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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.

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I thought the wikipedia article here was pretty good, but maybe it only makes sense if you have a little quantum mechanics background: en.wikipedia.org/wiki/… Were you able to get anything out of it? –  j.c. Nov 2 '10 at 19:13
    
i mostly thought this was a better place for the question than MO. –  Sean Tilson Nov 2 '10 at 19:16
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you may want to check out this question: math.stackexchange.com/questions/622/… –  Eric Nov 3 '10 at 19:22
    
Thanks Eric, Noah's comment is a really nice one to think about. –  Sean Tilson Nov 3 '10 at 21:23
    
I am accepting an answer because there is an answer on this page, and anyone looking for an answer to the question should consider what is here. Thank you JC, Igor, and Eric. –  Sean Tilson Nov 3 '10 at 21:36
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3 Answers 3

up vote 11 down vote accepted

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.

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See the Wigner theorem, it explain rigorously the relationship between a group of symmetries and states of a physical particle.

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The vector space that is being acted on typically is a Hilbert space of states in quantum mechanics; very roughly, there's a basis of this vector space which is in one-to-one correspondence with the set of possibilities for a physical system. The simplest example to try to get your head around is that of the spin 1/2 particle (2 dim representation of SU(2)), which is explained in any introductory quantum mechanics book.

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so the fundamental particle is acting on the quantum states? –  Sean Tilson Nov 2 '10 at 19:36
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No, the states of the fundamental particle are acted on by some symmetry operation, e.g. rotation in 3 dimensions. The "internal" symmetry groups in the standard model, e.g. the SU(3) corresponding to QCD color are a bit more abstract in that I can't imagine a way of actually physically performing an SU(3) rotation on a quantum field, but you might think of them as corresponding to different ways that a person might change coordinates when measuring a certain property of a quantum field. –  j.c. Nov 2 '10 at 20:17
    
By the way, if there's something I've said which is particularly unclear; let me know. At the moment I'm not sure if anything I've said makes any sense, or if it's complete nonsense to you. –  j.c. Nov 3 '10 at 14:55
    
Let me add that there might be an interpretation like the one in your comment for gauge theories, where the interaction with gauge bosons are interpreted as actions of the symmetry group, but I purposefully didn't go after that, since that's a much harder topic, that I'm not even sure that I have straight... –  j.c. Nov 3 '10 at 14:57
    
sorry, I have been really busy with other completely unrelated stuff. Your comment was more helpful and clear than the wikipedia article you linked to. I was exactly looking for such an explanation. Is the Gauge theoretic interpretation more complicated mathematically or physically? –  Sean Tilson Nov 3 '10 at 20:17
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