# Are the Møller wave operators $\Omega_\pm$ related to $\lim_{t\rightarrow\infty}U(t)$ from field theory?

When we want to compute correlation functions $\langle\Omega|\,T\hat{\phi}(x_1)\ldots|\Omega\rangle$ in an interacting quantum field theory, we relate it to the free-field objects $|0\rangle$ and $\hat\phi_I(x)$ using the interaction-picture time-evolution operator in the limit $T\rightarrow\infty$.

Eventually, we arrive at an expression like (see Peskin and Schroeder eqn. 4.30) $$\langle\Omega|\,T\hat{\phi}(x)\hat{\phi}(y)|\Omega\rangle=\lim_{T\rightarrow\infty}\mathcal{N}\langle 0|U(T,t_x)\phi_I(x)U(t_x,t_y)\phi_I(y)U(t_y,-T)|0\rangle.$$

The $U(T,t_x)$ and $U(t_y,-T)$ sitting at the end of the correlator look very much like the Møller wave operators of non-relativistic scattering theory

$$\Omega_\pm=\lim_{t\rightarrow\mp\infty}U(t)_\text{full}U_0(t),$$ that relate the in-asymptote and out-asymptote states to the actual state at $t=0$.

So my question is, are these two like the same thing, with the same properties? i.e. they are isometric, etc...

$$Ω_±=\lim_{{t→∓∞}}U(t)_\text{full}U{_{0}}(t),$$