**Author:**

(1) Ahmed Farag Ali, Essex County College and Department of Physics, Faculty of Science, Benha University.

## Table of Links

Space-time quanta and Becken Universal bound

Space-time quanta and Spectral mass gap

Conclusion, Acknowledgments, and References

## Abstract

Snyderâs quantum space-time which is Lorentz invariant is investigated. It is found that the quanta of space-time have a positive mass that is interpreted as a positive real mass gap of space-time. This mass gap is related to the minimal length of measurement which is provided by Snyderâs algebra. Several reasons to consider the space-time quanta as a 24-cell are discussed. Geometric reasons include its self-duality property and its 24 vertices that may represent the standard model of elementary particles. The 24-cell symmetry group is the Weyl/Coxeter group of the F4 group which was found recently to generate the gauge group of the standard model. It is found that 24-cell may provide a geometric interpretation of the mass generation, Avogadro number, color confinement, and the flatness of the observable universe. The phenomenology and consistency with measurements is discussed.

âThe knowledge at which geometry aims is knowledge of the eternalââ Plato.

## I. INTRODUCTION

In 1947, Snyder established a remarkable step that reconciles the minimal length of measurement with Lorentz symmetry by constructing quantum Lorentzian space-time [1]. The price was introducing non-commutative geometry and the generalized uncertainty principle (GUP) in Snyderâs algebra. For Non-commutative geometry part, it is found to emerge naturally at limits of M/string theory [2] as higher dimensional corrections of ordinary Yang-Mills theory [3]. Several implications of non-commutative geometry were investigated in quantum field theory and condensed matter systems [4, 5]. For the GUP part, it emerged in several approaches to quantum gravity such as string theory, loop quantum gravity, and quantum geometry [6â12]. Phenomenological and experimental implications of the GUP have been investigated in low and high-energy systems [13â25]. Useful reviews on quantum space-time and GUP can be found in [26â28]. Snyderâs algebra is generated by three main generators which are position xÂ”, momentum pÂ” and Lorentz generators JÂ”Îœ = xÂ”pÎœ â xÎœpÂ”. They satisfy the PoincarÂŽe commutation relations and suggest new commutation relations that provide a quantum/minimal length as follows:

where âP l is a Planck length, Îș is a dimensionless parameter that identifies the minimal measurable length, and Î·Â”Îœ = (â1, 1, 1, 1). Eq. (1) introduces the non-commutative geometry and Eq. (2) introduces a GUP. Both equations are invariant under Lorentz symmetry [1].

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