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A Equations of Motion for SU2 Field (Weinberg -Salam?)

  1. Mar 13, 2016 #1
    So I have the lagrangian density ## L = -\frac{1}{2} W^{\mu \nu}_i W_{\mu \nu}^i## where ##W^{\mu \nu}_i = \partial ^\mu W^\nu_i - \partial ^\nu W^\mu_i + \epsilon_i^{jk}W^\mu_j W^\nu_k## and I want to find the equations of motion.

    I have gotten to the stage using the EL equations ##\partial_\sigma \left(\frac{\partial L}{\partial(\partial_\sigma W_\lambda ^r)}\right)-\frac{\partial L}{\partial W_\lambda ^r}=0## that

    ##\partial_\sigma\left(W^{\sigma \lambda}_r-W^{\lambda \sigma}_r\right) - \frac{1}{2}\epsilon^i_{rk}\left(W_\nu^k W^{\lambda \nu}_i+W^{\lambda \nu}_i W_{\nu}^k-W_\nu^k W^{\nu \lambda}_i - W^{\nu \lambda}_iW_{\nu}^k\right)##

    So I am pretty sure I need to use commutation relations to get this stuff to cancel but I don't really know what those relations even are, I know W are non abelian and hence don't commute but I don't know what their commutation relations are or anything so any help would be appreciated. I also don't know if the W tensor is anti symmetric or not but I dont know what's the relation when you swap indices.
     
  2. jcsd
  3. Mar 14, 2016 #2

    ChrisVer

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    Gold Member

    Since you have the [itex]W_i[/itex]'s, they are supposed to commute, they are just components. What doesn't commute is the [itex]W^{\mu} \equiv T_i W^\mu_i[/itex]. For that reason you may see the antisymmetric tensor written:
    [itex]W_{\mu \nu} =\partial_\mu W_\nu - \partial_\nu W_\mu - [ W_\mu , W_\nu] [/itex]
    all these here have been contracted with the SU(2) generator [itex]T^i[/itex]. The antisymmetry under [itex]\mu \leftrightarrow \nu[/itex] is straightforward from this form.

    The [itex]W^{\mu \nu}[/itex] is antisymmetric; you can test it by changing [itex]\mu \nu[/itex]... you'll get that [itex]W^{\mu \nu} = - W^{\nu \mu}[/itex]. In components you write:
    [itex]W^{\nu \mu }_i = \partial^\nu W^\mu_i - \partial^\mu W^\nu_i + \epsilon_{ijk} W_j^\nu W_k^\mu[/itex]
    [itex]=- ( \partial^\mu W^\nu_i -\partial^\nu W^\mu_i ) + \epsilon_{ikj} W_j^\nu W_k^\mu[/itex]
    [itex] =- ( \partial^\mu W^\nu_i -\partial^\nu W^\mu_i ) - \epsilon_{ijk} W_k^\nu W_j^\mu[/itex]
    [itex]=- ( \partial^\mu W^\nu_i -\partial^\nu W^\mu_i + \epsilon_{ijk} W^\mu_j W^\nu_k )= - W^{\mu \nu}_i[/itex]

    1st line just wrote what you gave for ##W^{\mu \nu}_i## with just changing the notation of the indices.
    2nd line take a - out of the partial derivatives and writting it to a more obvious way, while also renaming the sum indices of k to j and j to k...
    3rd line swap those indices using antisymmetry of the epsilon
    4th line just put everything in the common overall -, and identifying what's in the parenthesis as the Wmunu.
     
    Last edited: Mar 14, 2016
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