Vector Field Dynamics: Apologies & Solutions

[slothwayne]

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?

 

[eys_physics]

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.