On the Equivalence between Euclidean and In-In Formalisms in de Sitter QFT

TL;DR

The paper shows that Euclidean (Hartle-Hawking) correlators and Lorentzian in-in correlators computed in the Poincaré patch of de Sitter space coincide for interacting scalar fields with $m^2>0$, diagram by diagram, at finite Pauli-Villars regulator masses. The authors present a three-step analytic framework: (i) relate Euclidean correlators to static-patch thermal correlators via factorization, (ii) establish equivalence between static-patch in-in and Poincaré patch in-in perturbation theory, and (iii) prove analyticity so that static-patch results extend to the full spacetime. They substantiate the claim with explicit one-loop checks for $ extphi^4$ and $ extphi^3$ interactions and discuss regulator-dependent convergence and renormalization. The result reinforces the reliability of Euclidean methods in de Sitter QFT, clarifies the role of horizons in perturbation theory, and suggests broader applicability to static regions of spacetimes with bifurcate Killing horizons.

Abstract

We study the relation between two sets of correlators in interacting quantum field theory on de Sitter space. The first are correlators computed using in-in perturbation theory in the expanding cosmological patch of de Sitter space (also known as the conformal patch, or the Poincaré patch), and for which the free propagators are taken to be those of the free Euclidean vacuum. The second are correlators obtained by analytic continuation from Euclidean de Sitter; i.e., they are correlators in the fully interacting Hartle-Hawking state. We give an analytic argument that these correlators coincide for interacting massive scalar fields with any $m^2 > 0$. We also verify this result via direct calculation in simple examples. The correspondence holds diagram by diagram, and at any finite value of an appropriate Pauli-Villars regulator mass M. Along the way, we note interesting connections between various prescriptions for perturbation theory in general static spacetimes with bifurcate Killing horizons.

Figures (13)

• Figure 1: Standard Carter-Penrose diagram of de Sitter space. Region I is the static patch, and the Poincaré patch consists of regions I and II. The causal pasts of points $X_1$ and $X_2$ are the shaded regions. See section \ref{['geom']} for details.
• Figure 2: Carter-Penrose diagram of de Sitter space with $\theta = {\rm const}$ surfaces (schematically) indicated by solid lines and $t={\rm const}$ surfaces by dashed lines.
• Figure 3: Carter-Penrose diagram of de Sitter space with timelike geodesics from point $O$ drawn with solid lines and spacelike geodesics from it drawn with dashed lines.
• Figure 4: A deformation of the Euclidean contour.
• Figure 5: Singularities and branch cuts of propagators.