Phase space holography with no strings attached

This note discusses the Wigner function representation from the standpoint of establishing a holography-like correspondence between the descriptions of a generic quantum system in the phase space ('bulk') picture versus its spacetime ('boundary') counterpart. Under certain circumstances the former might reduce to the classical dynamics of a local metric-like variable while the latter takes on the form of some bosonized collective field hydrodynamics. This generic pseudo-holographic duality neither relies on any particular symmetry of the system in question, nor does it require any connection to an underlying 'string theory', as in the various 'ad hoc' scenarios of applied holography.