Towards quantum gravity with neural networks: Solving the quantum Hamilton constraint of U(1) BF theory

Abstract

In the canonical approach of loop quantum gravity, arguably the most important outstanding problem is finding and interpreting solutions to the Hamiltonian constraint. In this work, we demonstrate that methods of machine learning are in principle applicable to this problem. We consider $U(1)$ BF theory in 3 dimensions, quantized with loop quantum gravity methods. In particular, we formulate a master constraint corresponding to Hamilton and Gauss constraints using loop quantum gravity methods. To make the problem amenable for numerical simulation we fix a graph and introduce a cutoff on the kinematical degrees of freedom, effectively considering $U_q(1)$ BF theory at a root of unity. We show that the Neural Network Quantum State (NNQS) ansatz can be used to numerically solve the constraints efficiently and accurately. We compute expectation values and fluctuations of certain observables and compare them with exact results or exact numerical methods where possible. We also study the dependence on the cutoff.

43 pages, 12 figures. Version now identical to the one in Class. Quantum Gravity

References in corpus (13)

• The density-matrix renormalization group in the age of matrix product states
• Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
• Computational complexity and fundamental limitations to fermionic quantum Monte Carlo simulations
• Quantum Nature of the Big Bang: Improved dynamics
• Weak binding between two aromatic rings: feeling the van der Waals attraction by quantum Monte Carlo methods
• Algebraic Quantum Gravity (AQG) I. Conceptual Setup
• Algebraic Quantum Gravity (AQG) II. Semiclassical Analysis
• $BF$ gravity
• Spinfoam on Lefschetz Thimble: Markov Chain Monte-Carlo Computation of Lorentzian Spinfoam Propagator
• An elementary introduction to loop quantum gravity
• Gauge Equivariant Neural Networks for 2+1D U(1) Gauge Theory Simulations in Hamiltonian Formulation
• Towards quantum gravity with neural networks: Solving quantum Hamilton constraints of 3d Euclidean gravity in the weak coupling limit
• Classical theta functions from a quantum group perspective

Cited by in corpus (3)

• Towards quantum gravity with neural networks: Solving quantum Hamilton constraints of 3d Euclidean gravity in the weak coupling limit
• Taming Thiemann's Hamiltonian constraint in canonical loop quantum gravity: reversibility, eigenstates and graph-change analysis
• Computing the graph-changing dynamics of loop quantum gravity