# Classification of out-of-time-order correlators

### Submission summary

 As Contributors: Mukund Rangamani Arxiv Link: https://arxiv.org/abs/1701.02820v4 (pdf) Date accepted: 2018-12-19 Date submitted: 2018-12-11 01:00 Submitted by: Rangamani, Mukund Submitted to: SciPost Physics Academic field: Physics Specialties: High-Energy Physics - Theory Quantum Physics Approach: Theoretical

### 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.

### Ontology / Topics

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Published as SciPost Phys. 6, 001 (2019)

### Author comments upon resubmission

We thank the referee for their comments. We have made some changes as requested. these are detailed in the "List of Changes" below.

### List of changes

The following changes have been made in the new version of the manuscript to incorporate the suggestions of the referee. The numbering refers to the referee's comments at the end of their report.

Re 1) We have moved some of the very mathematical details from three different parts of the paper to a new appendix (C). We also put boxes around some of the important results in the main text to highlight the essential points. We left the derivation of Eq. (5.20) in the main text since it is mostly about building intuition for the contours rather than complicated mathematics (we did, however, put a suggestion to skip to the main result, on page 33 before the derivation starts).

We did not move the example section (section 7) to the beginning of the paper. We believe that sections 1 and 2 are quite pedagogical and provide an extensive introduction to the subject. Furthermore, the way section 7 is written is largely motivated by the results of the paper, so we feel it is more appropriate to have it at the end rather than at the beginning.

Re 2) We have now included a citation to this paper as suggested.

Re 3) We have now spelled out w.l.o.g.'' everywhere (however, please see https://en.wikipedia.org/wiki/Without_loss_of_generality where the usage is noted as standard).