Revisiting Lagrangian Formulation of Stochastic inflation

TL;DR

This work reexamines the Lagrangian formulation of stochastic inflation by deriving the influence functional for a massless, minimally coupled $\lambda\varphi^4$ theory in exact de Sitter space using the Schwinger-Keldysh framework. By integrating out short-wavelength modes, it identifies both real (dissipative) and imaginary (stochastic) components of the effective action for the long-wavelength field, and recasts the imaginary part via a Hubbard–Stratonovich transformation into a stochastic description. While the leading (free) theory reproduces the standard stochastic-inflation results and matches quantum-field-theory predictions, an interacting case reveals additional, β-dependent terms arising from non-orthogonality between long- and short-wavelength sectors, which persist as the coarse-graining parameter σ → 0. The paper also exposes ambiguities in how to treat the imaginary part of the influence functional for general interaction terms, illustrating two perturbative pathways that yield different outcomes and emphasizing the need for a consistent theoretical framework. Overall, the results stress both the potential and the current limitations of the Lagrangian stochastic-inflation program, particularly in its treatment of imaginary action and mode-orthogonality in interacting theories.

Abstract

We revisit the Lagrangian formulation of stochastic inflation, where the path-integral approach is employed to derive the Langevin equation governing the dynamics of long-wavelength fields, in contrast to the standard method where the Langevin equation is derived directly from the equation of motion of the full quantum field. Focusing on a massless, minimally coupled scalar field with quartic self-interaction in a de Sitter background, we re-derive the formal expression for the influence functional that encapsulates the effects of short-wavelength fields up to second order in the coupling constant, and compare our results with those obtained in earlier works. In doing so, we highlight certain subtleties that have been previously overlooked, including the non-orthogonality between long- and short-wavelength modes, which we analyze in detail, as well as the absence of a consistent prescription for handling general interaction terms in the imaginary part of the influence functional. The latter issue points to a broader challenge: the lack of a universally accepted framework for treating the imaginary component of effective actions.