Coherent state path integral and Langevin equation of interacting fermions

Interacting fermions, electrons and holes in a semiconductor, are coupled to a thermal reservoir of bosons which yield the fluctuating noise. We use a coherent state path integral formulation on the time contour for non-equilibrium systems in terms of anticommuting variables which replace the fermionic creation- and annihilation operators in the time development operator. An auxiliary commuting field σx(tp), defined on the time contour, is introduced by a Hubbard–Stratonovich transformation. In terms of this new field, a Langevin equation is derived which is similar to a saddle-point equation with a random force fx(t). In comparison to the bosonic case of excitons in a semiconductor previously described, one obtains a different expression for the Langevin equation and, especially, a different relation for the probability distribution of the noise term fx(t). The calculation for density matrix elements with the Langevin approach can also be interpreted as an average over modified non-equilibrium Green functions with the appropriately derived probability distribution of the noise.