Coherent state path integral and Langevin equations of interacting bosons

Interacting excitons in a semiconductor coupled to a thermal reservoir are treated as bosons. We use a coherent state path integral formulation on the time contour and transform the quantum mechanical system of bosonic excitons to a classical Langevin equation with an inhomogenous random force which represents the influence of the thermal reservoir. However, this classical Langevin equation cannot take into account the different order of annihilation and creation operators or their correlations. This is achieved by an application of a Hubbard–Stratonovich transformation which introduces an auxiliary variable σx(tp) related to the self-energy. In terms of this new field, an additional Langevin equation is derived which is similar to a saddle-point equation with a random force fx(t). This equation contains the mean value of the auxiliary fields σx(t+), σx(t−) as a fluctuating potential or self-energy in matrix-form on the time contour.