# Dual Representation and anti-particles

## Main Question or Discussion Point

I am reading ahead for my Group Theory introduction to QFT and I have a question about the dual representation. If the dual representation is the same as the ordinary representation, that is to say the ordinary representation is "real", how do we represent anti-particles in this case? This seems to be the case for SU(2), which is the gauge group for weak interactions.
Thanks,
Ryan

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Anyone have any ideas? Maybe I posted this in the wrong forum... Or is my question completely non-sensical?
Cheers,
Ryan

Never mind. This is actually a pretty non-sensical question I think. (It is fun having an online conversation with myself) but in case someone else has this crazy question in the future, I will answer with what I think is the correct statement. SU(2) only talks about, say, weak isospin. But to take into account the charge of a particle we would need the group SU(2) X U(1). This group would account for electroweak theory and would allow the leptonic charge to change.
Cheers,
Ryan

CarlB
Homework Helper
Norman said:
I am reading ahead for my Group Theory introduction to QFT and I have a question about the dual representation. If the dual representation is the same as the ordinary representation, that is to say the ordinary representation is "real", how do we represent anti-particles in this case? This seems to be the case for SU(2), which is the gauge group for weak interactions.
Thanks,
Ryan
I'm not sure what you mean by requiring the ordinary representation to be "real". With that caveat, here's a stab:

If you take a look at $$\mathcal{SU}(3)$$ and its triplet and dual triplet irreps, you will find that the quantum numbers of the dual are complementary to the quantum numbers of the regular representation. So I'm not really sure what you mean when you say that they are the same. Sure you can rotate one to the other, but the antiparticles carry negated quantum numbers.

Carl

CarlB said:
I'm not sure what you mean by requiring the ordinary representation to be "real". With that caveat, here's a stab:

If you take a look at $$\mathcal{SU}(3)$$ and its triplet and dual triplet irreps, you will find that the quantum numbers of the dual are complementary to the quantum numbers of the regular representation. So I'm not really sure what you mean when you say that they are the same. Sure you can rotate one to the other, but the antiparticles carry negated quantum numbers.

Carl
SU(3) has a dual, but the dual to SU(2) is equal to the ordinary rep.

CarlB
Homework Helper
Norman said:
SU(3) has a dual, but the dual to SU(2) is equal to the ordinary rep.
I see your question now. So your answer to your own question is that if one wants an antiparticle associated with a particle collection that follows an SU(2) symmetry, then one must use a different symmetry to generate the relationship between the particles and antiparticles?

Carl

dextercioby
Homework Helper
By "dual" u mean "contragradient", i see, now. The fundamental and the contragradient representations of $\mbox{SU(2)}$ are not the same, but equivalent. This means there is a similarity transformation connecting the generators of the two representations. But you can't say "they're the same". Think of the Dirac algebra

$$\left[\gamma_{\mu},\gamma_{\nu}]_{+} =2g_{\mu\nu}\hat{1}_{V}$$

U can't say the Dirac representation is "the same" with the Weyl and the Majorana one.

Daniel.

P.S.Terminology is important.

CarlB said:
I see your question now. So your answer to your own question is that if one wants an antiparticle associated with a particle collection that follows an SU(2) symmetry, then one must use a different symmetry to generate the relationship between the particles and antiparticles?
Carl
Yes. I believe that is the answer to the question. Someone can correct me if I am wrong.

dextercioby said:
By "dual" u mean "contragradient", i see, now. The fundamental and the contragradient representations of $\mbox{SU(2)}$ are not the same, but equivalent. This means there is a similarity transformation connecting the generators of the two representations. But you can't say "they're the same". Think of the Dirac algebra

$$\left[\gamma_{\mu},\gamma_{\nu}]_{+} =2g_{\mu\nu}\hat{1}_{V}$$

U can't say the Dirac representation is "the same" with the Weyl and the Majorana one.

Daniel.

P.S.Terminology is important.
Of course. Terminology is always very important. But the lecture notes we were given stated things using the terms I used in my original post. Unfortunately I have to go with what is in there mainly. Oh well, a process is learning.
Cheers,
Ryan

George Jones
Staff Emeritus
Gold Member
Even though the representations are equivalent, the weights, i.e., quantum numbers, for the representations have different signs, and hence the connection between particles and antiparticles.

Regards,
George

George Jones said:
Even though the representations are equivalent, the weights, i.e., quantum numbers, for the representations have different signs, and hence the connection between particles and antiparticles.

Regards,
George
Yes but the weights for SU(2) are just opposite in sign, so changeing their sign maps the positive one onto the negative one and vice versa. This doesn't give you 2 new particles, this brings one particle to another in the same representation. That is just the way I understand it though. I believe this is exactly why a complete theory of weak interactions must incorporate electromagnetic interactions thus giving us a SU(2) X U(1) gauge group. (this may not be WHY EM must be incorporated into Weak interactions, but it is part of the reason I think.)

samalkhaiat