Johannes Aastrup

On spectral triples in quantum gravity: I

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 commutation relation between the Dirac operator and the algebra has a structure related to the Poisson bracket of general relativity. Moreover, the associated Hilbert space corresponds, up to a certain 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 loop quantum gravity, the form of a global area-squared operator. Furthermore, the spectral action functional 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.

A New Spectral Triple over a Space of Connections

A new construction of a semifinite spectral triple on an algebra of holonomy loops is presented. The construction is canonically associated to quantum gravity and is an alternative version of the spectral triple presented in [1].

Spectral triples of holonomy loops

The machinery of noncommutative geometry is applied to a space of connections. A noncommutative function algebra of loops closely related to holonomy loops is investigated. The space of connections is identified as a projective limit of Lie-groups composed of copies of the gauge group. A spectral triple over the space of connections is obtained by factoring out the diffeomorphism group. The triple consist of equivalence classes of loops acting on a hilbert space of sections in an infinite dimensional Clifford bundle. We find that the Dirac operator acting on this hilbert space does not fully comply with the axioms of a spectral triple.

On Spectral Triples in Quantum Gravity II

A semifinite spectral triple for an algebra canonically associated to canonical quantum gravity is constructed. The algebra is generated by based loops in a triangulation and its barycentric subdivisions. The underlying space can be seen as a gauge fixing of the unconstrained state space of Loop Quantum Gravity. This paper is the second of two papers on the subject. email: johannes.aastrup@uni-muenster.de email: grimstrup@nbi.dk email: rnest@math.ku.dk

On Semi-Classical States of Quantum Gravity and Noncommutative Geometry

We construct normalizable, semi-classical states for the previously proposed model of quantum gravity which is formulated as a spectral triple over holonomy loops. The semi-classical limit of the spectral triple gives the Dirac Hamiltonian in 3+1 dimensions. Also, time-independent lapse and shift fields emerge from the semi-classical states. Our analysis shows that the model might contain fermionic matter degrees of freedom.The semi-classical analysis presented in this paper does away with most of the ambiguities found in the initial semi-finite spectral triple construction. The cubic lattices play the role of a coordinate system and a divergent sequence of free parameters found in the Dirac type operator is identified as a certain inverse infinitesimal volume element.

Boutet de Monvel’s calculus and groupoids I

Can Boutet de Monvels algebra on a compact manifold with boundary be obtained as the algebra

Quantum Gravity coupled to Matter via Noncommutative Geometry

\Psi^0(G)

Holonomy loops, spectral triples and quantum gravity

of pseudodifferential operators on some Lie groupoid

From quantum gravity to quantum field theory via noncommutative geometry

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On a lattice-independent formulation of quantum holonomy theory

? If it could, the kernel