Intersecting Quantum Gravity with Noncommutative Geometry - a Review

TL;DR

This review explores how noncommutative geometry integrates with canonical quantum gravity, revealing new structures, spectral triples, and emergent fermionic fields within loop quantum gravity.

Contribution

It demonstrates the natural appearance of noncommutative structures in loop quantum gravity and constructs a spectral triple over holonomy loops, linking quantum gravity with fermionic fields.

Findings

Noncommutative structures are inherent in loop quantum gravity.

Spectral triples can encode quantum gravity kinematics.

Emergence of fermionic quantum field theories in semiclassical limits.

Abstract

We review applications of noncommutative geometry in canonical quantum gravity. First, we show that the framework of loop quantum gravity includes natural noncommutative structures which have, hitherto, not been explored. Next, we present the construction of a spectral triple over an algebra of holonomy loops. The spectral triple, which encodes the kinematics of quantum gravity, gives rise to a natural class of semiclassical states which entail emerging fermionic degrees of freedom. In the particular semiclassical approximation where all gravitational degrees of freedom are turned off, a free fermionic quantum field theory emerges. We end the paper with an extended outlook section.