Physical interpretations of Scattering Matrix $S$, Transition matrix $T$ and amplitude $M$

The S-matrix is the amplitude to transition from an initial state to some final state, so it evolves the initial state to the final state, $$ S_{fi}=langle f|U(infty, -infty)|irangle, $$ and since the evolution operator is unitary, the matrix (to all final states) is unitary, that is the initial state must scatter to some final state, including itself, for sure: it can't just disappear! That is, the transition probabilities to all final states sum to one, $$ S^dagger S= sum_f |langle f|U|irangle|^2 =1. $$

For a free theory, the final state is the undisturbed initial one, so the S-matrix is the identity: nothing happened. But if there is an interaction term, a potential, there is an amplitude to scatter to a different state, an off-diagonal state, and that amplitude increases with the "potential" or "interaction hamiltonian", the Dyson series of which yields an approximation to $U(t,t_0)$. Actually, in the interaction representation used most often, to $U_I(t,t_0)= e^{iH_0t/hbar} U(t,t_0) e^{-iH_0t_0/hbar}$.

In plain NR QM, the eigenstates of the free hamiltonian are plane waves, so the amplitude to transition from a plane wave to a distinctly different plane wave f is $$ T_{fi}=langle f|V|irangle , $$ so the probability of this distinct transition is $|langle f|V|irangle |^2$. That is, it is the nontrivial piece of the S-matrix, up to a convenience normalization in plane wave scattering, $$ S-1!! 1 = -2pi i T. $$ The unitarity of S then presents as $$ T^dagger -T - 2ipi T^dagger T =0 . $$ It is useful because in scattering experiments we normally ignore the forward stream of projectiles (which went through the target), and attend to the scattered products, which tell us something about the potential which scattered them. (Recall the Geige-Marsden experiment.)