The Feynman propagator for massive Klein-Gordon fields on radiative asymptotically flat spacetimes

TL;DR

This work defines a canonical Feynman propagator for massive Klein-Gordon fields on radiative asymptotically flat spacetimes by embedding the problem in Sussman’s de,sc calculus and a microlocal Fredholm framework. The authors extend the limiting absorption principle to this setting and prove a localized radial point estimate that propagates microlocal regularity into radial points across a complex radial set, accommodating radiation through null infinity. They develop a global microlocal analysis of the Klein-Gordon operator, including principal symbols, Hamilton flow, and radial sets, to establish global regularity and invertibility of the propagators under causality assumptions. Consequently, retarded, advanced, Feynman, and anti-Feynman propagators are constructed as inverses between carefully chosen weighted Sobolev spaces, with the Feynman propagator obtained as a limit of resolvents and satisfying a Hadamard-like microlocal structure at infinity. The results enable a robust, geometry-driven framework for quantum-field-theoretic propagators in curved spacetimes with radiation, improving the analytic control over their asymptotics and enabling principled definitions of in/out states in this setting.

Abstract

On a large class of asymptotically flat spacetimes which includes radiative perturbations of Minkowski space, we define a distinguished global Feynman propagator for massive Klein-Gordon fields by means of the microlocal approach to non-elliptic Fredholm theory, working in the de,sc-pseudodifferential algebra due to Sussman. We extend the limiting absorption principle (the "$i\varepsilon$ prescription" for the Feynman propagator) to this setting. Motivated by the complicated Hamilton flow structure arising in this problem, we also prove a new localized radial point estimate in the spirit of Haber-Vasy which, under appropriate nondegeneracy assumptions, allows one to propagate microlocal regularity into a single radial point belonging to a larger radial set which can be a source, sink, or saddle for the Hamilton flow.

Figures (8)

• Figure 1: $\tilde{\mathcal{M}}$ (left) and $\mathcal{M}$ (right) in the case of $d=1$ and trivial topology, with boundary hypersurfaces and corners and the neighborhoods defined above labeled. To get the $d=2$ picture, rotate about the central vertical axis.
• Figure 2: Schematic illustration of propagation setup in proof of Theorem \ref{['thm:pos']}. The slope of the upper boundary of $\tilde{W}$ ensures that the Hamilton flow only crosses it from the interior of $\tilde{W}$, and it similarly crosses all nearby level sets of the cutoff $\phi(\hat{z})$ in the same direction.
• Figure 3: Schematic illustration of Hamilton flow and commutant construction near a radial point $\alpha$ belonging to an extended radial set $R$. In this two-dimensional slice, $z_i$ represents a defining function of $R$ (in this case corresponding to a negative eigenvalue) and $w_j$ a coordinate along $R$. The commutant is localized to a neighborhood of $\alpha$ using a cutoff along $R$ which has level sets like the blue curves in the figure, so all bicharacteristics near $\alpha$ cross them in the same direction, the most relevant region being a neighborhood of an annular subset of $R$ around $\alpha$ (the "cutoff region"). In the figure on the left, the cutoff is constructed so that bicharacteristics do not enter the cutoff region from the sides; in the figure on the right, it is constructed so they do not exit through the sides. See Figure \ref{['fig:rad-pts-prop']} for more details.
• Figure 4: Schematic illustration of propagation setup in proof of Proposition \ref{['thm:localized-rp-estimates']}, for propagation into $\alpha$ in the forward direction (from $V_{\varepsilon}^-$, shaded blue). In these two two-dimensional slices of the same higher-dimensional region, $z_i$ represents a defining function of $R$ corresponding to a negative eigenvalue, $z_j$ a defining function corresponding to a positive eigenvalue, and $w_j$ a coordinate along $R$. The figure on the left illustrates the localization along $R$; the figure on the right illustrates a cross-section transverse to $R$ in which the flow is of saddle type. The regions $\tilde{W}_{\varepsilon}$ are constructed so that, for a range of values of $\varepsilon$, the flow only enters them through the top and bottom faces (cf. figure on the left in Figure \ref{['fig:localization']}). This choice ensures that bicharacteristics cross level sets of the cutoffs along $R$ in the same direction as they cross those of the cutoff on the unstable side of $R$ (left/right faces of $\tilde{W}_{\varepsilon}$ in the figure on the right) and opposite to the direction that they cross the cutoff on the stable side (top/bottom faces). This means that if a priori regularity on the unstable side is not required for the estimate (i.e. if one is propagating forward), then the cutoff along $R$ will also not contribute any new terms which require a priori assumptions. To propagate in the backward direction, one instead constructs the cutoff along $R$ as in the figure on the right in Figure \ref{['fig:localization']}, corresponding to opposite signs of the parameters $D_j$.
• Figure 5: Schematic illustration of the regions $V^-$, $\hat{V}^-$ defined in the proof of Theorem \ref{['thm:localized-rp-main']}. As in Figure \ref{['fig:rad-pts-prop']}, $z_i$ represents a defining function of $R$ corresponding to a negative eigenvalue, $z_j$ a defining function corresponding to a positive eigenvalue, and $w_j$ a coordinate along $R$. Note the flow only enters both regions through the top and bottom faces, and every integral curve in the regions either limits to $\alpha$ or exits the regions in finite time through the sides.

Theorems & Definitions (57)

• Theorem 1.1
• Lemma 2.1
• proof
• Example 2.1: Minkowski metric
• Example 2.2: Exterior Schwarzschild metric
• Example 2.3: Lorentzian scattering spaces
• Example 2.4: Perturbations of Minkowski space and Minkowski-like and asymptotically Minkowski metrics
• Remark 2.1
• Lemma 3.1
• proof