Group Theory

Symmetry has sung its siren song to Physicists since the beginning of time, or at least since before there were Physicists. Today the ideas of symmetry are incorporated into a subject with the less imaginative and suggestive name of Group Theory. This Chapter introduces many of the ideas of group theory that are important in the natural sciences. Natural philosophers in the past have come up with many imaginative arguments for estimating physical quantities. They have often used out-of-the-box methods that were proprietary to pull rabbits out of hats. When these ideas were made available to a wider audience they were often improved upon in unexpected and previously unimaginable ways. A number of these methods are precursors of group theory. These are Dimensional Analysis, Scaling Theory, and Dynamical Similarity. We review these three methods in Sec. II. In Sec. III we get down to the business at hand, introducing the definition of group and giving a small set of important examples. These range from finite groups to discrete groups to Lie groups. These also include transformation groups, which played an important if underrecognized rôle in the development of classical physics, in particular the theories of Special and General Relativity. The relation between these theories and group theory is indicated in Sec. IX. Despite this important rôle in the development of Physics, groups existed at the fringe of the Physics of the early 20 century. It was not until the theory of the linear matrix representations of groups was invented that the theory of groups migrated from the outer fringes to play a more central rôle in Physics. Important points in the theory of representations are introduced in Sec. X. Representations were used in an increasingly imaginative number of ways in Physics throughout the 20 century. Early on they were used to label states in Quantum systems with a symmetry group: for example, the rotation group SO(3). Once states were named, degeneracies could be predicted and computations simplified. Such applications are indicated in Sec. XI. Later, they were used when symmetry was not present, or just the remnant of a broken symmetry was present. When used in this sense, they are often called “dynamical groups.” This type of use greatly extended the importance of Group Theory in Physics. As a latest tour de force in the development of Physics, groups play a central rôle in the formulation of Gauge Theories. These theories describe the interactions between fermions and the bosons and lie at the heart of the Standard Model. We provide the simplest example of a gauge theory, based on the simplest compact one parameter Lie group U(1), in Sec. XIII. For an encore, in Sec. XIV we show how the theory of the special functions of mathematical physics (Legendre and associated Legendre functions, Laguerre and associated Laguerre functions, Gegenbauer, Chebyshev, Hermite, Bessel functions, and others) are subsumed under the theory of representations of some low-dimensional Lie groups. The classical theory of special functions came to fruition in the mid 19 century, long before Lie groups and their representations were even invented.