What are doublets, singletsin SU(N)?

[mjsd]

I was reading this thread on "representation and field theory" under
physics > quatnum physics
https://www.physicsforums.com/showthread.php?t=91764
which got me thinking...and perhaps there are experts out there who can clarify these for me:

(A.1) In particle physics we often speak of something as being a singlet, doublet, triplet...etc. in SU(3), SU(2) or the 5 in SU(5)... or whatever. What do they actually mean?

(A.2) eg. we put leptons into doublets, typically in a 2x1 vector... so I guess we are assuming the 2x2 matrix representation of SU(2)? My question is does the dimensionality of the vector space/tensor/representation (not sure which one I should be talking about) got anything to do with whether we call something a singlet, doublet, triplet... etc?
OR
does the fact that there are "two"(or three or one) items to be juggled with dictate whether something is called a "doublet" (or triplet or singlet) respectively?

(A.3) Related to this is the issue of irreducibility of tensors/representations. Do we always choose irreducible rep for putting in our particle contents? ie. say in SU(3), there are 1, 3, 6, 8 irreducible reps etc, does that mean one cannot have a SU(3) doublet (there is no 2 rep)? If not, how can this be written down?

(A.4) finally, when we introduce the Higgs doublet to create a mass term which is invariant under the group action, I noticed that in that context, we often say that we need a doublet Higgs because 2x2=1+3 contains a singlet! So are we now talking about combining two reps in such a way that we end up not having juggled anything? Or is there anything deeper than that? Doesn't a direct product of 2 and 2 give us a "bigger" space? or only the irreducible bit (related to A.3) matters?


I've got a feeling that I may have mixed up a lot of concenpts (irreducible representations, irreducible tensors, representation space, dim of rep and dim of matrix...etc.), pls feel free to correct me where applicable. Thanks in advance.

 

[CarlB]

Cool set of questions.

(A.2) eg. we put leptons into doublets, typically in a 2x1 vector... so I guess we are assuming the 2x2 matrix representation of SU(2)? My question is does the dimensionality of the vector space/tensor/representation (not sure which one I should be talking about) got anything to do with whether we call something a singlet, doublet, triplet... etc?
OR
does the fact that there are "two"(or three or one) items to be juggled with dictate whether something is called a "doublet" (or triplet or singlet) respectively?

It's because there's "two" (or three or one). The dimensionality of the vector space depends a little on the choice of representation. For example, every representation of an algebra that uses complex numbers is equivalent to a real representation with twice as many degrees of freedom. But people usually use a representation that matches the number of degrees of freedom to the number of things that are being juggled.

(A.3) Related to this is the issue of irreducibility of tensors/representations. Do we always choose irreducible rep for putting in our particle contents? ie. say in SU(3), there are 1, 3, 6, 8 irreducible reps etc, does that mean one cannot have a SU(3) doublet (there is no 2 rep)? If not, how can this be written down?

What symmetry buys you is that it allows you to compute the ratios of certain things. For example, if you know that a certain experiment produces a result at a given probability, then some other experiment should produce some other result at a rate that can be calculated.

As an example, Clebsch-Gordon coefficients are the ratios that apply to SU(2) symmetry. I use this as an example because I learned how to calculate with CG long before I realized that they were an example of the application of irreps, and I suppose someone else might not realize this.

Now it is important to note that there are representations of SU(2) that can be "reduced". If one had particles that just happened to follow such a reducible representation, then one could still use the irrep ratios to compute ratios of experiments in each of the irreps of that reducible representation; but it turns out that group theory will not give the ratios between irreps. That is, as far as the symmetry goes, one can only get information within an irrep. Because of this, physicists might as well assume that the symmetry is an irrep.

In a certain sense, symmetries are what you use to solve problems that you don't understand at a fundamental level. They're a mathematical trick that you use to extract information without having a lot of insight into what is going on.

Example, the excited states of the hydrogen have a 2 + 6 + 10 + 14 + 18 (i.e. s, p, d, f, ...) form. These are even because of spin-1/2 for the electron. Ignoring that, they form a 1+3+5+7+... These are the orbital spin 0 + spin 1 + spin 2 + spin 3... states, which are irreps of SU(2). If one didn't know Schroedinger's wave equation for a 1/r potential, one could still compute C-G coefficients for the relationships inside each of these multiplets, however, one could not compute ratios between the multiplets. (Another way of saying this is that the raising and lowering operators for SU(2) always stay within a multiplet.) Mathematically, what happens is that all central potentials have an SU(2) symmetry and end up with the 1+3+5+7 form. So to get the rest of the information, you have to guess the equation of motion, symmetry will only get you so far.

Note: I do not have a source for the above few paragraphs on (A.3). These are just things I've picked up over the years. I'd love to hear corrections. And your questions are good, I look forward to seeing other answers, especially to (A.4).

 

[mjsd]

Thanks for you ideas.

As an example, Clebsch-Gordon coefficients are the ratios that apply to SU(2) symmetry. I use this as an example because I learned how to calculate with CG long before I realized that they were an example of the application of irreps, and I suppose someone else might not realize this.

While I knew that CG coeffs are related to SU(2) symmetry (after all that's what describes spin) and application of irreps, the transition from thinking about it as adding two angular momenta/spins together to the mathematical abstraction is often hard (for me at least). I must add that when I was asking (A.4), I had this issue in mind.

As far as the CG case is concerned, this is how I visualise it (it may be wrong of course), we have two spin states (1,0) and (0,1). So I guess the up and down states form a doublet; it is in SU(2) because that's how spin seems to transform in nature; when we have a two-particles system, we need to expand the vector space in which the group acts (it is now SU(2)xSU(2)), and we construct this space via a tensor product operation. However, it turns out that there are invariant subspaces in this new and bigger representation space, that is under group actions some (or some linear combinations of the) components of the tensor/vector in this new space transform within its own set. In this case, we have 2x2=1+3.

So far all seem good, but the connection to irreps seems to have entered without me realising it (?) I guess, it is only a subtle point if one doesn't understand it fully.

In relation to (A.4), I guess I am confused by the fact that we use a similar language to discuss the situation for including the Higgs doublet...or at least on the surface, it seems to be similar language... yet another example would be the Majorana term for neutrinos making up of LEFT-handed fields. Neglecting the charge problem, one can have either a Higgs singlet or Triplet
because as I was told, 2x2x1 and 2x2x3 both contain a singlet in SU(2). That's what i meant by "similar language" but possibly different meaning from the CG case.
comments welcomed
thanks in advance