Stochastic Inflation:The Quantum Phase Space Approach

TL;DR

This paper develops a quantum phase space approach to stochastic inflation, deriving a quantum Liouville equation, solving it for specific expansion models, and analyzing the limitations of the stochastic formalism in explaining quantum-to-classical transition.

Contribution

It introduces a quantum phase space framework for inflationary fields, deriving the stochastic Liouville equation and analyzing its implications for quantum correlations and classicality.

Findings

Explicit solutions for phase space distribution in power law and exponential inflation.

Fair agreement with field theory results within certain coarse-graining scales.

The stochastic formalism faces difficulties in explaining decoherence and classicality.

Abstract

In this paper a quantum mechanical phase space picture is constructed for coarse-grained free quantum fields in an inflationary Universe. The appropriate stochastic quantum Liouville equation is derived. Explicit solutions for the phase space quantum distribution function are found for the cases of power law and exponential expansions. The expectation values of dynamical variables with respect to these solutions are compared to the corresponding cutoff regularized field theoretic results (we do not restrict ourselves only to $\VEV{\F^2}$). Fair agreement is found provided the coarse-graining scale is kept within certain limits. By focusing on the full phase space distribution function rather than a reduced distribution it is shown that the thermodynamic interpretation of the stochastic formalism faces several difficulties (e.g., there is no fluctuation-dissipation theorem). The coarse-graining does not guarantee an automatic classical limit as quantum correlations turn out to be crucial in order to get results consistent with standard quantum field theory. Therefore, the method does {\em not} by itself constitute an explanation of the quantum to classical transition in the early Universe. In particular, we argue that the stochastic equations do not lead to decoherence.