What is SU(2)xU(1) Unification in Weak and Electromagnetic Interactions?

[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]

What does it mean to have a model that is SU(2)xU(1)?

It means that the fields (particles) of your model form a representations (multeplets) of the (direct product) group SU(2)XU(1).

Does it have anything to do with the electro-weak unification?

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

I asking this because the weak interaction has 2 bosons

NO, there are three weak bosons W^{\pm},Z^{0}.
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 exp(i*\phi)[\TEX] multiplication that leaves the overall phase unchanged is a very common symetry e.g. the symetry of electromagnetism.<br /> 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.<br /> <br /> This symetry could be about the mixing of two particles for example e.g. you change the <br /> Amplitude for two particles beeing in a state where their amplitudes for manifestation are equal to one where one dominates or something.<br /> <br /> 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]

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 :)

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)xS_3 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 S_3 and this is freaking me out. I make no idea where to start! I just can't 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?