GROUP THEORY: FINITE DISCRETE GROUPS AND APPLICATIONS

This book deals with the role played by symmetry in the understanding of the physical world, beginning with the notion of geometric symmetries of the ancient Greek philosophers and mathematicians. The recognition of the existence of symmetries led to the notion of transformations, which led from one state of the system to another. It was then realized that such transformations, under the operation of multiplication, constitute an interesting set, whose study led to the branch of mathematics known as Group Theory. With the emergence of quantum mechanics, this theory became much more interesting and led to some additional applications. The theory got another boost with the need for of the internal degrees of freedom in describing physical systems. This way the notion of symmetry is no longer purely geometric and evolved into a useful tool in the study of all physical sciences.

For practical reasons as well as pedagogical reasons, group theory is usually split into two parts. The first deals with discrete groups, with the group elements being countable, usually finite in number, while the second deals with continuous groups, whose elements depend on continuous parameters. This volumefocuses the discussion on discrete groups.

Given that group theory should be presented from a unified perspective, involving not only the mathematical rigor and beauty of symmetries, but also the ability to use it as a tool for applications, either currently popular or expected to become so in the future, this approach will surely be more beneficial to the dedicated reader. It is not intended for those who would like to just look up a formula or use the results of a table, without understanding their derivation.

Contents:

• The Role of Symmetries in Physics — A Prelude
• Introduction to Discrete Groups
• Discrete Groups — Basic Theorems
• Elements of Representation Theory
• Representation Reduction — Schur's Lemmas — Character Tables
• Simple Applications in Quantum Mechanics
• Symmetries and Normal Modes of Oscillation
• Space Groups
• Irreducible Representations of Space Groups
• Normal Modes in Crystals
• The Symmetric Group S_n
• Further Applications in Molecular Physics and Crystal Structure — Crystal Harmonics
• Applications of Discrete Groups in Particle Physics
• Exotic Discrete Groups for Quantum Mechanics — Field Theory
• Appendices:
  • Proofs of Various Theorems
  • Representation Reduction Via a Chain of Group Operators
  • Generators and Character Tables of Point Groups

Readership: Undergraduate and graduate students in related field for their course and researchers in group theory.

Key Features:

• Unique treatment of the material. Thus, for example, it provides a good balance between mathematical rigor and utilizing the mathematical results for obtaining useful applications without too much demand on the student. It contains many suitable illustrative examples, accompanied with useful figures and tables
• Most existing books on this subject are either too advanced, appropriate for the mature researchers, or too elementary and, thus, not suitable for dealing with a sufficient number of applications
• Most existing textbooks on the subject of discrete groups are old and do not lead to the new developments in crystal structure, such as 2-dimensional crystals, graphene etc.