# The In-Out Formalism for In-In Correlators

**Authors:** Yaniv Donath, Enrico Pajer  
**arXiv:** [2402.05999](https://arxiv.org/abs/2402.05999)  
**Date:** 2024-02-08  
**Summary page:** https://www.summarizepaper.com/en/paper/2402.05999v1/

## Key points

- Cosmological correlators are essential observables of the primordial universe
- In-out formalism can be used to effectively compute correlators for non-dissipative closed systems
- Simplifies calculation process by reducing in-in propagators to a single time-ordered propagator
- Applicable to fields of any mass and spin under the assumption of no infrared divergences
- Key applications include representation of correlators through a sum over residues, algebraic recursion relation for computing Minkowski correlators, and deriving cutting rules from Veltman's largest time equation
- Enables a natural definition of a de Sitter scattering matrix with examples satisfying standard constraints
- Comparison between in-in and in-out formalisms reveals equivalence under specific conditions
- Development of in-out formalism opens up new avenues for research in computing unequal time correlators across various cosmological spacetimes

## Summary

In this study, we delve into the realm of cosmological correlators, which are essential observables of the primordial universe. Over the past two decades, these correlators have been extensively explored using the in-in formalism initially developed by Schwinger and Keldysh for dissipative open systems. Interestingly, while most applications in cosmology have focused on non-dissipative closed systems, we introduce a novel approach by demonstrating that for such systems, correlators can be effectively computed using the in-out formalism with familiar Feynman rules. By utilizing the in-out formalism, we simplify the calculation process by reducing the myriad of in-in propagators to a single time-ordered propagator without requiring a sum over vertex labelling. This approach is particularly relevant in de Sitter spacetime where we extend the expanding Poincaré patch with a contracting patch to prepare the bra from the future. Our results are applicable to fields of any mass and spin under the assumption of no infrared divergences. We present three key applications of the in-out formalism: a representation of correlators through a sum over residues of Feynman propagators in the energy-momentum domain; an algebraic recursion relation for computing Minkowski correlators based on lower-order ones; and deriving cutting rules from Veltman's largest time equation for two-vertex diagrams up to all loop orders. Furthermore, our formalism enables a natural definition of a de Sitter scattering matrix, which is discussed through simple examples and shown to satisfy standard optical theorem and positivity constraints. The comparison between in-in and in-out formalisms reveals their equivalence under specific conditions, providing insights into simplifying calculations while retaining accuracy. The development of an in-out formalism for computing unequal time correlators across various cosmological spacetimes opens up new avenues for research. By defining a scattering S-matrix in de Sitter akin to Minkowski space, exciting possibilities emerge for further exploration and analysis. Overall, our study contributes to advancing understanding and computation techniques within cosmological correlator studies.