Mathematical Methods for Physics and Engineering - Matematica.NET

29Representation theoryAs indicated at the start of the previous chapter, significant conclusions canoften be drawn about a physical system simply from the study of its symmetryproperties. That chapter was devoted to setting up a formal mathematical basis,group theory, with which to describe and classify such properties; the currentchapter shows how to implement the consequences of the resulting classificationsand obtain concrete physical conclusions about the system under study. Theconnection between the two chapters is akin to that between working withcoordinate-free vectors, each denoted by a single symbol, and working with acoordinate 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 interms of matrices; it will be clear that ordinary numbers would not be sufficient,as they make no provision for any non-commutation amongst the elementsof a group. Thus, in this chapter the group elements will be represented bymatrices 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 awritten description of the elements and their properties without recourse to suchrepresentations. Most of our applications will be concerned with representationsof the groups that consist of the symmetry operations on molecules containingtwo or more identical atoms.Firstly, in section 29.1, we use an elementary example to demonstrate the kindof conclusions that can be reached by arguing purely on symmetry grounds. Thenin sections 29.2–29.10 we develop the formal side of representation theory andestablish general procedures and results. Finally, these are used in section 29.11to tackle a variety of problems drawn from across the physical sciences.1076