Title: Extremal variety as the foundation of a cosmological quantum theory Show Chinese title

Title: 极值多样性作为宇宙量子理论的基础

Abstract: Dynamical systems of a new kind are described, which are motivated by the problem of constructing diffeomorphism invariant quantum theories. These are based on the extremization of a non-local and non-additive quantity that we call the variety of a system. In these systems all dynaqmical variables refer to relative coordinates or, more generally, describe relations between particles, so that they are invariant under discrete analogues of diffeomorphisms in which the labels of all particles are permutted arbitrarily. The variety is a measures of how uniquely each of the elements of the system can be distinguished from the others in terms of the values of these relative coordinates. Thus a system with extremal variety is one in which the parts are related to the whole in as distinct a way as possible. We study numerically several dynamical systems which are defined by setting the action of the system equal to its variety. We find evidence that suggests that such systems may serve as the basis for a new kind of pregeometry theories in which the geometry of low dimensional space emerges in the thermodynamic limit from a system which is defined without the use of any background space. The mathematical definition of variety may also provide a quantitative tool to study self-organizing systems, because it distinguishes highly structured, but asymmetric, configurations such as one finds in biological systems from both random configurations and highly ordered configurations. Show Chinese abstract

Abstract: 描述了一类新的动力系统，这些问题的动机来源于构建微分同胚不变量子理论的问题。 这些系统基于一个非局域且非可加的量的极值化，我们称这个量为系统的多样性。 在这些系统中，所有的动力学变量都对应于相对坐标，或者更一般地，描述粒子之间的关系，因此它们在离散版本的微分同胚下具有不变性，在这种情况下所有粒子的标记被任意置换。 多样性是对系统中的每个元素通过这些相对坐标的值来区分其他元素的独特程度的一种度量。 因此，具有极端多样性的系统是一个部分与整体以尽可能不同的方式相关的系统。 我们数值研究了几个由设定系统的作用量等于其多样性来定义的动力系统。 我们发现了一些证据，表明这样的系统可能作为新一类预几何理论的基础，在这种理论中，低维空间的几何性质在热力学极限中从一个没有使用任何背景空间定义的系统中出现。 多样性这一数学定义也可能提供一种定量工具来研究自组织系统，因为它可以区分生物系统中常见的高度结构化但不对称的构型，与随机构型和高度有序构型区分开来。