Classification of out-of-time-order correlators

TL;DR

This work develops a comprehensive framework for higher out-of-time-order (OTO) correlation functions via $k$-OTO timefold path integrals, generalizing the Schwinger-Keldysh formalism to arbitrary time-orderings. It introduces a generating functional $\\mathscr{Z}_{k-oto}$ and organizes observables into four bases—Wightman, nested correlators, LR correlators, and Av-Dif correlators—then derives a lattice of relations among them, including generalized sJacobi identities and extended Keldysh rules. The authors prove that any $n$-point function can be expressed in a canonical way using proper $q$-OTO contours with degeneracies described by $g_{n,q}$ and $h^{(q)}_{n,k}$, and provide explicit low-point examples and generating functions to enumerate these embeddings. The results offer a systematic, algebraic/combinatorial toolkit for computing and relating all Lorentzian $n$-point OTO correlators, with implications for chaos diagnostics, holography, BRST structure, and real-time quantum field theory perturbation theory.

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.

Figures (10)

• Figure 1: The timefolded contour necessary to compute the correlator with temporal ordering $t_1 > t_2$, $t_2< t_3$, $t_3 > t_4$ and $t_4 < t_5$. In the above we have drawn the time running upwards, but soon we will switch to a notation where forward evolution runs left to right.
• Figure 2: The k-OTO contour computing the out-of-time-ordered correlation functions encoded in the generating functional \ref{['eq:koto']}. (a) The contour drawn makes explicit the notion of depth; the segments with are nested inwards in order of increasing depth which is equivalent to the distance from the density matrix. (b) An alternate way of drawing the contour as e.g., used in Haehl:2016pec with contours of increasing depth going outwards from a central region. The second can be obtained from the first by turning the switchbacks inside-out.
• Figure 3: The allowed flips (denoted by vertical dashed lines) that give rise to $2^{2q-1}$ choices of placing the turning-point operators. There are $q$ future turning-point operators and $q-1$ past turning-point operators. Each of them can be chosen to come before or after the turning-point leading to an irreducible degeneracy of $2^{2q-1}$ contour correlators which evaluate to same single time correlator. In the 'canonical' arrangement we fix these $2^{2q-1}$ choices by demanding that the turning-point operators be placed always before the turning-point.
• Figure 4: Arranging $(k-q)$ empty timefolds between $(2q-1)$ turning-point operators (which are in the canonical arrangement). We have labeled the $i^{th}$ turning-point operator and the $j^{th}$ timefold is denoted by $(j)$. For the case of minimal $n$, i.e., $n=2q-1$ with all operators being turning-point operators, the counting reduces to the number of ways $(k-q)$ empty timefolds can be put into $2q$ boxes created by the legs with turning-point operators (and the last empty leg). This gives $\frac{(k-q+2q-1)!}{(k-q)! (2q-1)!}= \frac{(k+q-1)!}{(k-q)! (2q-1)!}$ number of ways of arranging empty timefolds.
• Figure 5: Wing spread $(w=1)$ configurations of the basic wing shown in fig (g) with the wing positions marked for every wing operator. Note that the future turning-point operator is always placed before the future turning-point,i.e., in canonical arrangement. Each wing configuration is completely specified by wing positions $\{x_1,x_2,x_3\}$ of the three wing operators. The wing-spread can be computed from the wing positions by using the formula $w=|x_1|+|x_2|+|x_3|$.