Classical Fermi Fluid and Geometric Action for c = 1

We formulate the c = 1 matrix model as a quantum fluid and discuss its classical limit in detail, emphasizing the h̄ corrections. We view the fermi fluid profiles as elements of w∞-coadjoint orbit and write down a geometric action for the classical phase space. In the specific representation of fluid profiles as ‘strings’ the action is written in a four-dimensional form in terms of gauge fields built out of the embedding of the ‘string’ in the phase plane. We show that the collective field action can be derived from the above action provided one restricts to quadratic fluid profiles and ignores the dynamics of their ‘turning points’.