A new diagrammatic representation for correlation functions in the in-in formalism

TL;DR

The paper addresses the computation of in-in correlation functions in cosmology, where loop corrections on the closed-time-path structure are challenging. It develops a perturbative scheme based on solving the operator equation of motion with retarded propagators $G_R$, yielding a local, tree-like diagrammatic expansion that is equivalent to standard in-in results but uses fewer graphs. A graphical representation with modified Feynman-like rules and generalized anticommutators is introduced, enabling automated, order-by-order construction of all contributions. This approach improves computational efficiency and clarity of physical interpretation, with particular usefulness for inflationary perturbations and non-Gaussianity studies.

Abstract

In this paper we provide an alternative method to compute correlation functions in the in-in formalism, with a modified set of Feynman rules to compute loop corrections. The diagrammatic expansion is based on an iterative solution of the equation of motion for the quantum operators with only retarded propagators, which makes each diagram intrinsically local (whereas in the standard case locality is the result of several cancellations) and endowed with a straightforward physical interpretation. While the final result is strictly equivalent, as a bonus the formulation presented here also contains less graphs than other diagrammatic approaches to in-in correlation functions. Our method is particularly suitable for applications to cosmology.