Quantum walks and exact RG in de Sitter space

The local physics of light scalar fields in de Sitter space is well described by classical random walks, as expressed through the framework of stochastic inflation. Recent work has clarified how this formalism arises from quantum field theory and the renormalization group (RG), allowing for corrections to this formalism to be determined order by order. Yet, this description is incomplete. For example, the quantum dynamics of these fields are expected to become important when determining the tail of the probability distribution for the fluctuations. In this paper, we develop the understanding of fields in de Sitter as a quantum walk in order to bridge the gap between the classical and quantum description. We use the framework of exact RG to calculate the evolution equation for the reduced density matrix of the long wavelength fields. This master equation provides the direct map from light fields to models of quantum walks. We show how to reduce the master equation to stochastic inflation, and provide a new understanding of how the higher-order corrections arise. In the process, we demonstrate that divergences and secular growth in de Sitter, for both light and heavy fields, can be absorbed by (dynamical) renormalization.