Holonomy Loops, Spectral Triples & Quantum Gravity

Abstract

We review the motivation, construction and physical interpretation of a semi-finite spectral triple obtained through a rearrangement of central elements of loop quantum gravity. The triple is based on a countable set of oriented graphs and the algebra consists of generalized holonomy loops in this set. The Dirac type operator resembles a global functional derivation operator and the interaction between the algebra of holonomy loops and the Dirac type operator reproduces the structure of a quantized Poisson bracket of general relativity. Finally we give a heuristic argument as to how a natural candidate for a quantized Hamiltonian might emerge from this spectral triple construction.

24 pages, 7 figures, based on talk given by J.M.G. at the QG2 conference, Nottingham, juli 2008; at the QSTNG conference in Rome in sept/oct 2008; at the AONCG conference, Canberra, dec. 2008

References in corpus (12)

• Background Independent Quantum Gravity: A Status Report
• Quantum Theory of Gravity I: Area Operators
• Gravity coupled with matter and foundation of non-commutative geometry
• Real Ashtekar Variables for Lorentzian Signature Space-times
• The Spectral Action Principle
• Barbero's Hamiltonian derived from a generalized Hilbert-Palatini action
• Differential Geometry on the Space of Connections via Graphs and Projective Limits
• A Universal Action Formula
• A new spectral triple over a space of connections
• Spectral triples of holonomy loops
• Intersecting Connes Noncommutative Geometry with Quantum Gravity
• On Spectral Triples in Quantum Gravity I

Cited by in corpus (5)

• Spin Foams and Noncommutative Geometry
• On Semi-Classical States of Quantum Gravity and Noncommutative Geometry
• Modular Theory, Non-Commutative Geometry and Quantum Gravity
• Intersecting Quantum Gravity with Noncommutative Geometry - a Review
• C*-algebras of Holonomy-Diffeomorphisms & Quantum Gravity I