Nonrelativistic hydrodynamics from quantum field theory: (I) Normal fluid composed of spinless Schr\"odinger fields

TL;DR

This paper derives nonrelativistic hydrodynamic equations from quantum field theory of spinless Schrödinger fields, systematically separating dissipative and nondissipative parts, and providing formulas for transport coefficients using a path-integral approach.

Contribution

It introduces a systematic derivation of hydrodynamics from quantum field theory, including dissipative and nondissipative parts, using symmetry principles and fluctuation theorems.

Findings

Derived first-order Navier-Stokes equations from quantum field theory.

Provided Green-Kubo formulas for transport coefficients.

Established a path-integral framework for local thermal equilibrium.

Abstract

We provide a complete derivation of hydrodynamic equations for nonrelativistic systems based on quantum field theories of spinless Schr\"odeinger fields, assuming that an initial density operator takes a special form of the local Gibbs distribution. The constructed optimized/renormalized perturbation theory for real-time evolution enables us to separately evaluate dissipative and nondissipative parts of constitutive relations. It is shown that the path-integral formula for local thermal equilibrium together with the symmetry properties of the resulting action---the nonrelativistic diffeomorphism and gauge symmetry in the thermally emergent Newton-Cartan geometry---provides a systematic way to derive the nondissipative part of constitutive relations. We further show that dissipative parts are accompanied with the entropy production operator together with two kinds of fluctuation theorems by the use of which we derive the dissipative part of constitutive relations and the second law of thermodynamics. After obtaining the exact expression for constitutive relations, we perform the derivative expansion and derive the first-order hydrodynamic (Navier-Stokes) equation with the Green-Kubo formula for transport coefficients.