Stochastic inflation as an open quantum system II: open effective field theory and stochastic matching

Abstract: We further develop the proposal in Phys.\ Rev.\ Lett.\ \textbf{136} 071501 that interprets stochastic inflation as an open quantum system, by constructing the open effective field theory for the reduced density matrix of long wavelength modes. We clarify that this open effective field theory enjoys two renormalization group flows: the conventional Wilsonian channel, and a stochastic channel arising from the openness that has no counterpart in ordinary Wilsonian effective field theory. Focusing on the stochastic channel in the hard cutoff scheme, we identify both Gaussian and non-Gaussian diffusion as effective operators in the influence functional, and show that they are required by matching onto correlators and form factors of the perturbative full theory through a method-of-region in time. Beyond Gaussian order, the matching data are no longer local Wilson coefficients but nonlocal and non-Markovian Wilson kernels. We then obtain the bare Hamiltonian density of this open effective field theory and derive its nonlocal functional master equations, including the Fokker-Planck equation for the diagonal density matrix and the Klein-Kramers equation for the Wigner functional, with their zero-modess simplifications discussed. Finally, we take a first step toward a continuum version of this open effective field theory, replacing the hard cutoff by an analytic regulator in the stochastic channel, and demonstrate stochastic renormalization using a massive scalar as an example.

• The paper introduces an open EFT for stochastic inflation that models long-wavelength modes as a system interacting with a short-wavelength environment.
• It employs the Schwinger–Keldysh formalism and time-dependent coarse-graining to derive nonlocal, non-unitary master equations.
• Matching techniques confirm that distinct Wilsonian and stochastic RG channels generate novel diffusion coefficients and non-Markovian dynamics.

Stochastic Inflation as an Open Quantum System: Open EFT and Stochastic Matching

Introduction and Motivation

This work extends the interpretation of stochastic inflation as an explicit open quantum system, following the proposal previously articulated in [Li:2025azq]. By viewing long-wavelength modes during inflation as a system coupled to an environment—that is, the short-wavelength modes—a non-unitary, nonlocal effective field theory (EFT) emerges for the reduced density matrix of the system. This open EFT fundamentally differs from standard Wilsonian EFTs: the key distinction is the presence of two renormalization group (RG) channels, one conventional and one intrinsically stochastic, arising from the time-dependent nature of environmental-to-system mode transfer in de Sitter spacetime.

Conceptual Framework

Path Integral and Schwinger–Keldysh (SK) Formalism

The density matrix for the system is constructed via the path integral along a SK contour, yielding a formalism that treats bra and ket as distinct replicas. Integrating out the environment (short modes) renders the system "open": unitarity is manifestly broken and dissipative and diffusive effects are encoded via the Feynman–Vernon influence functional, yielding a master equation for the reduced density matrix. The openness here is not simply a matter of tracing out heavy "invisible" matter, but results from a sharp coarse-graining in field theory, splitting $\phi = \phi_s + \phi_e$ at a time-dependent $k$-space boundary.

Approximations and Emergence of Openness

A precise system-environment split is imposed using a moving cutoff $\Lambda(t) = \varepsilon a(t) H_0$. The following approximations underlie the effective description:

• Born approximation: system–environment coupling is perturbatively weak, justifying diagrammatic expansions.
• Secular approximation: rapid oscillations relative to system timescales are suppressed, allowing multipole expansions (justified for $\phi_s^c$, but not $\phi_s^q$).
• Large spatial-scale separation: a spatial multipole expansion is justified for $k \ll aH$, but needs resummation near mode crossing.

Openness is fundamentally generated by the time-dependent nature of the system/environment split. Modes continually cross into the system sector as the universe expands, generating a dynamical non-unitary mapping absent in static Wilsonian EFTs.

Structure of the Open EFT

Two RG Channels: Wilsonian and Stochastic

The open EFT for the reduced density matrix shows two distinct RG flows:

1. Wilsonian channel ($\log b/\varepsilon$): Standard RG flow from integrating out heavy/short-wavelength modes. Affects the unitary (Hamiltonian) sector as well as diffusion operators via subdiagram dressings.
2. Stochastic channel ($\log\varepsilon$): Non-Wilsonian, unique to the open setting, arising from the cumulative transfer of modes into the system at the time-dependent horizon crossing. This drives the secular buildup of effective stochastic (diffusive) interactions.

