Application of Microlocal Analysis to the Theory of Quantum Fields Interacting with a Gravitational Field

Quantum field theory (QFT) is the theory of elementary particles and their fundamental interactions. The very successful standard model which describes the electromagnetic, weak and strong interactions of the observed elementary particles does however not incorporate gravity. The quantisation of gravity (“quantum gravity”) is a very difficult problem which is presently still far from its solution. An easier, but only approximate approach to the interaction of matter and gravitational fields is the “semiclassical theory” (or “QFT in curved spacetimes”) where the gravitational field is described classically as a 4-dim. Lorentzian manifold (M, g) and only the matter fields are quantized to operator valued Wightman fields. This theory, which we will describe in the following more closely, should have a wide range of physical applicability, from quantum effects in the early universe to the Hawking radiation of massive stars collapsing to a black hole, and should also lead to a better understanding of the local features of standard QFT (for a good introduction see e.g. [2]).