  Physical systems made of many interacting quantum particles can often be
described by Euler hydrodynamic equations in the limit of long wavelengths and
low frequencies. Recently such a classical hydrodynamic framework, now dubbed
Generalized Hydrodynamics (GHD), was found for quantum integrable models in one
spatial dimension. Despite its great predictive power, GHD, like any Euler
hydrodynamic equation, misses important quantum effects, such as quantum
fluctuations leading to non-zero equal-time correlations between fluid cells at
different positions. Focusing on the one-dimensional gas of bosons with delta
repulsion, and on states of zero entropy, for which quantum fluctuations are
larger, we reconstruct such quantum effects by quantizing GHD. The resulting
theory of quantum GHD can be viewed as a multi-component Luttinger liquid
theory, with a small set of effective parameters that are fixed by the
Thermodynamic Bethe Ansatz. It describes quantum fluctuations of truly
nonequilibrium systems where conventional Luttinger liquid theory fails.
