Light, Links and Causal Sets
Rafael D. Sorkin
在勾画了一个寻求时空离散性的可观测信号的背景之后，我简要地回顾了量子引力的因果集程序的现状，最后给出了一个简单的模型，用于背景因果集中运动电荷产生的场。
After sketching a context in which to seek observable signals of spatio-temporal discreteness, I briefly review the status of the causal set program for quantum gravity, concluding with a simple model for the field produced by a moving charge in a background causal set.
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Path integral suppression of badly behaved causal sets
本文证明引力路径积分会强烈抑制Kleitman-Rothschild序等最常见的非流形类因果集，从而支持连续时空可从因果集量子理论中自然涌现。
Causal sets and the deep structure of spacetime
量子引力的因果集方法体现了因果性和离散性的概念。本文探讨了因果集理论中的一些基础和概念问题。
Black Hole Entropy as Causal Links
我们将黑洞时空模型化为一个因果集，并在一定的定义下，计算穿过类空或零超曲面$Σ$视界的因果链的数目。我们发现这个数字与$Σ$上的视界面积成正比，从而支持将链接解释为解释其熵的“视界原子”。研究的情况不仅包括平衡的黑洞，但远离平衡。
While it is possible to build causal sets that approximate spacetime manifolds, most causal sets are not at all manifold-like. We show that a Lorentzian path integral with the Einstein-Hilbert action has a phase in which one large class of non-manifold-like causal sets is strongly suppressed, and suggest a direction for generalization to other classes. While we cannot yet show our argument holds for all non-manifold-like sets, our results make it plausible that the path integral might lead to emergent manifold-like behavior with no need for further conditions.
Causal Sets and an Emerging Continuum
本文探讨因果集理论中非类流形因果集在引力路径积分中被强烈抑制，从而向连续时空涌现迈出重要一步。
In this paper we will explore two different proposals for the action for causal sets: the Benincasa-Dowker action and a modified version of the chain action. We propose a variational principle for two-dimensional causal sets and use it for both actions to determine which causal sets at least on average satisfy a discrete version of the Einstein equation. Specifically, we test this method on causal sets embedded in 2d Minkowski, de Sitter, and anti-de Sitter spacetimes and compare these results to the most prominent nonmanifoldlike causal sets, Kleitman-Rothschild causal sets.
Causal Sets, a Possible Interpretation for the Black Hole Entropy,and Related Topics
因果集假说认为，时空最终是离散的，它的基本结构是局部有限的偏序集，宏观因果性反映了更深层次的有序概念，时空的所有几何结构都必须在这个概念中找到它们的最终表达。在回顾了因果集动力学的主要方面，以及最近发展起来的随机动力学。我们集中在宇宙学和黑洞领域的可能影响。在黑洞的背景下，我们提出了一个可能的解释熵的数量的链接跨越视界。
We investigate the spectral dimension obtained from non-local continuum d'Alembertians derived from causal sets. We find a universal dimensional reduction to 2 dimensions, in all dimensions. We conclude by discussing the validity and relevance of our results within the broader context of quantum field theories based on these nonlocal dynamics.
A Classical Sequential Growth Dynamics for Causal Sets
从一定的因果关系条件和离散形式的一般协方差，我们推导出一个非常一般的家庭的经典随机，因果集的顺序增长动力学。由此产生的理论为完全量子引力提供了一个相对容易的“中途之家”，可能包含后者的经典极限（广义相对论）。因为它们可以用伊辛自旋集合的状态模型来表达，而伊辛自旋的集合是基于因果集的关系，所以这些理论也说明了非引力物质如何从因果集动态地产生，而不必在基本水平上建立起来。此外，我们的研究结果带来了一些重要的解释问题的因果集动力学，更一般的量子引力。
Quantum Field Theory On Causal Sets
本文概述固定因果集背景上的物质场论构造，重点介绍 Sorkin-Johnston 形式体系及其从 Green 函数得到唯一两点函数的应用，并讨论相互作用理论与费米子等其他动力学方案。
We investigate the behavior of small subsets of causal sets that approximate Minkowski space in three, four, and five dimensions, and show that their effective dimension decreases smoothly at small distances. The details of the short distance behavior depend on a choice of dimensional estimator, but for a reasonable version of the Myrheim-Meyer dimension, the minimum dimension is $d \approx 2$, reproducing results that have been seen in other approaches to quantum gravity.
