Startseite Trace dynamics and division algebras: towards quantum gravity and unification
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

Trace dynamics and division algebras: towards quantum gravity and unification

  • Tejinder P. Singh EMAIL logo
Veröffentlicht/Copyright: 4. November 2020
Zeitschrift für Naturforschung A
Aus der Zeitschrift Zeitschrift für Naturforschung A Band 76 Heft 2

Abstract

We have recently proposed a Lagrangian in trace dynamics at the Planck scale, for unification of gravitation, Yang–Mills fields, and fermions. Dynamical variables are described by odd-grade (fermionic) and even-grade (bosonic) Grassmann matrices. Evolution takes place in Connes time. At energies much lower than Planck scale, trace dynamics reduces to quantum field theory. In the present paper, we explain that the correct understanding of spin requires us to formulate the theory in 8-D octonionic space. The automorphisms of the octonion algebra, which belong to the smallest exceptional Lie group G2, replace space-time diffeomorphisms and internal gauge transformations, bringing them under a common unified fold. Building on earlier work by other researchers on division algebras, we propose the Lorentz-weak unification at the Planck scale, the symmetry group being the stabiliser group of the quaternions inside the octonions. This is one of the two maximal sub-groups of G2, the other one being SU(3), the element preserver group of octonions. This latter group, coupled with U(1)em, describes the electrocolour symmetry, as shown earlier by Furey. We predict a new massless spin one boson (the ‘Lorentz’ boson) which should be looked for in experiments. Our Lagrangian correctly describes three fermion generations, through three copies of the group G2, embedded in the exceptional Lie group F4. This is the unification group for the four fundamental interactions, and it also happens to be the automorphism group of the exceptional Jordan algebra. Gravitation is shown to be an emergent classical phenomenon. Although at the Planck scale, there is present a quantised version of the Lorentz symmetry, mediated by the Lorentz boson, we argue that at sub-Planck scales, the self-adjoint part of the octonionic trace dynamics bears a relationship with string theory in 11 dimensions.

Keywords: spontaneous localisation; standard model; trace dynamics; unification; quantum gravity
Corresponding author: Tejinder P. Singh, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India, E-mail:

Acknowledgments

The author would like to thank Abhinash Kumar Roy and Anmol Sahu for collaboration and for intense and helpful discussions. The authors would also like to thank Stephen Adler, Cohl Furey, Niels Gresnigt, and Ovidiu Cristinel Stoica, for helpful correspondence and support and encouragement during the course of this project. Thanks also to Basudeb Dasgupta, Debajyoti Choudhury, Roberto Onofrio, Thanu Padmanabhan, Roberto Percacci and Carlos Perelman for helpful comments on an earlier version of the manuscript and for drawing my attention to related relevant research.

  1. Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: None declared.

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

References

[1] T. P. Singh, “The problem of time and the problem of quantum measurement,” in Re-thinking time at the interface of physics and philosophy, T. Filk, and A. von Muller, Eds. (arXiv:1210.81110), Berlin-Heidelberg, Springer, 2015.10.1007/978-3-319-10446-1_8Suche in Google Scholar

[2] A. Connes, in Visions in Mathematics – GAFA 2000 Special volume, Part II, chapter Non-commutative geometry 2000, N. Alon, J. Bourgain, A. Connes, M. Gromov, and V. Milman, Eds., Springer, arXiv:math/0011193, 2000, p. 481.10.1007/978-3-0346-0425-3_3Suche in Google Scholar

[3] S. L. Adler, Quantum Theory as an Emergent Phenomenon, Cambridge, Cambridge University Press, 2004.10.1017/CBO9780511535277Suche in Google Scholar

[4] S. L. Adler, “Generalized quantum dynamics,” Nucl. Phys. B, vol. 415, p. 195, 1994. https://doi.org/10.1016/0550-3213(94)90072-8.Suche in Google Scholar

[5] S. L. Adler and A. C. Millard, “Generalised quantum dynamics as pre-quantum mechanics,” Nucl. Phys. B, vol. 473, p. 199, 1996. https://doi.org/10.1016/0550-3213(96)00253-2.Suche in Google Scholar

[6] M. Palemkota and T. P. Singh, Black Hole Entropy from Trace Dynamics and Non-commutative Geometry, arXiv:1909.02434v2 [gr-qc], 2019, submitted for publication.Suche in Google Scholar

[7] A. Bassi, K. Lochan, S. Satin, T. P. Singh, and H. Ulbricht, “Models of wave function collapse, underlying theories, and experimental tests,” Rev. Mod. Phys., vol. 85, p. 471, 2013, arXiv:1204.4325 [quant-ph], https://doi.org/10.1103/revmodphys.85.471.Suche in Google Scholar

