An analogue of non-interacting quantum field theory in Riemannian signature

Abstract

In this paper, we define an analogue of non-interacting quantum fields satisfying $({\Delta }_{\mathbf{g}}-{\lambda }^2)\varphi =0$ on a Riemannian scattering space $(M,\mathbf{g})$ with two boundary components, i.e., a manifold with two asymptotically conic ends (meaning asymptotic to the “large end” of a cone). Thus, the Lorentzian spacetime of usual QFT constructions is replaced by a Riemannian manifold with two boundary components, which play a role analogous to the two components of the mass shell for the Klein–Gordon field on Lorentzian spacetimes. Our main result describes a canonical construction of two-point functions satisfying a version of the Hadamard condition.