On the Casimir Energy of Frequency Dependent Interactions

TL;DR

This paper investigates how frequency dependence in interactions affects the calculation of Casimir energies, highlighting modifications to scattering data relationships and exploring examples including dielectric materials and effective theories.

Contribution

It provides a detailed analysis of vacuum polarization energies with frequency-dependent interactions and proposes methods to account for dissipation and alternative energy formulations.

Findings

Frequency dependence modifies the scattering-Green's function relationship.

Dissipation requires a statistical mechanics approach for Casimir energy.

Alternative field theory formulations can handle non-dissipative frequency-dependent cases.

Abstract

Vacuum polarization (or Casimir) energies can be straightforwardly computed from scattering data for static field configurations whose interactions with the fluctuating field are frequency independent. In effective theories, however,such interactions are typically frequency dependent. As a consequence, the relationship between scattering data and the Green's function is modified, which may or may not induce additional contributions to the vacuum polarization energy. We discuss several examples that naturally include frequency dependent interactions: (i) scalar electrodynamics with a static background potential, (ii) an effective theory that emerges from integrating out a heavy degree of freedom, and (iii) quantum electrodynamics coupled to a frequency dependent dielectric material. In the latter case, we argue that introducing dissipation as required by the Kramers-Kronig relations requires the consideration of the Casimir energy within a statistical mechanics formalism, while in the absence of dissipation we can work entirely within field theory, using an alternative formulation of the energy density.