General equilibrium second-order hydrodynamic coefficients for free quantum fields

We present a systematic calculation of the corrections of the stress-energy tensor and currents of the free boson and Dirac fields up to second order in thermal vortic- ity, which is relevant for relativistic hydrodynamics. These corrections are non-dissipative because they survive at general thermodynamic equilibrium with non vanishing mean val- ues of the conserved generators of the Lorentz group, i.e. angular momenta and boosts. Their equilibrium nature makes it possible to express the relevant coefficients by means of correlators of the angular-momentum and boost operators with stress-energy tensor and current, thus making simpler to determine their so-called “Kubo formulae”. We show that, at least for free fields, the corrections are of quantum origin and we study several limiting cases and compare our results with previous calculations. We find that the axial current of the free Dirac field receives corrections proportional to the vorticity independently of the anomalous term.