# Usefulness of $| {\rm in}\rangle$ and $| {\rm out}\rangle$ states in S-matrix description of QFT

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 $$\vert {\rm in}\rangle := \exp(-iH_{\rm as} t_i)\vert i\rangle$$...

$$|in\rangle$$ and $$|out\rangle$$ states refer to the set of boundary conditions frequently employed in scattering problems, designed to mimic a classical scattering of a particle from an object - which is where the terminology in terms of behavior at $$t\rightarrow \pm \infty$$ comes from, although one may be actually working in Schrödinger picture. Using Heisenberg picture makes this boundary conditions more intuitive, and most naturally they probably come when using the Keldysh formalism (but by then one already feels comfortable with the scattering theory).