It has rightly been said that the mathematical theory of groups and group representations is a magnificent gift of nineteenth century mathematics to twentieth century physics. While this is particularly true within the framework of quantum mechanics, with the passage of time its relevance within classical physics has also become well understood and greatly appreciated. Today the importance of group theoretical ideas and methods for physics can hardly be overemphasised; and over the past century or so, a veritable profusion of books devoted to this theme, many of them gems of the literature, have appeared.

The present monograph is primarily based on lectures given by one of us (NM) at the Institute of Mathematical Sciences in Chennai, India, in the Fall of 2007. The lectures were prepared and presented at the invitation of Rajiah Simon, to whom both authors are indebted for his support and encouragement.

The course was titled ‘Continuous Groups for Physicists’ and consisted of about 45 extended lectures over a two month period. Its aim was to introduce the basic ideas of continuous groups and some of their applications to an audience of post graduate and doctoral students in theoretical physics. After an introduction to the basic ideas of groups and group representations (mainly in the context of finite groups and compact Lie groups), the course presented a selection of useful, interesting and quite sophisticated specific topics not often included in standard courses in physics curricula. The methods and concepts of quantum mechanics served as a backdrop for all the lectures.

The real rotation groups in two and three dimensions are followed by an account of the structures of Lie groups and Lie algebras, and then a description of the compact simple Lie groups. Their irreducible representations are described in some detail. Some of the ‘non standard’ topics that follow are: spinor representations of real orthogonal groups in both even and odd dimensions; the notion of the ‘Schwinger’ representation of a group with examples, induced representations, and systems of generalised coherent states; the properties and uses of the real symplectic groups, which are defined only in real even dimensions, and their metaplectic covering group, in a quantum mechanical setting; and the Wigner Theorem on the representation of symmetry operations in quantum mechanics.

The study of SO (3) and SU (2) has shown how elements of a continuous group can be labelled by (a certain number of) real independent continuous coordinates or parameters; how the composition law can be expressed using these coordinates; how in a representation we encounter generators, commutation relations, structure constants; the representation of finite group elements by (products of) exponentials in the generators; and so on. Now we will try to understand all this in a more basic manner and in a general situation.

The work of this chapter and the next one will lead us to a vast generalisation of SO (3) and SU (2) resulting in the so-called classical families of continuous groups which are all, like SO (3) and SU (2), compact. (The concept of compactness will be briefly described in a heuristic manner in Chapter 5.) These are mathematical results from the late nineteenth and early twentieth centuries, associated with the names of Killing, Cartan and Weyl and are truly beautiful.

Local Coordinates, Group Composition, Inverses

Let a Lie group G be given. The dimension of G , also called its order, will hereafter be denoted by r rather than n . (In the development of the theory of compact simple Lie groups, the order is traditionally denoted by r ; and another important property called the rank, which we will come to in Chapter 5, by l . These are the notations used, for instance, in the classic 1951 Princeton lectures by Giulio Racah on Group Theory and Spectroscopy.) In some neighborhood N of e ∈ G , we use r essential real independent parameters to label group elements:

It is understood that a , b , · · · ∈ N ⊂ G. As a convention we always assume

As a ∈ G runs over N , α runs over some open set around the origin in r -dimensional Euclidean space. So in this region and inN , coordinates and group elements determine one another uniquely.

In this chapter, we present the basic theory of finite groups and their representations as preparation for the discussion of continuous groups that starts from Chapter 3. It is assumed that readers know the basics of set theory, vector spaces, transformations, linear operators, matrix representations, inner products and such. These will be called upon as and when needed.

Definition of a Group

A group G is a set of elements a , b , c , · · · , g , g′ , · · · , e , · · · along with a composition (or ‘multiplication’) law obeying four conditions:

• (i) Closure: a , b ∈ G → ab = unique product element ∈ G .

• (ii) Associativity: for any a , b , c ∈ G ,

  a (bc ) = (ab )c = abc ∈ G .

• (iii) Identity: there is a unique element e ∈ G such that

  ae = ea = a , for any a ∈ G.

• (iv) Inverses: for each a ∈ G , there is a unique a−1 ∈ G , the inverse of a , such that

  a −1a = aa −1 = e . (1.1)

The composition rule or law can be called a binary law as the product is defined for any pair of elements. The conditions in (1.1) could be stated in more economical forms, for instance introducing only a left identity and left inverses, and then showing that the more general properties in (1.1) do hold.

One can immediately think of various qualitatively different possibilities. The number of (distinct) elements in G may be finite. Then this number, denoted by |G |, is called the order of G . Some other possibilities are that the number of elements may be a discrete infinity, or else a continuous infinity with G being a manifold of some dimension.

Some Examples

• (i) The symmetric group, the group of permutations on n objects, is finite, of order n !, and is denoted by Sn . We mention only a few pertinent properties now, and go into some more detail in Chapter 2.

The aim of this chapter is to look at the structures of Lie groups related to space and spacetime. We look very briefly at SO (3) which we have studied earlier quite extensively, then the Euclidean group E (3) which acts on physical space; the Galilei group on spacetime underlying nonrelativistic Galilean–Newtonian mechanics; the homogeneous Lorentz group SO (3, 1) and its double cover SL (2,ℂ); and finally the Poincaré group P basic to special relativity. In each case we look at the defining representation resulting from action on spacetime; useful descriptions of the group, the composition law and inverses; the structure of the Lie algebra and possible neutral elements which are permitted; and then a study of the UIR's of the concerned group. This involves constructing Casimir invariants, physical interpretation, etc. In the SO (3, 1) and SL (2,ℂ) cases, we also look at all their finite dimensional nonunitary representations. We will see similarities to the discussion of induced group representations in Chapter 7, in connection with representations of the group E(3) and the Poincaré group.

SO (3) andSU (2)

We studied these two groups and their UIR's in some detail in Chapter 3. We saw that SU (2) is a two-fold covering of SO (3). Here we first deal with the possible presence of neutral elements in a hermitian representation of the Lie algebra generators Jj . As we have seen in Chapter 9, Eq. (9.76), to handle ray representations of SO (3) and SU (2) in quantum mechanics we must allow for the presence of neutral elements djk in the basic angular momentum commutation relations:

But these are immediately and easily eliminated: antisymmetry djk = ∈dkj implies djk = ∈jkldl for some real dl ; then if we redefine J′jj = Jj + dj we get the standard commutation relations without any neutral elements:

This happens since physical space is three dimensional, there being no need to invoke the Jacobi identity explicitly. At the same time, no further shifts in J′l are permitted.