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

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Discussion Overview

The discussion centers on the concept of SU(2)xU(1) unification in the context of weak and electromagnetic interactions. Participants explore the implications of this model, its relation to electro-weak unification, and the representation of particles within this framework.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification, Debate/contested

Main Points Raised

  • Some participants inquire about the meaning of a model being SU(2)xU(1) and its connection to electro-weak unification.
  • One participant asserts that SU(2)xU(1) is the group used by Weinberg and Salam for unifying electromagnetic and weak interactions.
  • Another participant corrects an earlier claim about the number of weak bosons, stating there are three: W± and Z0.
  • A participant explains the nature of U(1) and SU(2) symmetries, suggesting that U(1) relates to phase changes while SU(2) is akin to rotational symmetry in three dimensions.
  • Discussion includes the mixing of W bosons and the photon, detailing how W1 and W2 mix to form W± and how W3 and B mix to produce the Z boson and photon.
  • One participant mentions the challenge of integrating non-Abelian SU(3) QCD gluons into the electroweak framework, suggesting this could lead to a Grand Unified Theory (GUT) and potentially a Theory of Everything (TOE).
  • Another participant references the Coleman-Mandula theorem, indicating it imposes restrictions on possible unifications.
  • A participant expresses uncertainty about the connection between particle doublets and the invariant subspace in the context of a specific paper on lepton doublets unification.
  • One participant seeks resources for learning about doublet representations and questions whether doublet and spinor representations are equivalent.

Areas of Agreement / Disagreement

Participants express differing views on the implications and interpretations of SU(2)xU(1) unification, with no consensus reached on several technical aspects, particularly regarding the mixing of particles and the implications for GUTs.

Contextual Notes

Some discussions involve complex mathematical concepts and assumptions about group theory that may not be fully articulated, leading to potential gaps in understanding among participants.

Who May Find This Useful

This discussion may be of interest to those studying particle physics, group theory, or the unification of fundamental forces, particularly in the context of electroweak interactions.

Magister
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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... :confused:
 
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Magister said:
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
 
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 already do that :)
 
Mr.Brown said:
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 already 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).
 
Yeah i guess Coleman-Mandula-Weinberg puts some pretty servery restrictions on what can be done and what can´t.
 
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.
 
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Where can I learn more about doublet representation? Are the doublet and spinor representations the same?
 
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