Boundary conditions and infrared divergences

TL;DR

This paper examines how boundary conditions affect the existence of ground states for scalar fields on static spacetimes, revealing that infrared divergences depend on the type of Robin boundary conditions applied.

Contribution

It demonstrates that infrared divergences in scalar field ground states are influenced by boundary conditions, providing specific examples on Bertotti-Robinson spacetime.

Findings

Infrared divergences occur only with Neumann boundary conditions in one coordinate patch.

Infrared divergences are absent with Dirichlet boundary conditions in another patch.

The existence of suitable bi-distributions depends on boundary conditions and spacetime geometry.

Abstract

We review the procedure to construct quasi-free ground states, for real scalar fields whose dynamics is dictated by the Klein-Gordon equation, on standard static Lorentzian manifolds with a time-like boundary. We observe that, depending on the assigned boundary condition of Robin type, this procedure does not always lead to the existence of a suitable bi-distribution $w_2\in \mathcal{D}'(M\times M)$ due to the presence of infrared divergences. As a concrete example we consider a Bertotti-Robinson spacetime in two different coordinate patches. In one case we show that infrared divergences do not occur only for Dirichlet boundary conditions as one might expect a priori, while, in the other case, we prove that they occur only when Neumann boundary conditions are imposed at the time-like boundary.