Spacetime entanglement entropy: covariance and discreteness

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.

Topical collection: In memory of Professor T Padmanabhan

“…Paddy was always interested in different approaches to quantum gravity and the conceptual issues related to the fundamental structure of spacetime. In this volume, the article by Abhishek Mathur, Sumati Surya, and Nomaan X [21] discusses structure of spacetime in the context of causal set theory. They review some recent results on Sorkin's spacetime formulation of the entanglement entropy for a free quantum scalar field on a manifold-like causal set and establish the existence of a volume scaling law instead of an area law.…”