Quantum Gravity coupled to Matter via Noncommutative Geometry

Abstract

We show that the principal part of the Dirac Hamiltonian in 3+1 dimensions emerges in a semi-classical approximation from a construction which encodes the kinematics of quantum gravity. The construction is a spectral triple over a configuration space of connections. It involves an algebra of holonomy loops represented as bounded operators on a separable Hilbert space and a Dirac type operator. Semi-classical states, which involve an averaging over points at which the product between loops is defined, are constructed and it is shown that the Dirac Hamiltonian emerges as the expectation value of the Dirac type operator on these states in a semi-classical approximation.

15 pages, 1 figure

References in corpus (7)

• Background Independent Quantum Gravity: A Status Report
• Gauge Field Theory Coherent States (GCS) : IV. Infinite Tensor Product and Thermodynamical Limit
• Gauge-invariant coherent states for Loop Quantum Gravity I: Abelian gauge groups
• Gauge-invariant coherent states for Loop Quantum Gravity II: Non-abelian gauge groups
• A new spectral triple over a space of connections
• Spectral triples of holonomy loops
• On Spectral Triples in Quantum Gravity I

Cited by in corpus (6)

• C*-algebras of Holonomy-Diffeomorphisms & Quantum Gravity I
• The Quantum Holonomy-Diffeomorphism Algebra & Quantum Gravity
• Canonical Noncommutativity Algebra for the Tetrad Field in General Relativity
• Quantum Holonomy Theory
• On a Lattice-Independent Formulation of Quantum Holonomy Theory
• A note on the fuzzy sphere area spectrum, black hole luminosity, and the quantum nature of spacetime