We describe numerical and analytical investigations of causal sets sprinkled into spacetime manifolds. The first part of the paper is a numerical study of finite causal sets sprinkled into Alexandrov subsets of Minkowski spacetime of dimensions $1 + 1$, $1 + 2$ and $1 + 3$. In particular we consider the rank 2 past of sprinkled causet events, which is the set of events that are two links to the past. Assigning one of the rank 2 past events as `preferred past' for each event yields a `preferred past structure', which was recently proposed as the basis for a causal set d'Alembertian. We test six…
Synge's world function applied to causal diamonds and causal sets
本文利用Synge世界函数开发了一种新的协变方法，将Poisson撒播因果集的组合学与时空几何联系起来，并直接证明了因果集的BDG作用量在连续极限下给出Einstein-Hilbert作用量。
An Algebraic Quantization of Causal Sets
本文给出了Sorkin等人因果集的一种代数量子化方案。建议的情况是沿着一个类似的代数和量子解释有限拓扑空间由于Zapatrin和这位作者。为了能够适用于后一个程序的因果集索尔金的“语义开关”从“偏序集有限拓扑空间”到“偏序集局部有限因果集”是雇用。其结果是“量子因果集”的定义。这样一个过程及其所产生的定义是物理合理的量子因果集的属性，满足芬克尔斯坦的要求，从“量子因果关系”是一个直接的，以及代数表示，事件之间的关系，为离散局域性的缘故。这里介绍的量子因果集具有这种性质直接使用的结果从代数有限拓扑空间由于Breslav，Parfionov和Zapatrin。
Universal homogeneous causal sets
因果集是一种特殊的偏序集，它是量子引力中离散时空的基本模型。我们证明了所有可数的过去有限因果集的类C包含唯一的泛因果集（U，<）（即，C的任何成员都可以嵌入（U，<）中）并且是均匀的（即，（U，<）具有最大对称度）。此外，（U，<）可以概率地和显式地构造。相反，所有可数因果集的较大类不包含泛对象。
In causal set theory the gravitational path integral is replaced by a path-sum over a sample space $Ω_n$ of $n$-element causal sets. The contribution from non-manifold-like orders dominates $Ω_n$ for large $n$ and therefore must be tamed by a suitable action in the low energy limit of the theory. We extend the work of Loomis and Carlip on the contribution of sub-dominant bilayer orders to the causal set path-sum and show that the "link action" suppresses the dominant Kleitman-Rothschild orders for the same range of parameters.
We explore whether the growth dynamics paradigm of Causal Set Theory is compatible with past-infinite causal sets. We modify the Classical Sequential Growth dynamics of Rideout and Sorkin to accommodate growth "into the past" and discuss what form physical constraints such as causality could take in this new framework. We propose convex-suborders as the "observables" or "physical properties" in a theory in which causal sets can be past-infinite and use this proposal to construct a manifestly covariant framework for dynamical models of growth for past-infinite causal sets.
We present a new method for embedding a causal set into an interval of Minkowski spacetime. The method uses spacetime volumes for causally related elements to define causal set analogs of Minkowski inner products. These are used to construct matrices of inner products which are then factored using the singular value decomposition to give coordinates in Minkowski spacetime. Results are presented showing good quality embeddings into Minkowski spacetime for dimensions d=2,3,4. The method applies in any dimension and does not require spacelike distances to be used as an input. It offers a new way …
The Einstein-Hilbert Action for Entropically Dominant Causal Sets
针对因果集量子引力中分层因果集熵贡献可能主导作用量的问题，证明任意维度下典型K层因果集的离散爱因斯坦-希尔伯特作用量主导阶可简化为简单链接作用量，结合前期工作证实离散标度足够大时分层集会被路径求和极强抑制。
Interacting Scalar Field Theory on Causal Sets
本文在因果集上引入 $\phi^4$ 相互作用的实标量量子场论，扩展规范与路径积分框架，并开发类似费曼图的算法计算相互作用两点函数，且因果性在图中显式体现。
Causal set quantum gravity is a Lorentzian approach to quantum gravity, based on the causal structure of spacetime. It models each spacetime configuration as a discrete causal network of spacetime points. As such, key questions of the approach include whether and how a reconstruction of a sufficiently coarse-grained spacetime geometry is possible from a causal set. As an example for the recovery of spatial geometry from discrete causal structure, the construction of a spatial distance function for causal sets is reviewed. Secondly, it is an open question whether the path sum over all causal se…
We propose a manifestly covariant framework for causal set dynamics. The framework is based on a structure, dubbed covtree, which is a partial order on certain sets of finite, unlabeled causal sets. We show that every infinite path in covtree corresponds to at least one infinite, unlabeled causal set. We show that transition probabilities for a classical random walk on covtree induce a classical measure on the $σ$-algebra generated by the stem sets.
