  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.
