Casimir effect on a finite lattice

Lattice quantum field theory is a well established branch of modern quantum field theory (QFT). However, it has only peripherally been used for the investigation of Casimir systems - i.e. for systems in which quantum fields are distorted by their interaction with classical background objects. This article presents a Hamiltonian lattice formulation of static Casimir systems at a level of generality appropriate for an introductory investigation. Background structure - represented by a lattice potential V(x) - is introduced along one spatial direction with translation invariance in all other spatial directions. It is simple to extend this formulation to include arbitrary background structure in more than one spatial direction. Following some general analysis two specific finite 1D lattice QFT systems are analyzed in detail. The first has three Dirichlet boundaries at the lattice sites x=0, 1 and L (L>l>0) with vanishing lattice potential V(x) everywhere in between. The vacuum energy and vacuum stress tensor Tμν for this system are calculated in 0l and beyond. The vacuum energy and Tμν are calculated for this far more complicated system in the region 0<x<L, again with very good results. The internal consistency of the lattice version of this system is carefully examined