On Spectral Triples in Quantum Gravity I - High Energy Physics - TheoryReport as inadecuate

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Abstract: This paper establishes a link between Noncommutative Geometry and canonicalquantum gravity. A semi-finite spectral triple over a space of connections ispresented. The triple involves an algebra of holonomy loops and a Dirac typeoperator which resembles a global functional derivation operator. Theinteraction between the Dirac operator and the algebra reproduces the Poissonstructure of General Relativity. Moreover, the associated Hilbert spacecorresponds, up to a discrete symmetry group, to the Hilbert space ofdiffeomorphism invariant states known from Loop Quantum Gravity.Correspondingly, the square of the Dirac operator has, in terms of canonicalquantum gravity, the form of a global area-squared operator. Furthermore, thespectral action resembles a partition function of Quantum Gravity. Theconstruction is background independent and is based on an inductive system oftriangulations. This paper is the first of two papers on the subject.

Author: Johannes Aastrup, Jesper M. Grimstrup, Ryszard Nest

Source: https://arxiv.org/

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