1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted