Operator Size for Holographic Field Theories

TL;DR

This paper introduces a state-dependent operator size in holographic theories, applies it to SYK and AdS3 geometries, and proposes a bulk dual capturing size growth, saturation, and backreaction effects.

Contribution

It generalizes the operator size concept to holographic CFTs, linking it to bulk scattering phases and black hole entropy, with new formulas for different geometries.

Findings

Operator size proportional to global Hamiltonian in AdS3 at leading order.

In BTZ geometries, size is given by the sum of Kruskal momenta.

Size growth saturates at the black hole entropy scale.

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$_3$ and BTZ black holes. In pure AdS$_3$, the operator size is proportional to the global Hamiltonian at leading order in $1/N$, mirroring the results of Lin-Maldacena-Zhao in AdS$_2$. For BTZ geometries, it is given by the sum of the Kruskal momenta. Higher $1/N$ 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.