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 VII. CONCLUSION We found that the GUP implied by Snyder's algebra vanishes at a specific energy scale. We define this energy scale as the scale of space-time quanta at which wavefunction collapses to form a mass. The mass of space-time quanta forms a mass gap of space-time. The covariance principle requires the space-time quanta to be a 4-dimensional object and to represent the elementary particles. Based on the geometric and symmetric analysis, we propose that the space-time quanta be represented by the 24-cell. First, it is highly-symmetric convex regular 4-polytope and self-dual. Second, The symmetry group of 24-cell is the Weyl/Coxeter group of Fs group that generates the gauge group of the standard model by the intersection of its two stabilizer groups. In addition, the 24-cell has a beautiful geometric property in which its vertices can be grouped into 3 different sets of eight vertices, each defining 16-cell with the rest defining the dual tesseract with 16 vertices. Therefore, we represent 8 vertices of 16-cell with the 8 gluons that may give a geometric interpretation of the color confinement. We represent 16 vertices of tesseract with the 12 fermions and 4 gauge bosons which may explain the flatness of the observable universe. The Higgs particle is represented as the tesseract which explains why Higgs only couple with 12 fermions and 4 gauge bosons and does not couple with 8 gluons. Vanishing uncertainty implies the creation of a mass of space-time quanta, so the length of 24-cell is identified by the electroweak length scale 10-18 m of mass creation and which is consistent with experimental measurements of the smallest measured value of charge radius of scattering for neutrinos, Higgs, Z-boson, and W boson. We think that a 24-cell symmetry group which is a solvable group of order "1152" could be useful in quantum computing. We hope to report on this application in the future. Acknowledgments I thank the Editor and referee for their meticulous review of this manuscript and express gratitude towards Klee Irwin, Raymond Aschheim, Fang Fang, Richard Clawson, and Dugan Hammock for their enlightening discussions on Polytopes and Quasicrystals. Special thanks to Ahmed Almheiri for his engaging lecture on the Path integral for chords. This manuscript is a tribute to my father, Farag Mohamed Ali, whose legacy illuminates my journey, and in memory of Mustafa Shalaby, whose teachings have profoundly inspired my exploration of physics’ core principles. 1] H. S. Snyder, Phys. Rev. 71, 38 (1947). [2] A. Connes, M. R. Douglas, and A. S. Schwarz, JHEP 02, 003 (1998), arXiv:hep-th/9711162. [3] N. Seiberg and E. Witten, JHEP 09, 032 (1999), arXiv:hep-th/9908142. [4] M. R. Douglas and N. A. Nekrasov, Rev. Mod. Phys. 73, 977 (2001), arXiv:hep-th/0106048. [5] R. J. Szabo, Phys. Rept. 378, 207 (2003), arXiv:hep-th/0109162 . [6] D. Amati, M. Ciafaloni, and G. Veneziano, Phys. Lett. B 216, 41 (1989). [7] L. J. Garay, Int. J. Mod. Phys. A 10, 145 (1995), arXiv:gr-qc/9403008. [8] F. Scardigli, Phys. Lett. B 452, 39 (1999), arXiv:hep-th/9904025. [9] K. Konishi, G. Paffuti, and P. Provero, Phys. Lett. B 234, 276 (1990). [10] A. Kempf et al., Phys. Rev. D 52, 1108 (1995), arXiv:hep-th/9412167 . [11] M. Maggiore, Phys. Lett. B 304, 65 (1993), arXiv:hep-th/9301067. [12] S. Capozziello et al., Int. J. Theor. Phys. 39, 15 (2000), arXiv:gr-qc/9910017. [13] S. Das and E. C. Vagenas, Phys. Rev. Lett. 101, 221301 (2008), arXiv:0810.5333 [hep-th]. [14] I. Pikovski et al., Nature Phys. 8, 393 (2012), arXiv:1111.1979 [quant-ph]. [15] F. Marin et al., Nature Phys. 9, 71 (2013). [16] L. Petruzziello et al., Nature Commun. 12, 4449 (2021), arXiv:2011.01255 [gr-qc]. [17] S. P. Kumar et al., Nature Commun. 11, 3900 (2020), arXiv:1908.11164 [quant-ph]. [18] H. Moradpour et al., Mon. Not. Roy. Astron. Soc. 488, L69 (2019), arXiv:1907.12940 [gr-qc]. [19] D. Gao and M. Zhan, Physical Review A 94, 013607 (2016). [20] M. Bawaj et al., Nature Commun. 6, 7503 (2015), arXiv:1411.6410 [gr-qc]. [21] P. Girdhar et al., New J. Phys. 22, 093073 (2020), arXiv:2005.08984 [quant-ph]. [22] A. Ashoorioon et al., Phys. Rev. D 71, 023503 (2005), arXiv:astro-ph/0410139. [23] R. Easther et al., Phys. Rev. D 67, 063508 (2003), arXiv:hep-th/0110226. [24] M. P. Dabrowski and F. Wagner, The European Physical Journal C 79, 1 (2019). [25] R. Easther et al., Phys. Rev. D 64, 103502 (2001), arXiv:hep-th/0104102. [26] A. Addazi et al., (2021), arXiv:2111.05659 [hep-ph] . [27] S. Hossenfelder, Living Rev. Rel. 16, 2 (2013), arXiv:1203.6191 [gr-qc]. [28] S. Mignemi, Ukr. J. Phys. 64, 991 (2019), arXiv:1911.06127 [hep-th]. [29] D. J. Gross and F. Wilczek, Phys. Rev. Lett. 30, 1343 (1973). [30] H. D. Politzer, Phys. Rev. Lett. 30, 1346 (1973). [31] J. R. Ellis, S. Mohanty, and D. V. Nanopoulos, Phys. Lett. B 221, 113 (1989). [32] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935). [33] A. F. Ali, (2022), arXiv:2210.13974 [quant-ph]. [34] A. F. Ali, (2022), 10.2139/ssrn.4292472. [35] J. D. Bekenstein, Phys. Rev. D 23, 287 (1981). [36] J. D. Bekenstein, Found. Phys. 35, 1805 (2005), arXiv:quant-ph/0404042. [37] J. D. Bekenstein, Phys. Lett. B 481, 339 (2000), arXiv:hep-th/0003058. [38] L. Lu and A. Stern, Nucl. Phys. B 860, 186 (2012), arXiv:1110.4112 [hep-th]. [39] R. Banerjee, K. Kumar, and D. Roychowdhury, JHEP 03, 060 (2011), arXiv:1101.2021 [hepth]. [40] H. S. M. Coxeter, Regular polytopes (Courier Corporation, 1973). [41] M. J. Wenninger, Dual models (Cambridge University Press, 2003). [42] G. C. Wick, Phys. Rev. 96, 1124 (1954). [43] G. W. Gibbons and S. W. Hawking, Euclidean quantum gravity (World Scientific, 1993). [44] S. Weinberg, Phys. Rev. Lett. 19, 1264 (1967). [45] I. Todorov and M. Dubois-Violette, Int. J. Mod. Phys. A 33, 1850118 (2018), arXiv:1806.09450 [hep-th] . [46] F. Bernardoni, S. L. Cacciatori, B. L. Cerchiai, and A. Scotti, Adv. Theor. Math. Phys. 12, 889 (2008), arXiv:0705.3978 [math-ph]. [47] J. C. Baez, The n-Category Cafe 27 (2018). [48] L. Boyle, (2020), arXiv:2006.16265 [hep-th]. [49] M. Dubois-Violette, Nucl. Phys. B 912, 426 (2016), arXiv:1604.01247 [math.QA]. [50] L. Boyle and S. Farnsworth, New J. Phys. 22, 073023 (2020), arXiv:1910.11888 [hep-th]. [51] I. Todorov, Springer Proc. Math. Stat. 335, 29 (2019), arXiv:1911.13124 [hep-th] . [52] V. Bhatt, R. Mondal, V. Vaibhav, and T. P. Singh, J. Phys. G 49, 045007 (2022), arXiv:2108.05787 [hep-ph]. [53] V. Vaibhav and T. P. Singh, (2021), arXiv:2108.01858 [hep-ph]. [54] I. Todorov and S. Drenska, Adv. Appl. Clifford Algebras 28, 82 (2018), arXiv:1805.06739 [hep-th]. [55] N. Furey and M. J. Hughes, Phys. Lett. B 831, 137186 (2022), arXiv:2210.10126 [hep-ph]. [56] K. Krasnov, J. Math. Phys. 62, 021703 (2021), arXiv:1912.11282 [hep-th]. [57] M. Dubois-Violette and I. Todorov, Nucl. Phys. B 938, 751 (2019), arXiv:1808.08110 [hep-th]. [58] N. Furey, Phys. Lett. B 785, 84 (2018), arXiv:1910.08395 [hep-th]. [59] V. Braun, JHEP 05, 101 (2012), arXiv:1102.4880 [hep-th]. [60] J. J. Blanco-Pillado, K. Sousa, M. A. Urkiola, and J. M. Wachter, JHEP 04, 149 (2021), arXiv:2007.10381 [hep-th]. [61] R. Aschheim, in Loops 11: International Conference on Quantum Gravity (2012) arXiv:1212.5473 [cs.IT]. [62] A. Jaffe and E. Witten, The millennium prize problems 1, 129 (2006). [63] P. Hege, M. Moscolari, and S. Teufel, Phys. Rev. B 106, 155140 (2022), arXiv:2205.10622 [quant-ph]. [64] L. Boyle, M. Dickens, and F. Flicker, Phys. Rev. X 10, 011009 (2020), arXiv:1805.02665 [hep-th]. [65] R. Aschheim, L. Bubuianu, F. Fang, K. Irwin, V. Ruchin, and S. I. Vacaru, Annals of Physics 394, 120 (2018). [66] K. Irwin, Activitas Nervosa Superior 62, 48 (2020). [67] F. Fang, S. Paduroiu, D. Hammock, and K. Irwin, Crystals 8, 416 (2018). [68] M. Amaral, D. Chester, F. Fang, and K. Irwin, Symmetry 14, 1780 (2022). [69] F. Fang, S. Paduroiu, D. Hammock, and K. Irwin, in Electron Crystallography (IntechOpen, 2019) p. 27. [70] K. Irwin, M. Amaral, R. Aschleim, and F. Fang, in Proceedings of the Fourth International Conference on the Nature and Ontology of Spacetime, Varna, Bulgaria, Vol. 30 (2016) pp. 117–160. [71] K. Irwin, “Simple programs - qc cycle clocks,” (2020). [72] S. Autti, V. B. Eltsov, and G. E. Volovik, Phys. Rev. Lett. 120, 215301 (2018), arXiv:1712.06877 [cond-mat.other]. [73] K. Fujii et al., (2017), arXiv:1710.07621 [hep-ex] . [74] R. J. Adler, P. Chen, and D. I. Santiago, Gen. Rel. Grav. 33, 2101 (2001), arXiv:gr-qc/0106080 . [75] G. Rudnick, H.-W. Rix, M. Franx, I. Labb´e, M. Blanton, E. Daddi, N. M. F. Schreiber, A. Moorwood, H. R¨ottgering, I. Trujillo, et al., The Astrophysical Journal 599, 847 (2003). [76] P. A. Zyla et al. (Particle Data Group), PTEP 2020, 083C01 (2020). [77] A. F. Ali, I. Elmashad, and J. Mureika, Phys. Lett. B 831, 137182 (2022), arXiv:2205.14009 [physics.gen-ph] . [78] P. T. P. Hutauruk, A. Sulaksono, and K. Tsushima, Nucl. Phys. A 1017, 122356 (2022), arXiv:2009.08781 [hep-ph] . [79] B. Lehnert, Progress in Physics 10, 5 (2014). 