Stochastic dynamics of large-scale inflation in de Sitter space.

Abstract

In this paper we derive exact quantum Langevin equations for stochastic dynamics of large-scale inflation in de Sitter space. These quantum Langevin equations are the equivalent of the Wigner equation and are described by a system of stochastic differential equations. We present a formula for the calculation of the expectation value of a quantum operator whose Weyl symbol is a function of the large-scale inflation scalar field and its time derivative. The quantum expectation value is calculated as a mathematical expectation value over a stochastic process in an extended phase space, where the additional coordinate plays the role of a stochastic phase. The unique solution is obtained for the Cauchy problem for the Wigner equation for large-scale inflation. The stationary solution for the Wigner equation is found for an arbitrary potential. It is shown that the large-scale inflation scalar field in de Sitter space behaves as a quantum one-dimensional dissipative system, which supports the earlier results of Graziani and of Nakao, Nambu, and Sasaki. But the analogy with a one-dimensional model of the quantum linearly damped anharmonic oscillator is not complete: the difference arises from the new time-dependent commutation relation for the large-scale field and its time derivative. It is found that, for the large-scale inflation scalar field, the large time asymptotics is equal to the ``classical limit.'' For the large time limit the quantum Langevin equations are just the classical stochastic Langevin equations (only the stationary state is defined by the quantum field theory). \textcopyright{} 1996 The American Physical Society.