Nonsmooth backgrounds in quantum field theory

TL;DR

This paper investigates how discontinuities in background fields affect heat kernel coefficients and Casimir energy in quantum field theory, highlighting the importance of singular potentials and counterterms for consistent renormalization.

Contribution

It introduces a method to calculate the effects of field discontinuities on heat kernel coefficients and clarifies the role of surface counterterms in renormalization with singular potentials.

Findings

Discontinuities influence heat kernel coefficients and Casimir energy.

Sign of contributions depends on the order of derivatives' discontinuity.

Surface tension counterterms are necessary for proper renormalization.

Abstract

The one-loop renormalization in field theories can be formulated in terms of the heat kernel expansion. In this paper we calculate leading contributions of discontinuities of background fields and their derivatives to the heat kernel coefficients. These results are then used to estimate contributions of the discontinuities to the Casimir energy. Sign of such contribution is defined solely by the order of discontinuous derivative. We also discuss renormalization in the presence of singular (delta-function) potentials. We show that an independent surface tension counterterm is necessary. This observation seems to resolve some contradictions in previous calculations.