I have seen the scattering matrix defined using initial ("in") and final ("out") eigenstates of the free hamiltonian, with

$$\left| \vec{p}_1 \cdots \vec{p}_n \; \text{out} \right\rangle
=
S^{-1}
\left| \vec{p}_1 \cdots \vec{p}_n \; \text{in} \right\rangle$$

so that

$$\left\langle \vec{p}_1 \cdots \vec{p}_n \; \text{out} \mid
\vec{q}_1 \cdots \vec{q}_m \; \text{in}
\right\rangle
=
\left\langle \vec{p}_1 \cdots \vec{p}_n \; \text{in}
\mid S \mid
\vec{q}_1 \cdots \vec{q}_m \; \text{in}
\right\rangle.$$

1) What *are* "in" and "out" states?

2) Are they Fock states?

3) In Schrödinger or Heisenberg or interaction representation?

4) How are they related? (I believe that I see what they handwavily represent physically, but not formally.)

My main issue is that, if "in" and "out" states are one-particle eigenstates of the free hamiltonian, i.e. if $\left| \vec{p}_1 \text{out} \right\rangle$ describes a free particle with momentum $\vec{p}_1$, and $\left| \vec{p}_1 \text{in} \right\rangle$ **also** describes a free particle with momentum $\vec{p}_1$, then $\left| \vec{p}_1 \text{out} \right\rangle = \left| \vec{p}_1 \text{in} \right\rangle$ ... which is false. Still, books (some at least) describe these "in" and "out" states like that.

Moreover, I have seen (e.g. in Wikipedia, but also on this answer) that the scattering matrix is a map between two different Fock states, and I don't understand that.

5) Do states of the interacting system live in the same Fock space that asymptotic free states?

6) And if not, where do they live?

Understandable references would be appreciated.

This post imported from StackExchange Physics at 2014-08-26 21:32 (UCT), posted by SE-user A. Zerkof