Thermal out-of-time-order correlators, KMS relations, and spectral functions. (arXiv:1706.08956v3 [hep-th] UPDATED)

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.

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