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Do non-Abelian gauge fields takes the same value in the Lie subalgebra

  1. Jun 4, 2013 #1
    Do the $\mathrm{SU(2)}$ gauge fields takes the same value in the Lie subalgebra spanned by the $\mathrm{SU(2)}$ field strength tensor?

    I will try to clarify my questions. Define the Lie algebra as:

    \begin{equation}
    [t^a,t^b] = \varepsilon_{abc}t^c
    \end{equation}

    where $\varepsilon$ is the usual Levi-Civita symbol. The field strength tensor:

    \begin{equation}
    F_{\mu \nu} = \partial_\mu A_\nu - \partial_\nu A_\mu + [A_\mu , A_\nu]
    \end{equation

    In components:

    \begin{equation}
    F_{\mu \nu}^a t^a = \partial_\mu A_\nu^a t^a - \partial_\nu A_\mu^a t^a + A_\mu^b A_\nu^c \varepsilon_{abc}t^a
    \end{equation}

    Now I will skip a lot of details, but basically in the Higgs vacuum $F_{\mu \nu}$ lies in the subalgebra spanned by the unbroken generators (in my example there is only one unbroken generator). I will denote the unbroken generators as $t^A$ (i.e. the group index of the unbroken generator is denoted by a capital letter), therefore:

    \begin{equation}
    \Rightarrow F_{\mu \nu}^A t^A = \partial_\mu A_\nu^A t^A - \partial_\nu A_\mu^A t^A + A_\mu^b A_\nu^c \varepsilon_{Abc} t^A
    \end{equation}

    Clearly, the gauge fields in the first two terms on the R.H.S. lie in the subalgebra. But I'm confused about the last term. To me it seems that those gauge field do not necessary live in the same subalgebra. However, the sources that I read seem to suggest that I'm wrong. If anybody can help me, then that would be greatly appreciated.

    P.s. I've recently learned Latex, so I hope that I have used it correctly.

    Edit: It seems that something is going wrong with my formulas, but I'm not sure how to fix it. I hope people can still understand what I've written.

    Thanks!
     
    Last edited: Jun 4, 2013
  2. jcsd
  3. Jun 4, 2013 #2
    Ok, I think I know where I went wrong in my thought process, the first two terms on the R.H.S. must be in the same subalgebra as the field strength tensor, and the gauge fields in the third term are the same gauge fields. Therefore, they must also lie in the subalgebra. However, if we only have one unbroken generator, then that mean that the last term vansishes and so it reduces to the electromagnetic field strength tensor. Is that correct?
     
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