Quantum Relativistic Gas in Arbitrary Spatial Dimensions

We consider the d-dimensional invariant phase space for a quantum ideal relativistic gas and derive the cluster decomposition for the grand canonical and canonical partition functions as well as for the microcanonical and grand microcanonical densities of states. The cluster decomposition for the latter density of states is rederived on the basis of the (anti-)symmetrization of the quantum states. As an application of the formalism for the ideal (without interaction) gas to the case of systems with interaction we choose the statistical bootstrap equation and study the effects of the dimensionality in such statistical bootstrap systems together with those of the quantum statistics.