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I'm self-studying Weinberg QFT. I'm confused by his treatment of scattering theory .

I have the following question:

He introduces the free particle states $\Phi_{\alpha}$ but I'm not sure what is it good for? It has the same shape of In and out states so why introduce it? So this equation

$$\tag{3.1.12} \int d\alpha e^{-iE_{\alpha}t}g(\alpha)\Psi^{\pm}_{\alpha}~\longrightarrow~\int d\alpha e^{-iE_{\alpha}t}g(\alpha)\Phi_{\alpha}$$

at the asymptotic time (eq 3.1.12) is not clear.

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Hi @Nabil. To merge accounts, go here. – Qmechanic Apr 23 '13 at 18:32
Another Phys.SE question that also involves eq. (3.1.12) in Weinberg: – Qmechanic Apr 23 '13 at 18:40

The idea here is, that you define in- and outgoing states that are eigenstates of the non-interacting theory. Only at a finite time in the past did the interaction start and at a finite time in the future, the interaction stops.

We know exactly how to deal with free states, and we want to use this knowledge even when we introduce interactions. By introducing free "in" and "out" states, we can separate the (exactly solvable) free theory from the interacting theory, whose solution can only be given by a perturbation series.

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