Lie theory, Representations and particle physics

This is a question that has been posted at many different forums, I thought maybe someone here would have a better or more conceptual answer than I have seen before:

Why do physicists care about representations of Lie groups? For myself, when I think about a representation that means there is some sort of group acting on a vector space, what is the vector space that this Lie group is acting on?

Or is it that certain things have to be invariant under a group action? maybe this is a dumb question, but i thought it might be a good start...

To clarify, I am specifically thinking of the symmetry groups that people think about in relation to the standard model. I do not care why it might be a certain group, but more how we see the group acting, what is it acting on? etc.

Answer

Let me give a try. When we construct a theory, we suspect that the objects it deals with can be rather complicated. It is natural that we want to find the simplest «building blocks» which the complicated objects are made of. If our theory were absolutely arbitrary, we won't be able to classify these simple building blocks at all. Fortunately, when constructing theories we note that the lagrangian we specify and the vacuum state have certain symmetries. Once we noted it, then it is pure math to show that the simple objects in our theory should be classified according to representations of the symmetry group of the lagrangian and the vacuum state.

Note that there are some symmetries which are obvious to us, which we perceive (like invariance under the Poincare group), and there are some symmetries which we invent (like non-abelian gauge symmetries). In the latter case we know that, by construction, all the macroscopic states (including the vacuum state) must be invariant under this new internal symmetry group. This gives us a short-cut to the assertion that the simple object in our theory must be classified according to the representations of the new group.

And what concerns the specific question:

so the fundamental particle is acting on the quantum states?

When we say that a particle or a field is in representation R of group G, we do not mean that the particles are associated with matrices of representation R acting on something else. We rather mean that the particle can be written in terms of eigenstates of matrices representing operators in R. So, it is the symmetry group transformations that act on the particles.