The Scalar Curvature of a Causal Set

TL;DR

This paper introduces a family of nonlocal, Lorentz-invariant operators on scalar fields in causal sets that approximate the continuum scalar D'Alembertian and Ricci scalar curvature, enabling a local action functional in quantum gravity models.

Contribution

It presents a new class of operators on causal sets that approximate the scalar D'Alembertian and Ricci scalar, bridging discrete causal structures with continuum curvature.

Findings

Operators approximate $ox$ in flat spacetime

Operators approximate $ox - rac{1}{2} R$ in curved spacetime

Enables defining a local action functional for causal sets

Abstract

A one parameter family of retarded linear operators on scalar fields on causal sets is introduced. When the causal set is well-approximated by 4 dimensional Minkowski spacetime, the operators are Lorentz invariant but nonlocal, are parametrised by the scale of the nonlocality and approximate the continuum scalar D'Alembertian, $\Box$, when acting on fields that vary slowly on the nonlocality scale. The same operators can be applied to scalar fields on causal sets which are well-approximated by curved spacetimes in which case they approximate $\Box - {{1/2}}R$ where $R$ is the Ricci scalar curvature. This can used to define an approximately local action functional for causal sets.