Negative temperature: further extensions

G. E. Volovik

*Low Temperature Laboratory, Aalto University, P.O. Box 15100, FI-00076 Aalto, Finland*

Landau Institute for Theoretical Physics, 142432 Chernogolovka, Russia

**Abstract**

This paper 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.