Commutative Evolution Laws in Holographic Cellular Automata: AdS/CFT, Near-Extremal D3-Branes, and a Deep Learning Approach
This work provides a theoretical framework for understanding how spacetime curvature and Poincaré symmetry can be encoded within holographic cellular automata, which is significant for researchers in theoretical physics and quantum gravity.
This paper explores how two commuting evolution laws, as proposed by 't Hooft for restoring Poincaré invariance in holographic cellular automata, are realized within the AdS/CFT framework. By assuming commutativity as a fundamental principle for encoding curvature, the authors demonstrate that the spatial evolution law connecting the stretched horizon and conformal boundary of a near-extremal black D3-brane solution inevitably encapsulates the bulk's curvature, including quantum effects.
According to 't Hooft, restoring Poincaré 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 Poincaré 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 Poincaré symmetry. We propose a deep learning algorithm that, given a known time evolution law and commutativity, deduces the corresponding spatial evolution law.