# Classification of out-of-time-order correlators

## TL;DR

This paper explores the structure and computation of higher out-of-time-order correlators using generalized path integral contours called timefolds, extending the Schwinger-Keldysh formalism to more complex time-orderings.

## Contribution

It provides a detailed framework for understanding and computing higher out-of-time-order correlators through generalized contours, expanding the Schwinger-Keldysh approach.

## Key findings

- Explicitly illustrates low point correlators (n=2,3,4)
- Describes the structure of higher OTO functional integrals
- Generalizes the Schwinger-Keldysh formalism

## Abstract

The space of n-point correlation functions, for all possible time-orderings of operators, can be computed by a non-trivial path integral contour, which depends on how many time-ordering violations are present in the correlator. These contours, which have come to be known as timefolds, or out-of-time-order (OTO) contours, are a natural generalization of the Schwinger-Keldysh contour (which computes singly out-of-time-ordered correlation functions). We provide a detailed discussion of such higher OTO functional integrals, explaining their general structure, and the myriad ways in which a particular correlation function may be encoded in such contours. Our discussion may be seen as a natural generalization of the Schwinger-Keldysh formalism to higher OTO correlation functions. We provide explicit illustration for low point correlators (n=2,3,4) to exemplify the general statements.