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Manifestly Causal In-In Perturbation Theory about the Interacting Vacuum

Matthew Baumgart, Raman Sundrum

TL;DR

The paper addresses the clash between the $i\epsilon$-prescription used to project the free vacuum onto the interacting vacuum in in-in perturbation theory and Weinberg's causality-focused nested-commutator formulation. It introduces a manifestly unitary $\epsilon$-deformation, validates its equivalence to the standard approach in Minkowski space, and extends the argument to De Sitter with BD vacuum, demonstrating that the unitary deformation reproduces the same perturbative correlators while preserving causality. Key contributions include a complete perturbative equivalence between the standard in-in expansion and the nested-commutator form, careful treatment of vacuum bubbles, and a De Sitter generalization that maintains a causal, unitary structure. This reconciles vacuum projection with strict causality in cosmological perturbation theory and clarifies the role of the $\epsilon$-prescription in in-in calculations for primordial correlations.

Abstract

In-In perturbation theory is a vital tool for cosmology and nonequilibrium physics. Here, we reconcile an apparent conflict between two of its important aspects with particular relevance to De Sitter/inflationary contexts: (i) the need to slightly deform unitary time evolution with an i*epsilon prescription that projects the free ("Bunch-Davies") vacuum onto the interacting vacuum and renders vertex integrals well-defined, and (ii) Weinberg's "nested commutator" reformulation of in-in perturbation theory which makes manifest the constraints of causality within expectation values of local operators, assuming exact unitarity. We show that a modified i*epsilon prescription maintains the exact unitarity on which the derivation of (ii) rests, while nontrivially agreeing with (i) to all orders of perturbation theory.

Manifestly Causal In-In Perturbation Theory about the Interacting Vacuum

TL;DR

The paper addresses the clash between the -prescription used to project the free vacuum onto the interacting vacuum in in-in perturbation theory and Weinberg's causality-focused nested-commutator formulation. It introduces a manifestly unitary -deformation, validates its equivalence to the standard approach in Minkowski space, and extends the argument to De Sitter with BD vacuum, demonstrating that the unitary deformation reproduces the same perturbative correlators while preserving causality. Key contributions include a complete perturbative equivalence between the standard in-in expansion and the nested-commutator form, careful treatment of vacuum bubbles, and a De Sitter generalization that maintains a causal, unitary structure. This reconciles vacuum projection with strict causality in cosmological perturbation theory and clarifies the role of the -prescription in in-in calculations for primordial correlations.

Abstract

In-In perturbation theory is a vital tool for cosmology and nonequilibrium physics. Here, we reconcile an apparent conflict between two of its important aspects with particular relevance to De Sitter/inflationary contexts: (i) the need to slightly deform unitary time evolution with an i*epsilon prescription that projects the free ("Bunch-Davies") vacuum onto the interacting vacuum and renders vertex integrals well-defined, and (ii) Weinberg's "nested commutator" reformulation of in-in perturbation theory which makes manifest the constraints of causality within expectation values of local operators, assuming exact unitarity. We show that a modified i*epsilon prescription maintains the exact unitarity on which the derivation of (ii) rests, while nontrivially agreeing with (i) to all orders of perturbation theory.

Paper Structure

This paper contains 3 sections, 29 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Minkowski Space
  3. De Sitter

Figures (1)

  • Figure 1: An in-in diagram with all $G^{\pm\pm}$ propagators where we demonstrate the structure of contributions to Eq. \ref{['eq:intschemev']}. The coarse, purple line is a partition that divides vertices into those whose time argument evaluates to the final correlation time, $\bar{t}$, from those set to the early-time cutoff, $T$. Since the original integral (Eq. \ref{['eq:intschem']}) is time-ordered, this dividing line occurs between the $i^{\rm th}$ and $(i+1)^{\rm th}$ vertices. Here we show it in a four vertex graph for $i=2$. The red labels on some propagators show the $\epsilon$ dependence, 1 or $e^{\epsilon T}$, that results from deformation ( cf. Eq. \ref{['eq:gsets']}). We see that the factor is 1 if the propagator connects two vertices on the same side of the partition. If, however, a propagator straddles the partition, then there is a suppressing $e^{\epsilon T}$. Furthermore, because the propagators that connect to the external correlation point have one end evaluated at $\bar{t}$ even before integration, the external correlation points are always above the partition line; the only way to avoid suppression is if every vertex evaluates to $\bar{t}$ (corresponding to the partition drawn below the entire graph and cutting no propagators), which is precisely the surviving contribution in Eq. \ref{['eq:intschemwd']}. We note that this straddler argument fails for the case of a vacuum bubble, as all vertices can evaluate to $T$ without any suppression, giving a finite contribution.