Canonical Momenta Action Electromagnetism

[Yoran91]

Hi everyone,

In one of the assignments in a course on classical field theory I'm given the action
$S = \int d^4 x \mathcal{L}$

where
$\mathcal{L} = -\frac{1}{16\pi} F_{\mu \nu} F^{\mu \nu} - A_{\mu}j^{\mu}$.


I'm now supposed to construct the canonical momenta $\pi_\mu = \frac{\delta S}{\delta \dot{A}^{\mu}}$,

but I have no idea how to. Is there any way to do this without loads and loads of algebra?

 

[Bill_K]

You mean π_k. Canonical quantization is a 3+1 dimensional approach and π_k is a 3-d variable. What you need to do is split all the index summations into 0 along with a sum over k. Then you'll see A_k,0 explicitly, making it easy to take the derivative.

 

[dextercioby]

I don't see other way to do it rather than brute force calculation using:

$$\frac{\partial\left(\partial_{\mu}A_{\nu}\right)}{ \partial\left(\partial_{0}A_{\sigma}\right)} = \delta_{\mu}^{0} \delta_{\nu}^{\sigma}$$

 

[Yoran91]

I guess I don't understand how to compute the variational derivative here, can anyone explain?

 

[Bill_K]

I guess I don't understand how to compute the variational derivative here, can anyone explain?

The functional derivative or variational derivative is the same thing you do when deriving the Euler-Lagrange equations. See Wikipedia. (I'm limiting my comments because you say this is an assignment. You need to work some of this out for yourself!)