# Thermal out-of-time-order correlators, KMS relations, and spectral functions

## TL;DR

This paper explores the structure of thermal correlation functions, especially out-of-time-order correlators, deriving relations from the KMS condition and constructing a causal basis for thermal n-point functions.

## Contribution

It introduces a novel formulation of KMS relations directly in terms of out-of-time-ordered correlators and constructs a causal basis for thermal n-point functions.

## Key findings

- KMS relations can be expressed using OTO correlators.
- A causal basis for thermal n-point functions is constructed.
- Explicit example with a quantum oscillator demonstrates the abstract results.

## Abstract

We describe general features of thermal correlation functions in quantum systems, with specific focus on the fluctuation-dissipation type relations implied by the KMS condition. These end up relating correlation functions with different time ordering and thus should naturally be viewed in the larger context of out-of-time-ordered (OTO) observables. In particular, eschewing the standard formulation of KMS relations where thermal periodicity is combined with time-reversal to stay within the purview of Schwinger-Keldysh functional integrals, we show that there is a natural way to phrase them directly in terms of OTO correlators. We use these observations to construct a natural causal basis for thermal n-point functions in terms of fully nested commutators. We provide several general results which can be inferred from cyclic orbits of permutations, and exemplify the abstract results using a quantum oscillator as an explicit example.