Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 8 (2012), 018, 25 pages      arXiv:1203.6164      http://dx.doi.org/10.3842/SIGMA.2012.018
Contribution to the Special Issue “Loop Quantum Gravity and Cosmology”

Intersecting Quantum Gravity with Noncommutative Geometry - a Review

Johannes Aastrup a and Jesper Møller Grimstrup b
a) Institut für Analysis, Leibniz Universität Hannover, Welfengarten 1, D-30167 Hannover, Germany
b) Wildersgade 49b, 1408 Copenhagen, Denmark

Received October 06, 2011, in final form March 16, 2012; Published online March 28, 2012

Abstract
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.

Key words: quantum gravity; noncommutative geometry; semiclassical analysis.

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