Casimir force in discrete scalar fields I: 1D and 2D cases

TL;DR

This paper investigates the Casimir force for a massless scalar field using a Hamiltonian lattice approach, demonstrating that the force is independent of lattice type and highlighting symmetry properties at different frequencies.

Contribution

It introduces a Hamiltonian lattice method for calculating the Casimir force, successfully reproducing subtle effects and analyzing symmetry behavior at various frequencies.

Findings

The method reproduces the Casimir effect accurately for square and triangular lattices.

High-frequency waves lose rotational symmetry, with group velocity approaching zero.

Casimir force is shown to be independent of lattice type.

Abstract

We calculate the Casimir force between parallel plates for a massless scalar field. When adding the energy of normal modes, we avoid infinities by using a discrete spacetime lattice; however, this approach proves ineffective as long as both space and time are kept discrete. Yet, when time is treated as continuous while the scalar field forms a spatial periodic lattice, our method succeeds, and we refer to this approach as Hamiltonian lattice theory. The dispersion relation for both square and triangular lattices accurately reproduces the subtle Casimir effect, providing evidence that the Casimir force is independent of the type of lattice used. At low frequencies, both lattices exhibit a high level of rotational symmetry. However, at high frequencies, they lose this symmetry, even though the propagation of high-frequency waves becomes limited as their group velocity approaches zero.