Negative Temperature: Further Extensions

This work is motivated by the recent paper [M. Baldovin, S. Iubini, R. Livi, and A. Vulpiani, Statistical mechanics of systems with negative temperature, arXiv:2103.12572]. The authors suggest that negative absolute temperatures are consistent with equilibrium thermodynamics. This is correct, if the environment also has negative temperature. Otherwise, such states represent the transient though maybe long-lived metastable effects. Here, we consider the extension of the negative temperatures to the relativistic systems. We show that the states with negative temperature are possible in the Dirac vacuum, which is obtained by the PT symmetry transformation from the conventional Dirac sea. If such vacuum with inverse population fills the whole Universe, its thermodynamics determined by negative temperatures becomes equilibrium.