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 VII. CONCLUSION We found that the GUP implied by Snyder's algebra vanishes at a specific energy scale. We define this energy scale as the scale of space-time quanta at which wavefunction collapses to form a mass. The mass of space-time quanta forms a mass gap of space-time. The covariance principle requires the space-time quanta to be a 4-dimensional object and to represent the elementary particles. Based on the geometric and symmetric analysis, we propose that the space-time quanta be represented by the 24-cell. First, it is highly-symmetric convex regular 4-polytope and self-dual. Second, The symmetry group of 24-cell is the Weyl/Coxeter group of Fs group that generates the gauge group of the standard model by the intersection of its two stabilizer groups. In addition, the 24-cell has a beautiful geometric property in which its vertices can be grouped into 3 different sets of eight vertices, each defining 16-cell with the rest defining the dual tesseract with 16 vertices. Therefore, we represent 8 vertices of 16-cell with the 8 gluons that may give a geometric interpretation of the color confinement. We represent 16 vertices of tesseract with the 12 fermions and 4 gauge bosons which may explain the flatness of the observable universe. The Higgs particle is represented as the tesseract which explains why Higgs only couple with 12 fermions and 4 gauge bosons and does not couple with 8 gluons. Vanishing uncertainty implies the creation of a mass of space-time quanta, so the length of 24-cell is identified by the electroweak length scale 10-18 m of mass creation and which is consistent with experimental measurements of the smallest measured value of charge radius of scattering for neutrinos, Higgs, Z-boson, and W boson. We think that a 24-cell symmetry group which is a solvable group of order "1152" could be useful in quantum computing. We hope to report on this application in the future. Acknowledgments I thank the Editor and referee for their meticulous review of this manuscript and express gratitude towards Klee Irwin, Raymond Aschheim, Fang Fang, Richard Clawson, and Dugan Hammock for their enlightening discussions on Polytopes and Quasicrystals. Special thanks to Ahmed Almheiri for his engaging lecture on the Path integral for chords. This manuscript is a tribute to my father, Farag Mohamed Ali, whose legacy illuminates my journey, and in memory of Mustafa Shalaby, whose teachings have profoundly inspired my exploration of physics’ core principles. 1] H. S. Snyder, Phys. Rev. 71, 38 (1947). [2] A. Connes, M. R. Douglas, and A. S. Schwarz, JHEP 02, 003 (1998), arXiv:hep-th/9711162. [3] N. Seiberg and E. Witten, JHEP 09, 032 (1999), arXiv:hep-th/9908142. [4] M. R. Douglas and N. A. Nekrasov, Rev. Mod. Phys. 73, 977 (2001), arXiv:hep-th/0106048. [5] R. J. Szabo, Phys. Rept. 378, 207 (2003), arXiv:hep-th/0109162 . [6] D. Amati, M. Ciafaloni, and G. Veneziano, Phys. Lett. B 216, 41 (1989). [7] L. J. Garay, Int. J. Mod. Phys. A 10, 145 (1995), arXiv:gr-qc/9403008. [8] F. Scardigli, Phys. Lett. B 