Do causal sets have symmetries?

TL;DR

This paper investigates the symmetries of finite causal sets and posets to understand their potential as discrete spacetime models, aiming to connect local symmetries with continuum spacetime properties in causal set theory.

Contribution

It systematically analyzes automorphism groups of finite posets to classify their symmetries and compare them with sprinkled causal sets, advancing the understanding of discrete spacetime models.

Findings

Automorphism groups of finite posets are characterized.

Comparison between generic poset symmetries and sprinkled causal sets.

Insights into which posets can serve as discrete spacetime models.

Abstract

Causal sets are locally finite, partially ordered sets (posets), which are considered as discrete models of spacetimes. On the one hand, causal sets corresponding to a spacetime manifold are commonly generated with a random process called sprinkling. This process keeps only a discrete set of points of the manifold and their causal relations (loosing the spacetime symmetries in each sprinkle). On the other hand, the main conjecture of causal set theory is that given an ensemble of causal sets there is a corresponding spacetime manifold and the continuum symmetries of it are like all manifold properties "reconstructable" from the partial orders of all the causal sets in the ensemble. But most generic finite posets have very few layers ("instances of time") in contrast to sprinkles with many layers in a sufficiently large spacetime region. In a recent project, I investigated the automorphism groups of (finite) posets in order to identify and classify their symmetries systematically. The comparison of local symmetries of generic posets (including Kleitmann-Rothschild orders) with sprinkled causal sets may help us to find those posets that can serve as discrete spacetime models in causal set theory.