Path-integral approach to Casimir effect with infinitely thin plates

TL;DR

This work analyzes how the BRW path-integral method imposes boundary conditions for Casimir-type problems using a massless scalar model, contrasting Dirichlet and Neumann cases and examining their compatibility with gauge-field duality. It demonstrates that Dirichlet BCs map cleanly to BRW, reproducing the standard Casimir pressure $P_{ ext{Cas}} = -\frac{\pi^2}{480 L^4}$, while Neumann BCs introduce divergences that the BRW framework does not naturally regularize. By testing cut-off, thick-plate, and Graham-style non-dynamical-background regulators, the paper shows that thick plates and Robin-type interpolations can regularize the problematic integrals and recover Dirichlet results in appropriate limits, whereas thin Neumann setups generally yield vanishing or regulator-sensitive forces. The results underscore that BRW-type boundary impositions for non-Dirichlet conditions require careful, duality-aware regularization, with implications for Casimir studies in settings like the MIT bag model and related QCD contexts.

Abstract

When studying the Casimir effect in a quantum field theory setting, one can impose the boundary conditions by adding appropriate Dirac-delta functions to the path integral. In this paper, the limits of this approach are explored under different boundary conditions.