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Geometry and the quantum: basics

Journal of High Energy Physics volume 2014, Article number: 98 (2014) Cite this article

A preprint version of the article is available at arXiv.

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 one-sided 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 non-commutative 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 4-manifold with quantized volume > 4 (in suitable units) appears as an irreducible representation of the two-sided 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.

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This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

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Affiliations

  1. Physics Department, American University of Beirut, Beirut, Lebanon

    Ali H. Chamseddine

  2. College de France, 3 rue Ulm, F75005, Paris, France

    Alain Connes

  3. I.H.E.S., F-91440, Bures-sur-Yvette, France

    Ali H. Chamseddine & Alain Connes

  4. Department of Mathematics, The Ohio State University, Columbus, OH, 43210, U.S.A.

    Alain Connes

  5. Theoretical Physics, Ludwig Maxmillians University, Theresienstr. 37, 80333, Munich, Germany

    Viatcheslav Mukhanov

  6. MPI for Physics, Foehringer Ring, 6, 80850, Munich, Germany

    Viatcheslav Mukhanov

Authors
  1. Ali H. Chamseddine
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  2. Alain Connes
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  3. Viatcheslav Mukhanov
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Corresponding author

Correspondence to Ali H. Chamseddine.

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ArXiv ePrint: 1411.0977

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Chamseddine, A.H., Connes, A. & Mukhanov, V. Geometry and the quantum: basics. J. High Energ. Phys. 2014, 98 (2014). https://doi.org/10.1007/JHEP12(2014)098

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  • DOI: https://doi.org/10.1007/JHEP12(2014)098

Keywords

  • Non-Commutative Geometry
  • Differential and Algebraic Geometry