Quantum fields in the static de Sitter universe

E. T. Akhmedov,D. V. Diakonov,U. Moschella,K. V. Bazarov

doi:10.1103/physrevd.102.085003

Abstract

We construct explicit mode expansions of various tree-level propagators in the Rindler -- de Sitter universe, also known as the static (or compact) patch of the de Sitter spacetime. We construct in particular the Wightman functions for thermal states having a generic temperature $T$. We give a fresh simple proof that the only thermal Wightman propagator that respects the de Sitter isometry is the restriction to the Rindler -- de Sitter wedge of the propagator for the Bunch--Davies state. It is the thermal state with $T = (2 \pi)^{-1}$ in the units of de Sitter curvature. We show that propagators with $T\ne(2\pi)^{-1}$ are only time translation invariant and have extra singularities on the boundary of the static patch. We also construct the expansions for the so-called alpha-vacua in the static patch and discuss the flat limit.

Highlights

Notwithstanding the existence of a vast literature on de Sitter quantum fields there is still no consensus as regard to their infrared behavior which is very much different from the behavior of Minkowski and anti–de Sitter fields
It is the thermal state with T 1⁄4 ð2πÞ−1 in the units of de Sitter curvature
We show that propagators with T ≠ ð2πÞ−1 are only time translation invariant and have extra singularities on the boundary of the static patch

Summary

INTRODUCTION

Notwithstanding the existence of a vast literature on de Sitter quantum fields there is still no consensus as regard to their infrared behavior which is very much different from the behavior of Minkowski and anti–de Sitter fields. Apart from the Gibbons-Hawking result, not very much is known about quantum fields in the Rindler–de Sitter wedge (the static patch) The relevance of this model for cosmology and black hole physics makes this lack of knowledge even more surprising. We produce new (to the best of our knowledge) formulas giving explicit mode expansions of the correlation functions where the coordinates of the static patch are separated. This is an obviously necessary preliminary step to apply perturbation theory and calculate loops as described above. The geodesic distances L and ζ are related as follows: ζ 1⁄4 − coshðLÞ for timelike geodesics, ζ 1⁄4 cosðLÞ for spacelike ones, ζ 1⁄4 −1 for lightlike separations or coincident points

CANONICAL QUANTIZATION

Thermal two-point functions

MODE EXPANSION OF THE HOLOMORPHIC PLANEPLANE WAVES

MODE EXPANSIONS OF THE MAXIMALLY ANALYTIC TWO-POINT FUNCTION

MORE ABOUT THE VACUUM OF THE STATIC GEODETIC OBSERVER

OTHER TIME-TRANSLATION INVARIANT STATES

VIII. ALPHA STATES IN THE STATIC CHART

MORE ABOUT THERMAL PROPAGATORS

Wightman propagators for large timelike separation

Large spacelike separation

Lightlike separation

Anomalous singularities at the horizon