Inconsistency between the Boltzmann distribution for relativistic free particles and the Planck spectrum for thermal radiation in quantum theory

Einstein's famous model of a two-level atom where radiation equilibrium is determined by the $A$ and $B$ coefficients of spontaneous and stimulated emission is here extended to relativistic particle velocities. We consider a massive particle containing a quantum two-level system and follow Einstein's analysis in the instantaneous rest frame of the particle. The changes of momentum due to spatially directed absorption and emission processes (photons) are obtained by using relativistic transformations between the laboratory frame and the frame of the particle, and these momentum changes are introduced into a suitable Fokker-Planck equation for the equilibrium velocity distribution of the massive particle. Although for a nonrelativistic analysis the Boltzmann distribution for free particles can be combined consistently with the Planck radiation spectrum, this is not true for a relativistic analysis. The Boltzmann distribution for relativistic free particles and the Einstein quantum analysis are inconsistent for any equilibrium spectrum of thermal radiation, and in particular for the Planck spectrum.