I am currently reading Niklas Beisert's lecture notes on QFT, Chapter 10, on the scattering matrix $S$.$^1$ My main confusion lies in the construction of $\vert \rm in \rangle$ and $\vert \rm out \rangle$ states and in what picture these states are in.
Let $\phi(x)$ be the interacting field, that can be decomposed into $$\phi(x) = \sqrt{Z}\phi_{\rm as}(x) + ...\tag{10.8},$$ where $\phi_{\rm as}$ is a canonically normalized free field of the physical mass $m$ with creation and annihilation operators $a(\vec{p})$ and $a^\dagger (\vec p)$. We define now $$H_{\rm as} := \frac{1}{2} \int \frac{d^3p}{(2\pi)^3}a^\dagger(\vec p)a(\vec p).\tag{10.10}$$
For the scattering setup we define two asymptotic regions of spacetime, one in thedistant past $t_{\rm i}\to -\infty$ and one in the distant future $t_{\rm f} \to \infty$. On the initial timeslice we construct a wave packet in the form of the initial state $\vert i\rangle$ and evolve this in time to the final state $$\vert f\rangle = \exp(-iH(t_{f}-t_i))\vert i\rangle .\tag{10.16}$$ If I'm not misunderstanding, we are working here in the Schrödinger picture, i.e. states are time independent and we use the interacting Hamiltonian $H$ to perform the time evolution. The lecture notes state:
It is hard to compare them to see what the effect of scattering is.
I don't understand the logic behind this statement. The Hilbert space in question is the multi-particle Hilbert space (Fock space). What does this space care about the time parameter in out theory? We just pic two random states of this space $\vert f \rangle$ and $\vert i \rangle$ and compare them.
The author then constructs the states:$^2$ $$\vert {\rm out}\rangle := \exp(iH_{\rm as} t_f)\vert f\rangle \quad \text{and}\quad \vert {\rm in}\rangle := \exp(iH_{\rm as} t_i)\vert i\rangle \tag{10.19}$$ and claims
The $\vert {\rm out}\rangle$ and $\vert {\rm in}\rangle$ states are both defined at time $t= 0$. Consequently,they are elements of the same Hilbert space and can be compared directly.
What does this "shift in time" using the free Hamiltonian do and why can I now compare states?
Last but not least, it seems the $\vert {\rm out}\rangle$ and $\vert {\rm in}\rangle$ states don't seem to be in the Schrödinger picture anymore.. Are they in the Interaction picture?
$^1$ All references with equations will be with respect to the above lecture notes.
$^2$ Is there a typo in this equation? I would have expected $\vert {\rm in}\rangle := \exp(-iH_{\rm as} t_i)\vert i\rangle$...