# Are there "physical fields" in non-abelian gauge theories?

In QED the field strength $$F_{\mu\nu}$$ is gauge-invariant. This is reasonable since its components are physical fields $$\vec{E}$$, $$\vec{B}$$, so it doesn't matter in which gauge you express it.

However, in non-abelian gauge theories, $$F_{\mu\nu}$$ is not gauge invariant $$F_{\mu\nu}'=VF_{\mu\nu}V^{-1}$$. A gauge-invariant object would be $$\text{Tr}(F_{\mu\nu})$$, but the number of independent components of this object has nothing to do with the gauge group. Is there any notion of "physical fields" (gauge-invariant,then) in non-abelian gauge theories?

• @CosmasZachos It's better now. You're right, there's a lot I have to read about.
– AFG
Jun 15, 2022 at 15:31
• The quantitities $F^a_{\mu\nu}$ are not gauge invariants, but when we consider the "gauge charges" $Q_a$ then the quantity $Q_a F^a_{\mu\nu}$ IS gauge invariant. In electromagnetism also $qF_{\mu\nu}$ is gauge invariant, because both electrical charge $q$ and the field $F_{\mu\nu}$ are gauge invariant by themselves. Jul 23, 2022 at 17:34