Vacuum polarization with zero-range potentials on a hyperplane

TL;DR

This paper studies how quantum vacuum fluctuations of a scalar field are affected by zero-range potentials on a hyperplane, analyzing boundary effects and divergences in a Minkowski spacetime setting.

Contribution

It provides a detailed calculation of vacuum polarization with general boundary conditions and introduces a local zeta regularization approach for zero and non-zero mass fields.

Findings

Boundary divergences are softened for Dirac delta potentials.

Vacuum polarization behavior is characterized at small and large distances.

The method applies to both reflecting and semitransparent surfaces.

Abstract

The quantum vacuum fluctuations of a neutral scalar field induced by background zero-range potentials concentrated on a flat hyperplane of co-dimension $1$ in $(d+1)$-dimensional Minkowski spacetime are investigated. Perfectly reflecting and semitransparent surfaces are both taken into account, making reference to the most general local, homogeneous and isotropic boundary conditions compatible with the unitarity of the quantum field theory. The renormalized vacuum polarization is computed for both zero and non-zero mass of the field, implementing a local version of the zeta regularization technique. The asymptotic behaviours of the vacuum polarization for small and large distances from the hyperplane are determined to leading order. It is shown that boundary divergences are soften in the specific case of a pure Dirac delta potential.