29

Representation theory

As indicated at the start of the previous chapter, significant conclusions can

often be drawn about a physical system simply from the study of its symmetry

properties. That chapter was devoted to setting up a formal mathematical basis,

group theory, with which to describe and classify such properties; the current

chapter shows how to implement the consequences of the resulting classifications

and obtain concrete physical conclusions about the system under study. The

connection between the two chapters is akin to that between working with

coordinate-free vectors, each denoted by a single symbol, and working with a

coordinate system in which the same vectors are expressed in terms of components.

The ‘coordinate systems’ that we will choose will be ones that are expressed in

terms of matrices; it will be clear that ordinary numbers would not be sufficient,

as they make no provision for any non-commutation amongst the elements

of a group. Thus, in this chapter the group elements will berepresentedby

matrices that have the same commutation relations as the members of the group,

whatever the group’s original nature (symmetry operations, functional forms,

matrices, permutations, etc.). For some abstract groups it is difficult to give a

written description of the elements and their properties without recourse to such

representations. Most of our applications will be concerned with representations

of the groups that consist of the symmetry operations on molecules containing

two or more identical atoms.

Firstly, in section 29.1, we use an elementary example to demonstrate the kind

of conclusions that can be reached by arguing purely on symmetry grounds. Then

in sections 29.2–29.10 we develop the formal side of representation theory and

establish general procedures and results. Finally, these are used in section 29.11

to tackle a variety of problems drawn from across the physical sciences.