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Lagrangian for electromagnetism

  1. Nov 19, 2007 #1
    In A. Zee's quantum field theory in a nutshell he assumes familiarity with Maxwell's lagrangian [itex]\mathcal{L} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}[/itex] where [itex]F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu[/itex] with A the vector potential.

    Although I've seen the magnetic vector potential, I've never seen the lagrangian formalism in either electrodynamics or lagrangian/hamiltonian dynamics courses.

    Could anyone point me in the direction of a suitable reference to allow me to familiarise myself with this?

  2. jcsd
  3. Nov 19, 2007 #2


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    Srednicki's QFT book explains this. You can download it for free from his web site.
  4. Nov 19, 2007 #3


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    Try chapter 12 in Griffiths, for a brief introduction into the field tensor and the four-vector potential. For an introduction to the lagrangian formalism in electrodynamics, try Goldstein's book on classical mechanics.
  5. Nov 20, 2007 #4
    I find Zee's notation a little bit confusing here. It seems like he is writing [itex]\partial_\mu[/itex] to mean [itex](\partial_t,\nabla)[/itex] and at the same time writing e.g. [itex]A_\mu = (V,-\mathbf{A})[/itex] and thus [itex]A^\mu = (V,\mathbf{A})[/itex]. Is this standard or am I misunderstanding his notation?

    This is the only way I could get Maxwell's equations out of

    [itex]F^{\mu\nu} = \partial^\mu A^\nu - \partial^\nu A^\mu[/itex].

    [itex]F^{0i} = \partial^0 A^i - \partial^i A^0 = -E^i[/itex]. etc
    Last edited: Nov 20, 2007
  6. Nov 20, 2007 #5
    After checking in another QFT text by Ryder it seems like this is indeed standard notation.
  7. Nov 20, 2007 #6


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    yes, for [itex] \partial_\mu [/itex] the sign is opposite to the other vectors. That's because
    [tex] \partial_\mu \equiv \frac{\partial}{\partial x^\mu} [/tex]
  8. Jul 16, 2009 #7
    never mind I'll make my own topic
  9. Jul 16, 2009 #8


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    The ultimate E&M reference: J. D. Jackson, "Classical Electrodynamics", 3rd ed., Chap. 12, Sec. 7.
    R. Shankar, "Principles of Quantum Mechanics", 2nd ed., Chap. 18, Sec. 5, Subsec. "Field Quantization".
    (more advanced) Peskin & Shroeder, "An Introduction to Quantum Field Theory", Chap. 15.
  10. Jul 16, 2009 #9


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    It is perhaps standard, but it is certainly just a convention. For example, in the "East Coast Metric" (η=diag(-1,+1,+1,+1)), that could be changed to [itex]A_\mu = (-V,\mathbf{A})[/itex], and for implicit metric: [itex]\partial_\mu=\partial^\mu=(\nabla,ic\partial_t)[/itex], [itex]A_\mu=A^\mu=(\mathbf{A},icV)[/itex].
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