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Like my title, what is the difference between $SU(2)_L$ and $SU(2)_R$? I know they transform differently, but I don't know how to write down the transformation. I'm not familiar with group theory, so I would very much appreciate a detailed answer.

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    $\begingroup$ in which context you are asking this question? is it left-right symmetric model? $\endgroup$ – AMS Oct 8 '16 at 15:01
  • $\begingroup$ Yes, I get why the left handed and right handed doublets look like that, but I don't know how to write down the covariant derivative for the newly introduced higgs field, and I guess this has some thing to do with the difference between $SU(2)_L$ and $SU(2)_R$. $\endgroup$ – LY3000 Oct 8 '16 at 15:08
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    $\begingroup$ To reopen this question (v1), consider to provide context and/or references. $\endgroup$ – Qmechanic Oct 8 '16 at 15:18
  • $\begingroup$ Related question, conceptually identical, 102575. $\endgroup$ – Cosmas Zachos Oct 9 '16 at 19:41
  • $\begingroup$ SU(2)_L means that the gauge transformation applies only to left-handed particles (treating right-handed particles as singlets under these transformations), SU(2)_R would correspondingly describe an (additional) interaction in which only right-handed fermions participate. The group structure of them (taken individually) is the same but the fact that SU(2)_R is often added in addition to SU(2)_L leads to additional gauge bosons etc. $\endgroup$ – Andre Holzner Oct 29 '16 at 12:08