Exact Results in Quantum Theory

In the construction of scalar quantum fields on a curved spacetime, the crucial step is to split the solution space of the Klein-Gordon equation into two parts corresponding to particles and anti-particles. The splittings need to satisfy the so-called Hadamard condition, originally formulated by Kay and Wald and incorporating the requirement that two-point functions of fields should be microlocally the same as the Minkowski vacuum. In this talk, I will present a solution to this problem on a class of asymptotically de Sitter spacetimes, derived in a recent joint work with András Vasy and relying on propagation estimates at radial sets. The crucial feature is the extendability of appropriately rescaled classical fields across the conformal horizon, to a region consisting of two asymptotically hyperbolic spaces. It turns out that non-interacting quantum fields follow the same behaviour and are uniquely determined by data in the asymptotically hyperbolic spaces.