The massive Feynman propagator on asymptotically Minkowski spacetimes

Abstract

We consider the massive Klein-Gordon equation on asymptotically Minkowski spacetimes, in the sense that the manifold is R 1+d and the metric approaches that of Minkowski space at infinity in a short-range way (jointly in time and space variables). In this setup we define Feynman and anti-Feynman scattering data and prove the Fredholm property of the Klein-Gordon operator with the associated Atiyah-Patodi-Singer type boundary conditions at infinite times. We then construct a parametrix (with compact remainder terms) for the Fredholm problem and prove that it is also a Feynman parametrix in the sense of Duistermaat and Hörmander.2010 Mathematics Subject Classification. 81T13, 81T20, 35S05, 35S35.

The Massive Feynman Propagator on Asymptotically Minkowski Spacetimes II

“…We proved in [GW1,Thm. 1.2] in the massive case that P : X m F → Y m is a Fredholm operator and we related its index to the index of some 'Feynman' wave operator W † F .…”

“…The present paper is a continuation of [GW1], which was devoted to the existence of the Feynman propagator for Klein-Gordon fields on asymptotically Minkowski spacetimes. Let us first briefly recall the motivation of [GW1].…”

“…In [GW1] we considered this problem for asymptotically Minkowski spacetimes. Inspired by works by Gell-Redman, Haber and Vasy [GHV,Va1] and Bär and Strohmaier [BS] on closely related non-elliptic Fredholm problems, we introduced Hilbert spaces Y m , X m F , where m ∈ R is an order of Sobolev regularity and the subscript F refers to what one can call Feynman boundary conditions, which are imposed at t = ±∞.…”

Microlocal Analysis of Quantum Fields on Curved Spacetimes

“…In this section we will describe the results of [GW4,GW6], devoted to this question on spacetimes which are asymptotically Minkowski, and hence have in general no global symmetries, only asymptotic ones. It turns out that it is possible in this case to define a canonical Feynman inverse G F , which is the inverse of P between some appropriate Sobolev type spaces.…”

“…It is shown in [GW4] that if (aMi) holds, then (aMii) is equivalent to the familiar non trapping condition for null geodesics of g, and if (aMi), ii), iii) hold there exists a Cauchy temporal functiont such thatt − t ∈ C ∞ 0 (M ). Replacing t by t − c,t byt − c for c 1 we can also assume that Σ · · = {t = 0} = {t = 0} is a Cauchy surface for (M, g), which can be canonically identified with R d .…”