Towards quantum gravity with neural networks: Solving quantum Hamilton constraints of 3d Euclidean gravity in the weak coupling limit

TL;DR

This paper demonstrates how neural networks can solve quantum constraints in 3d Euclidean gravity modeled as BF-theory, revealing the potential of machine learning in quantum gravity research.

Contribution

It introduces a neural network approach to solve quantum Hamilton constraints in 3d Euclidean gravity, a novel application in quantum gravity research.

Findings

Neural networks successfully solve quantum constraints with large overlap in solutions.

The quantum volume operator's behavior is analyzed in the solution space.

The neural network approach explores models beyond traditional numerical methods.

Abstract

We consider 3-dimensional Euclidean gravity in the weak coupling limit of Smolin and show that it is BF-theory with $\text{U(1)}^3$ as a Lie group. The theory is quantised using loop quantum gravity methods. The kinematical degrees of freedom are truncated, on account of computational feasibility, by fixing a graph and deforming the algebra of the holonomies to impose a cutoff on the charge vectors. This leads to a quantum theory related to $\text{U}_q \text{(1)}^3$ BF-theory. The effect of imposing the cutoff on the charges is examined. We also implement the quantum volume operator of 3d loop quantum gravity. Most importantly we compare two constraints for the quantum model obtained: a master constraint enforcing curvature and Gauss constraint, as well as a combination of a quantum Hamilton constraint constructed using Thiemann's strategy and the Gauss master constraint. The two constraints are solved using the neural network quantum state ansatz, demonstrating its ability to explore models which are out of reach for exact numerical methods. The solutions spaces are quantitatively compared and although the forms of the constraints are radically different, the solutions turn out to have a surprisingly large overlap. We also investigate the behavior of the quantum volume in solutions to the constraints.