The microcanonical ensemble of the ideal relativistic quantum gas

We derive the microcanonical partition function of the ideal relativistic quantum gas of spinless bosons in a quantum field framework as an expansion over fixed multiplicities. Our calculation generalizes well known expressions in literature in that it does not introduce any large volume approximation and it is valid at any volume. We discuss the issues concerned with the definition of the microcanonical ensemble for a free quantum field at volumes comparable with the Compton wavelength and provide a consistent prescription of calculating the microcanonical partition function, which is finite at finite volume and yielding the correct thermodynamic limit. Besides an immaterial overall factor, the obtained expression turns out to be the same as in the non-relativistic multi-particle approach. This work is introductory to derive the most general expression of the microcanonical partition function fixing the maximal set of observables of the Poincare' group.