Quantum Mechanics and the Continuum Limit of an Emergent Geometry

Philip Tee

doi:10.31526/lhep.2021.203

Abstract

Recent advances in emergent geometry have identified a new class of models that represent spacetime as the graph obtained as the ground state of interacting Ising spins. These models have many desirable features, including stable excitations possessing many of the characteristics of a quantum particle. We analyze the dynamics of such excitations, including a detailed treatment of the edge states not previously addressed. Using a minimal prescription for the interaction of defects we numerically investigate approximate bounds to the speed of propagation of such a `particle'. We discover, using numerical simulations, that there may be a Lieb-Robinson bound to propagation that could point the way to how a causal structure could be accommodated in this class of emergent geometry models.

Highlights

1.1 Background and motivationThe search for a consistent theory of Quantum Gravity (QG) has so far not produced a finite and self consistent theory
We proceed to show that the correspondence with non-relativistic dynamics in the continuum limit still holds once the edge states are incorporated, and discuss the details of how the interaction Hamiltonian can cause a defects interaction with remote vertices to propagate without recourse to the continuum limit
Our principle focus in this work was to investigate the dynamics of a defect in an emerged geometry obtained as the ground state of Eq (8)

Summary

Background and motivation

The search for a consistent theory of Quantum Gravity (QG) has so far not produced a finite and self consistent theory. We proceed to show that the correspondence with non-relativistic dynamics in the continuum limit still holds once the edge states are incorporated, and discuss the details of how the interaction Hamiltonian can cause a defects interaction with remote vertices to propagate without recourse to the continuum limit. This is the first contribution of this work. We discover that the simulation does indicate an approximate bound on the speed of propagation, and this is the second contribution of this work This is an intriguing result as the incorporation of causality in the Ising emergent geometry models is not immediately evident, and is an important open problem.

Ground state model and definitions

The interaction Hamiltonian

Quantum mechanics and the continuum limit

Lieb-Robinson bounds and causality

Conclusion