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.

这篇论文是受最近的论文[M. Baldovin，S。Iubini，R。Livi和A. Vulpiani，负温度系统的统计力学，arXiv：2103.12572]。作者认为负绝对温度与平衡热力学一致。如果环境也有负温度，这是正确的。否则，这些状态代表的是短暂的，虽然可能是长期存在的亚稳态效应。在这里，我们考虑将负温度扩展到相对论系统。我们表明，在狄拉克真空中，负温度状态是可能的，这是通过PT获得的常规狄拉克海的对称变换。如果这种具有反种群的真空充满整个宇宙，那么由负温度决定的其热力学就会达到平衡。