An Einstein equation for discrete quantum gravity

TL;DR

This paper develops a discrete quantum gravity framework inspired by Einstein's equations, using causal sets and operators to model curvature, metric, and mass-energy in a quantum context.

Contribution

It introduces a novel discrete analogue of Einstein's equation within the causal set approach to quantum gravity, decomposing curvature into metric and mass-energy operators.

Findings

Decomposition of curvature operator into metric and mass-energy parts

Formulation of a discrete Einstein-like equation for quantum gravity

Potential for assessing the approximation of general relativity by DQG

Abstract

The basic framework for this article is the causal set approach to discrete quantum gravity (DQG). Let $Q_n$ be the collection of causal sets with cardinality not greater than $n$ and let $K_n$ be the standard Hilbert space of complex-valued functions on $Q_n$. The formalism of DQG presents us with a decoherence matrix $D_n(x,y)$, $x,y\in Q_n$. There is a growth order in $Q_n$ and a path in $Q_n$ is a maximal chain relative to this order. We denote the set of paths in $Q_n$ by $\Omega_n$. For $\omega, \omega '\in\Omega_n$ we define a bidifference operator $\varbigtriangledown_{\omega, \omega '}^n$ on $K_n\otimes K_n$ that is covariant in the sense that $\varbigtriangledown_{\omega, \omega '}^n$ leaves $D_n$ stationary. We then define the curvature operator $\rscript_{\omega, \omega'}^n=\varbigtriangledown_{\omega, \omega '}^n-\varbigtriangledown_{\omega ', \omega}^n$. It turns out that $\rscript_{\omega, \omega '}^n$ naturally decomposes into two parts $\rscript_{\omega, \omega '}^n=\dscript_{\omega, \omega '}^n+\tscript_{\omega, \omega '}^n$ where $\dscript_{\omega, \omega '}^n$ is closely associated with $D_n$ and is called the metric operator while $\tscript_{\omega, \omega '}^n$ is called the mass-energy operator. This decomposition is a discrete analogue of Einstein's equation of general relativity. Our analogue may be useful in determining whether general relativity theory is a close approximation to DQG.