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Evaluating the operator of free photon lagragian

Physics Asked on March 13, 2021

I want to evaluate the operator from a free photon Lagrangian as a exercise and arrive into something with the form:

$$L = A_{mu}O^{mu nu}A_{nu};$$

parting from the free photon lagrangian:

$$L = -frac{1}{4}F_{mu nu}F^{mu nu};$$, where $$F_{mu nu} = partial_{mu}A_{nu} – partial_{nu}A_{mu};$$

The far I got was writing the lagrangian in this way:

$$L = -frac{1}{2}[ (partial_{nu}A_mu)(partial^nu A^mu) – (partial_{nu}A_mu)(partial^mu A^nu)]$$
$$= -frac{1}{2}[ partial_{nu}(A_mu partial^nu A^mu) – A_mu(partial_nu partial^nu A^mu) – partial_{nu}(A_mu partial^mu A^nu) + A_mu(partial_nu partial^mu A^nu)]$$

Could someone give me a insight of next steps?

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