Hi! I'm trying to show that the differential from equation(adsbygoogle = window.adsbygoogle || []).push({});

$$D \star F = 0$$ transform homogeneously under the adjoint action ##F \mapsto gFg^{-1}## of the lie group ##G##, where ##D## denotes the covariant exterior derivative ##D\alpha = d \alpha + A \wedge \alpha## for some lie algebra valued form ##\alpha##. Since

$$A \mapsto gAg^{-1} + gdg^{-1}$$

we get

$$D\star F \mapsto d(g\star Fg^{-1}) + (A \mapsto gAg^{-1} + gdg^{-1})\wedge \star F = dg \wedge \star F g^{-1} + g d\star F g^{-1} + g \star F \wedge dg^{-1} + g A \wedge \star F g^{-1} + g dg^{-1} \wedge g^{-1} \star F g^{-1}$$

where if we use that ##gdg^{-1} = - (dg) g^{-1}##, then the last term seem to cancel the first. However, if this is correct, then the third term has to be zero.

Is the third term zero, and if so why?

Am I going wrong somewhere else?

**Physics Forums - The Fusion of Science and Community**

# Showing that Yang-Mills equations transform homogeneousy

Know someone interested in this topic? Share a link to this question via email,
Google+,
Twitter, or
Facebook

- Similar discussions for: Showing that Yang-Mills equations transform homogeneousy

Loading...

**Physics Forums - The Fusion of Science and Community**