An irreducible representation of

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"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:

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.
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.

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:

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.
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)
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?)

The next part:

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
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.

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?
 

Answers and Replies

  • #2
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Hi.
I Think you are making confusion on what kind of simmetries we see in nature (experiments).
You should first distinguish between external and internal simmetries.
The first are the ones coming from the invariant of the Action under transformation of the coordinates (X'=Mx) where M belongs to Poincare Group and x are the space time coordinates.
The internal simmetries are the connected to the degree of freedom of a multiplet.
In any case what you are using are Groups. In the case you are using Lie Groups, there exsist a fantastic theorem (Noether) that tells you how to connect simmetries to conservation laws, say quantum Number if your theory is a quantum one.
But IMPORTANT is that you dont need a quantum theory to state so.
Another point: you were asking if Lorentz Group has something to do with SU(2). Well it is possible to show that SO(3.1)=SO(4)=SU(2)XSU(2) locally wich answer to your question. But the important thing is that: while in Math an SU(2) is just a Group, in Physics it has many differnt interpretation. Let me explain. SU(2) can be the Group of invariance of classical particle and also the isospin one. But mathematically talking it does have the same dimension and same structure. When you talk About the S.M. SU(3)XSU(2)XU(1) you are tralking about internal simmetries. Ponicare invariance is obviusly a request in this model, since you are talking abou relativistic quantum field.

I hope i answered to you.
bye Marco.

sorry for my bad english
 
  • #3
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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?
I suggest you a good reference: Hamermesh. Group theory and its application to... blah blah.
A rappresentation of A lie algebra is just this:
R: V--->End(V). And is homomorphic because you need that: R(g1)R(g2)=R(g1g2) and R(e)=Id. In other words a map wich preserve group structure.
 
  • #4
strangerep
Science Advisor
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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?
Poincare unirreps give just the mass, spin and parity quantum numbers. They don't give
charge, hypercharge, isospin, color, etc.

Wigner's approach is called the "method of little groups". One finds a maximal set of
mutually-commuting generators, in this case [itex]P^2, W^2, W \bullet P[/itex] and tries to
find all sets of eigenvalues for them. Let's suppose that we choose [itex]m^2 \ne 0[/itex] as
the eigenvalue of [itex]P^2[/itex]. (I'll leave out parity for now, because that's related to
the discrete parity-inversion transformation.)

But then... [itex]W^2[/itex] (Pauli-Lubansky vector squared) is a pain to work with
directly, so Wigner said "let's go the rest frame where 4-momentum is (m,0,0,0) and
then we need only work with the group of 3-space rotations, since these preserve (m,0,0,0).
This is SO(3), now called the "little group". Its double-cover is SU(2) and we need to use
it rather than just SO(3) if we want to find *all* the unirreps. Then, taking [itex]J^2,J_z[/itex]
as the remaining mutually-commuting generators we can find the various allowed
combinations of total spin and spin z-projection eigenvalues.

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
[...]
Only mass,spin,parity. Not the "internal" quantum numbers.

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?)
The set of mutually-commuting generators is quote different. The Casimir operators for
Poincare are the [itex]P^2, W^2[/itex] but for Lorentz generators [itex]J_{\mu\nu}[/itex] they are [itex]J_{\mu\nu} J^{\mu\nu}[/itex]
and [itex]\epsilon_{\mu\nu\rho\lambda}J^{\mu\nu}J^{\rho\lambda}[/itex]. The upshot is
that these enable you to distinguish invariantly between particles of opposite chirality,
whereas chiralities mix under Poincare translations, hence chirality is not one of the
quantum numbers we use to classify elementary particles under the full Poincare group
(well, actually, in the massless case it turns out to be the same helicity, but that's another
story - I only talked about the massive case above).

The bottom line is that restricting to a subgroup (here Poincare->Lorentz) does not
in general correspond in an intuitive way to "loss" of quantum numbers. Rather, the
set of quantum numbers changes drastically.

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?)
The quantum numbers arising from Poincare have nothing to with those of
SU(3)xSU(2)xU(1). To get a complete picture, we must look at all of them.
"Quantum Number" means something which is invariant under a physically-allowed
transformation, here either Poincare transformations or Gauge transformations.
This is a distinct concept from "conserved quantity", which just corresponds to a
generator which commutes with the Hamiltonian generator of time translations.

4. I'm finding I'm actually kind of confused as to what an "irreducible representation
is, or a "representation" for that matter.
I'm not sure if you're asking for a math explanation or a physically-intuitive
hand-waving explanation. I'll attempt the latter...

Consider just the everyday group of transformations in 3-space, i.e: E(3). Take a line, a plane
and a cube. If you hold the plane side-on it looks like a line, but as soon as you apply
some arbitrary rotations it becomes clear that they're two very different types of objects.
Similarly, if you look square-on at one face of the cube it looks like a plane. But applying
some rotations soon shows that they're different types of objects. OTOH, if you have
two planes, there's no possible set of rotations you can do that will make one look
like a plane and the other a cube. This is intuitively what it means for cubes,planes,and
lines to correspond to (some of) the irreducible representations of E(3). I.e., there is an
invariant distinction between the irreps.

About the subspace thing and irreducibility, one could think of a subspace of "lines and
planes", but it's pretty obvious that this can be invariantly decomposed into a set of
lines and a set of planes, but the latter two cannot be further decomposed invariantly.

HTH.
 
  • #5
560
1
Marco and Strangerep, thanks, that helps a lot!
 

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