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Quanta of Space-Time and Axiomatization of Physics

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Foundations of Mathematics and Physics One Century After Hilbert

Abstract

We consider Hilbert’s sixth problem on the axiomatization of physics starting with a higher degree Heisenberg commutation relation involving the Dirac operator and the Feynman slash of scalar fields. The two sided version of the commutation relation in dimension 4 implies volume quantization and determines a noncommutative space which is a tensor product of continuous and discrete spaces. This noncommutative space predicts the full structure of a unified model of all particle interactions based on Pati-Salam symmetries or, as a special case, the Standard Model. We study implications of this quantization condition on Particle Physics, General Relativity, the cosmological constant and dark matter. We demonstrate that, with little input, noncommutative geometry gives a compelling and attractive picture about the nature and structure of space-time.

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Notes

  1. 1.

    Due to a typographical error in the abstract of [12] the fermionic representation was listed incorrectly as \(\left( 2_{R},2_{L},4\right) \) while in the body of the paper the correct representation appears.

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Acknowledgements

I would like to thank Alain Connes for a fruitful and pleasant collaboration on the topic of noncommutative geometry for the last twenty years. I would also like to thank Walter van Suijlekom and Slava Mukhanov for essential contributions to this program of research. This research is supported in part by the National Science Foundation under Grant No. Phys-1518371.

Author information

Authors and Affiliations

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

    Ali H. Chamseddine

  2. Institut des Hautes Etudes Scientifique (I.H.E.S.), 91440, Bures-sur-Yvette, France

    Ali H. Chamseddine

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  1. Ali H. Chamseddine
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Correspondence to Ali H. Chamseddine .

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  1. Nice and Sophia Antipolis University, Nice, France

    Joseph Kouneiher

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Chamseddine, A.H. (2018). Quanta of Space-Time and Axiomatization of Physics. In: Kouneiher, J. (eds) Foundations of Mathematics and Physics One Century After Hilbert. Springer, Cham. https://doi.org/10.1007/978-3-319-64813-2_9

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