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

Abstract

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 pseu-dodifferential 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.

56 p., v3: some misprints corrected

References in corpus (5)

• Algebraic QFT in Curved Spacetime and quasifree Hadamard states: an introduction
• Quantum out-states holographically induced by asymptotic flatness: Invariance under spacetime symmetries, energy positivity and Hadamard property
• States of Low Energy on Robertson-Walker Spacetimes
• Complex powers of analytic functions and meromorphic renormalization in QFT
• Hadamard state in Schwarzschild-de Sitter spacetime

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