On the Casimir energy for a 2N-piece relativistic string

TL;DR

This paper calculates the Casimir energy for a relativistic, piecewise uniform closed string with 2N segments, introducing a recursive method for arbitrary N and analyzing its properties and temperature dependence.

Contribution

A novel recursive formula for calculating Casimir energy of a 2N-piece relativistic string, extending previous results to arbitrary N and including finite temperature effects.

Findings

Casimir energy is generally negative and increases in magnitude with N.

The recursive calculation method matches earlier results for small N.

Finite temperature effects are incorporated into the analysis.

Abstract

The Casimir energy for the transverse oscillations of a piecewise uniform closed string is calculated. The string consists of 2N pieces of equal length, of alternating type I and type II material, and is taken to be relativistic in the sense that the velocity of sound always equals the velocity of light. By means of a new recursion formula we manage to calculate the Casimir energy for arbitrary integers N. Agreement with results obtained in earlier works on the string is found in all special cases. As basic regularization method we use the contour integration method. As a check, agreement is found with results obtained from the \zeta function method (the Hurwitz function) in the case of low N (N = 1-4). The Casimir energy is generally negative, and the more so the larger is the value of N. We illustrate the results graphically in some cases. The generalization to finite temperature theory is also given.