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We study the power spectrum of super-Hubble fluctuations of an inflaton-likescalar field, the "system", coupled to another scalar field, the "environment"during de Sitter inflation. We obtain the reduced density matrix for theinflaton fluctuations by integrating out the environmental degrees of freedom.These are considered to be massless and conformally coupled to gravity as a\emph{proxy} to describe degrees of freedom that remain sub-Hubble allthroughout inflation. The time evolution of the density matrix is described bya quantum master equation, which describes the decay of the vacuum state, theproduction of particles and correlated pairs and quantum entanglement betweensuper and sub-Hubble degrees of freedom. The quantum master equation provides anon-perturbative resummation of secular terms from self-energy (loop)corrections to the inflaton fluctuations. In the case studied here these areSudakov-type double logarithms which result in the \emph{decay} of the powerspectrum of inflaton fluctuations upon horizon crossing with a concomitantviolation of scale invariance. The reduced density matrix and its quantummaster equation furnish a powerful non-perturbative framework to study theeffective field theory of long wavelength fluctuations by tracing shortwavelength degrees of freedom.

An effective field theory during inflation: reduced density matrix and its quantum master equation