An Einstein equation for discrete quantum gravity

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.