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

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.