So long, in my QFT courses, I've seen two definitions of the $S$-matrix:
The first, more elementary, definition is given in the interaction picture: $$S=\text T \lbrace \exp [-i\intop \text d ^4 x \,\mathscr H_{\text {int}} ^I(x)]\rbrace\qquad (1),$$where the operatore $\mathscr H _\text{int} ^I$ is the interaction hamiltonian density evaluated in the interaction picture. According to this definition, the probability amplitude for a state with in-asymptote $\vert \alpha \rangle$ to scatter into a state with out-asymptote $\vert \beta \rangle $ is given by: $$S_{\beta \alpha}=\langle \beta \vert S \vert \alpha \rangle \qquad (2).$$
The second definition is the one in terms of in/out asymptotic states: $$S_{\beta \alpha}=\langle\beta \,\text {out} \vert \alpha \,\text {in}\rangle \qquad (3).$$
In Non relativistic quantum mechanics, I understand how these two definitions are related, since there one can write the amplitude of scattering from a plane wave $\vert \mathbf k \rangle$ to a plane wave $\vert \mathbf p \rangle$ as:$$S_{\mathbf p \leftarrow \mathbf k } =\langle \mathbf p \vert S \vert \mathbf k \rangle=\langle \mathbf p -\vert \mathbf k +\rangle, $$ where $\vert \mathbf k \pm\rangle$ are the "in" and "out" solutions of the Lippmann-Schwinger equation, or generalized eigenfunctions of the full hamiltonian.
So, I guess (correct me if I'm wrong) that the "in" and "out" states in QFT are the analogues of the $\vert \mathbf k \pm \rangle $'s in potential scattering, and indeed it can be shown that $\vert \{\mathbf p_i\}\, \text {in/out}\rangle$ are eigenstates of the interacting fields hamiltonian.
However, those "in/out" states are constructed in a quite formal way, via ad-hoc constructed asymptotic in/out fields, and I'm really failing in seeing how this definition $(3)$ should relate to the more elementary $(1)$.
Just to clarify: I understand the physical idea behind the construction of "in/out" fields and, probably, if I hadn't seen def. $(1)$, I would accept $(3)$ as it stands. However I don't understand the mathematical relation beetween (1) and (3):
- Do they agree?
- Do they agree up to a phase?
- Do they agree if some assumption, like "adiabatic hypothesis" is made?
- How can I formally prove it?
Any help/thought would be appreciated, thank you for the attention.