  We consider the wave equation on asymptotically Minkowski spacetimes and the
Klein-Gordon equation on even asymptotically de Sitter spaces. In both cases we
show that the extreme difference of propagators (i.e. retarded propagator minus
advanced, or Feynman minus anti-Feynman), defined as Fredholm inverses, induces
a symplectic form on the space of solutions with wave front set confined to the
radial sets. Furthermore, we construct isomorphisms between the solution spaces
and symplectic spaces of asymptotic data. As an application of this result we
obtain distinguished Hadamard two-point functions from asymptotic data.
Ultimately, we prove that the corresponding Quantum Field Theory on
asymptotically de Sitter spacetimes induces canonically a QFT beyond the future
and past conformal boundary, i.e. on two even asymptotically hyperbolic spaces.
Specifically, we show this to be true both at the level of symplectic spaces of
solutions and at the level of Hadamard two-point functions.
