Holonomy Loops, Spectral Triples & Quantum Gravity

TL;DR

This paper reviews a spectral triple construction based on holonomy loops in loop quantum gravity, aiming to connect algebraic structures with quantum gravitational dynamics and proposing a candidate for a quantized Hamiltonian.

Contribution

It introduces a semi-finite spectral triple using holonomy loops, linking loop quantum gravity with noncommutative geometry, and suggests a way to derive a quantum Hamiltonian.

Findings

Spectral triple reproduces the Poisson structure of general relativity

Holonomy loop algebra is central to the construction

Heuristic argument for a quantized Hamiltonian emerging from the spectral triple

Abstract

We review the motivation, construction and physical interpretation of a semi-finite spectral triple obtained through a rearrangement of central elements of loop quantum gravity. The triple is based on a countable set of oriented graphs and the algebra consists of generalized holonomy loops in this set. The Dirac type operator resembles a global functional derivation operator and the interaction between the algebra of holonomy loops and the Dirac type operator reproduces the structure of a quantized Poisson bracket of general relativity. Finally we give a heuristic argument as to how a natural candidate for a quantized Hamiltonian might emerge from this spectral triple construction.