On Spectral Triples in Quantum Gravity I

TL;DR

This paper introduces a spectral triple framework linking Noncommutative Geometry with canonical quantum gravity, reproducing key structures like the Poisson bracket and diffeomorphism invariance, and providing a new perspective on quantum gravity dynamics.

Contribution

It constructs a semi-finite spectral triple over connections that captures quantum gravity features, integrating noncommutative geometry with loop quantum gravity concepts.

Findings

Reproduces the Poisson structure of General Relativity

Hilbert space aligns with diffeomorphism-invariant states

Spectral action resembles a quantum gravity partition function

Abstract

This paper establishes a link between Noncommutative Geometry and canonical quantum gravity. A semi-finite spectral triple over a space of connections is presented. The triple involves an algebra of holonomy loops and a Dirac type operator which resembles a global functional derivation operator. The interaction between the Dirac operator and the algebra reproduces the Poisson structure of General Relativity. Moreover, the associated Hilbert space corresponds, up to a discrete symmetry group, to the Hilbert space of diffeomorphism invariant states known from Loop Quantum Gravity. Correspondingly, the square of the Dirac operator has, in terms of canonical quantum gravity, the form of a global area-squared operator. Furthermore, the spectral action resembles a partition function of Quantum Gravity. The construction is background independent and is based on an inductive system of triangulations. This paper is the first of two papers on the subject.