The classical sequential growth model for causal sets provides a template for the dynamics in the deep quantum regime. This growth dynamics is intrinsically temporal and causal, with each new element being added to the existing causal set without disturbing its past. In the quantum version, the probability measure on the event algebra is replaced by a quantum measure, which is Hilbert space valued. Because of the temporality of the growth process, in this approach, covariant observables (or beables) are measurable only if the quantum measure extends to the associated sigma algebra of events. This is not always guaranteed. In this work we find a criterion for extension (and thence covariance) in complex sequential growth models for causal sets. We find a large family of models in which the measure extends, so that all covariant observables are measurable.

中文翻译：

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复杂连续增长模型中协方差的标准

因果集的经典顺序增长模型为深量子体系中的动力学提供了模板。这种增长动力本质上是时间性和因果性的，每个新元素都被添加到现有的因果集中，而不会影响其过去。在量子版本中，事件代数上的概率测度被一个量子测度取代，这是希尔伯特空间值。由于增长过程的时间性，在这种方法中，只有当量子度量扩展到事件的相关 sigma 代数时，协变可观测值（或 beables）才是可测量的。这并不总是有保证的。在这项工作中，我们在因果集的复杂顺序增长模型中找到了扩展（以及协方差）的标准。我们发现了一大类模型，其中的度量扩展，