In the second part of the 20th century, algebraic methods have emerged as a powerful tool to study theories of physical phenomena, especially those of quantal systems. The framework of Lie algebras, initially introduced by - phus Lie in the last part of the 19th century, has been considerably expanded to include graded Lie algebras, in?nite-dimensional Lie algebras, and other algebraic constructions. Algebras that were originally introduced to describe certainpropertiesofaphysicalsystem,inparticularbehaviorunderrotations and translations, have now taken center stage in the construction of physical theories. This book contains a set of notes from lectures given at Yale Univ- sity and other universities and laboratories in the last 20 years. The notes are intended to provide an introduction to Lie algebras at the level of a one-semester graduate course in physics. Lie algebras have been particularly useful in spectroscopy, where they were introduced by Eugene Wigner and Giulio Racah. Racah’s lectures were given at Princeton University in 1951 (Group Theory and Spectroscopy) and they provided the impetus for the initial applications in atomic and nuclear physics. In the intervening years, many other applications have been made. This book contains a brief account of some of these applications to the ?elds of molecular, atomic, nuclear, and particle physics. The application of Lie algebraic methods in Physics is so wide that often students are overwhelmed by the sheer amount of material to absorb.
This book, designed for advanced graduate students and post-graduate researchers, provides an introduction to Lie algebras and some of their applications to the spectroscopy of molecules, atoms, nuclei and hadrons. In the first part, a concise exposition is given of the basic concepts of Lie algebras, their representations and their invariants. The second part contains a description of how Lie algebras are used in practice in the treatment of bosonic and fermionic systems. Physical applications considered include rotations and vibrations of molecules (vibron model), collective modes in nuclei (interacting boson model), the atomic shell model, the nuclear shell model, and the quark model of hadrons. One of the key concepts in the application of Lie algebraic methods in physics, that of spectrum generating algebras and their associated dynamic symmetries, is also discussed. The book contains many examples that help to elucidate the abstract algebraic definitions. It provides a summary of many formulas of practical interest, such as the eigenvalues of Casimir operators and the dimensions of the representations of all classical Lie algebras.
