Quantum Stochastic Inflation

Abstract: We formulate stochastic inflation in an open quantum system framework. The field coarse-grained in a patch of fixed physical size, and the total momentum of that patch, form a canonical pair and act on a one-mode Fock space which we identify as the "bulk". At each time step, new comoving modes join the coarse-grained patch and the bulk has to be redefined. This redefinition produces an entangled mode that is traced over, yielding a non-unitary evolution equation for the bulk's density matrix. For a free test field in de Sitter, one obtains GKLS dynamics, generated by an effective Hamiltonian and a single non-Hermitian Lindblad operator, hence diffusion and Hubble friction originate from the same quantum channel. The Wigner-Weyl transform of the GKLS equation leads to a Fokker-Planck equation for the Wigner function, which matches the one that applies to the classical phase-space distribution of stochastic inflation. We also provide several schemes under which one can unravel the GKLS dynamics into stochastic Schrodinger equations when continuous measurements of the decoupled mode are performed, making contact with Langevin formulations of stochastic inflation. In the light-field regime, an additional overdamped reduction can be performed by integrating out the momentum variable in the Wigner distribution, leading to Starobinsky's slow-roll Fokker-Planck equation. In that regime, the purity of the patch is strongly suppressed. In contrast, for heavy fields, field diffusion is suppressed and the coarse-grained patch remains close to a pure underdamped oscillator, which prevents a classical stochastic treatment.

• The paper presents a canonical derivation of stochastic inflation as a quantum open system that unifies quantum fluctuations, dissipation, and decoherence.
• The methodology employs a canonical coarse-graining of field modes to construct a bulk Fock space whose evolution is governed by a GKLS master equation.
• It demonstrates a mass-dependent quantum-to-classical transition, reconciling the classical Fokker-Planck dynamics with underlying quantum open-system processes.

Quantum Stochastic Inflation: A Canonical Open-System Approach

Introduction and Motivation

Stochastic inflation provides a statistical framework to capture the super-Hubble dynamics of quantum fields during cosmic inflation. Traditionally, the Starobinsky formalism partitions field modes into infrared (IR) and ultraviolet (UV) sectors based on a moving physical cutoff, modeling the IR dynamics via classical stochastic differential equations (Langevin and Fokker-Planck). However, the explicit quantum-to-classical transition mechanisms and their implications for reduced phase-space dynamics and decoherence remain underexplored, particularly in connection with open quantum system theory. This work presents a canonical derivation of stochastic inflation as a quantum open system by tracking the time evolution of coarse-grained bulk variables in de Sitter, emphasizing how quantum fluctuations, dissipation, and decoherence are structurally connected in this context.

Canonical Coarse-Graining and Bulk Variable Construction

The authors construct a coarse-grained open system by defining canonical pairs $(\hat Q_N, \hat P_N)$, representing the field averaged over a finite physical region and its total momentum content. These operators act on a one-mode Fock space (the "bulk"), isolated by integrating modes below the cutoff $k_\sigma(N) = \sigma a H$. This construction clarifies the canonical structure: the canonical partner of the averaged field is not the momentum density but the total enclosed momentum, ensuring $[\hat Q_N, \hat P_N]=i$ is normalized for the bulk.

As new modes redshift below the cutoff, the effective bulk Fock space and canonical pair evolve. This time-dependent embedding is a nontrivial aspect: the coarse-grained system is continually redefined, entangling the redefined bulk with components of the newly included shell.

Open-System Dynamics: GKLS Structure and Single-Channel Dissipation

The redefinition of the bulk through mode entry generates entanglement between the updated bulk and a decoupled mode that is then traced out. This induces a non-unitary (decohering) Markovian evolution for the reduced bulk density matrix, formalized by a Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) equation. The effective generator contains a quadratic Hamiltonian and a single non-Hermitian Lindblad operator, which acts as a boundary-adapted annihilation operator absorbing the vacuum shell entering at the cutoff.

