Many-Body Contributions to Green's Functions and Casimir Energies

TL;DR

This paper develops a formalism to compute irreducible N-body Green's functions and Casimir energies, providing explicit calculations for three-body interactions involving scalar fields and various geometries.

Contribution

It introduces a method to extract irreducible N-body contributions to Green's functions and Casimir energies, including explicit solutions for three-body cases with complex geometries.

Findings

Irreducible three-body Casimir energy is finite and positive.

The formalism applies to multiple geometries including plates and wedges.

Green's functions decouple in certain boundary conditions.

Abstract

The multiple scattering formalism is used to extract irreducible N-body parts of Green's functions and Casimir energies describing the interaction of N objects that are not necessarily mutually disjoint. The irreducible N-body scattering matrix is expressed in terms of single-body transition matrices. The irreducible N-body Casimir energy is the trace of the corresponding irreducible N-body part of the Green's function. This formalism requires the solution of a set of linear integral equations. The irreducible three-body Green's function and the corresponding Casimir energy of a massless scalar field interacting with potentials are obtained and evaluated for three parallel semitransparent plates. When Dirichlet boundary conditions are imposed on a plate the Green's function and Casimir energy decouple into contributions from two disjoint regions. We also consider weakly interacting triangular--and parabolic-wedges placed atop a Dirichlet plate. The irreducible three-body Casimir energy of a triangular--and parabolic-wedge is minimal when the shorter side of the wedge is perpendicular to the Dirichlet plate. The irreducible three-body contribution to the vacuum energy is finite and positive in all the cases studied.