Say we want to calculate $\langle f(t_2)|O|i(t_1)\rangle$. Where $O$ is an arbitrary operator. We can treat the states as stationary and then evolve the operator
$$\langle f(0)|O(t)|i(0)\rangle\\O(t) = U^{\dagger}(t_2)OU(t_1)$$
Or we can split the unitary operator into a free part and an interacting part $U(t) = U_{f}(t)U_{i}(t)$ And then evolve the states with the interacting part and the operator with the free part
$$\langle f(t_2)|U_{f}OU_{i}|i(t_1)\rangle,$$
which is the interaction picture. Since the interacting field $\phi$ is an operator, we expect it to evolve as if it were free, so whenever we calculate the matrix element we just plug in the free field expressions into the Hamiltonian. However, this answer says that we expect $U^{\dagger}\phi U$ to give us an interacting field, that solves the interacting field equation. It turns out that it doesn’t due to Haag’s theorem. But why expect $U^{\dagger}\phi U$ to give us the time evolution of an interacting field when by the way we defined the interaction picture it should just give us the time evolution for the free field?
