  It is shown that the description of a relativistic fluid at local
thermodynamic equilibrium depends on the particular quantum stress-energy
tensor operator chosen, e.g., the canonical or symmetrized Belinfante
stress-energy tensor. We argue that the Belinfante tensor is not appropriate to
describe a relativistic fluid whose macroscopic polarization relaxes slowly to
thermodynamic equilibrium and that a spin tensor, like the canonical spin
tensor, is required. As a consequence, the description of a polarized
relativistic fluid involves an extension of relativistic hydrodynamics
including a new antisymmetric rank-two tensor as a dynamical field. We show
that the canonical and Belinfante tensors lead to different predictions for
measurable quantities such as spectrum and polarization of particles produced
in relativistic heavy-ion collisions.
