Anitsymmetric tensor/switching indices problem

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The discussion centers on the properties of an antisymmetric tensor and the implications of switching indices. It is noted that while the tensor components satisfy antisymmetry, the claim that a tensor with one index raised and one lowered is also antisymmetric is questioned. Participants suggest that additional steps are needed in the derivation to clarify the relationship between the indices and the Minkowski metric. There is consensus that the derivation appears correct, provided the metric remains symmetric. The conversation emphasizes the importance of understanding the underlying principles of tensor algebra to avoid confusion.
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Let's say that some non-operator (having only numbers as it's components) tensor is antisymmetric:

\omega^{\sigma\nu}=-\omega^{\nu\sigma}
and
\omega_{\sigma\nu}=-\omega_{\nu\sigma},

however, I have read in the Srednicki book that it is incorrect to say that the same tensor with one index down and one up would be antisymmetric as well.

Could you please point out, where and what are the errors of the derivation? Should I read something before asking such questions? g here is the Minkowski metric:

\omega^{\nu}\,_{\sigma}=\omega^{\nu\beta}g_{\beta\sigma}=-\omega^{\beta\nu}g_{\beta\sigma}=-\omega_{\sigma}\,^{\nu}<br />

Thank you.
 
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wasia said:
Could you please point out, where and what are the errors of the derivation? Should I read something before asking such questions? g here is the Minkowski metric:

-\omega^{\beta\nu}g_{\beta\sigma}=-\omega_{\sigma}\,^{\nu}<br />

Shouldn't there be another step in between?
 
wasia said:
Could you please point out, where and what are the errors of the derivation? Should I read something before asking such questions? g here is the Minkowski metric:

\omega^{\nu}\,_{\sigma}=\omega^{\nu\beta}g_{\beta\sigma}=-\omega^{\beta\nu}g_{\beta\sigma}=-\omega_{\sigma}\,^{\nu}<br />

This should be fine. Notice that you switched which index was up and which was down.
 
xboy, the steps in between might be something like
-\omega^{\beta\nu}g_{\beta\sigma}=-\omega^{\beta\nu}g_{\sigma\beta}=-g_{\sigma\beta}\omega^{\beta\nu}=-\omega_\sigma\,^\nu
I assume they are valid, as g is undoubtly symmetric (at least in my case) and also g commutes with omega, as they contain only numbers.

Ben Niehoff, I do have noticed, that positions have changed.

However, if anyone could point out a mistake, or tell if that's correct, as Ben says, please do it.
 
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Walia, your derivation seems correct to me. I can't think of a case of your derivation being invalid except for the metric being non-symmetric. But I don't know if the metric can be non-symmetric at all.
 
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