Generalized Neumann boundary condition for the scalar field

TL;DR

This paper generalizes the Neumann boundary condition for scalar fields by introducing a hyperplanar delta-like potential, analyzing its effects on the propagator and vacuum stability, and revealing conditions for pair creation akin to the Schwinger effect.

Contribution

It introduces a new model with a delta-like potential that generalizes Neumann boundary conditions and provides exact calculations of the propagator and interaction energy.

Findings

Modified Feynman propagator due to the potential

Exact expression for the interaction energy

Conditions for vacuum instability and pair creation

Abstract

In this paper, we explore the Klein-Gordon field theory in $(D+1)$ dimensions in the presence of a $(D-1)$-dimensional hyperplanar $\delta$-like potential that couples quadratically to the field derivatives. This model effectively generalizes the Neumann boundary condition for the scalar field on the plane, as it reduces to this condition in an appropriate limit of the coupling parameter. Specifically, we calculate the modifications to the Feynman propagator induced by the planar potential and analyze the interaction energy between a stationary point-like source and the potential, obtaining a general and exact expression. We demonstrate that, under certain conditions relating the field mass and the coupling constant to the external potential, the vacuum state becomes unstable, giving rise to a pair-creation phenomenon that resembles the Schwinger effect in quantum electrodynamics.