Wick rotation of the time variables for two-point functions on analytic backgrounds

TL;DR

This paper develops a framework for complex Laplacians via Wick rotation, showing their Green's functions can be analytically continued to produce two-point functions of Hadamard states without requiring stationarity.

Contribution

It introduces a general method for constructing two-point functions on analytic backgrounds using Wick rotation, extending previous approaches to non-stationary metrics.

Findings

Green's functions admit analytic continuation to produce Hadamard states

The framework applies to non-compact manifolds and non-stationary metrics

Thermal states are derived as a special case with time-independent coefficients

Abstract

We set up a general framework for Calder\'on projectors (and their generalization to non-compact manifolds), associated with complex Laplacians e.g. obtained by Wick rotation of a Lorentzian metric. In the analytic case, we use this to show that the Laplacian's Green's functions have analytic continuations whose boundary values are two-point functions of analytic Hadamard states. The result does not require the metric to be stationary. As an aside, we describe how thermal states are obtained as a special case of this construction if the coefficients are time-independent.