On the Causal Set-Continuum Correspondence

Abstract

We present two results which concern certain aspects of the question: when is a causal set well approximated by a Lorentzian manifold? The first result is a theorem which shows that the number-volume correspondence, if required to hold even for arbitrarily small regions, is best realized via Poisson sprinkling. The second result concerns a family of lattices in $1+1$ dimensional Minkowski space, known as Lorentzian lattices, which we show provide a much better number-volume correspondence than Poisson sprinkling for large volumes. We argue, however, that this feature should not persist in higher dimensions. We conclude by conjecturing a form of the aforementioned theorem that holds under weaker assumptions, namely that Poisson sprinkling provides the best number-volume correspondence in $3+1$ dimensions for spacetime regions with macroscopically large volumes.

References in corpus (3)

• The causal set approach to quantum gravity
• Causal Sets: Discrete Gravity (Notes for the Valdivia Summer School)
• Discreteness without symmetry breaking: a theorem

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• Local d'Alembertian for causal sets
• Semiclassical Length Measure from a Quantum-Gravity Wave Function
• Endeavours in Discrete Lorentzian Geometry; A thesis in five papers