Abstract
In [1] we gave a precise holographic calculation of chaos at the scrambling time scale. We studied the influence of a small perturbation, long in the past, on a two-sided correlation function in the thermofield double state. A similar analysis applies to squared commutators and other out-of-time-order one-sided correlators [2-6]. The essential bulk physics is a high energy scattering problem near the horizon of an AdS black hole. The above papers used Einstein gravity to study this problem; in the present paper we consider stringy and Planckian corrections. Elastic stringy corrections play an important role, effectively weakening and smearing out the development of chaos. We discuss their signature in the boundary field theory, commenting on the extension to weak coupling. Inelastic effects, although important for the evolution of the state, leave a parametrically small imprint on the correlators that we study. We briefly discuss ways to diagnose these small corrections, and we propose another correlator where inelastic effects are order one.
Highlights
The quantum consequences of these large boosts have been the object of extensive study
We studied the influence of a small perturbation, long in the past, on a twosided correlation function in the thermofield double state
Our analysis began with the thermofield double state (TFD) of two CFTs ‘L’ and ‘R’, dual to the eternal AdS Schwarzschild black hole [32, 33]
Summary
We would like to place the problem of computing φLφR W in a slightly more general context, and to introduce some new notation. (The discussion here has some overlap with [34].) The case we originally considered, in [1], was a two-sided correlation function of the form. (For simplicity of notation, we will assume the operators are Hermitian and that they have vanishing one-point functions.) In a suitably chaotic system, we expect correlation functions of this type to become small at large t, regardless of the specific choice of V, W. This is supported by the analysis of [1]. This implies that the behavior of the squared commutator [W (t), V ]2 is determined by (1.3) The fact that this correlator becomes small at late times indicates that the squared commutator becomes large, of order V V W W. See [34] for a recent explanation of the necessary continuation
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