In the S-matrix formulation of particle physics, which was pioneered by John Archibald Wheeler among others, all physical processes are modeled according to the following paradigm.
One begins with a non-interacting multiparticle state in the distant past.  Non-interacting does not mean that all of the forces have been turned off, in which case for example protons would fall apart, but rather that there exists an interaction-free Hamiltonian H, for which the bound states have the same energy level spectrum as the actual Hamiltonian .  This initial state is referred to as the in state.  Intuitively, it consists of elementary particles or bound states that are sufficiently well separated that their interactions with each other are ignored.
The idea is that whatever physical process one is trying to study may be modeled as a scattering process of these well separated bound states.  This process is described by the full Hamiltonian , but once its over, all of the new elementary particles and new bound states separate again and one finds a new noninteracting state called the out state'. The S-matrix is more symmetric under relativity than the Hamiltonian, because it does not require a choice of time slices to define.
This paradigm allows one to calculate the probabilities of all of the processes that we have observed in 70 years of particle collider experiments with remarkable accuracy.  But many interesting physical phenomena do not obviously fit into this paradigm. For example, if one wishes to consider the dynamics inside of a neutron star sometimes one wants to know more than what it will finally decay into.  In other words, one may be interested in measurements that are not in the asymptotic future.  Sometimes an asymptotic past or future is not even available.  For example, it is very possible that there is no past before the Big Bang.
In the 1960s, the S-matrix paradigm was elevated by many physicists to a fundamental law of nature. In S-matrix theory, it was stated that any quantity that one could measure should be found in the S-matrix for some process. This idea was inspired by the physical interpretation that S-matrix techniques could give to Feynman diagrams restricted to the mass-shell, and led to the construction of dual resonance models. But it was very controversial, because it denied the validity of quantum field theory based on local fields and Hamiltonians.