Vacuum fluctuations of a scalar field in a rectangular waveguide

An analysis of one-loop vacuum fluctuations associated with a scalar field confined in the interior of a infinite waveguide of rectangular cross section is presented. We first consider the massless scalar field defined in a four-dimensional Euclidean space. To identify the divergences of the vacuum fluctuations we use a combination of dimensional and zeta function analytic regularization procedures. The divergences which occur in the one-loop vacuum fluctuations fall into two distinct classes: local ones that are renormalized by the introduction of the usual bulk counterterms, and surface and edge divergences that demand counterterms concentrated on the boundaries. We present the detailed form of the surface and edge divergences. Finally we discuss how to generalize our calculations for a confined massive scalar field defined in a higher dimensional Euclidean space.