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Property antisymmetric tensors

  • #1

Homework Statement



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.

Homework 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]


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..
 
Last edited:

Answers and Replies

  • #2
hunt_mat
Homework Helper
1,739
18
The indices are incorrectly balanced, there is a [tex]/nu[/tex] on the LHS but none on the RHS
 
  • #3
The indices are incorrectly balanced, there is a [tex]/nu[/tex] on the LHS but none on the RHS
The \nu is contracted, only \mu and \lambda are free indices.
 
  • #4
Solved.
 

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