Semi-transparent Boundary Conditions in the Worldline Formalism

TL;DR

This paper extends the worldline formalism to handle quantum fields with semi-transparent boundary conditions modeled by Dirac delta functions, deriving heat-kernel asymptotics and Casimir forces for such configurations.

Contribution

It introduces a novel application of the worldline formalism to semi-transparent boundary conditions and computes related heat-kernel expansions and Casimir interactions.

Findings

Derived heat-kernel asymptotics with semi-transparent boundaries.

Calculated Casimir attraction between surfaces with delta-function boundary conditions.

Extended the applicability of the worldline formalism to new boundary interaction models.

Abstract

The interaction of a quantum field with a background containing a Dirac delta function with support on a surface of codimension 1 represents a particular kind of matching conditions on that surface for the field. In this article we show that the worldline formalism can be applied to this model. We obtain the asymptotic expansion of the heat-kernel corresponding to a scalar field on $\mathbb{R}^{d+1}$ in the presence of an arbitrary regular potential and subject to this kind of matching conditions on a flat surface. We also consider two such surfaces and compute their Casimir attraction due to the vacuum fluctuations of a massive scalar field weakly coupled to the corresponding Dirac deltas.