Energetic Causal Sets
我们提出了一种基于能量因果集的量子理论方法，该方法在Cortês和Smolin（2013）中引入。基本过程是因果集合，其事件携带动量和能量，这些动量和能量沿着因果联系传递，并在每个事件中保持不变。能量因果集模型不同于其他时空无关因果集方法，例如参考文献[2]提出的基于量子信息处理系统的因果集，以及参考文献[3]提出的由标准模型粒子构造的因果集。从根本上说，在能量因果集中，因果过程有振幅，但没有时空。因果过程在突现时空中的嵌入只在半经典层次上出现。因此，根本上不存在对易关系，不存在不确定原理，实际上也不存在$\hbar$。量子理论所剩下的就是复振幅的绝对值平方和概率之间的关系。因此，我们发现，无论是局部性，也不是非局部性，是主要的概念，只有因果关系存在于根本水平。
Link-based causal set propagators in $1+1$ dimensions
研究因果集中的延迟标量传播子是否可由链接矩阵L表示，通过渐近分析和数值模拟展示无质量延迟传播子与归一化指数exp(L)的关联，并扩展至有质量情况，与连续传播子一致。
Free Fermions on causal sets
我们构建了一个狄拉克理论的因果集;在建设的一个关键因素是，因果必须被视为紧急在适当的意义了。我们进一步注意到，混合范数空间出现在允许负范数粒子和“鬼”的构造中。
We examine the validity and scope of Johnston's models for scalar field retarded Green functions on causal sets in 2 and 4 dimensions. As in the continuum, the massive Green function can be obtained from the massless one, and hence the key task in causal set theory is to first identify the massless Green function. We propose that the 2-d model provides a Green function for the massive scalar field on causal sets approximated by any topologically trivial 2 dimensional spacetime. We explicitly demonstrate that this is indeed the case in a Riemann normal neighbourhood. In 4-d the model can again …
Do causal sets have symmetries?
本文研究因果集是否具有对称性，通过分析有限偏序集的自同构群来系统识别和分类其对称性，并将一般偏序集与撒点因果集的局部对称性进行比较，以寻找可作为离散时空模型的偏序集。
Dual structures on causal sets called timelets are introduced, being discrete analogs of global time coordinates. Algebraic and geometrical features of the set of timelets on a causal set are studied. A characterization of timelets in terms of incidence matrix of causal set is given. The connection between timelets and preclusive coevents is established, it is shown that any timelet has a unique decomposition over preclusive coevents. The equivalence classes of timelets with respect to reascaling are shown to form a simplicial complex.
Lorentzian Spectral Geometry with Causal Sets
研究离散洛伦兹谱几何，通过因果矩阵导出的算子谱等几何不变量识别因果集，提出两种分类方法并在多达9个元素的因果集上测试，其中结合扰动的新方法能完全区分因果集，并探讨了在量子引力路径积分中的应用。
Effective curvature coupling of link-based causal set propagators in $1+1$ dimensions
研究1+1维中基于链的因果集传播子对曲率的响应，发现其体积依赖性会产生主导曲率修正，大距离下等价于有效曲率耦合为ξR的连续标量传播子，该耦合由微观结构涌现，数值模拟支持预测。
Causal Sets: Discrete Gravity (Notes for the Valdivia Summer School)
这些是2002年1月为智利瓦尔迪维亚的暑期学校准备的关于因果集合论的讲义。遗憾的是，在某些地方，它们比较完整，而在另一些地方则不那么完整。在说明的末尾可以找到一套广泛的参考资料和术语表。
Insights on Entanglement Entropy in $1+1$ Dimensional Causal Sets
本文研究了1+1维因果集上的无质量与有质量标量场纠缠熵，揭示了截断贡献的涨落特性，扩展了Rényi熵和有质量理论的结果，并探讨了更一般设置下的应用意义。
Simpler embeddings of causal sets into Minkowski spacetime
本文提出一种更简单且效果更好的因果集嵌入闵可夫斯基时空的新方法，通过时空体积定义时间坐标和空间距离并利用三角不等式推导出距离矩阵，在维度 d=2,3,4 下实现了高质量嵌入。
We study the utility of chains defined on causal sets in estimating continuum properties like the curvature, the proper time and the spacetime dimension through a numerical analysis. In particular, we show that in $\text{dS}_2$ and $\text{FLRW}_3$ spacetimes the formalism of arXiv:1212.0631 with slight modifications gives the right continuum properties. We also discuss a possible test of manifoldlikeness using this formalism by considering two models of non-manifoldlike causal sets. This is a part of a broader idea of the geometrical reconstruction of continuum properties given a discrete sub …
