The continuum limit of a 4-dimensional causal set scalar d'Alembertian

TL;DR

This paper demonstrates that a discrete scalar d'Alembertian operator on causal sets converges to the standard continuum d'Alembertian in flat spacetime and to a modified operator involving Ricci scalar in curved spacetime, under certain conditions.

Contribution

It establishes the continuum limit of a causal set scalar d'Alembertian operator in both flat and curved spacetimes, connecting discrete causal set models to continuum physics.

Findings

The mean of the operator converges to the usual d'Alembertian in Minkowski spacetime.

In curved spacetime, the mean converges to the d'Alembertian minus half the Ricci scalar.

Convergence is valid when the scalar field varies slowly relative to the Poisson density.

Abstract

The continuum limit of a 4-dimensional, discrete d'Alembertian operator for scalar fields on causal sets is studied. The continuum limit of the mean of this operator in the Poisson point process in 4-dimensional Minkowski spacetime is shown to be the usual continuum scalar d'Alembertian $\Box$. It is shown that the mean is close to the limit when there exists a frame in which the scalar field is slowly varying on a scale set by the density of the Poisson process. The continuum limit of the mean of the causal set d'Alembertian in 4-dimensional curved spacetime is shown to equal $\Box - \frac{1}{2}R$, where $R$ is the Ricci scalar, under certain conditions on the spacetime and the scalar field.