Casimir Effect in $E^3$ closed spaces

TL;DR

This paper calculates the Casimir energy density for a massless scalar field in all compact orientable Euclidean 3-spaces, revealing independence from curvature coupling and providing numerical visualizations.

Contribution

It derives finite Epstein zeta function sums for Casimir energy in all compact orientable Euclidean 3-spaces, expanding understanding of topology's role in quantum field effects.

Findings

Casimir energy density is independent of curvature coupling.

Explicit finite sums for all compact orientable Euclidean 3-spaces.

Numerical plots illustrate energy distribution within Dirichlet regions.

Abstract

As it is well known the topology of space is not totally determined by Einstein's equations. It is considered a massless scalar quantum field in a static Euclidean space of dimension 3. The expectation value for the energy density in all compact orientable Euclidean 3-spaces are obtained in this work as a finite summation of Epstein type zeta functions. The Casimir energy density for these particular manifolds is independent of the type of coupling with curvature. A numerical plot of the result inside each Dirichlet region is obtained.