On the Entanglement Entropy of Quantum Fields in Causal Sets

TL;DR

This paper investigates how fundamental discreteness in causal set theory can lead to finite entanglement entropy for quantum fields, demonstrating the area law through numerical evidence and analyzing the role of double cutoffs.

Contribution

It introduces a double-cutoff prescription for causal set entanglement entropy and provides numerical evidence that the area law is recovered in this framework.

Findings

The area law is recovered with the double-cutoff prescription.

Two types of free massless scalar fields on causal sets are analyzed.

The necessity and role of the double cutoff are clarified.

Abstract

In order to understand the detailed mechanism by which a fundamental discreteness can provide a finite entanglement entropy, we consider 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 prescription proposed in arXiv:hep-th/1611.10281 is imposed. We discuss in detail the need for this double cutoff by studying the effect of two cutoffs on the quantum 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.