# Scalar Field Green Functions on Causal Sets

## TL;DR

This paper investigates scalar field Green functions on causal sets in 2 and 4 dimensions, validating models that approximate continuum Green functions and proposing generalizations to other spacetimes.

## Contribution

It validates Johnston's models for scalar Green functions on causal sets and extends their applicability to various spacetimes, including deSitter and anti deSitter.

## Key findings

- 2D model provides Green function for massive scalar fields in topologically trivial spacetimes.
- 4D model matches continuum Green functions in Riemann normal neighborhoods.
- Proposes a generalization for 3D flat spacetime causal sets.

## Abstract

We examine the validity and scope of Johnston's models for scalar field retarded Green functions on causal sets in 2 and 4 dimensions. As in the continuum, the massive Green function can be obtained from the massless one, and hence the key task in causal set theory is to first identify the massless Green function. We propose that the 2-d model provides a Green function for the massive scalar field on causal sets approximated by any topologically trivial 2 dimensional spacetime. We explicitly demonstrate that this is indeed the case in a Riemann normal neighbourhood. In 4-d the model can again be used to provide a Green function for the massive scalar field in a Riemann normal neighbourhood which we compare to Bunch and Parker's continuum Green function. We find that the same prescription can also be used for deSitter spacetime and the conformally flat patch of anti deSitter spacetime. Our analysis then allows us to suggest a generalisation of Johnston's model for the Green function for a causal set approximated by 3 dimensional flat spacetime.