The holographic Hadamard condition on asymptotically Anti-de Sitter spacetimes

TL;DR

This paper establishes a holographic Hadamard condition for quantum fields in asymptotically Anti-de Sitter spacetimes, linking bulk and boundary properties and proving existence, uniqueness, and propagation of two-point functions.

Contribution

It introduces a new boundary condition on two-point functions that extends the Hadamard condition holographically and proves key existence and uniqueness results.

Findings

Existence of two-point functions satisfying the holographic Hadamard condition

Uniqueness of these two-point functions modulo smooth kernels

Propagation of singularities results analogous to classical theorems

Abstract

In the setting of asymptotically Anti-de Sitter spacetimes, we consider Klein-Gordon fields subject to Dirichlet boundary conditions, with mass satisfying the Breitenlohner-Freedman bound. We introduce a condition on the b-wave front set of two-point functions of quantum fields, which locally in the bulk amounts to the usual Hadamard condition, and which moreover allows to estimate wave front sets for the holographically induced theory on the boundary. We prove the existence of two-point functions satisfying this condition, and show their uniqueness modulo terms that have smooth Schwartz kernel in the bulk and have smooth restriction to the boundary. Finally, using Vasy's propagation of singularities theorem, we prove an analogue of Duistermaat and H\"ormander's theorem on distinguished parametrices.