  We analyze a model system of fermions in a harmonic oscillator potential
under the influence of a fluctuating force generated by a bath of harmonic
oscillators. This represents an extension of the well-known Caldeira-Leggett
model to the case of many fermions. Using the method of bosonization, we
calculate Green's functions and discuss relaxation and dephasing of a single
extra particle added above the Fermi sea. We also extend our analysis to a more
generic coupling between system and bath, that results in complete
thermalization of the system.
