Infrared Divergence Separated for Stochastic Force - Langevin Evolution in the Inflationary Era

Abstract: Inflation in the early Universe is a grand phase transition which have produced the seeds of all the structures we now observe. We focus on the non-equilibrium aspect of this phase transition especially the inevitable infrared (IR) divergence associated to the the quantum and classical fields during the inflation. There is a long history of research for removing this IR divergence for healthy perturbation calculations. On the other hand, the same IR divergence is quite relevant and have developed the primordial density fluctuations in the early Universe. We develop a unified formalism in which the IR divergence is clearly separated from the microscopic quantum field theory but only appear in the statistical classical structure. We derive the classical Langevin equation for the order parameter within the quantum field theory through the instability of the de Sitter vacuum during the inflation. This separation process is relevant in general to develop macroscopic structures and to derive the basic properties of statistical mechanics in the quantum field theory.