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 behaviors of the vacuum polarization for small and large distances from the hyperplane are determined to leading order. It is shown that boundary divergences are softened in the specific case of a pure Dirac delta potential.

中文翻译：

---

超平面上具有零范围电势的真空极化

由背景零范围电势引起的中性标量场的量子真空涨落集中在余维1的平面超平面上 $（d+\mathrm{1个}）-$维Minkowski时空。完全反射和半透明的表面都被考虑在内，并参考了与量子场论的单一性相容的最一般的局部，均匀和各向同性的边界条件。针对零质量和非零质量的场计算重新归一化的真空极化，从而实现zeta正则化技术的局部版本。从超平面到远处的距离很小和很大时，真空极化的渐近行为被确定为超前阶。结果表明，在纯狄拉克δ电位的特定情况下，边界发散被软化了。