Intersecting Connes Noncommutative Geometry with Quantum Gravity

Intersecting Quantum Gravity with Noncommutative Geometry – a Review

We review applications of noncommutative geometry in canonical quantum gravity. First, we show that the framework of loop quantum gravity includes natural noncommutative structures which have, hitherto, not been explored. Next, we present the construction of a spectral triple over an algebra of holonomy loops. The spectral triple, which encodes the kinematics of quantum gravity, gives rise to a...

A Note on the Symmetries and Renormalisability of (Quantum) Gravity

We make some remarks on the group of symmetries in gravity; we believe that K-theory and noncommutative geometry inescepably have to play an important role. Furthermore we make some comments and questions on the recent work of Connes and Kreimer on renormalisation, the Riemann-Hilbert correspondence and their relevance to quantum gravity. PACS classification: 11.10.-z; 11.15.-q; 11.30.-Ly

Noncommutative geometry and number theory

Noncommutative geometry is a modern field of mathematics begun by Alain Connes in the early 1980s. It provides powerful tools to treat spaces that are essentially of a quantum nature. Unlike the case of ordinary spaces, their algebraof coordinates is noncommutative, reflecting phenomena like the Heisenberg uncertainty principle in quantum mechanics. What is especially interesting is the fact th...

Geometry and the Quantum: Basics

Motivated by the construction of spectral manifolds in noncommutative geometry, we introduce a higher degree Heisenberg commutation relation involving the Dirac operator and the Feynman slash of scalar fields. This commutation relation appears in two versions, one sided and two sided. It implies the quantization of the volume. In the one-sided case it implies that the manifold decomposes into a...

Noncommutative Geometry, Quantum Fields and Motives