# Varying the action with respect to A_\mu

1. Nov 30, 2014

### PhyAmateur

Was reading a paper and trying to work out the author's calculations: I am trying to vary the action,
$$S_A= \int d^3x \left[ \, -\frac{1}{4}F_{\mu\nu}(A)F^{\mu\nu}(A) +\frac{1}{2}m\, \epsilon^{\mu\nu\rho}A_\mu F_{\nu\rho}(A) \right]$$

with respect to $$A_\mu$$. I am finding difficulty deriving this because $$A_\mu$$ is embedded in $$F_{\mu \nu}$$. So my attempt was writing this all in terms of $$A_\mu$$:

$$S_A= \int d^3x \, \left[ -\frac{1}{4}(\partial_\mu A_\nu - \partial_\nu A_\mu) (\partial^\mu A^\nu - \partial^\nu A^\mu) +\frac{1}{2}m\, \epsilon^{\mu\nu\rho}A_\mu (\partial_\nu A_\rho - \partial_\rho A_\nu) \right]$$ and then I got stuck. If you could please lead me from here.

Reference, section 2: http://arxiv.org/pdf/hep-th/9705122.pdf

2. Dec 5, 2014