Short Introduction to and Motivation for Representation Theory

What may seem at a first glance like just another mathematical gimmick of group theory, is of incredible importance in physics. One can consider the Poincaré group (the set of all transformations that leave the speed of light constant) and use the framework of representation theory to construct the irreducible representations of this group. (The irreducible representations are the basic building blocks all other representations can be built of.) The straight-forward examination of the irreducible representations of the Poincaré group gives us physicists the appropriate mathematical tools needed to describe nature at the most fundamental level.

1. The lowest dimensional representation is trivial and called scalar or spin $0$ representation,  because the objects (scalars) the group acts on in this representation are used to describe elementary particles of spin $0$. (In this representation the group doesn’t changes the objects in question at all.)
2. The next higher-dimensional representation is called spin $\frac{1}{2}$ or spinor representation, because the objects (spinors) the group acts on in this representation are used to describe elementary particles of spin $\frac{1}{2}$.
3. The third representation is called spin $1$ or vector representation, because the objects (vectors) the group acts on in this representation are used to describe elementary particles of spin $1$.

But what exactly is a representation?

For theoretical considerations its often useful to regard any group as an abstract group. This means defining the group by its manifold structure and the group operation. For example $SU(2)$ is the three sphere $S^3$, the elements of the group are points of the manifold and the rule associating a product point $ab$ with any two points $b$ and $a$ satisfies the usual group axioms. In physical applications one is more interested in what the group actually does, i.e. the group action.

An important idea is that one group can act on many different kinds of objects (this will make much more sense in a moment). This idea motivates the definition of a representation: A representation is a map between any group element $g$ of a group $G$ and a linear transformation $T(g)$ of some vector-space $V$ in such a way that the group properties are preserved:

1. $T(e)=I$ (The identity element of the group transforms nothing at all)
2. $T(g^{-1})=\big ( T(g) \big )^{-1} $ (Every inverse element is mapped to the corresponding inverse transformation
3. $T(g)\circ T(h) = T(gh)$ (The combination of transformations corresponding to $g$ and $h$ is the same as the transformation corresponding to the point $gh$)

This concept can be formulated more general if one accepts arbitrary (not linear) transformations of an arbitrary (not vector) space. Such a map is called a realization.

In physics one is concerned most of the time with linear transformations of objects living in some vector space (for example, Hilbert space in Quantum Mechanics or Minkowski space for Special Relativity), therefore the concept of a representation is more relevant to physics than the general concept called realization.

A representation identifies with each point (abstract group element) of the group manifold (the abstract group) a linear transformation of a vector space. The framework of representation theory enables one to examine the group action on very different vector spaces.

One of the most important examples in physics is $SU(2)$. For example one can examine how $SU(2)$ acts on the complex vector space of dimension two: $C^2$ (the action on $C^1$ is trivial). The objects living in this space are complex vectors of dimension two. Therefore $SU(2)$ acts on these objects as $2\times2$ matrices. The matrices (=linear transformations) acting on $C^2$ are just the usual matrices one identifies with $SU(2)$. Nevertheless we can examine how $SU(2)$ acts on $C^3$. There is a well defined framework for constructing such representations and as a result $SU(2)$ acts on complex vectors of dimension three as $3\times 3$ matrices for which a basis is given by

\begin{equation} J_1 = \frac{1}{\sqrt{2}}  \begin{pmatrix}  0& 1 & 0 \\ 1&0 & 1 \\ 0 & 1 & 0  \end{pmatrix}  , \qquad  J_2 = \frac{1}{\sqrt{2}}  \begin{pmatrix}  0& -i & 0 \\ i&0 & -i \\ 0 & i & 0  \end{pmatrix}  , \qquad J_3 =  \begin{pmatrix}  1& 0 & 0 \\ 0&1 & 0 \\ 0 & 0 & 1  \end{pmatrix}  \end{equation}

One can go on and inspect how $SU(2)$ acts on higher dimensional vectors. This can be quite confusing and maybe its better to call the group $S^3$ instead of $SU(2)$, because usually $SU(2)$ is defined as the set of complex $2\times 2$ (!!) matrices satisfying

$U^\dagger U = 1$ and $\det(U)=1$

and now we write $SU(2)$ as $3 \times 3$ matrices. Therefore one must always keep in mind that one means the abstract group, instead of the $2 \times 2 $ definition, when one talks about higher dimensional representation of $SU(2)$ or any other group.

Typically a group is defined in the first place by a representation. This enables one to study the group properties concretely. After this initial study its often more helpful to regard the group as an abstract group, because its possible to find other, useful representations of the group.