I have been reading through the chapter on the LSZ Reduction Formula from Srednicki's Quantum Field Theory, and I have a few questions about which I'm sort of confused. The questions are referenced from the following copy of the textbook: http://web.physics.ucsb.edu/~mark/qft.html
On page 49, it is mentioned that if we time evolve a wave packet, it spreads out in time; thus, the wave packet (particle) is localized far from the origin (where it was initially prepared) as $t \longrightarrow \pm \; \infty$. I'm not sure how to interpret the $t = - \; \infty$ bit. As time progress from 0, doesn't it move to $\infty$? Does the other argument have something to do with time-reversal. I'm quite confused on this front. I probably don't get something elementary, I suppose,
As defined in $(5.8)$ and $(5.9)$, how does $\langle f \rvert i \rangle$ describe the scattering amplitude in the interacting theory? If I'm correct, in a free field theory for spin-0 bosons the excitations of the field describe particles with well-defined three-momentum (by dirac delta functions). The particles don't interact with other particles. So my question is, how does the notion of scattering come about in an interacting theory; what is the physical process governing it? As it stands, Srednicki's analysis simply seems to imply the annihilation and creation of particles (at $\pm \; \infty$), which technically is allowed for in a free field theory?
pp. 51-52 state: "the next (second) excited state is that of two particles. These two particles could form a bound state with energy less than $2m$. but, to keep things simple, let us assume that there are no such bound states. Then the lowest possible energy of a two-particle state is $2m$. However, a two-particle state with zero total three-momentum can have any energy above $2m$, because the two particles could have some relative momentum that contributes to their total energy. How and why can a two-particle state have an energy of less than or greater than $2m$ (in an interacting theory). Again, if I'm correct, in a free field theory, the energy is exactly $2m$ for a two-particle state.
In summary, I seem to be confused about some aspects of Srednicki's introduction of an interacting field theory in the context of the LSZ Reduction Formula. It'd be great if someone could help alleviate some of these concerns.