Vacuum Polarization with Zero-Range Potentials on a Hyperplane

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 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.

Zero-Range Potentials: Delta-Like Barrier Versus Self-Adjoint Extension

“…Encoding of the boundary conditions by a δ-like potential made it possible to investigate the Casimir interaction of two parallel perfectly conducting plates [3][4][5][6][7][8][9], two point or linear sources with a zero-range potential [10][11][12][13], and so on.…”

“…In recent literature, various singular configurations have been studied, including point-like interactions [28], delta potentials at finite temperature [29], Casimir interactions of manifolds with different codimensions [30], Casimir effect with a boundary hyperplane [32,33], and Casimir problems approached using Zeta-regularization [11,12].…”

Vacuum Polarization in a Zero-Width Potential: Self-Adjoint Extension

“…Models of this type have been widely discussed in physical and mathematical literature in connection with problems of solid-state physics (Kronig-Penney model [1]), atomic, and nuclear physics (description of short-range nuclear forces [2]). Encoding of the boundary conditions by a δ-like potential made it possible to investigate the Casimir interaction of two parallel perfectly conducting plates [3][4][5][6][7][8][9], of two pointlike or linear sources with a zero-range potential [10][11][12][13]. Similar situations arise in consideration of field-theoretical effects near topological defects.…”

“…Similar singular configurations were previously analyzed in the recent literature. Among the notable are the effective action of pointlike interactions [25], the delta-potentials at finite temperature [26], Casimir interactions of manifolds with diverse codimensionalities [27], the Casimir effect with a boundary hyperplane [28], zeta regularization approach to various Casimir problems [10][11][12]29], and so forth.…”