Spacetime Entanglement Entropy: Covariance and Discreteness

TL;DR

This paper reviews recent findings on spacetime entanglement entropy in quantum field theory, comparing continuum and causal set approaches, highlighting the role of UV cut-offs and non-locality in entropy scaling laws.

Contribution

It demonstrates how causal sets naturally regulate entanglement entropy and explores the transition from volume to area law through additional UV cut-offs.

Findings

SSEE for a causal diamond matches Calabrese-Cardy form in 2D

SSEE for horizons matches mode-wise von Neumann entropy in higher dimensions

Volume law observed in causal sets without additional UV cut-offs

Abstract

We review some recent results on Sorkin's spacetime formulation of the entanglement entropy (SSEE) for a free quantum scalar field both in the continuum and in manifold-like causal sets. The SSEE for a causal diamond in a 2d cylinder spacetime has been shown to have a Calabrese-Cardy form, while for de Sitter and Schwarzschild de Sitter horizons in dimensions $d>2$, it matches the mode-wise von-Neumann entropy. In these continuum examples the SSEE is regulated by imposing a UV cut-off. Manifold-like causal sets come with a natural covariant spacetime cut-off and thus provide an arena to study regulated QFT. However, the SSEE for different manifold like causal sets in $d=2$ and $d=4$ has been shown to exhibit a volume rather than an area law. The area law is recovered only when an additional UV cut-off is implemented in the scaling regime of the spectrum which mimics the continuum behaviour. We discuss the implications of these results and suggest that a volume-law may be a manifestation of the fundamental non-locality of causal sets and a sign of new UV physics.