C*-algebras of Holonomy-Diffeomorphisms & Quantum Gravity I

TL;DR

This paper introduces a novel algebraic framework combining noncommutative geometry and quantum gravity concepts, using holonomy-diffeomorphisms and a Dirac operator to model quantum spacetime, linking to loop quantum gravity.

Contribution

It constructs a new *-algebra based on holonomy-diffeomorphisms and a Dirac operator, connecting quantum gravity with noncommutative geometry and analyzing its spectral properties.

Findings

Spectrum contained in measurable connections modulo gauge transformations

Limitations on the non-separable spectrum part

Emergence of semi-classical and almost commutative structures

Abstract

A new approach to a unified theory of quantum gravity based on noncommutative geometry and canonical quantum gravity is presented. The approach is built around a *-algebra generated by local holonomy-diffeomorphisms on a 3-manifold and a quantized Dirac type operator; the two capturing the kinematics of quantum gravity formulated in terms of Ashtekar variables. We prove that the separable part of the spectrum of the algebra is contained in the space of measurable connections modulo gauge transformations and we give limitations to the non-separable part. The construction of the Dirac type operator -- and thus the application of noncommutative geometry -- is motivated by the requirement of diffeomorphism invariance. We conjecture that a semi-finite spectral triple, which is invariant under volume-preserving diffeomorphisms, arise from a GNS construction of a semi-classical state. Key elements of quantum field theory emerge from the construction in a semi-classical limit, as does an almost commutative algebra. Finally, we note that the spectrum of loop quantum gravity emerges from a discretization of our construction. Certain convergence issues are left unresolved. This paper is the first of two where the second paper is concerned with mathematical details and proofs concerning the spectrum of the holonomy-diffeomorphism algebra.