[8] A. Connes, and C. Rovelli, “von Neumann algebra automorphisms and time-thermodynamics relation in general covariant quantum theories,” Classical Quant. Grav., vol. 11, p. 2899, 1994.10.1088/0264-9381/11/12/007Suche in Google Scholar

[9] M. Takesaki, “Theory of operator algebras II,” in Encylopedia of Mathematical Sciences, vol. 125, Berlin, Springer Verlag, 2003.10.1007/978-3-662-10451-4Suche in Google Scholar

[10] M. Takesaki. “Tomita’s theory of modern Hilbert algebras and its applications,” in Lecture Notes in Mathematics, vol. 128, Berlin, Springer, 1970.10.1007/BFb0065832Suche in Google Scholar

[11] O. Nykodym, “Sur une généralisation des intégrales de M,” J. Radone. Fund. Math., vol. 15, pp. 131–179, 1930.10.4064/fm-15-1-131-179Suche in Google Scholar

[12] T. P. Singh, Spontaneous Quantum Gravity, arXiv:1912.03266v2, 2019 [submitted for publication].Suche in Google Scholar

[13] T. P. Singh, “From quantum foundations to spontaneous quantum gravity: an overview of the new theory,” Z. Naturforschung A, arXiv:1909.06340 [gr-qc], 2020, https://doi.org/10.1515/zna–2020–0073.10.1515/zna-2020-0073Suche in Google Scholar

[14] M. Palemkota and T. P. Singh, “Proposal for a new quantum theory of gravity III: equations for quantum gravity, and the origin of spontaneous localisation,” Z. Naturforschung A, vol. 75, p. 143, 2019, https://doi.org/10.1515/zna-2019-0267, arXiv:1908.04309.Suche in Google Scholar

[15] T. P. Singh. “Octonions, trace dynamics and non-commutative geometry: a case for unification in spontaneous quantum gravity,” Z. Naturforschung A, [to appear] arXiv:2006.16274v2, 2020.10.1515/zna-2020-0196Suche in Google Scholar

[16] A. H. Chamseddine and A. Connes, “The spectral action principle,” Commun. Math. Phys., vol. 186, p. 731, 1997, arXiv:hep-th/9606001. https://doi.org/10.1007/s002200050126.Suche in Google Scholar

[17] M. S. Meghraj, A. Pandey, and T. P. Singh, Why Does the Kerr–Newman Black Hole Have the Same Gyromagnetic Ratio as the Electron?, submitted for publication, 2020, arXiv:2006.05392.Suche in Google Scholar

[18] T. Jacobson, “Thermodynamics of spacetime: the Einstein equation of state,” Phys. Rev. Lett., vol. 75, no. 7, pp. 1260–1263, 1995, arXiv:grqc/9505004, https://doi.org/10.1103/physrevlett.75.1260.Suche in Google Scholar

[19] T. Padmanabhan, “Gravity and is thermodynamics,” Curr. Sci., vol. 109, p. 2236, 2015, arXiv:1512.06546, https://doi.org/10.18520/v109/i12/2236-2242.Suche in Google Scholar

[20] T. Schucker, “Spin group and almost commutative geometry,” hep-th/0007047, 2000.Suche in Google Scholar

[21] T. P. Singh, “Space-time from collapse of the wave-function,” Z. Naturforschung A, vol. 74, p. 147, 2019, arXiv.org:1809.03441, https://doi.org/10.1515/zna-2018-0477.Suche in Google Scholar

[22] G. M. Dixon, Division algebras, Octonions, Quaternions, Complex Numbers and the Algebraic Design of Physics, Dordrecht, Kluwer, 1994.Suche in Google Scholar

[23] C. H. Tze and F. Gursey, On the Role of Division, Jordan and Related Algebras in Particle Physics, Singapore, World Scientific Publishing, 1996.10.1142/3282Suche in Google Scholar

[24] C. Furey, Standard Model Physics from an Algebra?, PhD thesis, University of Waterloo, 2015, arXiv:1611.09182 [hep-th].Suche in Google Scholar

[25] C. Furey, “Three generations, two unbroken gauge symmetries, and one eight-dimensional algebra,” Phys. Lett. B, vol. 785, p. 1984, 2018. https://doi.org/10.1016/j.physletb.2018.08.032.Suche in Google Scholar

