Reducing Symmetry from SU(2) to U(1) | Dear PF

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SUMMARY

The discussion focuses on the process of reducing symmetry from the SU(2) symmetric Lagrangian to U(1). It is established that one cannot directly reduce SU(2) symmetry without breaking it entirely, resulting in three massive gauge bosons. Instead, the correct approach involves breaking the SU(2)xU(1) symmetry down to a nontrivial U(1) subgroup, a method utilized in Electroweak theory. This clarification was provided by a participant named Dimitri.

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Neitrino
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Dear PF,

I don't know whether this is the right place for such question, but
how to reduce symmetry suppose from SU(2) symmetric Lagrangian down to U(1)?

Thank you
 
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You can't. If you let the Higgs doublet acquire an VEV in a SU(2) invariant theory, you break the entire SU(2) symmetry and end up with with three massive gauge bosons.

You can however break down SU(2)xU(1) to a (nontrivial) subgroep U(1). This is what is done in the Electroweak theory.
 
Thank you Dimitri
 

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