The Action of a Causal Set by Dionigi
The Action of a Causal Set by Dionigi
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因果集的作用 作者：Dionigi
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发表时间：
2013
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通讯作者：
Maria Teofilo Benincasa
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作者：
Maria Teofilo Benincasa
因果集是一个离散时空的模型，其中的“时空原子”带有祖先的关系。这种顺序关系在数学上是由偏序给出的，并且被认为是之前和之后的宏观因果概念的基础。在这篇论文中提出的工作提出了一个定义的作用的因果集类似于连续爱因斯坦-希尔伯特作用。对这一行动的定义所采取的途径有些间接。我们首先构造一个推迟波算子的因果集近似的4维时空和证明，在一定的假设下，该运营商给出了通常的连续d'Alembertian和标量曲率的近似时空的连续极限。我们使用这个结果来定义标量曲率和因果集的作用。这个定义可以在任何维中工作，因此在所有维中都存在一个显式形式的动作。我们推测，在某些条件下，作用量的连续极限由爱因斯坦-希尔伯特作用量直到边界项给出，我们也推测其显式形式。我们通过分析和数值计算的各种时空区域的预期行动，这一猜想提供了证据。通过计算具有不同拓扑结构的二维时空中不同区域的作用量的期望值，证明了二维作用量具有拓扑性质。我们发现，2D行动的拓扑特征分解的因果凸区域的裤子时空包含的奇异性，并为非因果凸矩形。最后，我们提出了一个微观帐户的熵的因果视野的基础上的行动。它是一种“时空互信息”的形式，产生于视界对时空的分割。二维算例的分析结果和数值模拟结果为该建议提供了证据。进一步的证据提供了数值结果的林德勒和宇宙deSitter视界在3和4维，并为非平衡视界在一个坍缩壳时空在4维。3
A causal set is a model for a discrete spacetime in which the “atoms of spacetime” carry a relation of ancestry. This order relation is mathematically given by a partial order, and is is taken to underly the macroscopic causal notions of before and after. The work presented in this thesis proposes a definition for the action of a causal set analogous to the continuum Einstein-Hilbert action. The path taken towards the definition of this action is somewhat indirect. We first construct a retarded wave operator on causal sets well-approximated by 4-dimensional spacetimes and prove, under certain assumptions, that this operator gives the usual continuum d’Alembertian and the scalar curvature of the approximating spacetime in the continuum limit. We use this result to define both the scalar curvature and the action of a causal set. This definition can be shown to work in any dimension, so that an explicit form of the action exists in all dimensions. We conjecture that, under certain conditions, the continuum limit of the action is given by the Einstein-Hilbert action up to boundary terms, whose explicit form we also conjecture. We provide evidence for this conjecture through analytic and numerical calculations of the expected action of various spacetime regions. The 2-dimensional action is shown to possess topological properties by calculating its expectation value for various regions of 2-dimensional spacetimes with di↵erent topologies. We find that the topological character of the 2d action breaks down for causally convex regions of the trousers spacetime that contain the singularity, and for non-causally convex rectangles. Finally, we propose a microscopic account of the entropy of causal horizons based on the action. It is a form of “spacetime mutual information” arising from the partition of spacetime by the horizon. Evidence for the proposal is provided by analytic results and numerical simulations in 2dimensional examples. Further evidence is provided by numerical results for the Rindler and cosmic deSitter horizons in both 3 and 4-dimensions, and for a non-equilibrium horizon in a collapsing shell spacetime in 4-dimensions. 3