Particle propagators on discrete spacetime

TL;DR

This paper develops a quantum mechanical model for particle propagation on discrete causal set spacetime, using a path integral approach, and shows it aligns with known relativistic propagators in Minkowski spacetime.

Contribution

It introduces a discrete path integral method for particle propagation on causal sets, connecting causal set theory with quantum field theory.

Findings

Propagator on causal sets matches Klein-Gordon retarded propagator in Minkowski spacetime.

Uses matrix geometric series to sum over trajectories in discrete path integral.

Demonstrates potential for quantum field theory formulation on causal set spacetime.

Abstract

A quantum mechanical description of particle propagation on the discrete spacetime of a causal set is presented. The model involves a discrete path integral in which trajectories within the causal set are summed over to obtain a particle propagator. The sum-over-trajectories is achieved by a matrix geometric series. For causal sets generated by sprinkling points into 1+1 and 3+1 dimensional Minkowski spacetime the propagator calculated on the causal set is shown to agree, in a suitable sense, with the causal retarded propagator for the Klein-Gordon equation. The particle propagator described here is a step towards quantum field theory on causal set spacetime.