STOCHASTIC DYNAMICS OF LARGE-SCALE INFLATION IN DE~SITTER SPACE

TL;DR

This paper derives exact quantum Langevin equations for large-scale inflation in de Sitter space, revealing quantum dissipative behavior and the classical limit of the inflationary scalar field dynamics.

Contribution

It introduces a novel formulation of quantum stochastic dynamics for inflation in de Sitter space using Langevin equations and provides exact solutions for the Wigner function.

Findings

Quantum Langevin equations describe inflation dynamics.

Stationary solutions are obtained for arbitrary potentials.

Large-scale inflation field behaves as a quantum dissipative system.

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 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. 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).