Hadamard states for spacetimes with timelike boundaries
Abstract
We formulate and prove existence of pure quasifree Hadamard states for the Klein--Gordon field with Dirichlet boundary condition on a globally hyperbolic spacetime with timelike boundary. The microlocal condition is imposed on the one-particle Hilbert-space-valued distribution p∘ GD rather than on the two-point function. This is expressed by a b-wave-front set condition on the compressed future cone. The analysis on the manifold with corners M̅×M̅ is then almost completely avoided. We prove that our version of the Hadamard condition is universal in the class of quasi-free Hadamard states. We also prove a propagation theorem for positive-energy microlocal splittings which uses only a microlocal splitting in the small b-calculus and well-posedness of the Dirichlet problem, and in particular requires no propagation theorem for reflected or gliding rays; combined with an explicit computation in the ultrastatic case and a Fulling--Narcowich--Wald deformation argument this yields the construction of a pure quasi-free Dirichlet--Hadamard state.
Related
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- The Hadamard parametrix on globally hyperbolic spacetimes with Robin boundary conditions: Fundamental solutions and Hadamard states
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- Why Hadamard states?
BibTeX
@misc{contini2026hadamard,
title = {{Hadamard states for spacetimes with timelike boundaries}},
author = {Alessandro Pietro Contini and Alexander Strohmaier},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13358},
eprint = {2609.13358},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13358},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Hadamard states for spacetimes with timelike boundaries
AU - Alessandro Pietro Contini
AU - Alexander Strohmaier
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.13358
UR - https://arxiv.org/abs/2609.13358
AB - We formulate and prove existence of pure quasifree Hadamard states for the Klein--Gordon field with Dirichlet boundary condition on a globally hyperbolic spacetime with timelike boundary. The microlocal condition is imposed on the one-particle Hilbert-space-valued distribution p∘ G_D rather than on the two-point function. This is expressed by a b-wave-front set condition on the compressed future cone. The analysis on the manifold with corners M̅×M̅ is then almost completely avoided. We prove that our version of the Hadamard condition is universal in the class of quasi-free Hadamard states. We also prove a propagation theorem for positive-energy microlocal splittings which uses only a microlocal splitting in the small b-calculus and well-posedness of the Dirichlet problem, and in particular requires no propagation theorem for reflected or gliding rays; combined with an explicit computation in the ultrastatic case and a Fulling--Narcowich--Wald deformation argument this yields the construction of a pure quasi-free Dirichlet--Hadamard state.
ER -