[26] C. Furey, “SU(3)C × SU(2)L × U(1)Y(×U(1)X) as a symmetry of division algebraic ladder operators,” Euro. Phys. J. C, vol. 78, p. 375, 2018. https://doi.org/10.1140/epjc/s10052-018-5844-7.Suche in Google Scholar PubMed PubMed Central

[27] J. Chisholm and R. Farwell, Clifford Geometric Algebras: With Applications to Physics, Mathematics and Engineering, Boston, Birkhauser, 1996, p. 365, Ed. W. R. Baylis.10.1007/978-1-4612-4104-1_27Suche in Google Scholar

[28] G. Trayling and W. Baylis. A geometric basis for the standard-model gauge group. J. Phys. A: Math. Theor., 34:3309, 2001, https://doi.org/10.1088/0305-4470/34/15/309.Suche in Google Scholar

[29] M. Dubois-Violette, “Exceptional quantum geometry and particle physics,” Nucl. Phys. B, vol. 912, pp. 426–449, 2016. https://doi.org/10.1016/j.nuclphysb.2016.04.018.Suche in Google Scholar

[30] T. Ivan, “Exceptional quantum algebra for the standard model of particle physics,” Nucl. Phys. B, vol. 938, p. 751, 2019, arXiv:1808.08110 [hep–th].10.1016/j.nuclphysb.2018.12.012Suche in Google Scholar

[31] M. Dubois-Violette and I. Todorov, “Exceptional quantum geometry and particle physics II,” Nucl. Phys. B, vol. 938, pp. 751–761, 2019, arXiv:1808.08110 [hep–th]. https://doi.org/10.1016/j.nuclphysb.2018.12.012.Suche in Google Scholar

[32] I. Todorov and S. Drenska, “Octonions, exceptional Jordan algebra and the role of the group F4 in particle physics,” Adv. Appl. Clifford Algebras, vol. 28, no. 4, p. 82, 2018, arXiv:1911.13124 [hep–th]. https://doi.org/10.1007/s00006-018-0899-y.Suche in Google Scholar

[33] I. Todorov, “Jordan algebra approach to finite quantum geometry,” in PoS, volume CORFU2019, p. 163, 2020, https://doi.org/10.22323/1.376.0163.Suche in Google Scholar

[34] R. Ablamowicz, “Construction of spinors via Witt decomposition and primitive idempotents: a review,” in Clifford Algebras and Spinor Structures, R. Ablamowicz, and P. Lounesto, Eds., Dordrecht, Kluwer Acad. Publ., 1995, p. 113.10.1007/978-94-015-8422-7_6Suche in Google Scholar

[35] J. C. Baez, The Octonions, arXiv:math/0105155, 2001.10.1090/S0273-0979-01-00934-XSuche in Google Scholar

[36] J. C. Baez, “Division algebras and quantum theory,” Found. Phys., vol. 42, no. 7, pp. 819–855, 2011. https://doi.org/10.1007/s10701-011-9566-z.Suche in Google Scholar

[37] J. C. Baez and J. Huerta, The Algebra of Grand Unified Theories, 2009, arXiv:0904.1556 [hep-th].Suche in Google Scholar

[38] J. C. Baez and J. Huerta, “Division algebras and supersymmetry II,” Adv. Math. Theor. Phys., vol. 15, p. 1373, 2011. https://doi.org/10.4310/atmp.2011.v15.n5.a4.Suche in Google Scholar

[39] P. Jordan, J. von Neumann, and E. Wigner, “On an algebraic generalisation of the quantum mechanical formalism,” Ann. Math., vol. 35, p. 29, 1934. https://doi.org/10.2307/1968117.Suche in Google Scholar

[40] A. Adrien Albert, “On a certain algebra of quantum mechanics,” Ann. Math., vol. 35, no. 65, 1933.10.2307/1968118Suche in Google Scholar

[41] M. Gunaydin and F. Gursey, “Quark structure and octonions,” J. Math. Phys., vol. 14, p. 1651, 1973. https://doi.org/10.1063/1.1666240.Suche in Google Scholar

[42] Ovidiu Cristinel Stoica, “The standard model algebra (Leptons, quarks and gauge from the complex algebra Cl(6)),” Adv. Appl. Clifford Algebras, vol. 52, no. 28, p. 04336, 2018, arXiv:1702.Suche in Google Scholar

[43] A. B. Gillard and N. G. Gresnigt, “Three fermion generations with two unbroken gauge symmetries from the complex sedenions,” Eur. Phys. J. C, vol. 79, no. 5, p. 03186, 2019, arXiv:1904. https://doi.org/10.1140/epjc/s10052-019-6967-1.Suche in Google Scholar

