  Marginal Fermi liquid was originally introduced as a phenomenological
description of the cuprates in a part of the metallic doping range which
appears to be governed by fluctuations due to a quantum-critical point. An
essential result due to the form of the assumed fluctuation spectra is that the
large inelastic quasiparticle relaxation rate near the Fermi-surface is
proportional to the energy measured from the chemical potential,
$\tau_i^{-1}\propto\epsilon$. We present a microscopic long-wavelength
derivation of the hydrodynamic properties in such a situation by an extension
of the procedure that Eliashberg used for the derivation of the hydrodynamic
properties of a Landau-Fermi-liquid. In particular, the density-density and the
current-current correlations and the relation between the two are derived, and
the connection to microscopic calculations of the frequency dependence of the
optical conductivity with an additional fermi-liquid correction factor shown to
follow. The method used here may be necessary, quite generally, for the correct
hydrodynamic theory for any problem of quantum-critical fluctuations in
fermions.
