor and Wigner Operators

VI. Particle Classification, Masses, and Form Factors

VII. Some Remarks on R and SU(3)/Z3

VIII. Couplings and Decay Widths

IX. Weak Interactions

X. Appendix

References

De Sitter Space and Positive Energy

I. Introduction and Summary

II. Ambivalent Nature of the Classes of de Sitter Groups

III. The Infinitesimal Elements of Unitary Representations of the de Sitter Group

IV. Finite Elements of the Unitary Representations of Section III

V. Spatial and Time Reflections

VI. The Position Operators

VII. General Remarks about Contraction of Groups and Their Representations

VIII. Contraction of the Representations of the 2 + 1 de Sitter Group

References

Author Index

Subject Index

## Details

- No. of pages:
- 724

- Language:
- English

- Copyright:
- © Academic Press 1968

- Published:
- 1st January 1968

- Imprint:
- Academic Press

- eBook ISBN:
- 9781483264011