Extensivity, entropy current, area law, and Unruh effect

Abstract

We present a general method to determine the entropy current of relativistic matter at local thermodynamic equilibrium in quantum statistical mechanics. Provided that the local equilibrium operator is bounded from below and its lowest lying eigenvector is non-degenerate, it is proved that, in general, the logarithm of the partition function is extensive, meaning that it can be expressed as the integral over a 3D space-like hypersurface of a vector current, and that an entropy current exists. We work out a specific calculation for a non-trivial case of global thermodynamic equilibrium, namely a system with constant comoving acceleration, whose limiting temperature is the Unruh temperature. We show that the integral of the entropy current in the right Rindler wedge is the entanglement entropy.

Unruh effect for fermions from the Zubarev density operator

“…where, as before, we assume a = √ −a µ a µ , and ρ 0 , p 0 , A 1 , A 2 , A 3 , A 4 , B 1 , B 2 are coefficients to be defined. The formula (2.2) is the most general expression allowed by parity requirements in the fourth order of perturbation theory (look [13]). In [16] coefficients up to the second order of the perturbation theory were calculated.…”

“…In [8,9] it was shown that the hadronization process can be accompanied by accelerated particle motion, which can be a source of thermalization in elementary processes such as e + e − annihilation or pp and pp collisions, just due to the Unruh effect, and also allows to explain some features in the multiplicities of hadrons. So in the elementary processes of electron-positron annihilation or proton-(anti)proton collisions, tremendous accelerations can occur, which may open the way for observing the effects associated with acceleration, while in heavy ion collision also large vorticity may arise, and polarisation in heavy ion collisions provides information both about vorticity and acceleration [10,11].An interesting new look at the Unruh effect was recently obtained from the standpoint of quantum relativistic statistical mechanics [12,13]. In [12] it was shown by calculating the values of quantum correlators for scalar fields at a finite temperature that the average value of any local operator turns out to be zero after subtracting of the vacuum contribution at the proper temperature, measured by a comoving observer, equal to the Unruh temperature.…”

“…This fact means that the Minkowski vacuum is perceived by the accelerated observer as a medium filled with a thermal bath of particles with a Unruh temperature a 2π , which is the essence of the Unruh effect.The analysis in [12] is given for scalar particles. Our main result is a generalization of [12] for the case of massless fermions: we show that for gas of massless fermions with chemical potentials equal to zero, the energy-momentum tensor in the laboratory frame is zero at the proper temperature, measured by a comoving observer, equal to the Unruh temperature.We use a fundamental approach based on the quantum-statistical density operator of Zubarev [14-18] (see also in [19] the current review of Zubarev's approach and its connection with the Kubo formulas, and in [12,13] discussion of thermodynamic equilibrium with acceleration, as well as the derivation of the entropy current). The effects associated with acceleration in this operator are described by a term with the boost operator and can be investigated in the framework of quantum field perturbation theory.…”

“…An interesting new look at the Unruh effect was recently obtained from the standpoint of quantum relativistic statistical mechanics [12,13]. In [12] it was shown by calculating the values of quantum correlators for scalar fields at a finite temperature that the average value of any local operator turns out to be zero after subtracting of the vacuum contribution at the proper temperature, measured by a comoving observer, equal to the Unruh temperature.…”

“…We use a fundamental approach based on the quantum-statistical density operator of Zubarev [14-18] (see also in [19] the current review of Zubarev's approach and its connection with the Kubo formulas, and in [12,13] discussion of thermodynamic equilibrium with acceleration, as well as the derivation of the entropy current). The effects associated with acceleration in this operator are described by a term with the boost operator and can be investigated in the framework of quantum field perturbation theory.…”