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Quantum Field Theory-Gauge Transformations

  1. Apr 25, 2012 #1
    1. The problem statement, all variables and given/known data
    Given the Lagrangian density [tex] L(\phi^{\mu})=-\frac{1}{2}(\partial_{\mu}\phi^{\nu})(\partial^{\mu}\phi_{\nu}) + \frac{1}{2}(\partial_{\mu}\phi^{\mu})^2+\frac{m^2}{2}(\phi^{\mu}\phi_{\mu}) [/tex]
    and gauge transformation [tex] \phi^{\mu}\rightarrow \phi^{\mu} + \partial^{\mu}\alpha [/tex]

    (c) Introduce one extra real scalar field [tex] \sigma [/tex] and write some interacting Lagrangian [tex] L^{\prime}=L(\phi^{\mu})+L^2(\phi^{\mu},\sigma) [/tex] which is invariant under the gauge transformation and gives the original L for [tex] \sigma=0 [/tex].

    (d) Can we solve (c) with sigma that has a canonical kinetic term [tex] -\frac{1}{2}(\partial_{\mu}\sigma)^2 [/tex]
    2. Relevant equations

    3. The attempt at a solution
    The first parts of the question show that [tex] \partial_{\mu}\phi^{\mu}=0 [/tex] which simplifies the Lagrangian, and also that the initial Lagrangian is not invariant under the gauge transformation. I got those out.
    But parts (c) and (d) seems like the kind of thing you either know or you don't, is there a way of working it out?
  2. jcsd
  3. Dec 18, 2014 #2
    Bumping this because I would also like to know if (c) and (d) can be solved in a way that doesn't involve trial and error or just knowing the answer.
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