Quantum Gravity coupled to Matter via Noncommutative Geometry

TL;DR

This paper demonstrates how the Dirac Hamiltonian in 3+1 dimensions naturally arises from a noncommutative geometric framework that encodes quantum gravity kinematics, providing a bridge between quantum gravity and matter fields.

Contribution

It introduces a spectral triple over a space of connections that yields the Dirac Hamiltonian as a semi-classical limit, linking noncommutative geometry with quantum gravity and matter.

Findings

Dirac Hamiltonian emerges as a semi-classical expectation value

Spectral triple encodes quantum gravity kinematics

Construction involves holonomy loops and a Dirac type operator

Abstract

We show that the principal part of the Dirac Hamiltonian in 3+1 dimensions emerges in a semi-classical approximation from a construction which encodes the kinematics of quantum gravity. The construction is a spectral triple over a configuration space of connections. It involves an algebra of holonomy loops represented as bounded operators on a separable Hilbert space and a Dirac type operator. Semi-classical states, which involve an averaging over points at which the product between loops is defined, are constructed and it is shown that the Dirac Hamiltonian emerges as the expectation value of the Dirac type operator on these states in a semi-classical approximation.