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.
Highlights
Understanding thermal equilibrium in quantum systems is a necessary precursor to developing a comprehensive physical picture of out-of-equilibrium dynamics
We have primarily focused on synthesizing known features of thermal correlation functions and arguing that they are best understood in the space of out-of-time-order (OTO) observables
The KMS condition with time reversal or a CPT transformation to restore operator ordering, the generalization as discussed allows for a simpler interpretation
Summary
Understanding thermal equilibrium in quantum systems is a necessary precursor to developing a comprehensive physical picture of out-of-equilibrium dynamics. A remarkable fact about thermal states is that the real-time response functions are related to fluctuations about equilibrium thanks to the fluctuation-dissipation relations, which in turn follow from the Kubo-Martin-Schwinger (KMS) relations [1, 2]. The current work is aimed at synthesizing these developments by attempting to form a coherent picture that transcends the limitations of the Schwinger-Keldysh construction To appreciate this perspective, let us first note that the general FDT arises from the fact that thermal correlators are trace class observables in the Gibbs density matrix. The proper OTO number refers to the minimal number of forward-backward evolutions necessary to account for the time-ordering in the correlation function This set of relations has the effect of reducing the switchbacks in the path integral contours necessary for computing thermal Wightman functions. For completeness we check that the thermal identities we derive are consistent with the relations obtained in the thermal quantum field theory literature (appendix A)
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