We offer further evidence that discreteness of the sort inherent in a causal set cannot, in and of itself, serve to break Poincar{e} invariance. In particular we prove that a Poisson sprinkling of Minkowski spacetime cannot endow spacetime with a distinguished spatial or temporal orientation, or with a distinguished lattice of spacetime points, or with a distinguished lattice of timelike directions (corresponding respectively to breakings of reflection-invariance, translation-invariance, and Lorentz invariance). Along the way we provide a proof from first principles of the zero-one law on which our new arguments are based.

中文翻译：

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对称破缺和零一定律

我们提供了进一步的证据，证明因果集中固有的那种离散性本身并不能打破庞加莱{e}不变性。特别地，我们证明了闵可夫斯基时空的泊松喷洒不能赋予时空一个显着的空间或时间方向，或一个显着的时空点格，或一个显着的类时方向格（分别对应于反射不变性的破坏，平移-不变性和洛伦兹不变性）。在此过程中，我们从零一定律的第一原理提供了证据，我们的新论点基于该原理。