Canonical General Relativity and Emergent Geometry

TL;DR

This paper explores how emergent Ising models of geometry can be connected to canonical general relativity, proposing a discrete Hamiltonian that approximates the continuum theory.

Contribution

It introduces the Canonical Ising Model, linking emergent geometry models to discretized canonical general relativity using discrete differential geometry.

Findings

Ising models exhibit features of Ricci flat vacuum

Excitations replicate quantum particle dynamics

Proposed model approximates discretized canonical GR

Abstract

Ising models of emergent geometry are well known to possess ground states with many of the desired features of a low dimensional, Ricci flat vacuum. Further, excitations of these ground states can be shown to replicate the quantum dynamics of a free particle in the continuum limit. It would be a significant next step in the development of emergent Ising models to link them to an underlying physical theory that has General Relativity as its continuum limit. In this work we investigate how the canonical formulation of General Relativity can be used to construct such a discrete Hamiltonian using recent results in discrete differential geometry. We are able to demonstrate that the Ising models of emergent geometry are closely related to the model we propose, which we term the Canonical Ising Model, and may be interpreted as an approximation of discretized canonical general relativity.