Dynamics and the emergence of geometry in an information mesh

Abstract

The idea of a graph theoretical approach to modeling the emergence of a quantized geometry and consequently spacetime, has been proposed previously, but not well studied. In most approaches the focus has been upon how to generate a spacetime that possesses properties that would be desirable at the continuum limit, and the question of how to model matter and its dynamics has not been directly addressed. Recent advances in network science have yielded new approaches to the mechanism by which spacetime can emerge as the ground state of a simple Hamiltonian, based upon a multi-dimensional Ising model with one dimensionless coupling constant. Extensions to this model have been proposed that improve the ground state geometry, but they require additional coupling constants. In this paper we conduct an extensive exploration of the graph properties of the ground states of these models, and a simplification requiring only one coupling constant. We demonstrate that the simplification is effective at producing an acceptable ground state. Moreover we propose a scheme for the inclusion of matter and dynamics as excitations above the ground state of the simplified Hamiltonian. Intriguingly, enforcing locality has the consequence of reproducing the free non-relativistic dynamics of a quantum particle.

Canonical General Relativity and Emergent Geometry

“…We begin our treatment in the next section Section I C with an overview of the models previously studied, to enable detailed later comparison to the new discrete CIM model. In particular we focus upon the QMD model introduced in [7], as it has a simple Hamiltonian including a correction term to suppress triangles. This correction term will prove to be important and can be reinterpreted as approximating discrete curvature in the Hamiltonian.…”

“…Recently new theoretical models that capitalize on advances in network science have emerged that attempt to reconcile GR and QM using an emergent geometry. These 'Ising Geometry' (IG) models were originally proposed by Trugenberger [3], subsequently extended and applied to quantum gravity [4][5][6][7][8], and are central to the Combinatorial Quantum Gravity (CQG) program. A fundamental precept of these models is a discrete quantum structure, implicitly or explicitly assumed to be at the Planck length scale.…”

“…In this work we will focus upon the IG models so named because they propose that spacetime emerges from a disordered and independent collection of quantum information bits subject to a ferromagnetic and Ising like interaction. For the precise details we refer the reader to the cited literature [3,4,7], but in principle the interaction between information bits creates the fabric of a geometry expressed as a graph, with entangled spins sharing an edge. To prevent a fully connected graph emerging there are anti-ferromagnetic terms introduced into the Hamiltonians of these models that disfavor edges, and higher order structures such as triangles, that would impair locality.…”

“…To prevent a fully connected graph emerging there are anti-ferromagnetic terms introduced into the Hamiltonians of these models that disfavor edges, and higher order structures such as triangles, that would impair locality. The models possess many attractive features, including a regular and highly local ground state, the emergence of a low preferential dimension as evidenced by a convergence of extrinsic and intrinsic dimensions, and the capacity to support the modelling of matter by the presence of stable defects in the mesh [7].…”

Enhanced Forman curvature and its relation to Ollivier curvature

“…Recently an approach to discretize geometry by much "wilder" structures, like random graphs (for a review see [5]) has been proposed [6]. The idea of this combinatorial quantum gravity approach is for geometric manifolds to emerge from random graphs in a continuous network transition, for which there is indeed strong evidence [7][8][9]. Contrary to previous discrete approaches, combinatorial quantum gravity is agnostic as to the signature of the metric on the emerged manifolds.…”