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

TL;DR

This paper demonstrates the equivalence of two different formalisms for calculating correlators in interacting quantum field theory on de Sitter space, showing they produce identical results for massive scalar fields through analytic and direct methods.

Contribution

It provides an analytic proof and explicit verification that in-in and Euclidean correlators in de Sitter QFT are equivalent for massive scalars, bridging two common computational approaches.

Findings

Correlators from in-in and Euclidean formalisms coincide for m^2 > 0

The equivalence holds diagram by diagram at finite regulator mass M

Connections are noted between perturbation prescriptions in static spacetimes with 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 Poincar\'e 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.