SU(2)xU(1) unification

Magister

What does it mean to have a model that is SU(2)xU(1)? Does it have anything to do with the electro-weak unification? I asking this because the weak interaction has 2 bosons and the electromagnetic interaction has 1 boson...

samalkhaiat

[QUOTE=Magister;1530968]It means that the fields (particles) of your model form a representations (multeplets) of the (direct product) group SU(2)XU(1).

Yes, SU(2)XU(1) is the group used by Wienberg & Salam to unify electromagnatic interaction with the weak interaction.

NO, there are three weak bosons [itex]W^{\pm},Z^{0}[/itex].
Clearly, you need to know something about group theory.

regargs

Sam

Mr.Brown

I guess the easiest way to understand this is to say that a theory is U(1)xSU(2) if it symetric under an U(1) symetry and a SU(2) symetry.
An U(1) symetry is just a phase change some [TEX]exp(i*\phi)[\TEX] multiplication that leaves the overall phase unchanged is a very common symetry e.g. the symetry of electromagnetism.
The SU(2) symetry is a bit more abstract it´s very similar to an SO(3) symetry e.g. a symetry under rotations in 3D, you can read about that in many representation theory books.

This symetry could be about the mixing of two particles for example e.g. you change the
Amplitude for two particles beeing in a state where their amplitudes for manifestation are equal to one where one dominates or something.

I guess this would be the most elementary idea i guess it would be best if you start of with some good intro to classical mechanics and look up the noether stuff if you didn´t allready do that :)

mormonator_rm

And in electroweak theory, it sort of is. You have the W1, W2, W3, and B fields, where B operates only on hypercharge, and W3 only on isospin. What happens is that W1 and W2 mix to form W+ and W-, and W3 and B mix to produce Z and photon. W3 and B are both very massive, but the mixing to Z and photon leaves us with extremely massive Z and massless photon. The new fields Z and photon operate on linear combinations of hypercharge and isospin, giving us a Z boson that allows flavor-changing-neutral-currents and a photon that only operates on electric charge (which, itself, is a linear combination of hypercharge and isospin) in the Abelian sense.

The challenge now is to combine the non-Abelian SU(3) QCD gluons into the mix. If this can be done, it will give us a GUT, and adding gravitation would represent a possible TOE. If it can even be done (still debatable, I think).

Mr.Brown

Yeah i guess Coleman-Mandula-Weinberg puts some pretty servery restrictions on what can be done and what can´t.

Magister

I have being studying group theory but I am getting to it quite slowly. Please correct me if I am wrong. When we say that a particle theory is invariant for a given group we are saying that the particles form a representation of that group. So for instance the leptons doublets forms a representation of the SU(2) group and the photon a representation of the U(1) group.

Now I am asked to study the SU(2)xU(1)x[itex]S_3[/itex] lepton doublets unification (more precisely the paper of E. Derman, "Flavor unification, tao decay and b decay within the six-quark-six-lepton Weinberg-Salam model" Phys. Rev. D 19 (1979)). I am asked to write the Higgs potential (eq. 4.1 of that paper) in a new invariant subspace of [itex]S_3[/itex] and this is freaking me out. I make no idea where to start! I just cant make the connection between the particles doublets and the vector basis of the invariant subspace.

Thanks for any help.

Magister

Where can I learn more about doublet representation? Are the doublet and spinor representations the same?