- #1
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Hi, I've found this property of Strenght Field Tensors:
$$F_{\mu}^{\nu}\tilde{F}_{\nu}^{\lambda}=-\frac{1}{4}\delta_{\mu}^{\lambda}F^{\alpha\beta}\tilde{F}_{\alpha\beta}$$
Where $$\tilde{F}_{\mu\nu}=\frac{1}{2}\varepsilon_{\mu\nu\alpha\beta}F^{\alpha\beta}, \qquad \varepsilon_{0123}=1$$
I've tried to prove this relation but I can't find a way to do it. I have done specific cases for 4x4 antisymmetric matrices and it seems to work, but I would appreciate having a formal proof.
$$F_{\mu}^{\nu}\tilde{F}_{\nu}^{\lambda}=-\frac{1}{4}\delta_{\mu}^{\lambda}F^{\alpha\beta}\tilde{F}_{\alpha\beta}$$
Where $$\tilde{F}_{\mu\nu}=\frac{1}{2}\varepsilon_{\mu\nu\alpha\beta}F^{\alpha\beta}, \qquad \varepsilon_{0123}=1$$
I've tried to prove this relation but I can't find a way to do it. I have done specific cases for 4x4 antisymmetric matrices and it seems to work, but I would appreciate having a formal proof.