452, 39 (1999), arXiv:hep-th/9904025. [9] K. Konishi, G. Paffuti, and P. Provero, Phys. Lett. B 234, 276 (1990). [10] A. Kempf et al., Phys. Rev. D 52, 1108 (1995), arXiv:hep-th/9412167 . [11] M. Maggiore, Phys. Lett. B 304, 65 (1993), arXiv:hep-th/9301067. [12] S. Capozziello et al., Int. J. Theor. Phys. 39, 15 (2000), arXiv:gr-qc/9910017. [13] S. Das and E. C. Vagenas, Phys. Rev. Lett. 101, 221301 (2008), arXiv:0810.5333 [hep-th]. [14] I. Pikovski et al., Nature Phys. 8, 393 (2012), arXiv:1111.1979 [quant-ph]. [15] F. Marin et al., Nature Phys. 9, 71 (2013). [16] L. Petruzziello et al., Nature Commun. 12, 4449 (2021), arXiv:2011.01255 [gr-qc]. [17] S. P. Kumar et al., Nature Commun. 11, 3900 (2020), arXiv:1908.11164 [quant-ph]. [18] H. Moradpour et al., Mon. Not. Roy. Astron. Soc. 488, L69 (2019), arXiv:1907.12940 [gr-qc]. [19] D. Gao and M. Zhan, Physical Review A 94, 013607 (2016). [20] M. Bawaj et al., Nature Commun. 6, 7503 (2015), arXiv:1411.6410 [gr-qc]. [21] P. Girdhar et al., New J. Phys. 22, 093073 (2020), arXiv:2005.08984 [quant-ph]. [22] A. Ashoorioon et al., Phys. Rev. D 71, 023503 (2005), arXiv:astro-ph/0410139. [23] R. Easther et al., Phys. Rev. D 67, 063508 (2003), arXiv:hep-th/0110226. [24] M. P. Dabrowski and F. Wagner, The European Physical Journal C 79, 1 (2019). [25] R. Easther et al., Phys. Rev. D 64, 103502 (2001), arXiv:hep-th/0104102. [26] A. Addazi et al., (2021), arXiv:2111.05659 [hep-ph] . [27] S. Hossenfelder, Living Rev. Rel. 16, 2 (2013), arXiv:1203.6191 [gr-qc]. [28] S. Mignemi, Ukr. J. Phys. 64, 991 (2019), arXiv:1911.06127 [hep-th]. [29] D. J. Gross and F. Wilczek, Phys. Rev. Lett. 30, 1343 (1973). [30] H. D. Politzer, Phys. Rev. Lett. 30, 1346 (1973). [31] J. R. Ellis, S. Mohanty, and D. V. Nanopoulos, Phys. Lett. B 221, 113 (1989). [32] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935). [33] A. F. Ali, (2022), arXiv:2210.13974 [quant-ph]. [34] A. F. Ali, (2022), 10.2139/ssrn.4292472. [35] J. D. Bekenstein, Phys. Rev. D 23, 287 (1981). [36] J. D. Bekenstein, Found. Phys. 35, 1805 (2005), arXiv:quant-ph/0404042. [37] J. D. Bekenstein, Phys. Lett. B 481, 339 (2000), arXiv:hep-th/0003058. [38] L. Lu and A. Stern, Nucl. Phys. B 860, 186 (2012), arXiv:1110.4112 [hep-th]. [39] R. Banerjee, K. Kumar, and D. Roychowdhury, JHEP 03, 060 (2011), arXiv:1101.2021 [hepth]. [40] H. S. M. Coxeter, Regular polytopes (Courier Corporation, 1973). [41] M. J. Wenninger, Dual models (Cambridge University Press, 2003). [42] G. C. Wick, Phys. Rev. 96, 1124 (1954). [43] G. W. Gibbons and S. W. Hawking, Euclidean quantum gravity (World Scientific, 1993). [44] S. Weinberg, Phys. Rev. Lett. 19, 1264 (1967). [45] I. Todorov and M. Dubois-Violette, Int. J. Mod. Phys. A 33, 1850118 (2018), arXiv:1806.09450 [hep-th] . [46] F. Bernardoni, S. L. Cacciatori, B. L. Cerchiai, and A. Scotti, Adv. Theor. Math. Phys. 12, 889 (2008), arXiv:0705.3978 [math-ph]. [47] J. C. Baez, The n-Category Cafe 27 (2018). [48] L. Boyle, (2020), arXiv:2006.16265 [hep-th]. [49] M. Dubois-Violette, Nucl. Phys. B 