Thermodynamical inequivalence of quantum stress-energy and spin tensors

TL;DR

This paper demonstrates that different quantum stress-energy and spin tensor pairs, which are classically equivalent, become thermodynamically inequivalent in quantum relativistic fields, with measurable consequences in rotating systems at equilibrium.

Contribution

It shows that the choice of stress-energy and spin tensors affects thermodynamic predictions, highlighting the physical significance of specific tensor pairs in quantum fields.

Findings

Canonical and Belinfante tensors differ in angular momentum density predictions.

Quantum effects persist in the non-relativistic limit, affecting particle polarization.

The difference could be experimentally measurable in rotating quantum systems.

Abstract

It is shown that different couples of stress-energy and spin tensors of quantum relativistic fields, which would be otherwise equivalent, are in fact inequivalent if the second law of thermodynamics is taken into account. The proof of the inequivalence is based on the analysis of a macroscopic system at full thermodynamical equilibrium with a macroscopic total angular momentum and a specific instance is given for the free Dirac field, for which we show that the canonical and Belinfante stress-energy tensors are not equivalent. For this particular case, we show that the difference between the predicted angular momentum densities for a rotating system at full thermodynamical equilibrium is a quantum effect, persisting in the non-relativistic limit, corresponding to a polarization of particles of the order of \hbar \omega/KT (\omega being the angular velocity) and could in principle be measured experimentally. This result implies that specific stress-energy and spin tensors are physically meaningful even in the absence of gravitational coupling and raises the issue of finding the thermodynamically right (or the right class of) tensors. We argue that the maximization of the thermodynamic potential theoretically allows to discriminate between two different couples, yet for the present we are unable to provide a theoretical method to single out the "best" couple of tensors in a given quantum field theory. The existence of a non-vanishing spin tensor would have major consequences in hydrodynamics, gravity and cosmology.