I am working on Chapter 5.6 of Weinberg QFT book vol1.(adsbygoogle = window.adsbygoogle || []).push({});

In page 231 and 232, he said (1,0) or (0,1) field corresponds to an antisymmetric tensor

[itex]

F^{\mu\nu}

[/itex]

that satisfies the further irreducibilitry 'duality' conditions

[itex]

F^{\mu\nu}= \pm {{i}\over{2}} \epsilon ^{\mu\nu\lambda\rho}F_{\lambda \rho}

[/itex]

for (1,0) and (0,1) fields, respectively.

I check that Self-dual antisymmetrc tensor has three independent components and so does Anti Self-dual antisymmetric tensor. So, I can guess that there might be correspondence between Self-dual(or Anti Self-dual) antisymmetric tensor and (1,0) field (or (0,1)field ).

But, I could not show how they are related to each other mathematically.

I would like you guys to show me that how we correspond Self-dual antisymmetric tensor with (1,0) field and

Anti Self-dual antisymmetric tensor with (0,1) field.

Thank you so much.

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# Self-dual and Anti Self-dual antisymmetric tensor

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