Thermodynamics of an Ultrarelativistic Ideal Bose Gas

We discuss the properties of an ideal relativistic Bose gas with nonzero chemical potential $\mu$ (i.e., with net charge) but differ from all previous discussions by including the effects of antiparticles. We obtain for the first time the relevant high-temperature expansion, emphasizing the constraint $\left|\mu \right|<~m$, and show that a box of massless particles can have a net charge even though $\mu =0\mathrm{}$. Finally, we discuss the properties of Bose-Einstein condensation for both massless and massive bosons in $d$ space dimensions.