Measuring spatial distances in causal sets via causal overlaps
本文提出基于因果重叠测量空间分离事件距离的方法，证明在连续极限下测量误差消失，为因果集几何生成和动态因果集模型发展提供理论支持。
"Observables" in causal set cosmology
因果集的经典序列增长动力学的"类属"家族提供了因果集的宇宙学模型，这是对迄今未知的量子理论的想法的试验场。特别是，我们可以研究一般协方差如何表现自己，并解决在量子引力中识别和解释协变"可观测"的问题。在这种情况下，问题变成了确定因果集的可测量的协变集合，每个集合对应的问题是：“发生的因果集属于这个集合吗？”它的答案是集合的概率测度。因此，可测协变问题对应于独立于因果集的标记的因果集的可测集合。然而，经典序贯增长动力学的转移概率直接提供的是{\it labeled}因果集空间上的测度，因而模糊了协变可测集合的物理解释.我们表明，有一个物理意义的表征类的可测协变集工会和差异的“干集”。
Indications of de Sitter Spacetime from Classical Sequential Growth Dynamics of Causal Sets
由经典的连续增长过程描述的因果集的一大类动力学定律产生了一个循环宇宙，其膨胀和收缩的周期被因果集的单个“起源元素”打断。我们提出的证据表明，这些起源元素之一，在连续增长动力学的背景下，在不久的将来的有效动力学，产生一个初始阶段的德西特一样的指数膨胀，并认为由此产生的图片有许多有吸引力的功能作为一个模型的早期宇宙，有可能解决一些标准模型难题没有任何微调。
Ph.D. Thesis: Quantum Field Theory and Gravity in Causal Sets
这是我为博士学位答辩而提交的论文的副本。这是我的论文，从那以后我做了一些小的修改。该项目的目标是将物质场及其拉格朗日从规则时空推广到因果集。
The Scalar Curvature of a Causal Set
引入了因果集上标量场上的一类单参数滞后线性算子族。当因果集被四维闵可夫斯基时空很好地近似时，算子是洛伦兹不变的但非定域的，由非定域尺度参数化，当作用于在非定域尺度上缓慢变化的场时，近似连续标量D'Alembertian，$\Box$。同样的算子也可以应用于因果集上的标量场，这些标量场被弯曲时空很好地近似，在这种情况下，它们近似于$\Box - {1/2}R$，其中$R$是里奇标量曲率。这可以用来定义因果集的近似局部作用泛函。
Discrete mechanics: a kinematics for a particular case of causal sets
该模型是因果集的一个特例。这是一个微观层次上的时空离散模型。在文件中的模型的最一般的属性进行了调查，没有任何参考的动态。模型的动力学在[arXiv：1004.5077]中介绍。这两篇论文介绍了一个一致的描述模型。
We show that the Feynman propagator (or the d'Alembertian) of a causal set contains the complete information about the causal set. Intuitively, this is because the Feynman propagator, being a correlator that decays with distance, provides a measure for the invariant distance between pairs of events. Further, we show that even the spectra alone (of the self-adjoint and anti-self-adjoint parts) of the propagator(s) and d'Alembertian already carry large amounts of geometric information about their causal set. This geometric information is basis independent and also gauge invariant in the sense th…
We describe a new form of retrocausality, which is found in the behaviour of a class of causal set theories, called energetic causal sets (ECS). These are discrete sets of events, connected by causal relations. They have three orders: (1) a birth order, which is the order in which events are generated; this is a total order which is the true causal order, (2) a dynamical partial order, which prescribes the flows of energy and momentum amongst events, (3) an emergent causal order, which is defined by the geometry of an emergent Minkowski spacetime, in which the events of the causal sets are emb…
The structure of causal sets
最近流行的结构主义对物理理论的解释往往使结构的中心概念不够精确。量子引力的初始因果集方法提供了一个注定要用结构主义术语解释的物理理论的范例。它表明如何采用结构主义本身的因果集理论的物理意义的自然解释。相反，因果集的概念异常清晰的情况下，被用来作为一个陪衬，以说明如何在数学上知情的严格概念化的结构，以确定在物理理论的结构。此外，一些困扰结构主义对物理理论的解释的技术问题，例如在高度对称的物理结构的关系轮廓中建立其位置的身份的困难，以及可以解决这些困难的方法，可以用因果集生动地说明。
