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Vector field dynamics

  1. Oct 30, 2016 #1
    Currently working through some exercises introducing myself to quantum field theory, however I'm completely lost with this problem.

    Let $$L$$ be a Lagrangian for for a real vector field $$A_\mu$$ with field strength $$F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$$ gauge parameter $$\alpha$$ and external current $$J^\mu$$

    $$L = -\frac 14 F_{\mu\nu}F^{\mu\nu} - \frac \alpha2 (\partial_\mu A^\mu)^2 - J_\mu A^\mu.$$
    Derive the equations of motion for $$A_\mu$$ for arbitrary $$\alpha$$ and show that they give the same field equations in Lorenz gauge $$\partial_\mu A^\mu = 0.$$

    Apologies if my formatting is difficult to read.

    Obviously the Euler-Lagrange equation is required however I'm not sure how to apply the equation correctly on this particular Lagrangean. Can anybody help me figure the method and/or the solution?
     
  2. jcsd
  3. Nov 2, 2016 #2
    In your problem ##A_\mu## is the field. Consequently, you have to compute the derivatives ##\partial_\nu(\frac{\partial \mathcal{L}}{\partial(\partial_\nu A_\mu)})## and ##\frac{\partial\mathcal{L}}{A_\mu}## for your Lagrangian density ##\mathcal{L}##. I'm not sure where you are stuck. Maybe, you should first try the case ##\alpha=0##, i.e. the "usual" electromagnetic Lagrangian density. This is covered in many books on QFT and/or relativistic QM.
     
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