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

Abstract

We calculate the Casimir force between parallel plates for a discrete massless scalar field. The scalar field forms a periodic lattice in continuous spacetime. The dispersion relation for both the square and triangular lattices allows for the accurate reproduction of the subtle Casimir effect without encountering infinite quantities. Our findings demonstrate that the Casimir force is independent of the type of periodic lattice used. At low frequencies, we observe a high level of rotational symmetry in both lattices. However, at high frequencies, both lattices lose their rotational symmetry, although the propagation of high-frequency waves becomes significantly limited as their group velocity approaches zero. We claim that the behavior of physics in discrete fields becomes similar to that in the continuous case, with the exception that we now have a natural and smooth cutoff mechanism that proves useful in cases requiring regularization. It appears that we have found an alternative approach to regularization using lattices with different symmetries in the background of continuous spacetime.Comment: 10 pages, 9 figure