In order to understand the detailed mechanism by which a fundamental discreteness can provide a finite entanglement entropy, we consider the entanglement entropy of two classes of free massless scalar fields on causal sets that are well approximated by causal diamonds in Minkowski spacetime of dimensions 2,3 and 4. The first class is defined from discretised versions of the continuum retarded Green functions, while the second uses the causal set's retarded nonlocal d'Alembertians parametrised by a length scale $l_k$. In both cases we provide numerical evidence that the area law is recovered wh…
Spin foam models as energetic causal sets
能量因果集是由因果相关事件之间的能量-动量流赋予的因果集。这些包含了一种新的机制，从因果关系中出现时空。在这里，我们构建了一个自旋泡沫模型，这也是一个充满活力的因果集模型。该模型与Wolfgang Wieland在arXiv：1407.0025中并行引入的模型密切相关，并且该构造利用了那里使用的结果。使自旋泡沫模型也成为一个充满活力的因果集的是维兰德对新动量的识别，在事件（或四个单形）中守恒，其规范不是质量，而是四面体的体积。这实现了扭转约束，这在以前的自旋泡沫模型中是缺失的，并且需要将连接动力学与度规的动力学联系起来，就像在广义相对论中一样。这一发现使得将时空出现的新机制应用于自旋泡沫模型成为可能。
We study the distribution of maximal-chain lengths between two elements of a causal set and its relationship with the embeddability of the causal set in a region of flat spacetime. We start with causal sets obtained from uniformly distributed points in Minkowski space. After some general considerations we focus on the 2-dimensional case and derive a recursion relation for the expected number of maximal chains $n_k$ as a function of their length $k$ and the total number of points $N$ between the maximal and minimal elements. By studying these theoretical distributions as well as ones generated …
Local d'Alembertian for causal sets
本文提出因果集的局部d'Alembert算子，解决了非局部性与连续极限之间的矛盾，实现了局部性与洛伦兹不变性的统一。
The framework of perturbative algebraic quantum field theory (pAQFT) is used to construct QFT models on causal sets. We discuss various discretised wave operators, including a new proposal based on the idea of a `preferred past', which we also introduce, and show how they may be used to construct classical free and interacting field theory models on a fixed causal set; additionally, we describe how the sensitivity of observables to changes in the background causal set may be encapsulated in a relative Cauchy evolution. These structures are used as the basis of a deformation quantization, using…
Boundary Terms for Causal Sets
我们提出了一个家庭的边界项的因果集与类空边界的行动。我们表明，在连续极限恢复Gibbons-Hawking-York边界项的平均值。我们还计算了平坦时空中亚历山德罗夫区间的平均因果集作用量的连续极限。我们发现它等于区间的两个光锥边界的余维2交点的体积。
Quantum Fields on Causal Sets
因果集理论提供了一个离散时空的模型，其中时空事件由因果集的元素表示-局部有限的偏序集，其中偏序表示事件之间的因果关系。这里介绍的工作描述了一个因果集上的物质模型，特别是因果集时空背景上的量子标量场理论。工作开始于一个离散的路径积分模型的因果集上的粒子。在这里，量子力学振幅被分配给因果集内的轨迹。通过对两个时空事件之间的所有轨迹求和，我们得到一个因果集粒子传播子。通过选择适当的振幅，这与闵可夫斯基时空中克莱因-戈登方程的延迟传播子一致（在适当的意义上）。这个因果集传播，然后用来定义一个因果集类似的保利-约旦功能，出现在连续量子场论。一个量子标量场，然后由满足三个简单条件（包括玻色子交换规则）的算子代数建模。通过因果集的线性扩展定义时间排序，这些场算子用于定义因果集费曼传播算子。证据表明协议（在适当的意义上）之间的因果集费曼传播和连续费曼传播的克莱因-戈登方程在闵可夫斯基时空。Feynman传播子是使用Pauli-Jordan函数的特征分解获得的，该方法也可以应用于基于连续性的理论。将自由场论推广到相互作用的标量场。这导致了一个建议的非微扰S-矩阵的因果集。基于连续的现象学和自旋半粒子的因果集模型也提出。
