Vacuum polarization and energy conditions at a planar frequency-dependent dielectric-to-vacuum interface

Abstract

The form of the vacuum stress tensor for a quantized scalar field at a dielectric-to-vacuum interface is studied. The dielectric is modeled to have an index of refraction that varies with frequency. We find that the stress-tensor components, derived from the mode function expansion of the Wightman function, are naturally regularized by the reflection and transmission coefficients of the mode at the boundary. Additionally, the divergence of the vacuum energy associated with a perfectly reflecting mirror is found to disappear for the dielectric mirror at the expense of introducing a new energy density near the surface which usually has the opposite sign. Thus the weak energy condition is always violated in some region of spacetime. For a two-dimensional dielectric mirror, the mean vacuum energy density in a constant time hypersurface is always found to be zero and the averaged weak energy condition is proved to hold for all observers with non-zero velocity. Both results are found to be generic features of the vacuum stress tensor and generally do not depend on the frequency behavior of the dielectric or the value of the curvature coupling constant, $\ensuremath{\xi},$ in the stress tensor. In four dimensions, it is shown that the mean vacuum energy density per unit plate area will be positive and the averaged weak energy condition will hold for all observers with non-zero velocity along the normal direction to the interface with the coupling constant $\ensuremath{\xi}<~1/4.$ Again, both results are generic features of the vacuum stress tensor.