Gauge invariance of the path integral measure

Question

When we have an abelian group the gauge transformation of gauge field is given by

$$ A^\mu \rightarrow A^\mu + \partial^\mu \alpha \equiv A^{\prime \mu} $$

Here it's easy to see that the path integral is gauge invariant since

$$ DA^\mu \rightarrow \Big|\det\Big(\frac{DA}{DA^\prime}\Big)\Big|DA^{\prime \mu} = DA^{\prime \mu}, \quad \frac{DA^{\prime \mu}}{DA^\nu} = \delta^\mu_\nu + \frac{D\partial^\mu\alpha}{DA^\nu} = \delta^\mu_\nu, \quad \frac{D\partial^\mu\alpha}{DA^\nu} = 0 $$

Nevertheless, for the non-abelian case we have the field tranformation

$$ A^{\prime \mu} = UA^\mu U^\dagger + \frac{i}{g}U\partial^\mu U^\dagger $$

So,

$$ \frac{DU\partial^\mu U^\dagger}{DA^\nu} = 0,\ \mbox{but}\ \frac{DUA^\mu U^\dagger}{DA^\nu} = U\frac{DA^\mu }{DA^\nu}U^\dagger $$

The last equation shows that

$$ DA^{\prime \mu} = UDA^\mu U^\dagger \neq DA^\mu,\ \mbox{due to the group is non-abelian} $$

My problem comes form the fact that my Professor said in class that they had to be equal (as assumption), but I have proven contrary. What do you think?

Answer 1

Since $\det(U) = \det(U^{\dagger})^{-1}$ and the determinant is multiplicative, you get $$\det \left[U \frac{DA^{\mu}}{DA^{\nu}} U^{\dagger}\right] = \det\left[\frac{DA^{\mu}}{DA^{\nu}}\right]$$