Differential Geometry Seminar: The Feynman propagator on asymptotically Minkowski spacetimes

The Feynman propagator is a fundamental object in Quantum Field Theory which is also interesting when viewed from the perspective of analysis of linear hyperbolic PDEs.

Thanks to the work of Baer-Strohmaier it has also been shown to finally provide an index theory of hyperbolic Dirac operators on some globally hyperbolic Lorentzian manifolds.

We will discuss a new construction of the Feynman propagator for the Klein-Gordon (KG) equation on asymptotically Minkowski spacetimes. Our construction generalises existing work, and in particular gives a construction which works for naturally occurring families of Callias-type operators.

In particular, for large classes of potential perturbations, including magnetic potentials in the case of Dirac-Klein-Gordon, we construct global in time solutions to hyperbolic equations whose wavefront sets satisfy a Hadamard condition. Such solutions were shown to exist locally by Duistermaat-Hormander. To accomplish this, we prove a global Fredholm estimate and show that these non-elliptic operators are in fact Fredholm operators between appropriate Sobolev spaces. This is joint work with Dean Baskin and Moritz Doll.

School of Computer and Mathematical Sciences