Exact results for the Casimir force of a three-dimensional model of relativistic Bose gas in a film geometry

It has recently recently been suggested that a relativistic Bose gas of some type may play a role in issues such as dark matter, dark energy, and in some other cosmological problems. In this article, we investigate one known exactly solvable model of a three-dimensional statistical-mechanical model of relativistic Bose gas which takes into account the existence of both particles and antiparticles. We derive exact expressions for the behavior of the Casimir force for a system subjected to film geometry under periodic boundary conditions. We show that the Casimir force between the plates is attractive, monotonic as a function of the temperature scaling variable, and has a scaling function that, at low temperatures, approaches a universal negative constant equal to the corresponding one for two-component three-dimensional Gaussian system. The force decays with the distance in a power-law near and below the bulk critical temperature T _c of the Bose condensate, and exponentially above T _c. We obtain a closed-form exact expression for the Casimir amplitude ${\Delta }_{\text{Cas}}^{\text{RBG}}=-4\zeta \left(3\right)/\left(5\pi \right)$. We establish the precise correspondence of the scaling function of the free energy of the model with the scaling functions of two other well-known models of statistical mechanics: the spherical model, and the imperfect Bose gas model. (paper)