1. Not finding help here? Sign up for a free 30min 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!

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.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted

Similar Discussions: Vector field dynamics
  1. Weak Field Dynamics (Replies: 10)

  2. Electric field vector (Replies: 17)