Hadamard states for a scalar field in anti-de Sitter spacetime with arbitrary boundary conditions

TL;DR

This paper classifies boundary conditions for a scalar field in anti-de Sitter spacetime, constructs corresponding two-point functions, and demonstrates their Hadamard property, advancing the understanding of quantum fields in AdS with arbitrary boundary conditions.

Contribution

It provides a complete classification of boundary conditions, explicit two-point functions, and a new Hadamard state definition for scalar fields in PAdS spacetime.

Findings

Explicit form of two-point functions satisfying boundary conditions

Identification of boundary conditions allowing bound states

Verification of Hadamard property in PAdS regions

Abstract

We consider a real, massive scalar field on ${\rm PAdS}_{d+1}$, the Poincar\'e domain of the $(d+1)$-dimensional anti-de Sitter (AdS) spacetime. We first determine all admissible boundary conditions that can be applied on the conformal boundary, noting that there exist instances where "bound states" solutions are present. Then, we address the problem of constructing the two-point function for the ground state satisfying those boundary conditions, finding ultimately an explicit closed form. In addition, we investigate the singularities of the resulting two-point functions, showing that they are consistent with the requirement of being of Hadamard form in every globally hyperbolic subregion of ${\rm PAdS}_{d+1}$ and proposing a new definition of Hadamard states which applies to ${\rm PAdS}_{d+1}$.