  We investigate a one dimensional quantum fluid coupled to a dissipative bath.
The quantum fluid is captured by the canonical Luttinger liquid; the bath is
given by the model of Caldeira and Leggett, i.e. a tower of oscillators coupled
linearly to the fluid density, $\rho$. The bath can be integrated out exactly,
producing an effective interaction for the fluid that is nonlocal in time; we
argue that the form corresponding to Ohmic dissipation is generic. Compared to
previous works, we compute correlation functions for this minimal model without
approximation, including at finite temperature $T>0$. From these and a Kubo
calculation, we conclude that arbitrary dissipation destroys the perfect
conductivity of the Luttinger liquid via Zeno localization, even in the absence
of a spatial potential; from RG analysis of harmonic terms, we also find that
the open Luttinger liquid is significantly more prone to localization by such
potentials, in contrast to the usual intuition that baths make systems less
localized.
