Origin of the form of symmetry operation It is well known that the symmetry operations $U$acting on the operators could be written as
$$U AU^{-1}$$
Now I want to know the logical origin or motivation of this form of operation, my thought came from my QM class that the symmetry operators acting on states like $U |\Psi \rangle$, so acting on operators is just like transforming from Schodinger picture to the Heisenberg picture, like the construction of my another question How to apply anti-unitary symmetry operators? (I only have a speculation for the antu-unitary operator case.)
Any other way to derive or motivate this form of operation?
 A: Well it is because the way we defined what is called "symmetry".
According to Weinberg's QFT Page 50,

A symmetry transformation is a change in our point of view that does not change the results of possible experiments.If an observer $O$ sees a system in a state represented by a ray $\mathscr{R}$ or $\mathscr{R}_{1}$ or $\mathscr{R}_{2} \ldots,$ then an equivalent observer $O^{\prime}$ who looks at the same system will observe it in a different state, represented by a ray $\mathscr{R}^{\prime}$ or $\mathscr{R}_{1}^{\prime}$ or $\mathscr{R}_{2}^{\prime} \ldots,$ respectively, but the two observers must find the same probabilities
$$
P\left(\mathscr{R} \rightarrow \mathscr{R}_{n}\right)=P\left(\mathscr{R}^{\prime} \rightarrow \mathscr{H}_{n}^{\prime}\right)
$$

Using dirac's notation, the probability of observing the experiment's outcome $\psi$ when using operator $O$ acting on the state $\phi$ is
$$
\langle\psi|\hat{O}| \phi\rangle
$$
so we should have
$$
\langle g \psi|\widehat{g O}| g \phi\rangle=\langle\psi|\hat{O}| \phi\rangle
$$
which simply means
$$
\widehat{g O}=\hat{g} \hat{O} \hat{g}^{-1}
$$
thus giving us the form of symmetry operators acting on operators.
