Property antisymmetric tensors

[phoenixofflames]

Homework Statement

I was wondering how I could prove the following property of 2 antisymmetric tensors $$F_{1\mu \nu}$$ and $$F_{2\mu \nu}$$ or at least show that it is correct.

Homework Equations

$$\frac{1}{2}\epsilon^{\mu \nu \rho \sigma} F_{1\rho \sigma}F_{2\nu \lambda} + \frac{1}{2}\epsilon^{\mu \nu \rho \sigma} F_{2\rho \sigma}F_{1\nu \lambda} = - \frac{1}{4} \delta^{\mu}_{\lambda} \epsilon^{\rho \sigma \alpha \beta} F_{1 \alpha \beta}F_{2\rho \sigma}$$

The Attempt at a Solution

If \mu = \lambda, the left side gives $$- \epsilon^{\rho \sigma \alpha \beta} F_{1 \alpha \beta}F_{2\rho \sigma}$$ and the right side also ( summing over \mu )

But how can I see that if \mu is different from \lambda, that this relation is true? The right hand side is zero, but how can I proof that the left side is also zero? I have no idea..

 

[hunt_mat]

The indices are incorrectly balanced, there is a $$/nu$$ on the LHS but none on the RHS

 

[phoenixofflames]

The indices are incorrectly balanced, there is a $$/nu$$ on the LHS but none on the RHS

The \nu is contracted, only \mu and \lambda are free indices.

 

[phoenixofflames]

Solved.