Zeta function regularization technique in the electrostatics context for discrete charge distributions

TL;DR

This paper applies zeta function regularization to compute electrostatic potentials for infinite discrete charge distributions, effectively handling infinities and revealing asymptotic behaviors similar to continuous cases.

Contribution

It introduces a novel application of zeta function regularization in electrostatics for discrete charges, bridging spectral functions with classical electrostatics.

Findings

Successfully regularized infinite charge distributions

Demonstrated asymptotic behavior similar to continuous distributions

Provided educational insights for spectral function applications

Abstract

Spectral functions, such as the zeta functions, are widely used in Quantum Field Theory to calculate physical quantities. In this work, we compute the electrostatic potential and field due to an infinite discrete distribution of point charges, using the zeta function regularization technique. This method allows us to remove the infinities that appear in the resulting expression. We found that the asymptotic behavior dependence of the potential and field is similar to the cases of continuous charge distribution. Finally, this exercise can be useful for graduate students to explore spectral and special functions.