Consider an ideal relativistic Bose gas composed of \(N_{1}\) particles and \(N_{2}\) antiparticles, each of rest mass \(m_{0}\), with occupation numbers

\[
\frac{1}{\exp \left[\beta\left(\varepsilon-\mu_{1}\right)\right]-1} \quad \text { and } \quad \frac{1}{\exp \left[\beta\left(\varepsilon-\mu_{2}\right)\right]-1}
\]

respectively, and the energy spectrum \(\varepsilon=c \sqrt{ }\left(p^{2}+m_{0}^{2} c^{2}\right)\). Since particles and antiparticles are supposed to be created in pairs, the system is constrained by the conservation of the number \(Q\left(=N_{1}-N_{2}\right)\), rather than of \(N_{1}\) and \(N_{2}\) separately; accordingly, \(\mu_{1}=-\mu_{2}=\mu\), say.

Set up the thermodynamic functions of this system in three dimensions and examine the onset of Bose-Einstein condensation as \(T\) approaches a critical value, \(T_{c}\), determined by the "number density" \(Q / V\). Show that the nature of the singularity at \(T=T_{c}\) is such that, regardless of the severity of the relativistic effects, this system falls in the same universality class as the nonrelativistic one.