We develop a bosonization technique for one-dimensional fermions out of equilibrium. The approach is used to study a quantum wire attached to two electrodes with arbitrary energy distributions. The non-equilibrium electron Green function is expressed in terms of functional determinants of a single-particle"counting" operator with a time-dependent scattering phase. The result reveals an intrinsic relation of dephasing and energy redistribution in Luttinger-liquids to "fractionalization" of electron-hole excitations in the tunneling process and at boundaries with leads.
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Gutman et al. (2010) studied this question.
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