Group Theory and Its Applications

Volume III

Ernest M. Loebl

• Publisher: Academic Press
• Year: 2014
• Print ISBN: 9780124551534
• eBook ISBN: 9781483263779
• Topic: Mathematics
• Subtopic: Algebra
• Index: Mathematics

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

• Front Cover
• Group Theory and its Applications
• Copyright Page
• Table of Contents
• List of Contributors
• Preface
• Contents of Other Volumes
• Chapter 1. Finite Groups and Semisimple Algebras in Quantum Mechanics
• Chapter 2. Semisimple Subalgebras of Semisimple Lie Algebras: The Algebra a5 (SU(6)) as a Physically Significant Example
• Chapter 3. Frobenius Algebras and the Symmetric Group
• Chapter 4. The Heisenberg–Weyl Ring in Quantum Mechanics
• Chapter 5. Complex Extensions of Canonical Transformations and Quantum Mechanics
• Chapter 6. Quantization as an Eigenvalue Problem
• Chapter 7. Elementary Particle Reactions and the Lorentz and Galilei Groups
• Author Index
• Subject Index