  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.
