Neumann Casimir effect: a singular boundary-interaction approach

TL;DR

This paper introduces a nonlocal boundary-interaction approach to impose Neumann conditions in Casimir effect calculations, overcoming UV divergences by incorporating a minimal length scale, and extends the method to arbitrary surfaces.

Contribution

It proposes a novel regularization technique using nonlocal terms with a minimal length scale to define Neumann boundary conditions for Casimir energy computations.

Findings

Defined meaningful reflection coefficients for nonlocal boundary conditions

Calculated Casimir energies for flat parallel mirrors using the new approach

Extended the method to arbitrary surface geometries

Abstract

Dirichlet boundary conditions on a surface can be imposed on a scalar field, by coupling it quadratically to a $\delta$-like potential, the strength of which tends to infinity. Neumann conditions, on the other hand, require the introduction of an even more singular term, which renders the reflection and transmission coefficients ill-defined because of UV divergences. We present a possible procedure to tame those divergences, by introducing a minimum length scale, related to the non-zero `width' of a {\em nonlocal} term. We then use this setup to reach (either exact or imperfect) Neumann conditions, by taking the appropriate limits. After defining meaningful reflection coefficients, we calculate the Casimir energies for flat parallel mirrors, presenting also the extension of the procedure to the case of arbitrary surfaces. Finally, we discuss briefly how to generalize the worldline approach to the nonlocal case, what is potentially useful in order to compute Casimir energies in theories containing nonlocal potentials; in particular, those which we use to reproduce Neumann boundary conditions.