Chamseddine, Ali H.; Connes, Alain; Mukhanov, Viatcheslav
(2014):
Geometry and the quantum: basics.
In: Journal of High Energy Physics, Vol. 12, 98

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Abstract
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 onesided case it implies that the manifold decomposes into a disconnected sum of spheres which will represent quanta of geometry. The two sided version in dimension 4 predicts the two algebras M 2(ℍ) and M 4(ℂ) which are the algebraic constituents of the Standard Model of particle physics. This taken together with the noncommutative algebra of functions allows one to reconstruct, using the spectral action, the Lagrangian of gravity coupled with the Standard Model. We show that any connected Riemannian Spin 4manifold with quantized volume > 4 (in suitable units) appears as an irreducible representation of the twosided commutation relations in dimension 4 and that these representations give a seductive model of the “particle picture” for a theory of quantum gravity in which both the Einstein geometric standpoint and the Standard Model emerge from Quantum Mechanics. Physical applications of this quantization scheme will follow in a separate publication.