Intersecting quantum gravity with noncommutative geometry - a review. (English) Zbl 1251.83017
Summary: 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 natural class of semiclassical states which entail emerging fermionic degrees of freedom. In the particular semiclassical approximation where all gravitational degrees of freedom are turned off, a free fermionic quantum field theory emerges. We end the paper with an extended outlook section.
MSC:
83C45 | Quantization of the gravitational field |
46L52 | Noncommutative function spaces |
46L87 | Noncommutative differential geometry |
46L89 | Other “noncommutative” mathematics based on \(C^*\)-algebra theory |
58B34 | Noncommutative geometry (à la Connes) |
81R60 | Noncommutative geometry in quantum theory |
81T75 | Noncommutative geometry methods in quantum field theory |
83C65 | Methods of noncommutative geometry in general relativity |
70S15 | Yang-Mills and other gauge theories in mechanics of particles and systems |
81T20 | Quantum field theory on curved space or space-time backgrounds |