  Bosonization technique for one-dimensional fermions out of equilibrium is
developed in the framework of the Keldysh action formalism. We first
demonstrate how this approach is implemented for free fermions and for the
problem of non-equilibrium Fermi edge singularity. We then employ the technique
to study an interacting quantum wire attached to two electrodes with arbitrary
energy distributions. The non-equilibrium electron Green functions, which can
be measured via tunneling spectroscopy technique and carry the information
about energy distribution, zero-bias anomaly, and dephasing, are expressed in
terms of functional determinants of single-particle "counting" operators. The
corresponding time-dependent scattering phase is found to be intrinsically
related to "fractionalization" of electron-hole excitations in the tunneling
process and at boundaries with leads. Results are generalized to the case of
spinful particles as well to Green functions at different spatial points
(relevant to the problem of dephasing in Luttinger liquid interferometers). For
double-step distributions, the dephasing rates are oscillatory functions of the
interaction strength.
