Automorphic scalar fields in two-dimensional de Sitter space

TL;DR

This paper investigates automorphic scalar fields in two-dimensional de Sitter space, revealing the uniqueness of de Sitter invariant Hadamard states and exploring thermal properties of non-invariant states.

Contribution

It extends the understanding of automorphic scalar fields by characterizing invariant states and constructing two-point functions for non-invariant states in 2D de Sitter space.

Findings

No de Sitter invariant Hadamard states except in the periodic case

Constructed two-point functions for non-Hadamard states

Non-invariant Hadamard state approximates a thermal state at Gibbons-Hawking temperature

Abstract

We study non-interacting automorphic quantum scalar fields with positive mass in two-dimensional de Sitter space. We find that there are no Hadamard states which are de Sitter invariant except in the periodic case, extending the result of Epstein and Moschella for the anti-periodic case. We construct the two-point Wightman functions for the non-Hadamard de Sitter-invariant states by exploiting the fact that they are functions of the geodesic distance between the two points satisfying an ordinary differential equation. We then examine a certain Hadamard state, which is not de Sitter invariant, and show that it is approximately a thermal state with the Gibbons-Hawking temperature when restricted to a static region of the spacetime.