On Spectral Triples in Quantum Gravity I

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.

84 pages, 8 figures

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Cited by in corpus (8)

• Spin Foams and Noncommutative Geometry
• On Semi-Classical States of Quantum Gravity and Noncommutative Geometry
• On a Classification of Irreducible Almost-Commutative Geometries V
• The Quantum Holonomy-Diffeomorphism Algebra & Quantum Gravity
• Quantum Gravity coupled to Matter via Noncommutative Geometry
• Holonomy Loops, Spectral Triples & Quantum Gravity
• Quantum Holonomy Theory
• The Inverse Seesaw Mechanism in Noncommutative Geometry