[Paper Review] Non-Gaussianity from Schwinger-Keldysh Effective Field Theory

Shu Lin, Yanyan Bu+1 more|arXiv (Cornell University)|Jan 17, 2023

Advanced Thermodynamics and Statistical Mechanics4 citations

TL;DR

This paper establishes a systematic, non-perturbative framework for non-Gaussianity in stochastic systems using Schwinger-Keldysh effective field theory (SKEFT), deriving exact stochastic formulations via Langevin and Fokker-Planck equations with nonlinear noise and higher-order diffusive terms. It demonstrates full equivalence between SKEFT and stochastic formulations for arbitrary non-Gaussian parameters, revealing a critical ambiguity in the simultaneous limit of vanishing coarse-graining scale and non-Gaussianity that underlies unphysical divergences in perturbative treatments.

ABSTRACT

We present a systematic treatment of non-Gaussianity in stochastic systems using the Schwinger-Keldysh effective field theory framework, in which the non-Gaussianity is realized as nonlinear terms in the fluctuation field. We establish two stochastic formulations of the Schwinger-Keldysh effective field theory, with those nonlinear terms manifested as multiple non-Gaussian noises in the Langevin equation and as higher order diffusive terms in the Fokker-Planck equation. The equivalence of the stochastic formulations with the original Schwinger-Keldysh effective field theory is demonstrated with non-trivial examples for arbitrary non-Gaussian parameters. The stochastic formulations will be more flexible and effective in studying non-equilibrium dynamics. We also reveal an ambiguity when coarse-graining time scale and non-Gaussian parameters vanish simultaneously, which may be responsible for the unphysical divergence found in perturbative analysis.

Motivation & Objective

• To develop a systematic, non-perturbative treatment of non-Gaussianity in stochastic dynamics beyond Gaussian noise approximations.
• To establish exact stochastic formulations—Langevin and Fokker-Planck—derived from SKEFT with arbitrary non-Gaussian parameters.
• To resolve the unphysical divergences found in perturbative analyses by identifying a critical ambiguity when coarse-graining time scale and non-Gaussian parameters vanish simultaneously.
• To demonstrate the equivalence between the original SKEFT and the derived stochastic formulations through explicit correlation function computations.

Proposed method

• Formulate a Schwinger-Keldysh effective field theory (SKEFT) Lagrangian with quartic nonlinear terms in fluctuation fields to encode non-Gaussianity.
• Derive two stochastic formulations: a Langevin equation with multiple nonlinear noise terms and a Fokker-Planck equation with higher-order diffusive terms.
• Use diagrammatic field theory to compute correlation functions (⟨Δ²⟩ and ⟨Δ⁴⟩_c) up to next-to-leading order (NLO), including one- and two-loop diagrams with both reducible and irreducible topologies.
• Apply scaling analysis in discrete time steps to derive the continuous limit of the stochastic equations, confirming consistency with the original SKEFT.
• Perform explicit perturbative computations in the non-Gaussian parameters and compare results across formulations to verify equivalence.
• Identify a non-trivial ambiguity in the simultaneous limit of vanishing coarse-graining time scale and non-Gaussian parameters, linking it to unphysical divergences in perturbation theory.

Experimental results

Research questions

• RQ1 How can non-Gaussianity in stochastic systems be systematically incorporated into the Schwinger-Keldysh effective field theory framework beyond Gaussian approximations?
• RQ2 What are the exact stochastic formulations—Langevin and Fokker-Planck—corresponding to a general SKEFT with nonlinear terms?
• RQ3 Why do perturbative analyses of non-Gaussian systems exhibit unphysical divergences, and what is the underlying origin of this pathology?
• RQ4 To what extent are the SKEFT, Langevin, and Fokker-Planck formulations equivalent when non-Gaussian parameters are arbitrary (non-perturbative)?
• RQ5 How can higher-point correlation functions be computed consistently in the presence of nonlinear non-Gaussian noise?

Key findings

• The Langevin and Fokker-Planck formulations derived from SKEFT exactly reproduce the correlation functions of the original SKEFT for arbitrary non-Gaussian parameters, confirming full equivalence.
• The NLO corrections to ⟨Δ²⟩ and ⟨Δ⁴⟩_c computed via diagrammatic field theory match exactly with the continuous limits of the stochastic formulations, validating the framework.
• The unphysical divergences in perturbative analyses are traced to a critical ambiguity in the simultaneous limit of vanishing coarse-graining time scale and non-Gaussian parameters.
• The non-Gaussian terms in the SKEFT Lagrangian—specifically ε₁Δₐ⁴, ε₂Δₐ³Δᵣ, ε₃Δₐ²Δᵣ², and εΔₐΔᵣ³—generate nonlinear noise and higher-order diffusive terms in the stochastic equations.
• The framework is robust and generalizable to more complex systems, such as hydrodynamic models, with straightforward adaptation of the stochastic formulations.
• The scaling analysis of discrete-time equations confirms that the continuous limit correctly reproduces the original SKEFT dynamics, even in the presence of nonlinear non-Gaussian terms.