Affiliation:
1. Centro de Álgebra de Universidade de Lisboa, Av. Prof. Gama Pinto, 2, 1649-003 Lisboa, Portugal
Abstract
In this paper, we present some of the concepts underlying a program of non-commutative quantum gravity and recall some of the results. This program includes a novel approach to spectral triple categorification and also a precise connection between Fell bundles and Connes' non-commutative geometry. Motivated by topics in quantization of the non-commutative standard model and introduction of algebraic techniques and concepts into quantum gravity (following for example Crane, Baez and Barrett), we define spectral C*-categories, which are deformed spectral triples in a sense made precise. This definition gives to representations of a C*-category on a small category of Hilbert spaces and bounded linear maps, the interpretation of a topological quantum field theory. The construction passes two mandatory tests: (i) there is a classical limit theorem reproducing a Riemannian spin manifold manifesting Connes' and Schücker's non-commutative counterpart of Einstein's equivalence principle, and (ii) there is consistency with the experimental fermion mass matrix. We also present an algebra invariant taking the form of a partition function arising from a C*-bundle dynamical system in connection with C*-subalgebra theory.
Publisher
World Scientific Pub Co Pte Lt
Subject
Physics and Astronomy (miscellaneous)
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