Hydrodynamics is a low-energy effective theory which describes a long-distance and long-time behavior of many-body systems. It is applicable not only to a non-relativistic weakly-interacting dilute gas but also a relativistic strongly-interacting dense liquid such as the quark-gluon plasma created in ultra relativistic heavy-ion collision experiments. Although relativistic hydrodynamics itself is well-established formalism, its foundation from underlying microscopic theories, or quantum field theories, remains unclear. In this study, based on the recent development of non-equilibrium statistical mechanics, we provide the field-theoretical derivation of the relativistic Navier-Stokes equation [1]. We show that the procedure to derive hydrodynamic equations is similar to the so-called renormalized/optimized perturbation theory. Furthermore, we give a path-integral formula for local thermal equilibrium which results in the emergence of thermally induced curved spacetime [2]. Based on these results, we perform the derivative expansion and derive the first-order hydrodynamic equation (the Navier-Stokes equation) with the Green-Kubo formulas for transport coefficients.

References:
[1] T. Hayata, Y. Hidaka, M. Hongo, and T. Noumi, Phys. Rev. D 92, 065008 (2015).
[1] M. Hongo, Annals of Physics, 383, 1 (2017)