Irreducible Scalar Many-Body Casimir Energies: Theorems and Numerical Studies

TL;DR

This paper introduces the concept of irreducible N-body spectral functions and Casimir energies for scalar fields, providing theoretical theorems and numerical studies that reveal their properties, finiteness, and dependence on geometry.

Contribution

It defines irreducible N-body spectral functions and Casimir energies, proving their properties and computing them for specific geometries using analytical and numerical methods.

Findings

Irreducible N-body spectral functions are conditional probabilities of random walks.

Irreducible Casimir energies are finite and depend on the number of objects.

Force between objects separated by a plane is always attractive.

Abstract

We define irreducible N-body spectral functions and Casimir energies and consider a massless scalar quantum field interacting locally by positive potentials with classical objects. Irreducible N-body spectral functions in this case are shown to be conditional probabilities of random walks. The corresponding irreducible contributions to scalar many-body Casimir energies are finite and positive/negative for an odd/even number of objects. The force between any two finite objects separable by a plane is always attractive in this case. Analytical and numerical world-line results for the irreducible four-body Casimir energy of a scalar with Dirichlet boundary conditions on a tic-tac-toe pattern of lines are presented. Numerical results for the irreducible three-body Casimir energy of a massless scalar satisfying Dirichlet boundary conditions on three intersecting lines forming an isosceles triangle are also reported. In both cases the symmetric configuration (square and isosceles triangle) corresponds to the minimal irreducible contribution to the Casimir energy.