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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.

References

  1. J. Alexander, Note on Riemann spaces, Bull. Amer. Math. Soc. 26 (1920) 370.

    Article  MATH  MathSciNet  Google Scholar 

  2. A.L. Carey, A. Rennie, F. Sukochev and D. Zanin, Universal measurability and the Hochschild class of the Chern character, arXiv:1401.1860.

  3. A.H. Chamseddine and A. Connes, The uncanny precision of the spectral action, Commun. Math. Phys. 293 (2010) 867 [arXiv:0812.0165] [INSPIRE].

    ADS  Article  MATH  MathSciNet  Google Scholar 

  4. A.H. Chamseddine and A. Connes, Why the standard model, J. Geom. Phys. 58 (2008) 38 [arXiv:0706.3688] [INSPIRE].

    ADS  Article  MATH  MathSciNet  Google Scholar 

  5. A.H. Chamseddine, A. Connes and W.D. van Suijlekom, Beyond the spectral standard model: emergence of Pati-Salam unification, JHEP 11 (2013) 132 [arXiv:1304.8050] [INSPIRE].

    ADS  Article  Google Scholar 

  6. A.H. Chamseddine, A. Connes and V. Mukhanov, in preparation.

  7. A. Connes, Noncommutative geometry, Academic Press, U.S.A. (1994).

    MATH  Google Scholar 

  8. A. Connes, A short survey of noncommutative geometry, J. Math. Phys. 41 (2000) 3832 [hep-th/0003006] [INSPIRE].

    ADS  Article  MATH  MathSciNet  Google Scholar 

  9. A. Connes and H. Moscovici The local index formula in noncommutative geometry, Geom. Funct. Anal. 5 (1995) 174.

    Article  MATH  MathSciNet  Google Scholar 

  10. M. Golubitsky and V. Guillemin, Stable mappings and their singularities, Graduate Texts in Mathematics volume 14. Springer, Germany (1973).

    Book  Google Scholar 

  11. J. Gracia-Bondia, J. Varilly and H. Figueroa, Elements of noncommutative geometry, Birkhauser Boston Inc., Boston U.S.A. (2001).

    Book  MATH  Google Scholar 

  12. W. Greub, S. Halperin and R. Vanstone, Connections, curvature and cohomology, Academic Press, U.S.A. (1976).

    MATH  Google Scholar 

  13. H. Hilden, J. Montesinos and T. Thickstun, Closed oriented 3-manifolds as 3-fold branched coverings of S 3 of special type, Pacific J. Math. 65 (1976) 65.

    Article  MATH  MathSciNet  Google Scholar 

  14. M. Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959) 242.

    Article  MATH  MathSciNet  Google Scholar 

  15. M. Iori and R. Piergallini, 4-manifolds as covers of the 4-sphere branched over non-singular surfaces, Geom. Topol. 6 (2002) 393.

    Article  MATH  MathSciNet  Google Scholar 

  16. J. Montesinos, A representation of closed orientable 3-manifolds as 3-fold branched coverings of S 3, Bull. Amer. Math. Soc. 80 (1974) 845.

    Article  MATH  MathSciNet  Google Scholar 

  17. J. Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965) 286.

    Article  MATH  MathSciNet  Google Scholar 

  18. V. Poenaru, Sur la théorie des immersions, Topology 1 (1962) 81.

    Article  MATH  MathSciNet  Google Scholar 

  19. A. Phillips, Submersions of open manifolds, Topology 6 (1967) 171.

    Article  MATH  MathSciNet  Google Scholar 

  20. A. Ramirez, Sobre un teorema de Alexander, Ann. Inst. Mat. Univ. Nac. Autonoma Mexico 15 (1975) 77.

    MathSciNet  Google Scholar 

  21. B. Sevennec, Janus immersions in the product of two n-spheres (2014).

  22. S. Smale, The classification of immersions of spheres in Euclidean spaces, Ann. Math. 69 (1959) 327.

    Article  MATH  MathSciNet  Google Scholar 

  23. H. Whitney, On the topology of differentiable manifolds, Lectures in Topology, University of Michigan Press, Ann Arbor, U.S.A. (1941).

    Google Scholar 

  24. H. Whitney, The self-intersection of a smooth n-manifold in 2n-space, Ann. Math. 45 (1944) 220.

    Article  MATH  MathSciNet  Google Scholar 

Download references

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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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Authors and 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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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