Introduction to causal sets: an alternate view of spacetime structure
本文针对学生和其他感兴趣的人，提供了一个全面的介绍因果集假设，与广义相对论和非相对论量子力学的一些知识。我阐明了为什么因果集结构可能是量子引力理论的适当结构的论点。因果集理论的逻辑和形式的发展以及一些启发性的例子也提供了。
Causal Sets: Quantum gravity from a fundamentally discrete spacetime
为了构建引力的量子理论，我们可能不得不放弃我们所做的某些假设。特别是，时空作为连续体底层的概念受到质疑。因果集是一个试图从基本离散的时空开始构建引力的量子理论的尝试。在这方面的贡献，我们回顾了整个方法，专注于一些最近的事态发展的运动学和动力学的方法。
On the Uniqueness of Embeddings of Causal Sets
研究因果集嵌入洛伦兹流形的唯一性，引入良态嵌入概念，证明若因果集能良态嵌入两个流形，其内部通过ε-近似等距相关，还表明高密度极限下泊松散布产生的因果集有良态嵌入，给出趋于零的误差ε。
Lorentzian Manifolds and Causal Sets as Partially Ordered Measure Spaces
我们考虑洛伦兹流形作为偏序测度空间的例子，集合被赋予相容的偏序关系和测度，在这种情况下由因果结构和每个洛伦兹度量定义的体积元给出。这将引力几何理论中通常用来描述时空的结构放在了更一般的背景下，其中包括量子引力因果集方法中的局部有限偏序集。然后，我们引入一个函数的特点之间的任何两个偏序测度空间的接近，并表明，当限制到满足一个简单的可分性条件的紧空间，它是一个距离。特别是，这提供了一个定量的，协变的方式来描述如何接近两个流形与洛伦兹度量，或如何manifoldlike因果集。
Quantum Spacetime Leaps: Higher Dimensional Energetic Causal Sets
该研究提出能量因果集（ECS）的2+1维模拟，探索其三种序的不一致性，尝试从基本时间不可逆定律中获得可逆动力学。
A general solution for classical sequential growth dynamics of Causal Sets
因果集的完全量子动力学的一个经典先驱已经被公式化为随机顺序增长过程，其中因果集的元素出现在一种吸积过程中。马尔可夫增长过程的转移概率满足一定的因果关系和广义协方差的物理要求，并且所有转移概率都不为零的通解已经被找到，这里我们去掉概率不为零的假设，定义一个合理的物理要求的扩展以覆盖概率为零的情况，并且找到这些物理条件的完全通解。由此产生的生长过程家族有一个有趣的结构，让人想起“无限的乌龟塔”宇宙学。
Causal sets and conservation laws in tests of Lorentz symmetry
许多关于洛伦兹对称性的最重要的天体物理学测试也假设所观测到的粒子的能量动量是完全守恒的。在量子引力的因果集方法中，一种特殊的洛伦兹对称性成立，但能量动量守恒可能被违反。我们表明，不正确地假设精确的守恒可以引起一个虚假的信号的洛伦兹对称性违反因果集。然而，这种虚假信号的大小比目前可以检测到的要小得多，因此目前进行的天体物理洛伦兹对称性测试是安全的，不受因果集引起的能量动量守恒的破坏。
The internal description of a causal set: What the universe looks like from the inside
我们描述了一种代数编码的因果信息的离散时空。因果集合C被转换成一个描述，它是根据C中事件的因果过去来描述的。这是通过一个演化集合来完成的，一个函子，它为C的每个事件分配其因果过去。发展集服从海廷代数的特点是由一个非标准的概念补充。因果集的因果结构可以通过计算演化集的补集得出。因果量子理论可以基于演化集合的量子版本，我们对此进行了简要讨论。
This is a chapter of the planned monograph "Out of Nowhere: The Emergence of Spacetime in Quantum Theories of Gravity", co-authored by Nick Huggett and Christian Wüthrich and under contract with Oxford University Press. (More information at www.beyondspacetime.net.) This chapter sketches how spacetime emerges in causal set theory and demonstrates how this question is deeply entangled with genuinely philosophical concerns.
We study some kinematical aspects of quantum fields on causal sets. In particular, we are interested in free scalar fields on a fixed background causal set. We present various results building up to the study of the entanglement entropy of de Sitter horizons using causal sets. We begin by obtaining causal set analogs of Green functions for this field. First we construct the retarded Green function in a Riemann normal neighborhood (RNN) of an arbitrary curved spacetime. Then, we show that in de Sitter and patches of anti-de Sitter spacetimes the construction can be done beyond the RNN. This all…
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