Power counting is thus threefold: Wilsonian powers, $k/(aH)$ expansions, and time-derivative expansions.

Influence Functional and Operator Content

• Gaussian sector: The leading order introduces quadratic $\left(\phi^q\right)^2$ diffusion terms, with coefficients fixed by matching to Keldysh correlators. Even free theory yields a nontrivial diffusion functional in de Sitter due to the nonlocal nature of the system/environment crossing.
• Non-Gaussian sector: Higher-point response field insertions ($k$0) encode non-Gaussian stochastic effects. Beyond Gaussian order, influence functional coefficients are Wilson kernels—nonlocal, history-dependent, and non-Markovian, tracking not just field amplitudes but entire response histories.

The explicit open EFT, in Liouville space, is characterized by a nonlocal, time-asymmetric (dissipative) master action, whose structure is dictated by both RG channels.

Diagrammatic Matching and Method of Regions

The effective coefficients are not phenomenological: they are fixed by a diagrammatic matching procedure—the method of regions—where time integrals in full theory diagrams are split at the crossing time $k$1. The stochastic sector thus encodes early-time contributions missing from late-time dynamics, in full analogy to the separation of hard and soft momenta in standard EFTs. Explicit calculation shows precise agreement for key observables:

• Keldysh correlators
• Equal-time form factors
• Squeezed- and general four-point functions

In the non-Gaussian sector, the matching dictates that Wilson kernels encode entire histories, rendering the open EFT non-Markovian and nonlocal in time and space.

Functional Master Equations and Continuum Limit

The open EFT leads to functional master equations for the reduced density matrix:

• Functional Fokker-Planck equation: Governs the diagonal components using the detailed influence functional structure, including higher-order functional derivatives for non-Gaussian sectors.
• Klein-Kramers equation for the Wigner functional: Controls phase-space distributions with both drift and nonlocal dissipation.

A simplification occurs in the "zero-mode limit" (IR projection onto spatially homogeneous modes), where the functional equations become those of Starobinsky's stochastic inflation with matched diffusion coefficients, and projection onto phase space yields the appropriate Klein-Kramers equation.

The framework is further extended towards a continuum EFT scheme, replacing the hard cutoff with an analytic regulator. This enables closed-form stochastic RG equations for the Wilson coefficients, explicitly demonstrated for $k$2 interactions.

Numerical and Structural Results

• The stochastic RG channel generates secular $k$3 enhancements in long-time diffusion coefficients, independent of the UV cutoff.
• Non-Gaussian diffusion, e.g., $k$4 and $k$5, is unambiguously fixed by matching, and the Wilson kernels are structurally necessary and calculable.
• Bare Hamiltonian densities and nonlocal master equations are derived, including explicit Lindblad forms in the Gaussian limit.
• The continuum RG structure supports mixing of stochastic operators, introducing stochastic anomalous dimensions and entirely novel flow equations compared to standard Wilsonian settings.

Implications and Future Directions

Theoretical implications:

• This formalism unifies stochastic inflation, open quantum system theory, and quantum field theoretic matching. It clarifies which stochastic features are fixed by first principles, separating universal stochastic dynamics from scheme artifacts.
• Non-Gaussian, non-Markovian stochastic effects are shown to be robust—required for full matching, not just model-dependent corrections.
• The necessity for two RG channels suggests novel universality classes for the long-wavelength sector of de Sitter QFTs.

Practical implications:

• A systematic tool to compute late-time statistical properties of inflationary scalar fields beyond the Fokker-Planck/Langevin paradigm, including quantum memory and higher-point statistics.
• The open EFT framework provides a basis for exploring decoherence, phase-space dynamics, and eternal inflation, as well as the production of primordial black holes where non-Gaussian quantum fluctuations and memory effects are consequential.

Outlook:

• Full continuum regularization—combining stochastic and Wilsonian channels—is needed for renormalization at arbitrary orders.
• The operator mixing structure for non-Gaussian stochastic diffusion awaits further elucidation.
• Generalization to multifield models, curved backgrounds (e.g., global de Sitter, $k$6 vacua), and connections to exact RG and soft de Sitter EFT are promising directions.

The present work thus systematizes open quantum system methods for cosmological IR physics and elucidates the EFT structure, matching procedures, and RG flows necessary for a complete statistical description of stochastic inflation (2605.21929).