# Hadamard states for the Klein-Gordon equation on Lorentzian manifolds of bounded geometry

February 02, 2016

We consider the Klein-Gordon equation on a class of Lorentzian manifolds with Cauchy surface of bounded geometry, which is shown to include examples such as exterior Kerr, Kerr-de Sitter and Kerr-Kruskal spacetimes. In this setup, we give an approximate diagonalization and a microlocal decomposition of the Cauchy evolution using a time-dependent version of the pseudodifferential calculus on Riemannian manifolds of bounded geometry. We apply this result to construct all pure regular Hadamard states (and associated Feynman inverses), where regular refers to the state's two-point function having Cauchy data given by pseudodifferential operators. This allows us to conclude that there is a one-parameter family of elliptic pseudodifferential operators that encodes both the choice of (pure, regular) Hadamard state and the underlying spacetime metric.

open access link
Commun. Math. Phys. 352(2), 519-583 (2017)

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%%% add \usepackage[utf8]{inputenc} to your latex preamble
@article{Gerard:2016jqj,
author = "Gérard, Christian and Oulghazi, Omar and Wrochna,
Michal",
title = "{Hadamard states for the Klein-Gordon equation on
Lorentzian manifolds of bounded geometry}",
journal = "Commun. Math. Phys.",
volume = "352",
year = "2017",
number = "2",
pages = "519-583",
doi = "10.1007/s00220-017-2847-4",
eprint = "1602.00930",
archivePrefix = "arXiv",
primaryClass = "math-ph",
SLACcitation = "%%CITATION = ARXIV:1602.00930;%%"
}

Keywords:

quantum field theory on curved spacetimes, Hadamard states