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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