Vacuum polarization on three-dimensional anti-de Sitter space-time with Robin boundary conditions

TL;DR

This paper analyzes the vacuum polarization of a quantum scalar field in three-dimensional anti-de Sitter space with various boundary conditions, revealing how Robin conditions affect the expectation values and their dependence on space-time location.

Contribution

It provides a detailed study of vacuum and thermal expectation values of scalar fields in AdS3 with Robin boundary conditions, highlighting their dependence on boundary parameters and space-time location.

Findings

Finite vacuum polarization for Robin boundary conditions below a critical parameter

Constant expectation values for Neumann and Dirichlet boundary conditions

Robin boundary conditions lead to location-dependent expectation values

Abstract

We study a quantum scalar field, with general mass and coupling to the scalar curvature, propagating on three-dimensional global anti-de Sitter space-time. We determine the vacuum and thermal expectation values of the square of the field, also known as the vacuum polarisation (VP). We consider values of the scalar field mass and coupling for which there is a choice of boundary conditions giving well-posed classical dynamics. We apply Dirichlet, Neumann and Robin (mixed) boundary conditions to the field at the space-time boundary. We find finite values of the VP when the parameter governing the Robin boundary conditions is below a certain critical value. For all couplings, the vacuum expectation values of the VP with either Neumann or Dirichlet boundary conditions are constant and respect the maximal symmetry of the background space-time. However, this is not the case for Robin boundary conditions, when both the vacuum and thermal expectation values depend on the space-time location. At the space-time boundary, we find that both the vacuum and thermal expectation values of the VP with Robin boundary conditions converge to the result when Neumann boundary conditions are applied, except in the case of Dirichlet boundary conditions.