# Analytic Hadamard states, Calder\'on projectors and Wick rotation near analytic Cauchy surfaces

## TL;DR

This paper establishes the existence of pure analytic Hadamard states for the Klein-Gordon equation on analytic spacetimes by employing Wick rotation and Calderón projectors, advancing quantum field theory in curved spacetime.

## Contribution

It introduces a novel method using Wick rotation and Calderón projectors to construct Hadamard states analytically on curved spacetimes.

## Key findings

- Proves existence of pure analytic Hadamard states.
- Constructs Cauchy data via Calderón projectors for elliptic operators.
- Extends methods to non-compact hypersurfaces.

## Abstract

We consider the Klein-Gordon equation on analytic spacetimes with an analytic Cauchy surface. In this setting, we prove the existence of pure analytic Hadamard states. The proof is based on considering an elliptic operator obtained by Wick rotating the Klein-Gordon operator in a neighborhood of a Cauchy hypersurface. The Cauchy data of Hadamard two-point functions are constructed as Calder\'{o}n projectors (suitably generalized if the hypersurface is non-compact) for the elliptic operator.