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Canonical momenta action electromagnetism

  1. Jun 10, 2013 #1
    Hi everyone,

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

    [itex] \mathcal{L} = -\frac{1}{16\pi} F_{\mu \nu} F^{\mu \nu} - A_{\mu}j^{\mu}[/itex].

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

    but I have no idea how to. Is there any way to do this without loads and loads of algebra?
  2. jcsd
  3. Jun 10, 2013 #2


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    You mean πk. Canonical quantization is a 3+1 dimensional approach and πk 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 Ak,0 explicitly, making it easy to take the derivative.
  4. Jun 10, 2013 #3


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

    [tex] \frac{\partial\left(\partial_{\mu}A_{\nu}\right)}{ \partial\left(\partial_{0}A_{\sigma}\right)} = \delta_{\mu}^{0} \delta_{\nu}^{\sigma} [/tex]
  5. Jun 11, 2013 #4
    I guess I don't understand how to compute the variational derivative here, can anyone explain?
  6. Jun 11, 2013 #5


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    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!)
    Last edited: Jun 11, 2013
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