Noncommutative Geometry and Quantum Gravity

Specialeforsvar ved Jarl Sidelmann

Titel: Noncommutative Geometry and Quantum Gravity

 

Abstract: We present an overview of noncommutative geometry and its applications in theoretical physics, with a view towards quantum gravity. We summarize the Connes-Chamseddine programme, showing how one can formulate gauge theories coupled to Einstein gravity in terms of spectral triples, and how this formalism allows for a geometric unification of both (at the classical level) on an almost-commutative manifold. We build the configuration space of quantum connections and the kinematical Hilbert space of loop quantum gravity (LQG). Based on this construction, we show a result, due to Aastrup-Grimstrup-N´est, that it is possible to formulate a spectral triple, via the algebra of holonomy loops over a projective system of nested lattices, which encodes the fundamental relations of the holonomy-flux algebra from LQG.We touch upon recent advancements that have been inspired by this approach. Our contribution with this thesis is to provide an overarching and pedagogical presentation, so that both subjects – with the aim of studying their intersections – are collected in a single source.

  

Vejledere: Ryszard Nest, Mat
                   Jan Ambjørn, Fysik
Censor:     Poul Hjorth, DTU