Complexity and Momentum

TL;DR

This paper explores the relationship between operator size, complexity, and bulk radial momentum in higher-dimensional holographic models, using a toy model and empirical formulas to extend previous lower-dimensional findings.

Contribution

It introduces a higher-dimensional generalization of the connections between operator growth, complexity, and radial momentum, supported by circuit analysis and empirical formulas.

Findings

Operator growth relates to complexity increase rate.

Empirical formula links complexity with bulk radial momentum.

Connections extend from JT gravity and SYK to higher dimensions.

Abstract

Previous work has explored the connections between three concepts -- operator size, complexity, and the bulk radial momentum of an infalling object -- in the context of JT gravity and the SYK model. In this paper we investigate the higher dimensional generalizations of these connections. We use a toy model to study the growth of an operator when perturbing the vacuum of a CFT. From circuit analysis we relate the operator growth to the rate of increase of complexity and check it by complexity-volume duality. We further give an empirical formula relating complexity and the bulk radial momentum that works from the time that the perturbation just comes in from the cutoff boundary, to after the scrambling time.