Timelike boundary and corner terms in the causal set action

TL;DR

This work analyzes the Benincasa-Dowker-Glaser causal set action for Poisson sprinklings into subregions of flat Minkowski space. It establishes that for manifolds with timelike boundaries the mean action diverges as $l^{-1}$ in the continuum limit and is governed by a timelike boundary term, while also introducing and testing a joint contribution from codimension-2 joints. The authors propose a Lorentzian-angle dependent joint term, extend the analysis to higher dimensions, and validate the leading predictions through analytic calculations and 3D simulations in both embedded and isolated regimes. The findings illuminate how boundary and joint structures shape the causal set path integral and offer concrete predictions for how nonmanifold-like spacetimes might be suppressed, providing a step toward a continuum GR-like regime from causal sets.

Abstract

The causal set action of dimension $d$ is investigated for causal sets that are Poisson sprinklings into submanifolds of $d$-dimensional Minkowski space. Evidence, both analytic and numerical, is provided for the conjecture that the mean of the causal set action over sprinklings into a manifold with a timelike boundary, diverges like $l^{-1}$ in the continuum limit as the discreteness length $l$ tends to zero. A novel conjecture for the contribution to the causal set action from co-dimension 2 corners, also known as joints, is proposed and justified.

Figures (35)

• Figure 1: As an example, the blue region is $M$, a non-causally convex manifold. $x,y$ are elements of $\mathcal{C}$ but $z$ is an element only of ${\Tilde{C}}$. Whether the order interval $[x,y]$ counts as having cardinality 0 or 1 depends on whether we are working in the isolated or embedded regime, respectively.
• Figure 2: A d=2 example, showing how by cloning the manifold and displacing it by the defining vector $\vec{c}=(\Delta t, \Delta x)$, the volume of realisation can be found from the overlapping area. The dark blue region on the left has an identical area to the overlapping area on the right.
• Figure 3: Triangular manifold (left) and its area of realisation (= area of overlapping region) with $u> v$ (middle) and $v> u$ (right).
• Figure 4: Left: a spacetime region with infinite timelike boundaries. Right: cloning the region to obtain the volume of realisation, shown in dark blue.
• Figure 5: Left: the area of integration in $\Delta t,\Delta r$ space. Right: the area of integration in $u,v$ space.