912, 426 (2016), arXiv:1604.01247 [math.QA]. [50] L. Boyle and S. Farnsworth, New J. Phys. 22, 073023 (2020), arXiv:1910.11888 [hep-th]. [51] I. Todorov, Springer Proc. Math. Stat. 335, 29 (2019), arXiv:1911.13124 [hep-th] . [52] V. Bhatt, R. Mondal, V. Vaibhav, and T. P. Singh, J. Phys. G 49, 045007 (2022), arXiv:2108.05787 [hep-ph]. [53] V. Vaibhav and T. P. Singh, (2021), arXiv:2108.01858 [hep-ph]. [54] I. Todorov and S. Drenska, Adv. Appl. Clifford Algebras 28, 82 (2018), arXiv:1805.06739 [hep-th]. [55] N. Furey and M. J. Hughes, Phys. Lett. B 831, 137186 (2022), arXiv:2210.10126 [hep-ph]. [56] K. Krasnov, J. Math. Phys. 62, 021703 (2021), arXiv:1912.11282 [hep-th]. [57] M. Dubois-Violette and I. Todorov, Nucl. Phys. B 938, 751 (2019), arXiv:1808.08110 [hep-th]. [58] N. Furey, Phys. Lett. B 785, 84 (2018), arXiv:1910.08395 [hep-th]. [59] V. Braun, JHEP 05, 101 (2012), arXiv:1102.4880 [hep-th]. [60] J. J. Blanco-Pillado, K. Sousa, M. A. Urkiola, and J. M. Wachter, JHEP 04, 149 (2021), arXiv:2007.10381 [hep-th]. [61] R. Aschheim, in Loops 11: International Conference on Quantum Gravity (2012) arXiv:1212.5473 [cs.IT]. [62] A. Jaffe and E. Witten, The millennium prize problems 1, 129 (2006). [63] P. Hege, M. Moscolari, and S. Teufel, Phys. Rev. B 106, 155140 (2022), arXiv:2205.10622 [quant-ph]. [64] L. Boyle, M. Dickens, and F. Flicker, Phys. Rev. X 10, 011009 (2020), arXiv:1805.02665 [hep-th]. [65] R. Aschheim, L. Bubuianu, F. Fang, K. Irwin, V. Ruchin, and S. I. Vacaru, Annals of Physics 394, 120 (2018). [66] K. Irwin, Activitas Nervosa Superior 62, 48 (2020). [67] F. Fang, S. Paduroiu, D. Hammock, and K. Irwin, Crystals 8, 416 (2018). [68] M. Amaral, D. Chester, F. Fang, and K. Irwin, Symmetry 14, 1780 (2022). [69] F. Fang, S. Paduroiu, D. Hammock, and K. Irwin, in Electron Crystallography (IntechOpen, 2019) p. 27. [70] K. Irwin, M. Amaral, R. Aschleim, and F. Fang, in Proceedings of the Fourth International Conference on the Nature and Ontology of Spacetime, Varna, Bulgaria, Vol. 30 (2016) pp. 117–160. [71] K. Irwin, “Simple programs - qc cycle clocks,” (2020). [72] S. Autti, V. B. Eltsov, and G. E. Volovik, Phys. Rev. Lett. 120, 215301 (2018), arXiv:1712.06877 [cond-mat.other]. [73] K. Fujii et al., (2017), arXiv:1710.07621 [hep-ex] . [74] R. J. Adler, P. Chen, and D. I. Santiago, Gen. Rel. Grav. 33, 2101 (2001), arXiv:gr-qc/0106080 . [75] G. Rudnick, H.-W. Rix, M. Franx, I. Labb´e, M. Blanton, E. Daddi, N. M. F. Schreiber, A. Moorwood, H. R¨ottgering, I. Trujillo, et al., The Astrophysical Journal 599, 847 (2003). [76] P. A. Zyla et al. (Particle Data Group), PTEP 2020, 083C01 (2020). [77] A. F. Ali, I. Elmashad, and J. Mureika, Phys. Lett. B 831, 137182 (2022), arXiv:2205.14009 [physics.gen-ph] . [78] P. T. P. Hutauruk, A. Sulaksono, and K. Tsushima, Nucl. Phys. A 1017, 122356 (2022), arXiv:2009.08781 [hep-ph] . [79] B. Lehnert, Progress in Physics 10, 5 (2014). 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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