  A hydrodynamic theory of transport in quantum mechanically phase-disordered
superconductors is possible when supercurrent relaxation can be treated as a
slow process. We obtain general results for the frequency-dependent
conductivity of such a regime. With time-reversal invariance, the conductivity
is characterized by a Drude-like peak, with width given by the supercurrent
relaxation rate. Using the memory matrix formalism, we obtain a formula for
this width (and hence also the dc resistivity) when the supercurrent is relaxed
by short range Coulomb interactions. This leads to a new -- effective field
theoretic and fully quantum -- derivation of a classic result on flux flow
resistance. With strong breaking of time-reversal invariance, the optical
conductivity exhibits what we call a `hydrodynamic supercyclotron' resonance.
We obtain the frequency and decay rate of this resonance for the case of
supercurrent relaxation due to an emergent Chern-Simons gauge field. The
supercurrent decay rate in this `topologically ordered superfluid vortex
liquid' is determined by the conductivities of the normal component of the
liquid. Our work gives a controlled framework for low temperature metallic
phases arising from phase-disordered superconductivity.
