When the Casimir energy is not a sum of zero-point energies

TL;DR

This paper calculates the first radiative correction to the Casimir force in a scalar field theory, revealing limitations of the zero-point energy sum approach and providing a refined perturbative method.

Contribution

It introduces a corrected approach to compute Casimir energy beyond zero-point sums and extends the calculation to next-to-leading order in a scalar field model.

Findings

Heuristic zero-point energy sum yields incorrect results

Amended method avoids infrared singularities in massless limit

Next-to-leading order correction proportional to λ^{3/2}

Abstract

We compute the leading radiative correction to the Casimir force between two parallel plates in the $\lambda\Phi^4$ theory. Dirichlet and periodic boundary conditions are considered. A heuristic approach, in which the Casimir energy is computed as the sum of one-loop corrected zero-point energies, is shown to yield incorrect results, but we show how to amend it. The technique is then used in the case of periodic boundary conditions to construct a perturbative expansion which is free of infrared singularities in the massless limit. In this case we also compute the next-to-leading order radiative correction, which turns out to be proportional to $\lambda^{3/2}$.