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## Main Question or Discussion Point

**"An irreducible representation of..."**

So I was reading this paper by Max Tegmark linked from another thread, and one particular thing he said-- although didn't really have anything to do specifically with the paper it was part of-- caught my eye:

So this is something I've actually seen expressed before, but this is the clearest expression of it I've ever seen. I wanted to ask a few (possibly stupid) questions about it.Indeed, when asked "What is a particle?", many theoretical physicists like to smugly reply "An irreducible representation of the Poincaré group". This refers to the famous insight by Wigner and others [3, 35-38] that any mathematical property that we can assign to a quantum-mechanical object must correspond to a ray representation of the group of spacetime symmetries. Let us briefly review this argument...

Wigner and others studied the special case of quantum mechanics, where the description [tex]\psi[/tex] corresponded to a complex ray, i.e., to an equivalence class of complex unit vectors where any two vectors were defined as equivalent if they only differed by an overall phase. They realized for this case, the symmetry transformation [any symmetry transformation] [tex]\psi[/tex] must be linear and indeed unitary, which means that it satisfies the definition of being a so-called ray representation (a regular unitary group representation up to a complex phase) of the symmetry group G. Finding all such representations of the Poincaré group thus gave a catalog of all possible transformation properties that quantum objects could have (a mass, a spin=0, 1/2, 1, ... , etc), essentially placing an upper bound on what could exist in a Poincaré-invariant world. This cataloging effort was dramatically simplified by the fact that all representations that can be decomposed into a simple list of irreducible ones, whereby degrees of freedom in [tex]\psi[/tex] can be partitioned into disjoint groups that transform without mixing between the groups.

1. A variation on this claim that I've seen is that each quantum-mechanical spin value corresponds to an irreducible representation of SU(2). Is this linked to Tegmark's claim that the irreducible representations of Poincare provide all the quantum numbers of a particle; and if so, how? I.E., is the idea that SU(2) is a subgroup of the Poincaré group or something (...err, is it?) and there is some way we can decompose the Poincare group into SU(2) and some other subgroups where each of the subgroups has irreducible representations which correspond to a different quantum number?

2. In general, would it be correct to say that the set of quantum numbers accessible to us is a consequence of the poincare group being the fundamental symmetry of our spacetime-- and if we lived in a spacetime with some other fundamental symmetry there would be different quantum numbers? Or is the way the quantum numbers line up with the poincare group largely coincidental? Say, what if we were to construct a toy model universe that had Lorentz group (which IS a subgroup of the Poincare group) symmetry, but not full Poincare symmetry-- would we "lose" some quantum number? (Which one?)

3. Are the quantum numbers resulting from the poincare group in any way fundamentally different from the quantum numbers resulting from the SU(3)xSU(2)xU(1) symmetries of the standard model gauge groups? (For example when we say "quantum numbers" in this context we just mean a conserved quantity right?)

4. I'm finding I'm actually kind of confused as to what an "irreducible representation" is, or a "representation" for that matter. I'd been under the impression that a "representation" of a lie group was when you take a lie group and then you find a type of affine transformation (i.e. a set of matrices, or rather a subspace of some GL(n)) that has the exact same properties as the lie group you want to model. Now I've been reading some stuff and I'm not so sure. This is kind of shifting from physics to math, and perhaps this is too big a question to be stuffing at the end of the post?!?, but...

I'm looking at this "Representation Theory of Lie Algebras" text by Clara Loeh which Garrett Lisi linked in a previous thread here. This gives a very compact description of representations and irreducible representations, but I want to make sure I understand it. Loeh says on page 3, only in pretty TeX:

So, looking at this closer:A representation of the Lie algebra g is a (finite dimensional) complex vector

space V together with a homomorphism g -> gl(V) of lie algebras.

A subrepresentation of a representation R: g -> gl(V) consists of a subspace W satisfying

For all x which are a member of g:

(R(x))(W) [is a proper subset of] W

A representation of the lie algebra g is called irreducible if it contains no proper subrepresentations.

This sounds basically like what I said before-- we have a vector space, and there's a homomorphism from the lie algebra into* the general linear group of that vector space. (* I assume the homomorphism can be injective?)A representation of the Lie algebra g is a (finite dimensional) complex vector

space V together with a homomorphism g -> gl(V)

The next part:

This... is... a little more confusing to me? I'm not entirely sure I properly understand the operation being described by (W) being in parenthesis like that. I think what this is saying is: we have a representation R defined as described above, and also a W which is a proper subspace of V. For each item in g, when you pick the transformation out of gl(V) corresponding to that item under the representation, and apply that transformation to the space W, the resulting space is a [proper?] subspace of W.A subrepresentation of a representation R: g -> gl(V) consists of a subspace W satisfying

For all x which are a member of g:

(R(x))(W) [is a proper subset of] W

Is that right so far? And if so, why must (R(x))(W) be a

*proper*subspace of W, couldn't it be equal (for example under the identity transformation)? Or is the idea that when we are talking about subspaces you are allowed to omit the bar under the subset symbol and have it be assumed?

And if this is all more or less correct: Would it be accurate to summarize the Loeh text as saying that a representation of a lie algebra is a space of linear transformations to which the lie algebra has a homomorphism, and a subrepresentation of that representation would be a subspace of that space which is closed under its own transformations?