2008/02/13 by Johannes Aastrup, Jesper M. Grimstrup, Jesper Møller Grimstrup +1
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Advanced Operator Algebra Research #Diffeomorphism #Dirac operator #Hilbert space #Holonomy #Noncommutative and Quantum Gravity Theories #Noncommutative geometry #Operator (biology) #Operator algebra #Partition function (quantum field theory) #Quantum spacetime #Spectral triple #gr-qc #hep-th
paper · pdf · doi:10.1088/0264-9381/26/6/065011
84 pages, 8 figures
arxiv created 2008/02/13 · openalex publication_date 2009/02/25 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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.