Integrable Quantum Hydrodynamics in Two Dimensional Phase Space

Quantum liquids in two dimensions represent interesting dynamical quantum systems for several reasons, among them the possibility of the existence of infinite hidden symmetries, such as conformal symmetry or the symmetry associated with area preserving diffeo-morphisms. It is known that when the symmetry algebra is large enough, symmetry may fully prescribe the dynamics. However, the way this is borne out in two dimensional hydrodynamics, both classical and quantum, is not fully understood. Here we take a step in clarifying this issue, by focusing on a particular example, namely that of a two dimensional phase space liquid which emerges when one considers the Calogero model, a many-body one-dimensional system interacting through an inverse square law potential. We demonstrate how the symmetry algebra of conserved quantities of the one dimensional system is expressed in terms of the incompressible Euler hydrodynamics of point vortices in phase space. Due to a formal relation between quantum hydrodynamics and classical stochastic hydrodynamics, which is inherent in the method of stochastic quantization, the ideas and methods developed here may also have application in the study of stochastic classical hydrodynamics, after suitable modifications.