Group Theory

Abstract

Symmetry can be seen as the most basic and important concept in physics. Momentum conservation is a consequence of translational symmetry of space. More generally, every process in physics is governed by selection rules that are the consequence of symmetry requirements. On a given physical system, the eigenstate properties and the degeneracy of eigenvalues are governed by symmetry considerations. The beauty and strength of group theory applied to physics resides in the transformation of many complex symmetry operations into a very simple linear algebra. The concept of representation, connecting the symmetry aspects to matrices and basis functions, together with a few simple theorems, leads to the determination and understanding of the fundamental properties of the physical system, and any kind of physical property, its transformations due to interactions or phase transitions, are described in terms of the simple concept of symmetry changes. The reader may feel encouraged when we say group theory is “simple linear algebra.” It is true that group theory may look complex when either the mathematical aspects are presented with no clear and direct correlation to applications in physics, or when the applications are made with no clear presentation of the background. The contact with group theory in these terms usually leads to frustration, and although the reader can understand the specific treatment, he (she) is unable to apply the knowledge to other systems of interest. What this book is about is teaching group theory in close connection to applications, so that students can learn, understand, and use it for their own needs.