Exact Results in Quantum Theory & Gravity

I will introduce two different operator-theoretic settings which can be used to describe various two-point functions of quantum fields. These settings are not common points of view but they have several advantages. For example, they can be used to give rigorous meaning to expressions that are in the usual interpretation zero divided by zero, or they can serve to distinguish certain states in Quantum Field Theory on curved spacetimes. As example, I will discuss is a scalar field on de Sitter space. This example at the same time exhibits a rich structure and is comparably simple.