Spectral Representation of Thermal OTO Correlators

TL;DR

This paper develops a spectral representation framework for finite temperature out-of-time-ordered correlators on multi-time-fold contours, extending classical spectral function results to the OTO case.

Contribution

It introduces a basis of column vectors for Wightman correlators and expresses contour-ordered correlators as sums over permutations, generalizing spectral functions to OTO correlators.

Findings

Decomposition of Wightman array into basis vectors

Expression of correlators as sums over permutations

Generalized spectral functions as Fourier transforms of nested commutators

Abstract

We study the spectral representation of finite temperature, out of time ordered (OTO) correlators on the multi-time-fold generalised Schwinger-Keldysh contour. We write the contour-ordered correlators as a sum over time-order permutations acting on a funda- mental array of Wightman correlators. We decompose this Wightman array in a basis of column vectors, which provide a natural generalisation of the familiar retarded-advanced basis in the finite temperature Schwinger-Keldysh formalism. The coefficients of this de- composition take the form of generalised spectral functions, which are Fourier transforms of nested and double commutators. Our construction extends a variety of classical results on spectral functions in the SK formalism at finite temperature to the OTO case.