# An Einstein Equation for Discrete Quantum Gravity

## Publication Date

2012

## Document Type

Article

## Keywords

Causal set approach, Discrete quantum gravity

## Abstract

The basic framework for this article is the causal set approach to discrete quantum gravity (DQG). Let Qn be the collection of causal sets with cardinality not greater than n and let Kn be the standard Hilbert space of complex-valued functions on Qn. The formalism of DQG presents us with a decoherence matrix Dn(x, y), x, y ∈ Qn. There is a growth order in Qn and a path in Qn is a maximal chain relative to this order. We denote the set of paths in Qn by Ωn. For ω, ω0 ∈ Ωn we define a bidifference operator 5n ω,ω0 on Kn ⊗ Kn that is covariant in the sense that 5n ω,ω0 leaves Dn stationary. We then de- fine the curvature operator Rn ω,ω0 = 5n ω,ω0 − 5n ω0 ,ω. It turns out that Rn ω,ω0 naturally decomposes into two parts Rn ω,ω0 = Dn ω,ω0 +T n ω,ω0 where Dn ω,ω0 is closely associated with Dn and is called the metric operator while T n ω,ω0 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.

## Recommended Citation

Gudder, S. (2012). An Einstein equation for discrete quantum gravity. Mathematics Preprint Series. Retrieved from https://digitalcommons.du.edu/math_preprints/30