property antisymmetric tensors

phoenixofflames

1. The problem statement, all variables and given/known data

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

2. Relevant equations
[tex]\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}[/tex]


3. The attempt at a solution
If \mu = \lambda, the left side gives [tex]- \epsilon^{\rho \sigma \alpha \beta} F_{1 \alpha \beta}F_{2\rho \sigma}[/tex] 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 [tex]/nu[/tex] on the LHS but none on the RHS

phoenixofflames

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

phoenixofflames

Solved.