Abstract
Continuing a program of examining the behavior of the vacuum expectation value of the stress tensor in a background which varies only in a single direction, we here study the electromagnetic stress tensor in a medium with permittivity depending on a single spatial coordinate, specifically, a planar dielectric half-space facing a vacuum region. There are divergences occurring that are regulated by temporal and spatial point-splitting, which have a universal character for both transverse electric and transverse magnetic modes. The nature of the divergences depends on the model of dispersion adopted. And there are singularities occurring at the edge between the dielectric and vacuum regions, which also have a universal character, depending on the structure of the discontinuities in the material properties there. Remarks are offered concerning renormalization of such models, and the significance of the stress tensor. The ambiguity in separating "bulk" and "scattering" parts of the stress tensor is discussed.
Highlights
Most studies of the Casimir effect deal with quantum fluctuation forces between rigid bodies separated by vacuum
III we show how the Green’s dyadic for this problem breaks up into transverse electric (TE) and transverse magnetic (TM) parts
In this paper we have extended our previous calculations on the soft wall problem to the electromagnetic case
Summary
Most studies of the Casimir effect deal with quantum fluctuation forces between rigid bodies separated by vacuum. There we show, using the uniform (Debye) asymptotic expansions for the modified Bessel functions, that there are two types of singularities in the normal-normal component of the stress tensor occurring at the edge between the vacuum and dielectric region: a cubic singularity if there is a discontinuity in the permittivity, and a quadratic one For the plasma model, the divergences arising from the bulk term in the Green’s function coincide with those found for the scalar situation for both TE and TM modes, and the edge singularity for the TE mode for the energy density coincides with that found for the canonical scalar energy density in Ref.
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