On the Entanglement Entropy of Quantum Fields in Causal Sets

In order to understand the detailed mechanism by which a funda mental discreteness can provide a finite entanglement entropy, we con sider the entanglement entropy of two classes of free massless scalar fields on causal sets that are well approximated by causal diamonds in Minkowski spacetime of dimensions 2,3 and 4. The first class is defined from discretised versions of the continuum retarded Green functions, while the second uses the causal set’s retarded nonlocal d’Alembertians parametrised by a length scale l k . In both cases we provide numerical evidence that the area law is recovered when the double-cutoff pre scription proposed in [24] is imposed. We discuss in detail the need for this double cutoff by studying the effect of two cutoffs on the quan tum field and, in particular, on the entanglement entropy, in isolation. In so doing, we get a novel interpretation for why these two cutoff are necessary, and the different roles they play in making the entanglement entropy on causal sets finite