Stochastic dynamics of large-scale inflation in de Sitter space

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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). © 1996 The American Physical Society.
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