Gravity and Matter in Causal Set Theory

TL;DR

This paper develops a framework for formulating the dynamics of causal sets coupled with matter fields, translating continuum actions into discrete causal set variables and proposing a discretized action for quantum gravity research.

Contribution

It introduces a novel method to express continuum matter-gravity actions in terms of causal set variables and discretizes these actions for use in causal set quantum gravity.

Findings

Reformulation of Klein-Gordon coupled to gravity in causal set terms

Discretization of the continuum action for causal sets

Proposal of a discrete action for causal set dynamics

Abstract

The goal of this paper is to propose an approach to the formulation of dynamics for causal sets and coupled matter fields. We start from the continuum version of the action for a Klein-Gordon field coupled to gravity, and rewrite it first using quantities that have a direct correspondent in the case of a causal set, namely volumes, causal relations, and timelike lengths, as variables to describe the geometry. In this step, the local Lagrangian density $L(f;x)$ for a set of fields $f$ is recast into a quasilocal expression $L_0(f;p,q)$ that depends on pairs of causally related points $p \prec q$ and is a function of the values of $f$ in the Alexandrov set defined by those points, and whose limit as $p$ and $q$ approach a common point $x$ is $L(f;x)$. We then describe how to discretize $L_0(f;p,q)$, and use it to define a discrete action.