  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.
