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Different representation of Laplacian

  1. Apr 3, 2010 #1

    Oddbio

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    Gold Member

    I am trying to show that the laplacian:

    [tex]L = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}[/tex]
    can also be represented as:
    [tex]L = \frac{1}{2}(\vec{E}^{2}-\vec{B}^{2})[/tex]

    where [tex]F^{\mu\nu} = \partial{}^{\mu}A^{\nu} - \partial{}^{\nu}A^{\mu}[/tex]
    and
    [tex]F_{\mu\nu} = g_{\mu\alpha}F^{\alpha\beta}g_{\beta\nu}[/tex]

    A is the scalar potential.

    [itex]F^{\mu\nu}[/itex] is the antisymmetric field strength tensor.

    But I cannot see how they are able to represent the first equation as the second equation.
    Any advice would really help me a lot.
     
  2. jcsd
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