Scattering in the static patch of de Sitter space

We study the scattering problem in the static patch of de Sitter space, i.e. the problem of field evolution between the past and future horizons of a de Sitter observer. We calculate the leading-order scattering for a conformally massless scalar with cubic interaction, as both the simplest case and a warmup towards Yang-Mills and gravity. Our strategy is to decompose the static-patch evolution problem into a pair of more symmetric evolution problems in two Poincare patches, sewn together by a spatial inversion. To carry this out explicitly, we end up developing formulas for the momentum-space effect of inversions in flat spacetime. The geometric construction of an electron's 4-momentum and spin vectors from a Dirac spinor turns out to be surprisingly relevant.