On the continuum limit of Benincasa-Dowker-Glaser causal set action

TL;DR

This paper investigates the continuum limit of the Benincasa-Dowker-Glaser causal set action, demonstrating it converges to the Einstein-Hilbert action plus boundary terms, supporting the inclusion of boundary contributions in causal set quantum gravity.

Contribution

It provides the first explicit calculation showing the causal set action converges to the Einstein-Hilbert action with boundary terms in a curved spacetime setting.

Findings

The causal set action admits a finite continuum limit.

The limit includes an Einstein-Hilbert bulk term.

A boundary term proportional to the joint volume appears.

Abstract

We study the continuum limit of the Benincasa-Dowker-Glaser causal set action on a causally convex compact region. In particular, we compute the action of a causal set randomly sprinkled on a small causal diamond in the presence of arbitrary curvature in various spacetime dimensions. In the continuum limit, we show that the action admits a finite limit. More importantly, the limit is composed by an Einstein-Hilbert bulk term as predicted by the Benincasa-Dowker-Glaser action, and a boundary term exactly proportional to the codimension-two joint volume. Our calculation provides strong evidence in support of the conjecture that the Benincasa-Dowker-Glaser action naturally includes codimension-two boundary terms when evaluated on causally convex regions.