I am studying the article "A Formal Optical Model", of Bell and Squires (the article is here https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.3.96) where they proof that the self-energy of the Green function is equal to the optical potential. They consider the scattring of a particle by a target (both being identical fermions) and they say that the scattering state can be write as

$$| \alpha \rangle = \int_{-\infty}^{t} dt' e^{-iEt'} \hat{\psi} (\textbf{r}, t') | 0 \rangle $$

where $E$ is the energy of the incident particle and $| 0 \rangle $ is the state of the target. What is the justification or deduction for this relation? I have never seen it before... and it's weird, because the ket is depending on the position $\textbf{r}$. Can someone enlighten me what they are actually doing?


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