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Higgs Mechanism and Lorentz transformation of currents

  1. Mar 23, 2016 #1

    CAF123

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    Gold Member

    1. The problem statement, all variables and given/known data
    Consider the Higgs mechanism lagrangian, $$\mathcal L = (D_{\mu} \phi)^* (D^{\mu} \phi) -\mu^2 (\phi^* \phi) - \lambda (\phi^* \phi)^2 - \frac{1}{4}F_{\mu \nu}F^{\mu \nu},$$ with ##F_{\mu \nu} = \partial_{\mu} A_{\nu} - \partial_{\nu} A_{\mu}## and ##D_{\mu} = \partial_{\mu} + iqA_{\mu}##. One can show that then ##\partial_{\mu}F^{\mu \nu} = J^{\nu}##, where ##J_{\nu} = iq ((D^{\nu} \phi)^* \phi - \phi^* D^{\nu} \phi)##. Following spontaneous symmetry breaking, the massless gauge field acquires mass and let's suppose it has the following oscillatory behaviour $$A_{\mu} = \cos (M t) \epsilon^{1}_{\mu}$$ where ##\epsilon^1_{\mu} = (0,1,0,0)## What is the four vector current and what is it if I apply a lorentz boost in the z direction?

    2. Relevant equations


    lorentz boost in z direction is $$t' = \gamma \left( t -\frac{vz}{c^2}\right)$$ and ##z' = \gamma(z-vt)##

    3. The attempt at a solution

    I can evaluate the components of ##A_{\mu}## to get ##A_{\mu} = (0,\cos (Mt), 0, 0)## and then I could just evaluate the four current by evaluating each of its four components separately. When I am taking a lorentz transformation, would I just send ##t \rightarrow t'## and use the equation given above in relevant equations? Or would I also need to consider transformation of the differential operator as well?

    Thanks!
     
  2. jcsd
  3. Mar 28, 2016 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
     
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