Crucially, the resulting Kossakowski matrix is positive and rank one. This implies Hubble friction and phase-space diffusion both arise from the same quantum channel—they are not independent, disjoint effects. This provides a rigorous quantum prescription for the origin of stochasticity in inflation and ensures the correct preservation of phase-space commutators and uncertainty relations throughout the dynamics.

Wigner-Fokker-Planck and Langevin Correspondence

By considering the Wigner-Weyl transform of the Lindblad equation, the authors demonstrate that the evolution of the Wigner function for the bulk coincides with the phase-space Fokker-Planck equation used in classical stochastic inflation. For Gaussian states, the Wigner function is positive and can be identified with the classical probability density; in the overdamped (slow-roll) regime, integrating out the bulk momentum leads to the expected one-dimensional Starobinsky Fokker-Planck equation for the field variable.

The framework also allows for exact stochastic unravelings of the GKLS master equation into stochastic Schrödinger equations, contingent on the measurement protocol applied to the decoupled mode. In the field-adapted homodyne measurement, the stochastic equation for the quantum expectation value of the field reduces to the standard Starobinsky Langevin equation in the massless, super-Hubble limit.

Quantum-to-Classical Transition and Stationary Purity Diagnostics

The mass of the field fundamentally divides the regime of stochastic inflationary dynamics. For light fields ($m < 3H/2$), the system displays strong overdamping, leading to rapid decoherence and phase-space diffusion: the field behaves classically with the marginal probability distribution strongly mixed (purity $\gamma \to 0$ as $\sigma \to 0$). The variance diverges as the cutoff is tightened, and the classical random-walk limit is attained.

Figure 2: Phase-space diagrams of the Gaussian stationary state derived explicitly in Eq. (StationaryCovMatrixMain), showing convergence towards classicality for light fields.

At the critical mass $m=3H/2$ and for heavy fields $m>3H/2$, the field diffusion coefficient is suppressed as $\sigma \to 0$, and the reduced bulk instead approaches an underdamped quantum state of finite purity. Thus, strong classical stochasticity emerges only in the light-field overdamped regime, while heavier fields persist as nearly pure quantum oscillators, precluding a valid classical stochastic limit.

Contrasts with Prior Approaches

While other works (e.g., Li 2026) also derive a quantum open-system picture for inflationary perturbations via Schwinger-Keldysh and influence functional methods, the present construction, anchored in canonical variables and covariance structure, resolves ambiguities in the normalization of effective Lindblad channels and clarifies the connection between canonical pairs and momentum density variables used in previous literature. The single-channel structure and identification of diffusion and friction as aspects of the same quantum map is unique to this formulation.

Theoretical and Practical Implications

This canonical open quantum system treatment of stochastic inflation bridges rigorous quantum field theoretic treatments and phenomenological stochastic descriptions. The explicit identification of the quantum underpinnings of diffusion, dissipation, and decoherence elucidates the mass dependence of phase-space classicalization and clarifies the role of decoherence in suppressing quantum coherences on super-Hubble scales. The approach provides a natural framework for extending stochastic inflation to systems with additional interactions, non-Gaussianities, or inflaton self-couplings, where non-Markovian effects and higher-order Moyal terms would alter the Wigner function dynamics. Future generalizations are expected to affect predictions for rare fluctuation statistics, PBH formation, and non-Gaussian signatures in the inflationary perturbation spectrum.

Conclusion

This work systematically derives stochastic inflation as a canonical quantum open system, making explicit the quantum source of stochasticity and decoherence for coarse-grained fields in de Sitter. The identification of a single quantum channel generating both friction and diffusion, the precise reduction to the classical Fokker-Planck regime in the light-field limit, and the explicit structure of the quantum-to-classical transition by field mass are principal findings. The canonical approach provides a mathematically controlled foundation for exploring the emergence and limitations of classical stochastic dynamics from underlying quantum field theoretic principles in cosmological inflation.