A question on <PERSON>'s QFT vol 1
In <PERSON>'s QFT volume 1, chapter 3.3, just below equation 3.3.19, he says $\vec P=\vec P_0$ and $\vec J=\vec J_0$ can be(easily) concluded from the definition of <PERSON> wave operator or equivalently the Lippmann-Schwinger equations. However, I don't see how.
In virtually all known field theories, the effect of interactions is to add an interaction term V to the Hamiltonian, while leaving the momentum and angular momentum unchanged: $$H =H_0+ V, \vec P =\vec P_0, \vec J =\vec J_0. \tag{3.3.18}$$ (The only known exceptions are theories with topologically twisted fields, such as those with magnetic monopoles, where the angular momentum of states depends on the interactions.) Eq. (3.3.18) implies that the commutation relations (3.3.11), (3.3.14), and (3.3.16) are satisfied provided that the interaction commutes with the free-particle momentum and angular-momentum operators $$[V,\vec P_0] = [V,\vec J_0] = 0.\tag{3.3.19}$$ It is easy to see from the Lippmann-Schwinger equation (3.1.16) or equivalently from (3.1.13) that the operators that generate translations and rotations when acting on the 'in' (and 'out') states are indeed simply $\vec P_0 $ and $\vec J_0$.
Related commutators:
$$[J^i,J^j]=i\epsilon_{ijk}J^k\tag{3.3.11}$$ $$[J^i,P^j]=i\epsilon_{ijk}P^k\tag{3.3.14}$$ $$[J^i,H]=[P^i,H]=[P^i,P^j]=0\tag{3.3.16}$$
Definition of the wave operators:
The 'in' and 'out' states can now be defined as eigenstates of $H$, not $H_0$, $$H\Psi^\pm_\alpha=E_\alpha\Psi^\pm_\alpha\tag{3.1.11}$$ which satisfy the condition $$\int d\alpha e^{-iE_\alpha\tau}g(\alpha)\Psi^\pm_\alpha\to \int d\alpha e^{-iE_\alpha\tau}g(\alpha)\Phi_\alpha\tag{3.1.12}$$ for $\tau\to-\infty$ or $\tau\to\infty$, respectively. 346
Eq. (3.1.12) can be rewritten as the requirement that: $$e^{-iH\tau}\int d\alpha g(\alpha)\Psi^\pm_\alpha\to e^{-iH_0\tau}\int d\alpha g(\alpha)\Phi_\alpha$$ for $\tau\to-\infty$ or $\tau\to\infty$, respectively. This is sometimes rewritten as a formula for the 'in' and 'out' states: $$\Psi^\pm_\alpha=\Omega(\pm \infty)\Phi_\alpha\tag{3.1.13}$$ where $$\Omega(\tau)=e^{i\tau H}e^{-i\tau H_0}\tag{3.1.14}$$
The Lippmann-Schwinger equations are given by:
$$\Psi^\pm_\alpha=\Phi_\alpha+(E_\alpha-H_0\pm i\epsilon)^{-1}V\Psi_\alpha^\pm\tag{3.1.16}$$
So, my questions are,
1. Is (3.3.19) assumed to be true in that statement? If yes, then can we not just follow the previous argument backwards and prove 3.3.18 from 3.3.19? Or is he just trying to see if the definitions of wave operators are consistent with 3.3.18 and 3.3.19? If no, then:
2. How can one conclude $\vec P =\vec P_0, \vec J =\vec J_0$ from 3.1.16 or 3.1. 394