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Classification of out-of-time-order correlators

by Felix M. Haehl, R. Loganayagam, Prithvi Narayan, Mukund Rangamani

This is not the latest submitted version.

This Submission thread is now published as SciPost Phys. 6, 001 (2019)

Submission summary

As Contributors: Mukund Rangamani
Arxiv Link: (pdf)
Date submitted: 2018-06-07 02:00
Submitted by: Rangamani, Mukund
Submitted to: SciPost Physics
Academic field: Physics
  • High-Energy Physics - Theory
  • Quantum Physics
Approach: Theoretical


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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Correlation functions Schwinger-Keldysh method
Current status:
Has been resubmitted

Submission & Refereeing History

Published as SciPost Phys. 6, 001 (2019)

Resubmission 1701.02820v4 on 11 December 2018

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Submission 1701.02820v3 on 7 June 2018

Reports on this Submission

Anonymous Report 1 on 2018-11-26 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1701.02820v3, delivered 2018-11-26, doi: 10.21468/SciPost.Report.670


1) Nice and clear formalism developed to calculate general out of time ordered correlation functions.

2) Thorough analysis that will likely be of use to the community in the future.


1) Main body of the paper hard to read at the moment. Too many details in the main text, that can otherwise be moved to the appendices.

2) Beyond the formalism developed here (which is useful in its own right; see above), no fundamentally "new" results/insights at the moment.


I have gone through the manuscript by Haehl et al. with a great deal of interest. In this work the authors have formally set up a framework for computing general (higher) OTO correlation functions using a timefolded path integral. I believe that the present manuscript will be of some use to those working on the subject in the future. I will recommend publication of this manuscript in SciPost.

However I do have some questions and comments for the authors that they might want to take into consideration before the paper is formally published.

To the best of my knowledge, most of the field-theoretic computations of the OTO correlation functions work with the "regularized" version of the correlators as opposed to the usual "unregularized" version. The latter object is physically more relevant. If the late time behavior (especially the exponential growth and Lyapunov exponent) are the same for both sets of correlators, then this regularization is harmless. Is this something that the authors can shed some light on using their formalism?

I bring up this question primarily since there has been some recent claim ( that the two sets of correlators can lead to different Lyapunov exponents for a specific problem. Perhaps the authors can use their formalism to comment on what might lead to such differences in a more general setting even at late times?

Requested changes

1) The manuscript, in my opinion, is quite hard to read through at the moment. Even though the formalism and steps involved in the formal manipulations are relatively straightforward, I feel that by including an excessive amount of the technical details in the main text, the authors have made the manuscript harder to read. If the authors could move some of these details to the appendices, I feel that it will be much easier to read the main body of the paper.

On a related note, I think it would have been nicer for the authors to give some examples while setting up their formalism. At the moment, the few examples that they discuss appears in section 7, which is practically at the end of the paper. From the point of view of the young readers and newcomers to the field, I think a restructuring of the paper would be nicer (but is not mandatory).

2) In section 7.2, when the authors discuss the example of the scalar field theory, I think it would be appropriate for them to cite Phys. Rev. D 96, 065005 (2017).

3) A minor comment: I found it quite non-standard to use the acronym "W.l.o.g." for without loss of generality. I don't think it is that standardized and it only appears three times in the text. Perhaps the authors can refrain from using this acronym.

  • validity: high
  • significance: high
  • originality: good
  • clarity: ok
  • formatting: good
  • grammar: perfect

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