Wikipedia (see here) says perfect fluid may be quantized. I do find an article (arXiv 1011.6396) about this, and the procedure is straight forward. What I do not understand is whether this quantization really makes sense, i.e. whether the quantized liquid dynamics corresponds to actual systems.
For example, can a quantum many-body system - like electrons in a metal - be described using a quantized version fluid dynamics? Does this give a universal scheme of bosonization?
Liquid helium can be described as superfluid.
the first liquid in which superfluidity was recognized was helium-4 back in 1938. It loses its viscosity below 2.12 K. At the time Fritz London suggested that the phenomenon was associated with Bose-Einstein condensation (BEC), which occurs when an assembly of bosons (particles with zero or integer “spin”) is cooled below a critical temperature. In Bose-Einstein condensation a large fraction of all the particles in the assembly congregate in the zero-momentum ground state.
....
Although it is generally agreed that there is a close association between Bose-Einstein condensation and superfluidity, the exact relationship has yet to be established.
The review starts with"
Recent work at Göttingen has revealed convincing evidence for superfluidity in liquid hydrogen, the only liquid other than helium to exhibit this quantum behaviour.
and ends with:
How does this result relate to more conventional types of experiment? It should be emphasized that superfluidity in a shell containing about 15 hydrogen molecules does not necessarily mean that the phenomenon will ever be observed in bulk liquid. Indeed, it probably will not, due to the difficulty of maintaining the liquid state at low enough temperatures. .....
So it seems there is research going on.
After three years I still don't really have a clear answer, but I'll write down what I have now. The idea is documented in Sec. 24 in Statistical Physics II in Landau's Course of Theoretical Physics, where it is assumed that in "liquids", the most important degrees of freedom include the density and the current, and the commutation relation between the two is Eq. (24.6): $$ j(\mathbf{r}) \rho(\mathbf{r}') - \rho(\mathbf{r}') j(\mathbf{r}) = - \mathrm{i} \hbar \rho \nabla\delta(\mathbf{r} - \mathbf{r}'), $$ which is equivalent to $$ v(\mathbf{r}) \rho(\mathbf{r}') - \rho(\mathbf{r}') v(\mathbf{r}) = - \mathrm{i} \hbar \delta(\mathbf{r} - \mathbf{r}'). $$ The Hamiltonian is then given in Eq. (24.11), where by intuition, we have Eq. (24.11) $$ H = \int \mathrm{d}^3x \left( \frac{\mathbf{v} \cdot \rho \mathbf{v}}{2} + \rho e(\rho) \right). $$ An earlier source for this theory is https://journals.aps.org/pr/pdf/10.1103/PhysRev.60.356, written by, of course, Landau.
So we indeed have some sort of quantum hydrodynamics (the same word is also used to refer to several things different, like rewriting the single-electron Schrödinger's equation into a hydrodynamics-like equation, and deriving hydrodynamics-like equations from quantum master equations; see my other question for the latter sense). The question is what system gives rise to this kind of quantum hydrodynamics.
The model Landau proposes in Sec. 24 seems to be intended as a model for quantum liquids in Course. But there is not derivation provided to justify this. In the framework of bosonization, we can indeed derive these equations from certain Hamiltonians, as in, say, this paper. Landau's quantum hydrodynamics is also the same as Luttinger liquid without spin (see e.g. https://arxiv.org/pdf/cond-mat/0307033). But a generic discussion on what systems' effective theories are within the framework of quantum hydrodynamics seems missing (which of course is likely because I haven't encountered the right paper yet). As a comparison, at least we know certain self-energy approximations, if they work, lead to Fermi liquid behaviors.
A final remark: Landau's quantum hydrodnamics gives us Euler's equation, which doesn't have viscosity. This goes against some intuitions about "hydrodynamics" in everyday life, which are however always related to viscosity (see e.g. discussion in Feynman's lecture notes). Hydrodynamics, with viscocity, described by Navier-Stokes equation, of course can't be quantized as a pure-state quantum theory, and should be conceived within a Langevin-like framework.
