An exact evaluation of the Casimir energy in two planar models

TL;DR

This paper precisely calculates the Casimir energy for two planar models using the method of images, revealing negative energy in a non-relativistic case and non-negative energy in a relativistic scalar field theory.

Contribution

It provides an exact evaluation of Casimir energy in two planar models with Dirichlet boundary conditions, including explicit expressions involving special functions.

Findings

Non-relativistic Landau problem yields negative Casimir energy.

Relativistic scalar field model results in non-negative Casimir energy.

Explicit formulas involve Lerch transcendent and polylogarithm functions.

Abstract

The method of images is used to calculate the Casimir energy in Euclidean space with Dirichlet boundary conditions for two planar models, namely: i. the non-relativistic Landau problem for a charged particle of mass m for which - irrespective of the sign of the charge - the energy is negative, and ii. the model of a real, massive, noninteracting relativistic scalar field theory in 2 + 1 dimensions, for which the Casimir energy density is non-negative and is expressed in terms of the Lerch transcendent xxx and the polylogarithm xxx with 0 < xxx < 1 and n = 2, 3.