Operator Size for Holographic Field Theories
arXiv:1911.05089
Abstract
We formulate a state-dependent definition of operator size that captures the effective size of an operator acting on a reference state. We apply our definition to the SYK model and holographic 2-dimensional CFTs, generalizing the Qi-Streicher formula to a large class of geometries which includes pure AdS and BTZ black holes. In pure AdS, the operator size is proportional to the global Hamiltonian at leading order in , mirroring the results of Lin-Maldacena-Zhao in AdS. For BTZ geometries, it is given by the sum of the Kruskal momenta. Higher corrections become relevant when backreaction gets large, and we expect a transition in the growth pattern that depends on the transverse profile of the excitation. We propose a bulk dual that captures this profile dependence and exhibits saturation at a size of order the black hole entropy. This bulk dual is an averaged eikonal phase over a class of scattering events, and it can be interpreted as the "number of virtual gravitons" in the gravitational field created by an infaller.
arXiv comment60 pages, 5 figures
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