Two-point Functions and Quantum Fields in de Sitter Universe

TL;DR

This paper develops a framework for two-point functions and free quantum fields in de Sitter space, introducing a geodesic spectral condition and holomorphic functions to parallel Minkowski space theory.

Contribution

It introduces a novel approach using complex geometry and perikernels to analyze quantum fields in de Sitter space, including a new integral representation and a Fourier-Laplace transform.

Findings

Established a geodesic spectral condition for de Sitter quantum fields

Derived a new integral representation of two-point functions using de Sitter plane waves

Proved the Reeh-Schlieder property and developed a substitute for Wick rotation

Abstract

We present a theory of general two-point functions and of generalized free fields in d-dimensional de Sitter space-time which closely parallels the corresponding minkowskian theory. The usual spectral condition is now replaced by a certain geodesic spectral condition, equivalent to a precise thermal characterization of the corresponding ``vacuum''states. Our method is based on the geometry of the complex de Sitter space-time and on the introduction of a class of holomorphic functions on this manifold, called perikernels, which reproduce mutatis mutandis the structural properties of the two-point correlation functions of the minkowskian quantum field theory. The theory contains as basic elementary case the linear massive field models in their ``preferred'' representation. The latter are described by the introduction of de Sitter plane waves in their tube domains which lead to a new integral representation of the two-point functions and to a Fourier-Laplace type transformation on the hyperboloid. The Hilbert space structure of these theories is then analysed by using this transformation. In particular we show the Reeh-Schlieder property. For general two-point functions, a substitute to the Wick rotation is defined both in complex space-time and in the complex mass variable, and substantial results concerning the derivation of Kallen-Lehmann type representation are obtained.