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Norman

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Thanks,

Ryan

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- Thread starter Norman
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- #1

Norman

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Thanks,

Ryan

- #2

Norman

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Cheers,

Ryan

- #3

Norman

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Cheers,

Ryan

- #4

CarlB

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Norman said:

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 [tex]\mathcal{SU}(3)[/tex] 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

- #5

Norman

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SU(3) has a dual, but the dual to SU(2) is equal to the ordinary rep.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 [tex]\mathcal{SU}(3)[/tex] 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

- #6

CarlB

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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

- #7

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

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

Daniel.

P.S.Terminology is important.

- #8

Norman

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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.

- #9

Norman

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dextercioby said:notthe 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

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

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

- #10

George Jones

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Regards,

George

- #11

Norman

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George Jones said:

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.)

- #12

samalkhaiat

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Norman 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 said:take a fundamental doublet of fields, u and d, write them as a column su(2)-vector (the {2}). The conjugate doublet of antiparticle fields transforms as a row vector;(u*,d*). Now with aid of the totally antisymmetric tensor (the Levi-Civita SYMBOL) which is invariant under su(2) transformations, we may write the above as a column vector also, with (-d*) as 1st element and (u*) the 2nd.i.e the antiparticle doublet(-d*,u*) transforms exactly as the particle doublet(u,d),and belongs to the same {2}.

regards

sam

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