Quantum probability distributions in the early Universe. III. A geometric representation of stochastic systems

We present a geometric interpretation of the dynamics of quantum stochastic systems obeying the Wigner equation. We apply this representation to the quantum fluctuations of the homogeneous long-range free scalar field in de Sitter space (m≪H). This geometric picture leads to the concept of trajectories in moment [i.e., 〈φ(t)〉, 〈${\mathrm{\phi }}^2$(t)〉, 〈${\mathrm{\phi }}^3$(t)〉, ...] space. Up to initial conditions, each trajectory uniquely characterizes the evolution of the system. The stationary states of the system act like fixed points or attractors in the moment space. We present the evolution of the scalar field as a flow in the moment space and show that the Bunch-Davies vacuum is the fixed point.