Wick rotation of linearized gravity in Gaussian time and Calder\'on projectors

TL;DR

This paper explores the Wick rotation of linearized gravity equations in Gaussian time, demonstrating how Calderón projectors relate to Hadamard bi-solutions and identifying boundary-related obstructions affecting quantization.

Contribution

It introduces a novel analysis of Wick-rotated linearized Einstein equations and their boundary effects, advancing understanding of quantum gravity formulations.

Findings

Calderón projectors induce Hadamard bi-solutions on Lorentzian spacetime.

Boundary terms create obstructions to gauge-invariance and positivity in quantization.

Obstructions depend on boundary conditions and arise from Wick-rotation boundary terms.

Abstract

Motivated by the quantization of linearized gravity, we consider gauge-fixed linearized Einstein equations and their Wick rotation near a Cauchy surface. We show that Calder\'on projectors for the Wick-rotated equations induce Hadamard bi-solutions on the Lorentzian level. On the other hand, we find smoothing obstructions to gauge-invariance and positivity conditions needed in quantization. These obstructions are primarily due to boundary terms arising in the Wick-rotated theory and depend on the boundary conditions.