Vacuum Polarization and Energy Conditions at a Planar Frequency Dependent Dielectric to Vacuum Interface

TL;DR

This paper investigates the vacuum stress-tensor at a dielectric-vacuum interface with frequency-dependent refractive index, revealing regularization effects, energy condition violations, and conditions under which energy densities are positive or violate weak energy conditions.

Contribution

It introduces a detailed analysis of vacuum stress-tensor regularization at a dielectric interface with frequency dependence, and explores energy condition violations and positivity in this context.

Findings

Vacuum stress-tensor components are regularized by reflection and transmission coefficients.

Divergence at a perfect mirror disappears for dielectric mirrors, replaced by a localized energy density.

Weak energy condition is violated in some regions, but the averaged weak energy condition holds for certain observers.

Abstract

The form of the vacuum stress-tensor for the 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 has the opposite sign. Thus the weak energy condition is always violated in some region of the spacetime. For the dielectric mirror, the mean vacuum energy density per unit plate area in a constant time hypersurface is always found to be positive (or zero) and the averaged weak energy condition is proven to hold for all observers with non-zero velocity along the normal direction to the boundary. Both results are found to be generic features of the vacuum stress-tensor and not necessarily dependent of the frequency dependence of the dielectric.