On the definition of dielectric permittivity for media with temporal dispersion in the presence of free charge carriers

TL;DR

This paper investigates the challenges in defining frequency-dependent dielectric permittivity in media with free charge carriers, proposing a regularization method and highlighting differences from insulators, with implications for Casimir force theory.

Contribution

It introduces a regularization approach for defining dielectric permittivity in media with free charge carriers and compares it to insulators, clarifying conceptual limitations.

Findings

Regularization is necessary for dielectric permittivity with free carriers.

Standard Fourier integral representation fails for media with free carriers.

Differences in permittivity definitions impact Casimir force calculations.

Abstract

We show that in the presence of free charge carriers the definition of the frequency-dependent dielectric permittivity requires additional regularization. As an example, the dielectric permittivity of the Drude model is considered and its time-dependent counterpart is derived and analyzed. The respective electric displacement cannot be represented in terms of the standard Fourier integral. The regularization procedure allowing to circumvent these difficulties is suggested. For purpose of comparison it is shown that the frequency-dependent dielectric permittivity of insulators satisfies all rigorous mathematical criteria. This permits us to conclude that in the presence of free charge carriers the concept of dielectric permittivity is not as well defined as for insulators and we make a link to widely discussed puzzles in the theory of thermal Casimir force which might be caused by the use of such kind permittivities.