Casimir forces in a quantum exactly solvable model with long - range interaction

A d–dimensional quantum model system confined to a general hypercubical geometry with linear spatial size L and " temporal size " 1/T (T-temperature of the system) is considered in the spherical approximation under periodic boundary conditions. For a film geometry in different space dimensions 1 2 σ < d < 3 2 σ , where 0 < σ ≤ 2 is a parameter controlling the decay of the long–range interaction, an analysis of the free energy and the Casimir forces is given. We have proven that, if d = σ the Casimir amplitude of the model, characterizing the leading temperature corrections to its ground state is ∆ = −16ζ(3)/[5σ(4π) σ/2 Γ(σ/2)]. The last implies that the universal constant˜c = 4/5 of the model remains the same for both the short, as well as long–range interactions if one takes the normalization factor for the Gaussian model to be such that˜c = 1 for it. This is a generalization of the well known result due to Sachdev for the case of long–range interaction.