Different representation of Laplacian

  • Thread starter Oddbio
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  • #1
Oddbio
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Main Question or Discussion Point

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.
 

Answers and Replies

  • #2
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I would use the matrix representations of ##F^{\mu \nu}## and ##F_{\mu \nu}## and compute it.
 

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