Calculating Casimir Energies in Renormalizable Quantum Field Theory

TL;DR

This paper defends the validity of conventional Casimir energy calculations in quantum field theory, clarifying misconceptions about divergences and demonstrating that these calculations are reliable for various geometries and dimensions.

Contribution

The paper clarifies misconceptions about divergences in Casimir energy calculations and confirms the validity of conventional methods across different geometries and dimensions.

Findings

Casimir energy is finite for massless fields in non-even dimensions.

Divergences occur for massive fields when dimension D ≠ 1.

Conventional perturbative techniques reliably compute Casimir energies.

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 studied 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 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 $D\ne1$. These conclusions are reinforced by a calculation of the relevant leading Feynman diagram in $D$ and three dimensions. There is therefore no doubt of the validity of the conventional finite Casimir calculations.