# Why Irreducibel Representations?

## Main Question or Discussion Point

Hallo,

I would like to know why physicists are always seeking for irreducible representation of a given group. I know that a reducible one is decomposable into irreducible representations (under special circumstances), but what is the physical motivation that irreducible reps are fundamental? Is the calculation just easier or is there any deeper thought behind it?

thank you all for some hints!

- philipp

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CompuChip
Homework Helper
Well, as you said, reducible representations are made up of irreducible ones. That is always the case (not under special circumstances, as you say) though in some circumstances it may be more obvious.
But consider for example this case: you have an irreducible representation with nxn matrices M. Then you can make another representation of k n x k n by making new matrices, which have k times M on their diagonal. Clearly, this representation doesn't add any new physical information, it's just bigger and therefore more unpleasant to work with.

Similarly, all the information in a representation is stored in its irreducible parts, which are always smaller (or as large, but never bigger) than the reducible representation. Therefore, to understand the physics, it's much easier to look at the smaller blocks and understand each of them separately. Then a reducible representation contains just the same information. I also think that there are only finitely many irreducible representations (up to some obvious isomorphisms, maybe) while there are infinitely many irreducible ones (in the previous paragraph, there is a recipe to make infinitely many from just one )

Also, view the converse situation. You are doing a calculation, and out comes some representation in big matrices. It is a hell of a job to find all the physics in there. But it may turn out, that you can split it into smaller parts, of which you know exactly what they do, and you actually already know everything that makes up the representation. From this point of view, it's something like diagonalizing a matrix to read off the eigenvalues: if you have the right diagonalization technique its easier than solving the characteristic polynomial - and the matrix may look very complicated, but it will become extremely easy if you choose the right basis (OK, so maybe that was a bad analogy, but it was the first to come to mind and it seems appropriate at first sight ).

By the way, mathematicians do the same. Give a mathematician some large group, and the first thing (s)he'll do is try to find all the (normal) subgroups to see if it's made up of pieces that (s)he already knows.

Dr Transport said it just right, and much shorter [/edit]

Dr Transport
Gold Member
Think of irreducible representations as the eigenvectors of a system. By reducing, we are actually expanding a set of vectors into the fundamental eigenvectors.

so to understand you right; the reason to look for irred.reps (in physics!) is just something according to their great (mathematical) properties.....theres nothing like a 1:1 correspondance in nature that allows us only to look for irreducibel reps.

It is sometimes tricky in physics to find out if this is wether a mathematical reasoning or has something to do with intuitive observations of nature only to look for this or that property.

But I think I gotcha concerning what you have said.....thanks for that!

CompuChip
Homework Helper
I'd prefer to look at it this way; all the physics is locked in the irreducible representations. You can just as well use reducible ones, but then there is redundancy in your mathematical description (which might make the problem easier, or harder, depending on the situation - we just try to choose the most convenient mathematical formulation all the time). We do this all the time, for example: we also use 4-component vectors to describe physical particles with only two degrees of freedom, just because it is "convenient".

Fredrik
Staff Emeritus
Gold Member
I have only had to think about irreducible representations in the case of the Poincaré group, and in that case it's definitely more than just a convenience. Once we have decided to trust the postulate of quantum mechanics that says that states are represented by vectors in a Hilbert space, we need to start thinking about representations of the Poincaré group. For example, if I use a vector X to represent the state of a system, I would have used another vector Y=U(R)X to represent the state of the system if I had been rotated relative to my current orientation as described by the rotation matrix R. The function U is a representation. (Actually it's either a "projective representation" of the Poincaré group or an ordinary representation of its universal covering group).

If you now start thinking about breaking the representation up into irreducible representations, you will end up with invariant subspaces of the original Hilbert space that correspond to the states of non-interacting elementary particles. Each invariant subspace is characterized by a particular value of mass and spin.

So the mathematical representation of the state of a non-interacting elementary particle is always a vector in the representation space of an irreducible representation.

Chapter 2 of vol. 1 of Weinbergs QFT book explains this stuff pretty well.

If you now start thinking about breaking the representation up into irreducible representations, you will end up with invariant subspaces of the original Hilbert space that correspond to the states of non-interacting elementary particles. Each invariant subspace is characterized by a particular value of mass and spin.

So the mathematical representation of the state of a non-interacting elementary particle is always a vector in the representation space of an irreducible representation.

Chapter 2 of vol. 1 of Weinbergs QFT book explains this stuff pretty well.
But this sounds more like a definition to me, something grown out of experience or being a convention.

I hope that I am right in thinking of irreducible representations as the fundamental object for any representation(?) And that is the reason for looking such structures if we are doing axiomatic QM. You consider what a state is (ray in hilbert space), observable is treated as an operator acting on hilbert space and so on.

Then you must realize what to call a symmetry in QM, in general the theorem of wigner gives the answer. By choosing the symmetry group (for non-relativistic QM the Galilei group) you need to find a representation. This might be an irreducible one, but I think we are looking for those structures since we know from mathematics that irreducible reps got some fine properties.

because ad hoc I dont see really that an elementary particle is described as a physical system its corresponding hilbert space enables a irreducibel rep. of the galilei group.

So I will try to ask slightly different; why is this a good defintion of an "elementary particle" (given by fredrik)? Would other definitions of an elementary particle lead to contradiction? Or this defintion something like an axiom and by intuitive reasoning (that irr. reps are so fundamental) it is somewhat clear to look for those reps?

Fredrik
Staff Emeritus
Gold Member
I don't think their "fine properties" is the only reason to look for irreducible representations. A physical system is represented by a Hilbert space. Different observers may disagree about what vector to use to represent the state of the system, but they will agree about what Hilbert space to use. So if an elementary particle is a physical system at all, it must correspond to a Hilbert space, and if that space is the representation space of a reducible representation, it contains eigenstates of different mass, or eigenstates of different spin. That doesn't sound very "elementary" to me.

But you are still right that this definition of one-particle states should be considered an axiom (if that's what you're saying), or rather, it's the identification of these "mathematical particles" with particles in the real world (defined roughly as "those things that cause a signal in a detector") that should be considered an axiom.

I can't answer the question about other definitions. I think I would at least have to see such a definition to know if it leads to contradictions.

strangerep
[...] why is this a good defintion of an "elementary particle" [...]
We're trying to classify the _types_ of elementary particles. If one type could become
indistinguishable from another type merely via a symmetry transformation, then that's
not a very good definition of "type". The irreducible representations are good for classifying
types because they tend to "remain themselves" under the action of the symmetry group.

HTH.

Is there some indication that elementary particles such as quarks and leptons are irreducible or are physicists holding out how that it doesn't stop there?

Fredrik
Staff Emeritus