Classical and quantum scattering of massive Klein--Gordon fields in asymptotically flat spacetimes

We use microlocal tools to define distinguished incoming and outgoing vacuum states of the quantum theory of massive Klein–Gordon fields in non-trapping asymptotically flat spacetimes, including radiative perturbations of Minkowski space. The asymptotic vacuum states satisfy the Hadamard condition, and the incoming and outgoing particle representations are unitarily equivalent, or in other words the quantum S-matrix is well-defined. These results are based on a description of scattering of classical fields which keeps track of the asymptotic distinction between positive- and negative-frequency solutions of the Klein–Gordon equation. Our emphasis is on developing the theory without recourse to explicit asymptotics for classical waves.