Path Sums for Propagators in Causal Sets

TL;DR

This paper investigates conditions under which path sums in causal sets accurately reproduce scalar field propagators, ensuring consistency with continuum limits and exploring solutions verified through numerical analysis.

Contribution

It introduces a family of solutions for path sums in causal sets that match scalar propagators and verifies these solutions numerically.

Findings

Identified conditions for path sums to match scalar propagators.

Developed a family of solutions for path sums.

Numerically verified solutions in specific cases.

Abstract

A major challenge in Causal Set research is that theories need only to match general relativity and quantum field theory in the appropriate limits. This means that there should be many different ways to calculate a scalar field propagator in a causal set that match the known limits, but may give significantly different results on the small scale. In this work, we explore under what conditions a path sum will correspond to a scalar field propagator in such a way that it matches the known value in the continuum limit. A family of solutions for the path sum is found and is verified numerically in a few specific cases.