Hard and soft walls

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 $z^α$ for $z>0$ is considered. If $α>2$, there are no surface divergences, which in any case vanish if the conformal stress tensor is employed. Divergences within the wall are also considered.

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