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\leq 4n≤4) to exemplify the general statements.
Highlights
Our discussion may be seen as a natural generalization of the Schwinger-Keldysh formalism to higher OTO correlation functions
That we have identified the four classes of correlation functions that we will deal with, let us summarize the basic set of statements relating them to each other
The elements of the Wightman basis are obtained by analytically continuing τi → i ti +εi, with εi ordered according to the permutation of interest, viz., Gσ(t1, t2, · · ·
Summary
Euclidean quantum field theories are completely defined by their vacuum correlation functions, sometimes referred to as Schwinger functions [1]. Expanding out a Heisenberg operator using the Baker-Campbell-Hausdorff formula, we will note a series of nested commutators, which can be taken to be a proxy for ever increasing complexity of the precursor operator [13] Motivated by this intuition, [14] studied the behaviour of precursors and higher out-oftime-order correlation functions, as a diagnostic of quantum chaos in the context of black hole physics and holography. Their primary goal was to understand how black holes scramble information.. Some useful technical steps which aid our analysis are collected in the Appendices
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