Author:
(1) Ahmed Farag Ali, Essex County College and Department of Physics, Faculty of Science, Benha University. Table of Links Abstract and Introduction Space-time quanta and Becken Universal bound Shape of space-time quanta Symmetry of space-time quanta Space-time quanta and Spectral mass gap Phenomenological implications Conclusion, Acknowledgments, and References V. SPACE-TIME QUANTA AND SPECTRAL MASS GAP The solution of the mass gap problem as described in [62] requires proving that Yang-Mills theory exists and that the mass of all particles predicted by the theory is strictly positive. Both conditions are satisfied by the space-time quanta. According to Eq. (9), we find that the mass of space-time quanta is a real positive value. The space-time quanta is quatified by the parameter κ in Snyder algebra that was found to generate both non-commutative geometry and GUP at the same time. 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]. Since the space-time is locally flat, the space-time quanta must be described by a 4-polytope. We introduce a geometric and symmetric reason to consider the 24-cell as the space-time quanta. The most important reason is that 24-cell is the Weyl/Coxeter group of F4 group that can generate the standard model gauge symmetry as shown in recent studies [45]. Therefore, Yang-Mill’s theory exists as a F4 group and is explained by the space-time quanta from the first geometric principles as 24-cell. The gluon masses should be related to the 16-cell which is related to the 24-cell we explained in previous sections. We conclude that the space-time quanta introduces a geometric origin of the spectral mass gap [62]. The spectral mass gap is entirely determined by the length/radius of 24-cell according to Eq. (9). Recently, it was shown that spectral gaps exist in Hamiltonian with quasicrystal line order [63]. Quasicrystal considerations in Holography, the basic structure of nature, and cosmology are discussed in [64–71]. We think the quantum space-time may be a quasicrystal with a fundamental structure of a 24-cell. Experimental observations of quantum time quasicrystal are reported in [72]. This quasicrystal order is expected to follow from simulating Snyder’s algebra with considering 24-cell as its fundamental structure. This needs further investigation. This paper is available on arxiv under CC BY 4.0 DEED license. Author: (1) Ahmed Farag Ali, Essex County College and Department of Physics, Faculty of Science, Benha University. Author: Author: (1) Ahmed Farag Ali, Essex County College and Department of Physics, Faculty of Science, Benha University. Table of Links Abstract and Introduction Abstract and Introduction Space-time quanta and Becken Universal bound Space-time quanta and Becken Universal bound Shape of space-time quanta Shape of space-time quanta Symmetry of space-time quanta Symmetry of space-time quanta Space-time quanta and Spectral mass gap Space-time quanta and Spectral mass gap Phenomenological implications Phenomenological implications Conclusion, Acknowledgments, and References Conclusion, Acknowledgments, and References V. SPACE-TIME QUANTA AND SPECTRAL MASS GAP The solution of the mass gap problem as described in [62] requires proving that Yang-Mills theory exists and that the mass of all particles predicted by the theory is strictly positive. Both conditions are satisfied by the space-time quanta. According to Eq. (9), we find that the mass of space-time quanta is a real positive value. The space-time quanta is quatified by the parameter κ in Snyder algebra that was found to generate both non-commutative geometry and GUP at the same time. 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]. Since the space-time is locally flat, the space-time quanta must be described by a 4-polytope. We introduce a geometric and symmetric reason to consider the 24-cell as the space-time quanta. The most important reason is that 24-cell is the Weyl/Coxeter group of F4 group that can generate the standard model gauge symmetry as shown in recent studies [45]. Therefore, Yang-Mill’s theory exists as a F4 group and is explained by the space-time quanta from the first geometric principles as 24-cell. The gluon masses should be related to the 16-cell which is related to the 24-cell we explained in previous sections. We conclude that the space-time quanta introduces a geometric origin of the spectral mass gap [62]. The spectral mass gap is entirely determined by the length/radius of 24-cell according to Eq. (9). Recently, it was shown that spectral gaps exist in Hamiltonian with quasicrystal line order [63]. Quasicrystal considerations in Holography, the basic structure of nature, and cosmology are discussed in [64–71]. We think the quantum space-time may be a quasicrystal with a fundamental structure of a 24-cell. Experimental observations of quantum time quasicrystal are reported in [72]. This quasicrystal order is expected to follow from simulating Snyder’s algebra with considering 24-cell as its fundamental structure. This needs further investigation. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

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