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Feynman spectral action of the wave operator on asymptotically de Sitter spaces

Ruben Zeitoun

TL;DR

This work develops a Lorentzian spectral theory for the wave operator on non-trapping, even asymptotically de Sitter spaces by constructing a Feynman propagator framework. It combines microlocal and semiclassical methods, a variable-order pseudo-differential calculus, and a Hadamard parametrix to define complex powers of the Feynman propagator and a meromorphic spectral action whose residues encode the scalar curvature. The main contributions include proving uniform microlocal estimates, establishing Fredholm properties for the transformed operator $P(\lambda)$, and deriving explicit curvature terms from Feynman propagator residues, thereby extending spectral-action ideas to physically relevant Lorentzian geometries. The results offer a rigorous link between geometry and spectral-type invariants in a dynamical, cosmological setting and provide a pathway for Lorentzian spectral actions beyond the Riemannian, compact case.

Abstract

In this paper, we investigate the wave operator $\square_g$ on non-trapping (at all energies) even asymptotically de Sitter spaces. We construct a Feynman operator on the conformal extension of asymptotically de Sitter spaces and give a proof of uniform microlocal estimates for the Feynman operator in this setting. This enables the study of the Lorentzian "spectral" zeta functions in asymptotically de Sitter and the construction of a "spectral" action of the Feynman propagator.

Feynman spectral action of the wave operator on asymptotically de Sitter spaces

TL;DR

This work develops a Lorentzian spectral theory for the wave operator on non-trapping, even asymptotically de Sitter spaces by constructing a Feynman propagator framework. It combines microlocal and semiclassical methods, a variable-order pseudo-differential calculus, and a Hadamard parametrix to define complex powers of the Feynman propagator and a meromorphic spectral action whose residues encode the scalar curvature. The main contributions include proving uniform microlocal estimates, establishing Fredholm properties for the transformed operator , and deriving explicit curvature terms from Feynman propagator residues, thereby extending spectral-action ideas to physically relevant Lorentzian geometries. The results offer a rigorous link between geometry and spectral-type invariants in a dynamical, cosmological setting and provide a pathway for Lorentzian spectral actions beyond the Riemannian, compact case.

Abstract

In this paper, we investigate the wave operator on non-trapping (at all energies) even asymptotically de Sitter spaces. We construct a Feynman operator on the conformal extension of asymptotically de Sitter spaces and give a proof of uniform microlocal estimates for the Feynman operator in this setting. This enables the study of the Lorentzian "spectral" zeta functions in asymptotically de Sitter and the construction of a "spectral" action of the Feynman propagator.

Paper Structure

This paper contains 37 sections, 40 theorems, 222 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Motivations
  3. State of the art
  4. Main result
  5. Structure of the proofs
  6. Content
  7. Microlocal analysis for variable order
  8. Pseudo-differential calculus with variable order
  9. Sobolev spaces
  10. Microlocalization
  11. Propagation of singularities and radial estimates
  12. Semiclassical analysis for variable order
  13. Semi-classical pseudo-differential operators
  14. Sobolev spaces
  15. Microlocalization
  16. ...and 22 more sections

Key Result

Theorem 1.1

Let $(M,g)$ be a non-trapping even asymptotically de Sitter space of even dimension $n$. Then the Schwartz kernel $K_{\alpha,k}(.,.)$ of $R_F^{(\alpha,k)}(\pm i\epsilon)$ exists as a family of distributions near the diagonal depending holomorphically in $\alpha$ on the half-plane $\mathop{\mathrm{Re where $R_{ F}^{(\alpha,k)}(\mu+i\epsilon,.)=\frac{1}{2\pi i}P_0(\lambda(\mu+i\epsilon))^k\int_{\gam

Figures (2)

  • Figure 1: The map $x\mapsto x_{\mathbb S}\in{\mathbb S}^d$, here for $x\in{\rm dS}^d$. In reality ${\rm dS}$ is connected for $d\geqslant 2$.
  • Figure 2: Extended de Sitter space. The 'equatorial belt' part of the sphere ${\mathbb S}^d$ (i.e., the region between ${Y}_-$ and ${Y}_+$) is identified with ${\rm dS}^d$, and the two 'caps' resp. below ${Y}_-$ and above ${Y}_+$ are identified with resp. ${\mathbb H}^d_-$ and ${\mathbb H}^d_+$.

Theorems & Definitions (89)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Remark 2.5
  • Proposition 2.6
  • Definition 2.7
  • Lemma 2.8
  • Definition 2.9
  • ...and 79 more