Mode-by-mode summation for the zero point electromagnetic energy of an infinite cylinder

TL;DR

This paper calculates the zero point electromagnetic energy of an infinite cylinder using mode-by-mode summation, demonstrating divergence cancellation under specific conditions and confirming results with previous methods.

Contribution

It introduces a mode-by-mode summation approach for cylindrical geometries and verifies divergence cancellation and energy calculations with established techniques.

Findings

Casimir energy vanishes for a dilute dielectric cylinder.

Casimir energy matches previous results for a perfect conductor.

Divergences are successfully regulated using zeta function techniques.

Abstract

Using the mode-by-mode summation technique the zero point energy of the electromagnetic field is calculated for the boundary conditions given on the surface of an infinite solid cylinder. It is assumed that the dielectric and magnetic characteristics of the material which makes up the cylinder $(\epsilon_1, \mu_1)$ and of that which makes up the surroundings $(\epsilon_2, \mu_2)$ obey the relation $\epsilon_1\mu_1= \epsilon_2\mu_2$. With this assumption all the divergences cancel. The divergences are regulated by making use of zeta function techniques. Numerical calculations are carried out for a dilute dielectric cylinder and for a perfectly conducting cylindrical shell. The Casimir energy in the first case vanishes, and in the second is in complete agreement with that obtained by DeRaad and Milton who employed a Green's function technique with an ultraviolet regulator.