Local d'Alembertian for causal sets

Abstract

Causal set theory is an intrinsically nonlocal approach to quantum gravity, inheriting its nonlocality from Lorentzian nonlocality. This nonlocality causes problems in defining differential operators -- such as the d'Alembert operator, a cornerstone of any relativistic field theory -- within the causal set framework. It has been proposed that d'Alembertians in causal sets must themselves be nonlocal to respect causal set nonlocality. However, we show that such nonlocal d'Alembertians do not converge to the standard continuum d'Alembertian for some basic fields. To address this problem, we introduce a local d'Alembert operator for causal sets and demonstrate its convergence to its continuum counterpart for arbitrary fields in Minkowski spacetimes. Our construction leverages recent developments in measuring both timelike and spacelike distances in causal sets using only their intrinsic structure. This approach thus reconciles locality with Lorentz invariance, and paves a way toward defining converging discrete approximations to locality-based differential operators even in theories that are inherently nonlocal.