On the Imaginary Part of the Effective Action in de Sitter Spacetime with Different Regularization Schemes

TL;DR

The paper tackles the mismatch between the Bogoliubov and Green's function methods for extracting the imaginary part of the effective action in de Sitter spacetime. It introduces explicit time and momentum cutoffs, encapsulated by a single trade-off parameter $\lambda = -\tau\Lambda$, to unify the two regularization schemes into one framework. By analyzing the $\lambda$-dependent limits, it shows that $\lambda \to 0^+$ reproduces the Bogoliubov result while $\lambda \to +\infty$ yields the Green's function result, clarifying that the discrepancy arises from different limiting procedures. The work emphasizes the importance of regularization order in dynamical backgrounds and suggests that more robust observables may be better accessed via Keldysh–Schwinger approaches. These insights have broad implications for understanding vacuum instability and particle production in curved spacetimes where UV/IR limits do not commute.

Abstract

The imaginary part of the effective action encodes vacuum instability and particle production in the background field. Two standard approaches are commonly used to derive it: the Bogoliubov method and the Green's function method, which are usually expected to agree. However, in de Sitter spacetime they yield different results. We revisit this problem by introducing explicit time and momentum cutoffs in the Green's function representation of the effective action. The apparent discrepancy is found to be due to the different limiting procedures in regularization, which reproduces the Bogoliubov result and the Green's function result respectively. Therefore, the two approaches are understood to be different regularization limits of the same expression, which clarifies the origin of their disagreement.