Link-based causal set propagators in $1+1$ dimensions

TL;DR

This paper explores expressing retarded scalar propagators on causal sets in 1+1 dimensions using the link matrix, proposing a novel exponential formulation that aligns with continuum results.

Contribution

It introduces a new exponential-based representation of propagators on causal sets and extends it to massive cases, connecting discrete structures with continuum physics.

Findings

Averaged massless propagator corresponds to exp( L )

Good agreement between discrete and continuum propagators after averaging

Inverse kernel exp( - L ) may serve as a discrete d'Alembertian

Abstract

We investigate whether retarded scalar propagators on causal sets can be expressed in terms of the link matrix $\mathbf{L}$. For Poisson sprinklings into $1+1$ dimensional Minkowski spacetime, we show by asymptotic analysis and supporting numerical simulations that the averaged massless retarded propagator is naturally associated with a normalized exponential exp$(\mathbf{L})$. We then extend the construction to the massive case via the usual mass-scattering series and obtain good agreement with the continuum propagator after averaging. Finally, we discuss the inverse kernel exp$(-\mathbf{L})$ as a possible candidate for a discrete d'Alembertian.