Group Theory: A Physicist's Survey

by Pierre Ramond

Group theory has long been an important computational tool for physicists, but, with the advent of the Standard Model, it has become a powerful conceptual tool as well. This book introduces physicists to many of the fascinating mathematical aspects of group theory, and mathematicians to its physics applications. Designed for advanced undergraduate and graduate students, this book gives a comprehensive overview of the main aspects of both finite and continuous group theory, with an emphasis on applications to fundamental physics. Finite groups are extensively discussed, highlighting their irreducible representations and invariants. Lie algebras, and to a lesser extent Kac-Moody algebras, are treated in detail, including Dynkin diagrams. Special emphasis is given to their representations and embeddings. The group theory underlying the Standard Model is discussed, along with its importance in model building. Applications of group theory to the classification of elementary particles are treated in detail.

ebook

Published March 5, 2013

About the author

Ramond played a major role in the development of superstring theory. In 1971, Ramond generalized Dirac's work for point-like particles to stringlike ones. In this process he discovered two-dimensional supersymmetry and laid the ground for supersymmetry in four spacetime dimensions. He found the spectrum of fermionic modes in string theory and the paper started superstring theory. From this paper André Neveu and John Schwarz developed a string theory with both fermions and bosons.

According to quantum mechanics, particles can be divided into two types: bosons and fermions. The distinction between bosons and fermions is basic. Fermions are particles which have half integer spin (1/2, 3/2, 5/2 and so on), measured in units of Planck's constant and bosons are particles which have integer spin (0, 1, 2 and so on), measured in units of Planck's constant. Examples of fermions are quarks, leptons and baryons. Quantum of fundamental forces such as gravitons, photons, etc. are all bosons. In quantum field theory, fermions interact by exchanging bosons.

Early string theory proposed by Yoichiro Nambu and others in 1970 was only a bosonic string. Ramond completed the theory by inventing a fermionic string to accompany the bosonic ones. The Virasoro algebra which is the symmetry algebra of the bosonic string was generalized to a superconformal algebra (the Ramond algebra, an example of a super Virasoro algebra) including anticommuting operators also.