Hard and soft walls

TL;DR

This paper investigates the local energy densities near different types of walls confining quantum scalar fields, analyzing divergences and their regulation depending on the wall's potential profile.

Contribution

It provides a detailed analysis of divergences in energy densities near walls with various potentials, including new insights into how potential shape affects divergences.

Findings

Infinite Dirichlet wall shows known divergences regulated by point-splitting.

Softened divergences occur with linear potential walls.

No surface divergences for walls with potential $z^eta$ when $eta>2$.

Abstract

In a continuing effort to understand divergences which occur when quantum fields are confined by bounding surfaces, we investigate local energy densities (and the local energy-momentum tensor) in the vicinity of a wall. In this paper, attention is largely confined to a scalar field. If the wall is an infinite Dirichlet plane, well known volume and surface divergences are found, which are regulated by a temporal point-splitting parameter. If the wall is represented by a linear potential in one coordinate $z$, the divergences are softened. The case of a general wall, described by a potential of the form $z^\alpha$ for $z>0$ is considered. If $\alpha>2$, there are no surface divergences, which in any case vanish if the conformal stress tensor is employed. Divergences within the wall are also considered.