Recent studies of out-of-time ordered thermal correlation functions (OTOC) in
holographic systems and in solvable models such as the Sachdev-Ye-Kitaev (SYK)
model have yielded new insights into manifestations of many-body chaos. So far
the chaotic behavior has been obtained through explicit calculations in
specific models. In this paper we propose a unified description of the
exponential growth and ballistic butterfly spreading of OTOCs across different
systems using a newly formulated "quantum hydrodynamics," which is valid at
finite ℏ and to all orders in derivatives. The scrambling of a generic
few-body operator in a chaotic system is described as building up a
"hydrodynamic cloud," and the exponential growth of the cloud arises from a
shift symmetry of the hydrodynamic action. The shift symmetry also shields
correlation functions of the energy density and flux, and time ordered
correlation functions of generic operators from exponential growth, while leads
to chaotic behavior in OTOCs. The theory also predicts an interesting
phenomenon of the skipping of a pole at special values of complex frequency and
momentum in two-point functions of energy density and flux. This pole-skipping
phenomenon may be considered as a "smoking gun" for the hydrodynamic origin of
the chaotic mode. We also discuss the possibility that such a hydrodynamic
description could be a hallmark of maximally chaotic systems.Comment: 48 pages, 9 figures. v2: references added, various clarifications
made including an expanded discussion of predictions in the introduction and
an expanded discussion of four-point functions, v3: journal versio
