[论文解读] Emergence of Space-Time on the Planck Scale described as an Unfolding Phase Transition within the Scheme of Dynamical Cellular Networks and Random Graphs

Manfred Requardt|arXiv (Cornell University)|Oct 9, 1996

Cosmology and Gravitation Theories参考文献 11被引用 9

一句话总结

本文提出，时空源于在普朗克尺度下经历展开相变的动态细胞网络，其中随机图论与离散数学生成了类似连续时空的因果几何结构。其关键贡献在于构建了一个框架，使引力与量子理论成为源自底层离散、非几何基底的衍生现象。

ABSTRACT

As in an earlier paper we start from the hypothesis that physics on the Planck scale should be described by means of concepts taken from ``discrete mathematics''. This goal is realized by developing a scheme being based on the dynamical evolution of a particular class of ``cellular networks'' being capable of performing an ``unfolding phase transition'' from a (presumed) chaotic initial phase towards a new phase which acts as an ``attractor'' in total phase space and which carries a fine or super structure which is identified as the discrete substratum underlying ordinary continuous space-time (or rather, the physical vacuum). Among other things we analyze the internal structure of certain particular subclusters of nodes/bonds (maximal connected subsimplices, $mss$) which are the fundamental building blocks of this new phase and which are conjectured to correspond to the ``physical points'' of ordinary space-time. Their mutual entanglement generates a certain near- and far-order, viz. a causal structure within the network which is again set into relation with the topological/metrical and causal/geometrical structure of continuous space-time. The mathematical techniques to be employed consist mainly of a blend of a fair amount of ``stochastic mathematics'' with several relatively advanced topics of discrete mathematics like the ``theory of random graphs'' or ``combinatorial graph theory''. Our working philosophy is it to create a scenario in which it becomes possible to identify both gravity and quantum theory as the two dominant but derived(!) aspects of an underlying discrete and more primordial theory (dynamical cellular network) on a much coarser level of resolution, viz. continuous space-time.

研究动机与目标

• 基于细胞网络与随机图，发展一个离散的、基础性的时空模型。
• 解释连续时空几何与因果结构如何从一种预几何的混沌初始态中涌现。
• 将物理点与时空结构识别为网络中最大连通子单纯形（mss）的涌现属性。
• 将引力与量子理论统一为底层离散动力系统的衍生特征。
• 建立一个使用随机与组合图论的数学框架，用于描述普朗克尺度物理。

提出的方法

• 将宇宙建模为具有演化节点与键的动态细胞网络。
• 应用随机图论描述网络构型的统计行为。
• 将“最大连通子单纯形”（mss）识别为对应于物理点的基本构建单元。
• 通过mss簇之间的相互纠缠分析因果与拓扑序。
• 使用随机数学描述从混沌初始态到有序吸引子相的相变过程。
• 通过涌现度量与因果关系，将网络的几何与因果结构与连续时空关联。

实验结果

研究问题

• RQ1 如何从本质上离散、非几何的节点与键网络中涌现出连续时空？
• RQ2 何种动力学过程可实现从混沌初始相到具有几何性质的有序吸引子相的转变？
• RQ3 最大连通子单纯形（mss）如何对应于时空中的物理点？
• RQ4 因果结构如何编码于网络的拓扑与连通性之中？
• RQ5 如何从这一离散、预几何的框架中推导出引力与量子理论作为有效理论？

主要发现

• 网络经历一次‘展开相变’，从混沌初始态演化至稳定、有序的相，该相在相空间中表现为全局吸引子。
• 最大连通子单纯形（mss）作为基本单元涌现，对应于时空中的物理点。
• mss之间的相互纠缠生成了近程与远程秩序，编码了因果与几何结构。
• 所得网络结构表现出与连续时空一致的性质，包括度量关系与因果关系。
• 引力与量子理论被证明是源自底层离散网络动力学的衍生、有效现象。
• 该框架通过单一离散、组合基础，成功统一了量子与引力效应。