Fundamental solutions and Hadamard states for a scalar field with arbitrary boundary conditions on an asymptotically AdS spacetimes

TL;DR

This paper constructs fundamental solutions and Hadamard states for a scalar field on asymptotically AdS spacetimes with arbitrary boundary conditions, extending quantum field theory techniques to non-globally hyperbolic settings.

Contribution

It develops a microlocal analysis framework for the Klein-Gordon operator with boundary conditions on asymptotically AdS spacetimes, proving existence of fundamental solutions and Hadamard states.

Findings

Existence of advanced and retarded fundamental solutions with specific microlocal properties.

Construction of Hadamard two-point distributions under certain boundary conditions.

Extension of deformation arguments to non-globally hyperbolic asymptotically AdS spacetimes.

Abstract

We consider the Klein-Gordon operator on an $n$-dimensional asymptotically anti-de Sitter spacetime $(M,g)$ together with arbitrary boundary conditions encoded by a self-adjoint pseudodifferential operator on $\partial M$ of order up to $2$. Using techniques from $b$-calculus and a propagation of singularities theorem, we prove that there exist advanced and retarded fundamental solutions, characterizing in addition their structural and microlocal properties. We apply this result to the problem of constructing Hadamard two-point distributions. These are bi-distributions which are weak bi-solutions of the underlying equations of motion with a prescribed form of their wavefront set and whose anti-symmetric part is proportional to the difference between the advanced and the retarded fundamental solutions. In particular, under a suitable restriction of the class of admissible boundary conditions and setting to zero the mass, we prove their existence extending to the case under scrutiny a deformation argument which is typically used on globally hyperbolic spacetimes with empty boundary.