关于因果集合论的公理
On the Axioms of Causal Set Theory
摘要：
本文对离散引力中的因果集程序提出了改进建议，该程序将时空几何视为基本尺度上因果结构的一种紧急表现。这个观点，我称之为因果度量假说，可以用拉斐尔·索尔金的短语来概括，“顺序加上数字等于几何。“建议的改进包括承认一个一般非传递的因果关系比因果顺序更基本，一个改进的局部图片的因果结构，发展和使用关系空间方法，和一个新的背景独立版本的历史方法的量子理论。除了因果集合论，如邦贝利、李、迈耶和索尔金，这项工作还借鉴了伊沙姆的物理学拓扑理论框架、索尔金的量子测度理论、芬克尔斯坦的因果网络和格罗滕迪克的结构原理。这种方法避免了因果集理论中不希望有的结构特征，例如Major、Rideout和Surya研究的最大反链的渗透性，以及Kleitman和Rothschild的渐近枚举所产生的配置空间病理学。本文结合因果度规假设、量子理论的历史方法和Grothendieck的相对论观点，以相关历史理论和运动学方案为高潮。这导致了因果薛定谔方程作为离散量子时空的动力学定律的推导。
英文摘要：
This paper offers suggested improvements to the causal sets program in discrete gravity, which treats spacetime geometry as an emergent manifestation of causal structure at the fundamental scale. This viewpoint, which I refer to as the causal metric hypothesis, is summarized by Rafael Sorkin's phrase, "order plus number equals geometry." Proposed improvements include recognition of a generally nontransitive causal relation more fundamental than the causal order, an improved local picture of causal structure, development and use of relation space methods, and a new background-independent version of the histories approach to quantum theory. Besides causal set theory, à la Bombelli, Lee, Meyer, and Sorkin, this effort draws on Isham's topos-theoretic framework for physics, Sorkin's quantum measure theory, Finkelstein's causal nets, and Grothendieck's structural principles. This approach circumvents undesirable structural features in causal set theory, such as the permeability of maximal antichains, studied by Major, Rideout, and Surya, and the configuration space pathology arising from the asymptotic enumeration of Kleitman and Rothschild. The paper culminates in the theory of co-relative histories and kinematic schemes, combining the causal metric hypothesis, the histories approach to quantum theory, and Grothendieck's relative viewpoint. This leads to the derivation of causal Schrödinger-type equations as dynamical laws for discrete quantum spacetime.
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