Renormalisation in maximally symmetric spaces and semiclassical gravity in Anti-de Sitter spacetime

TL;DR

This paper develops methods for renormalising quantum fields in maximally symmetric spacetimes and applies them to find semiclassical gravity solutions in Anti-de Sitter space with various boundary conditions.

Contribution

It introduces a simplified Hadamard renormalisation approach in maximally symmetric spaces and constructs semiclassical solutions in AdS with different boundary conditions.

Findings

Explicit semiclassical solutions in PAdS4 with various boundary conditions.

Simplified Hadamard renormalisation leveraging spacetime symmetries.

Graphical results for specific mass and curvature coupling values.

Abstract

We obtain semiclassical gravity solutions in the Poincar\'e fundamental domain of $(3+1)$-dimensional Anti-de Sitter spacetime, PAdS$_4$, with a (massive or massless) Klein-Gordon field (with possibly non-trivial curvature coupling) with Dirichlet or Neumann boundary. Some results are explicitly and graphically presented for special values of the mass and curvature coupling (e.g. minimal or conformal coupling). In order to achieve this, we study in some generality how to perform the Hadamard renormalisation procedure for non-linear observables in maximally symmetric spacetimes in arbitrary dimensions, with emphasis on the stress-energy tensor. We show that, in this maximally symmetric setting, the Hadamard bi-distribution is invariant under the isometries of the spacetime, and can be seen as a `single-argument' distribution depending only on the geodesic distance, which significantly simplifies the Hadamard recursion relations and renormalisation computations.