Quantum probability distributions in the early Universe. II. The quantum Langevin equation

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Abstract
In this paper, we construct a stochastic differential equation for the quantum evolution of the large-scale or coarse-grained (>causal horizon) scalar field (inflaton) in de Sitter space that is valid to all orders in ħ. This quantum Langevin equation is the equivalent of the Wigner equation for quantum probability distributions. We show that in general quantum fluctuations are associated with multicomponent multiplicative non-Gaussian Markovian noise. However, to order ħ this noise becomes simple white noise. This is the origin behind the observations of Linde, Starobinsky, and Vilenkin that the large-scale quantum evolution of the inflaton is similar to Brownian motion. In addition, we show that Starobinsky’s Langevin equation arises from our quantum Langevin equation as an order-ħ–slow-rolling approximation. Finally we compute the random-number distribution associated with noise of the quantum Langevin equation. We conclude that the Wigner description based on quasiprobability distributions is probably more useful computationally than the quantum Langevin equation.

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