Irreducible representation in physics

Question

I am confused about something.

Group theory books written for physicists say that any reducible representation can be decomposed in terms of irreducible representations (so correct me if I am wrong, to me irreducible representations are like the unit vectors i j k in terms of which any 3D vector can be expanded, or they are like sines and cosines in terms of which any periodic function can be Fourier expanded.)

Now at the same time they say that any bigger representation of a group can be built out of irreducible ones.

What is unclear to me is the physical motivation for each direction. Of course those books contain physical applications but the big picture is never obvious (they get to applications after 200-300 pages of abstract details).

If someone could answer the following questions I would be really appreciated:

1-what is the physical motivation to write a representation in terms of irreducible ones?

2-what is the physical motivation to build bigger representations using irreducible ones?

Answer 1

I guess you mean unitary representations, as for nonunitary reps, your introductory statement is wrong.

Arbitrary representations exists in scores, while irreducible representations are fairly few. For many groups one knows and understands in detail the irreducible representations. If one knows the decomposition of a representation under study into irreps, one can usually answer questions about this rep by using the knowledge about the irrep.

For example if a Hamiltonian is invariant under a symmetry group, spectral calculations simplify by looking at the (often easy to determine) spectrum of the irreducible parts.

If a symmetry group acts on a space (a very typical situation), one knows that it is some rep of the group, but to know which one, it must be build from the irreducible ones.

But building reps from smaller reps can be done in a number of ways. For example, by taking tensor products of irreps one gets reducible reps, and the reducible parts may be new irreps. For example, in the simplest case U(2), all irreps can be obtained from the defining 2D rep by taking tensor products followed by splitting these into the irreducible reps. Thus bulding up and splitting are complementary, and benefit each other.

In the applications, one needs to understand arbitrary reps. To understand them one breaks them down into irreducible ones and studies these first (like the factorization of integers reduces integers to primes). After having answered in the irreducible case the questions that usually arise one can go back to the general situation and see how much information one can lift from the irreps to the general case. (Usually everything of importance.)

Thus the natural emphasis is on studying the properties of irreps first, and then looking at what this implies for the remaining reps.

Answer 2

The physical motivation is pretty simple. In quantum mechanics this means that if Hamiltonian is invariant under all $g\in G$ you may use the fact that the solutions of this Hamiltonian form a (reducible) representation of this group (as any other full system). Here comes representation theory. It is pretty easy to show that the energies of the states corresponding to the same irreducible representation are equal. Furthermore, there is Wigner-Eccart theorem which allows one to reduce number of values which describe the system by using its symmetry properties.

Thus, knowing irreducible representations of $G$ one may say something about levels degeneracy, selection rules, and so on. As a result, it turns out that it is constructive to classify states of the system in accordance with irreducible representations of this system symmetry group.

It is exactly how energy levels in atoms are classified. This idea may be easily generalized from $SO(3)$ to any other group of symmetry.

For the second question, I do not really get it. Usually one has the number of the states which form a basis of the reducible representation given by number of particles in the system, etc. So it is not very natural to build up reducible representations by purpose.