There is obvious utility for developing techniques to compute scattering amplitudes nonperturbatively: while currently distant targets, one can easily imagine applications to meson spectroscopy, QCD corrections to precision QED observables, Pomeron physics, the high density and far from equilibrium conditions present in heavy ion collisions, etc. Unfortunately, nonperturbative methods like the lattice or bootstrap only access partial information. In this talk, we present a method for computing the full, honest-to-goodness amplitude M(s,t) nonperturbatively using Hamiltonian data. Such data is the natural output of an old, but recently revived, technique broadly known as Hamiltonian truncation. We first answer the general question “given (approximate) eigenstates, how do you compute the S-matrix?”, and then demonstrate our method on a strongly coupled system in three dimensions. We conclude by touching upon extensions towards theories with more intricate phenomenology. Based on work done in collaboration with: Hitoshi Murayama, Francesco Riva, Jed Thompson, and Matt Walters
