Finite Casimir Energies in Renormalizable Quantum Field Theory

TL;DR

This paper investigates the finiteness of Casimir energies in various geometries within renormalizable quantum field theory, clarifying when divergences occur and challenging recent claims of universal divergences.

Contribution

It demonstrates that Casimir energies are finite in certain dimensions and geometries, and refutes recent assertions that all such energies are divergent, providing detailed calculations and analysis.

Findings

Casimir energy is finite in 2D for a circular boundary

For non-even dimensions, massless fields' Casimir energy is finite

Divergences occur mainly at third order, but are often irrelevant to physical shells

Abstract

Quantum vacuum energy has been known to have observable consequences since 1948 when Casimir calculated the force of attraction between parallel uncharged plates, a phenomenon confirmed experimentally with ever increasing precision. Casimir himself suggested that a similar attractive self-stress existed for a conducting spherical shell, but Boyer obtained a repulsive stress. Other geometries and higher dimensions have been considered over the years. Local effects, and divergences associated with surfaces and edges have been investigated by several authors. Quite recently, Graham et al. have re-examined such calculations, using conventional techniques of perturbative quantum field theory to remove divergences, and have suggested that previous self-stress results may be suspect. Here we show that most of the examples considered in their work are misleading; in particular, it is well-known that in two dimensions a circular boundary has a divergence in the Casimir energy for massless fields, while for general dimension $D$ not equal to an even integer the corresponding Casimir energy arising from massless fields interior and exterior to a hyperspherical shell is finite. It has also long been recognized that the Casimir energy for massive fields is divergent for curved boundaries. These conclusions are reinforced by a calculation of the relevant leading Feynman diagram in $D$ dimensions. Divergences do occur in third order, as has been recognized for many years, but this logarithmic divergence is of questionable relevance to real shells.