全息元胞自动机中的对易演化定律：AdS/CFT、近极端D3-膜与深度学习方法
Commutative Evolution Laws in Holographic Cellular Automata: AdS/CFT, Near-Extremal D3-Branes, and a Deep Learning Approach
AI总结：
本文研究全息元胞自动机中恢复庞加莱不变性所需的对易演化定律，利用近极端D3-膜使视界与边界近似为闵可夫斯基空间，并提出深度学习算法从已知时间演化定律推导空间演化定律。
摘要：
根据't Hooft的观点，在全息元胞自动机（CA）中恢复庞加莱不变性需要两个互相对易的不同演化定律。我们探索了在AdS/CFT框架中如何实现这一点，假设对易性作为编码曲率的基本原理——就像曾经的广义协变性一样。在我们的设置中，给定时空中的物理过程被编码在一个CA中；为了保持庞加莱对称性，时空曲率必须有效消失，因此我们考虑一个近极端黑洞D3-膜解，其中拉伸视界和共形边界都近似为闵可夫斯基空间。AdS/CFT暗示了一个连接这些超曲面的空间演化定律。对易性意味着最终状态不依赖于每个超曲面上的时间演化与它们之间的空间演化的顺序，这迫使视界和边界上的时间演化定律一致。为了满足所有这些条件，我们旨在证明空间演化定律不可避免地包含了体时空的曲率，包括量子效应。对于计算模型，我们将超平面紧致化为环面，将自由度减少到有限数量；然后将这些环面取为无限尺寸以恢复庞加莱对称性。我们提出了一种深度学习算法，在给定已知时间演化定律和对易性的情况下，推导出相应的空间演化定律。
英文摘要：
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.
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