Commutative Evolution Laws in Holographic Cellular Automata: AdS/CFT, Near-Extremal D3-Branes, and a Deep Learning Approach

TL;DR

This paper investigates how commutative evolution laws in holographic cellular automata relate to AdS/CFT, near-extremal D3-branes, and uses deep learning to deduce spatial evolution laws from known laws and symmetry constraints.

Contribution

It introduces a framework connecting commutative evolution laws in holographic CAs with AdS/CFT and develops a deep learning method to infer spatial evolution laws from known temporal laws.

Findings

Demonstrates the necessity of curvature encoding in spatial evolution laws.

Proposes a deep learning algorithm to deduce spatial laws from temporal laws.

Shows how Poincaré symmetry restoration relates to infinite torus limits.

Abstract

According to 't Hooft, restoring Poincar\'e invariance in a holographic cellular automaton (CA) requires two distinct evolution laws that commute. We explore how this is realized in the AdS/CFT framework, assuming commutativity as a fundamental principle--much like general covariance once did--for encoding curvature. In our setup, physical processes in a given spacetime are encoded in a CA; to preserve Poincar\'e symmetry, the spacetime curvature must effectively vanish, so we consider a near-extremal black D3-brane solution, in which both the stretched horizon and the conformal boundary are approximated by Minkowski space. AdS/CFT implies a spatial evolution law connecting these hypersurfaces. Commutativity means the final state does not depend on the order of time evolution on each hypersurface and spatial evolution between them, forcing the time evolution law on the horizon and boundary to coincide. To satisfy all these conditions, we aim to demonstrate that the spatial evolution law inevitably encapsulates the curvature of the bulk, including quantum effects. For a computational model, we compactify the hyperplanes to tori, reducing the degrees of freedom to a finite number; taking these tori to infinite size then restores Poincar\'e symmetry. We propose a deep learning algorithm that, given a known time evolution law and commutativity, deduces the corresponding spatial evolution law.