In Ryder's text, he defines the dual tensor as the anti-symmetric [tex] \tilde F^{\nu \mu} = \epsilon^{\nu \mu \alpha \beta} F_{\alpha \beta}[/tex]. Later he plops down the complex scalar field Lagrangian as(adsbygoogle = window.adsbygoogle || []).push({});

[tex] L = (D_\mu \phi)(D^\mu \phi *) - m^2 \phi * \phi - \frac{1}{4} F^{\nu \mu}F_{\nu \mu}[/tex]

where [tex] D_\mu[/tex] is the covariant derivative. So the thing i was wondering is why can't you have terms like [tex] \tilde F^{\nu \mu} \tilde F_{\nu \mu}[/tex]? I did the work to figure out why you can't have terms like [tex] F^{\nu \mu} \tilde F_{\nu \mu}[/tex], but I just wanted to see what happens to the scalar term using two dual tensors.

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# Dual-tensors not in Lagrangian

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