Hyperbolic groups, 4-manifolds and Quantum Gravity
arXiv:1811.04464 · doi:10.1088/1742-6596/1194/1/012009
Abstract
4-manifolds have special topological properties which can be used to get a different view on quantum mechanics. One important property (connected with exotic smoothness) is the natural appearance of 3-manifold wild embeddings (Alexanders horned sphere) which can be interpreted as quantum states. This relation can be confirmed by using the Turaev-Drinfeld quantization procedure. Every part of the wild embedding admits a hyperbolic geometry uncovering a deep connection between quantum mechanics and hyperbolic geometry. Then the corresponding symmetry is used to get a dimensional reduction from 4 to 2 for infinite curvatures. Physical consequences will be discussed. At the end we will obtain a spacetime representation of a quantum state of geometry by a non-singular fractal space (wild embedding) which is stable in the limit of infinite curvatures.
10 pages, 1 figure, Proceedings of Group 32 (Prague, July 2018). arXiv admin note: text overlap with arXiv:1211.3012, arXiv:1601.06436