Complexity and Momentum

TL;DR

The paper extends the known links between operator growth, complexity, and radial momentum from $AdS_2$/$SYK$ to higher dimensions, introducing a vacuum AdS regime and a gluon-splitting toy model to describe operator growth under perturbations. It shows that operator size grows linearly in time in vacuum and that complexity growth matches this behavior via both circuit-time and complexity-volume dualities, yielding $\mathcal{C}(t)\approx \mathcal{C}_0+2\pi E t$ in the vacuum regime. It then proposes and tests an empirical relation $\frac{\lambda}{2\pi}\frac{d\mathcal{C}}{dt}=2\pi P$ (with $\lambda$ the boundary wavelength) that links complexity growth to bulk radial momentum, finding consistency from the cutoff boundary to after scrambling in both vacuum AdS and BTZ black holes. The work offers a unified perspective on UV–IR flow, operator growth, and bulk dynamics, while inviting a first-principles derivation of the central empirical relation.

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.

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