The rank-2 Lie algebra and associated group SU(3) were introduced in Ch. 8. Although the algebra and group structure of SU(3) can be developed without reference to physics, it is easiest to illustrate its features through concrete applications to important physical problems. In this chapter we examine the pivotal contribution of SU(3) symmetry in imposing order on the phenomenology of strongly interacting elementary particles.

Chapter 6 described how to use the technology of Clebsch–Gordan coefficients and the Wigner–Eckart theorem to calculate matrix elements for the groups SU(2) and SO(3). In this chapter we wish to extend those methods to a more complicated group and illustrate some general means for calculating matrix elements when a symmetry and associated group structure of physical interest can be attached to a problem.

In Ch. 19 we saw that the weak, electromagnetic, and strong interactions are all described by local gauge theories and, if gravity is neglected, the fundamental interactions correspond to a gauge symmetry $\text{SU}{(2)}_{\text{w}}\times \text{U}{(1)}_y\times \text{SU}{{(}_3)}_{\text{c}}$ called the Standard Model.

Chapter 2 introduced some basic concepts relevant to an understanding of group theory and its application in physics. Often these concepts have been illustrated with finite groups, although most apply with suitable modification both to finite and to continuous groups. This chapter continues our introductory survey but now the emphasis will be on continuous groups, in particular on a certain kind of continuous group called a Lie group. We have already met some examples of these groups in the preceding chapter, but now their properties will be considered more systematically.

The integer quantum Hall effect and the fractional quantum Hall effect represent extraordinary physics that appears when a strong magnetic field is applied to a low-density electron gas confined in two dimensions at very low temperature. As we shall see, these effects can be interpreted in terms of quantum numbers that are topological, and hint at the existence of whole new classes of topological matter, which will be the subject of Ch. 29.

The three-dimensional harmonic oscillator is important in many areas of physics and it is well known that the 3D quantum oscillator exhibits an unusually high level of degeneracy. We have learned that non-accidental degeneracy is typical of a Hamiltonian invariant with respect to some symmetry. The degeneracy of the oscillator exceeds that deriving from rotational invariance, so it may be expected that the group associated with this symmetry is larger than the SO(3) of angular momentum.

Dynamical symmetry applied to nuclear structure physics has a long history tracing back to Wigner supermultiplet theory, SU(2) quasispin models, and the Elliott SU(3) model described in Ch. 10. These models have had broad conceptual influence but more limited practical application in the full context of nuclear structure physics because the conditions for their application are realized only in some nuclei.

In this chapter we consider the group SO(3) of continuous rotations in 3D space and the closely related special unitary group SU(2). These groups are of practical significance because of the importance of angular momentum in quantum mechanics, and they serve as examples of techniques that may be adapted to the analysis of more complicated groups. As part of this discussion we will investigate the relationship between the groups SO(3) and SU(2). They will be found to obey the same Lie algebra, so they are locally identical but differ in the global structure of the group manifold. Hence, we will also introduce in this chapter a distinction between the local and global properties of Lie groups.