While I'm reading Hong Liu's notes, it says:
Now we have introduced two theories:
(a)$$\mathcal{L}=-\frac{1}{g^2}Tr[\frac{1}{2}(\partial \Phi )^2+\frac{1}{4}\Phi^4]$$
(b)$$\mathcal{L}=\frac{1}{g^2_{YM}}[-\frac{1}{4}Tr F_{\mu\nu}F^{\mu\nu}-i\bar{\Psi}(\not{D}-m)\Psi]$$
(a) is invariant under the global $U(N)$ transformation, and (b) is invariant under local $U(N)$ transformation.
What confused me is the description:
On the other hand, consider allowed operators in the two theories. In (a), operators like $\Phi^a_b$ are allowed, although it is not invariant under global $U(N)$ symmetry. But in (b), allowed operators must be gauge invariant, so $\Phi^a_b$ is not allowed.
I remember that in electrodynamics the operator must be gauge invariant, but I don't know why in (a) can allow the operator which is not invariant under global $U(N)$ symmerty.
Furthermore, how do we decide on which operator should be invariant under what transformation? Are the operators invariant under gauge transformation because how the real world performs?
