# Different representation of Laplacian

#### Oddbio

Gold Member
I am trying to show that the laplacian:

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

where $$F^{\mu\nu} = \partial{}^{\mu}A^{\nu} - \partial{}^{\nu}A^{\mu}$$
and
$$F_{\mu\nu} = g_{\mu\alpha}F^{\alpha\beta}g_{\beta\nu}$$

A is the scalar potential.

$F^{\mu\nu}$ 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.

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#### fresh_42

Mentor
2018 Award
I would use the matrix representations of $F^{\mu \nu}$ and $F_{\mu \nu}$ and compute it.

"Different representation of Laplacian"

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