Abstract
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.
Highlights
The measurement of a finite global polarization of particles in relativistic heavy-ion collisions [1], in agreement with the predictions of relativistic hydrodynamics [2,3], has opened a new perspective in the phenomenology of these collisions as well as in the theory of relativistic matter, showing for the first time a direct manifestation of quantum features in this realm
The problem of the physical significance of the spin tensor — mostly in relativistic gravitational theories, notably in the Einstein–Cartan theory — is a long-standing one [16] and has been rediscussed more recently in Refs. [17,18], where it was demonstrated that well known thermodynamic formulae for transport coefficients such as viscosity do depend on the presence of a spin tensor
The density operator describing local thermodynamic equilibrium in quantum field theory was obtained in ref. [19,20] and has been rederived more recently in refs. [21,22]; we briefly summarize the derivation
Summary
The measurement of a finite global polarization of particles in relativistic heavy-ion collisions [1], in agreement with the predictions of relativistic hydrodynamics [2,3] (see [4,5,6,7,8,9,10,11]), has opened a new perspective in the phenomenology of these collisions as well as in the theory of relativistic matter, showing for the first time a direct manifestation of quantum features in this realm. Our conclusion is that for a general relativistic fluid the spin tensor is significant, if spin density “slowly” relaxes towards equilibrium, where slowly means on a time scale which is comparable to the familiar hydrodynamic time scale of evolution of charge and momentum densities. In this case, we will see that the description of the polarization of particles and the calculation of its final value requires a new thermodynamic potential, akin to chemical potential or temperature, coupled to the spin tensor. We work in flat space-time, in many equations we use covariant derivative ∇μ instead of partial derivative ∂μ to emphasize the validity of the relations in general coordinates
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