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

Recently it has been suggested that relativistic Bose gas of some type can be playing role in issues like dark matter, dark energy, and in some cosmological problems. In the current article we investigate one known exactly solvable model of three-dimensional statistical-mechanical model of relativistic Bose gas that takes into account the existence of both particles and antiparticles. We derive exact expressions for the behavior of the Casimir force for the 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, with a scaling function that approaches at low temperatures 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 closed form exact expression for the Casimir amplitude $\Delta_{\rm Cas}^{\rm RBG} =-4\zeta(3)/(5\pi)$. 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.