[44] I. Yokota, “Exceptional lie groups,” arXiv:0902.043 [math.DG], 2009.Suche in Google Scholar

[45] I. Todorov and M. Dubois-Violette, “Deducing the symmetry of the standard model fom the automorphism and structure groups of the exceptional Jordan algebra,” arXiv:1806.9450 [hep-th], 2018.10.1142/S0217751X1850118XSuche in Google Scholar

[46] A. K. Roy, A. Sahu, and T. P. Singh, Trace Dynamics, and a Ground State in Spontaneous Quantum Gravity, 2020, www.tifr.res.in/∼tpsingh/q1q2uni.pdf, Submitted for publication [available at home page of TPS].Suche in Google Scholar

[47] I. Agricola, “Old and new in the exceptional group {G2},” Not. AMS, vol. 55, p. 922, 2008.Suche in Google Scholar

[48] R. Onofrio, “On weak interactions as short distance manifestations of gravity,” Mod. Phys. Lett. A, vol. 28, p. 1350022, 2013, arXiv:1412.4513 [hep-ph]. https://doi.org/10.1142/s0217732313500223.Suche in Google Scholar

[49] R. Onofrio, “Proton radius puzzle and quantum gravity at the Fermi scale,” Europhys. Lett., vol. 104, p. 20002 2013, arXiv:1312.3469 [hep-ph]. https://doi.org/10.1209/0295-5075/104/20002.Suche in Google Scholar

[50] F. Nesti and R. Percacci, “Gravi-weak unification,” J. Phys. A, vol. 41, p. 075405, 2008, arXiv:0706.3307. https://doi.org/10.1088/1751-8113/41/7/075405.Suche in Google Scholar

[51] K. Krasnov and R. Percacci, “Gravity and unification: a review,” Classical Quant. Grav., vol. 35, p. 143001, 2018, arXiv:1712.03006 [hep-th]. https://doi.org/10.1088/1361-6382/aac58d.Suche in Google Scholar

[52] T. P. Singh, “A basic definition of spin in the new matrix dynamics,” Z. Naturforschung A, 2020, arXiv:2006.16274v1, https://doi.org/10.1515/zna–2020–0183.10.1515/zna-2020-0183Suche in Google Scholar

[53] K. Cahill, “Is the local Lorentz invariance of general relativity implemented by gauge bosons that have their own Yang–Mills-like action?,” Phys. Rev. D, vol. 102, p. 065011, 2020, To appear:arXiv:2008.10381 [gr-qc].10.1103/PhysRevD.102.065011Suche in Google Scholar

[54] A. Borel and J. de Siebenthal, “Le sou groupes fermes de rang maximum des groupes de lie clos,” Comment Math. Helv., vol. 23, no. 200, 1949. https://doi.org/10.1007/bf02565599.Suche in Google Scholar

[55] A. K. Roy and A. Sahu. (private communication). 2020.Suche in Google Scholar

[56] T. P. Singh, “Space-time from collapse of the wave-function,” Z. Naturforschung A, vol. 74, p. 147, 2019, arXiv:1809.03441. https://doi.org/10.1515/zna-2018-0477.Suche in Google Scholar

[57] G. Landi, “Eigenvalues as dynamical variables,” Lect. Notes Phys., vol. 596, p. 299, 2002, gr-qc/9906044. https://doi.org/10.1007/3-540-46082-9_16.Suche in Google Scholar

[58] G. Landi and C. Rovelli, “General relativity in terms of Dirac eigenvalues,” Phys. Rev. Lett., vol. 78, p. 3051, 1997, arXiv:gr-qc/9612034. https://doi.org/10.1103/physrevlett.78.3051.Suche in Google Scholar

[59] M. A. Zubkov, “Gauge theory of Lorentz group as a source of the dynamical electroweak symmetry breaking,” JHEP, vol. 1309, p. 044, 2013, arXiv:1301.6971.10.1007/JHEP09(2013)044Suche in Google Scholar

[60] S. L. Adler, “Gravitation and the noise needed in objective reduction models,” arXiv:1401.0353 [gr-qc] 2014.Suche in Google Scholar

[61] L. P. Horwitz, Relativistic Quantum Mechanics, Springer Netherlands, 2015.10.1007/978-94-017-7261-7Suche in Google Scholar

[62] F. Karolyhazy, “Gravitation and quantum mechanics of macroscopic objects,” Magy. Fiz. Foly., vol. 42, no. 23, p. 390, 1966.10.1007/BF02717926Suche in Google Scholar

[63] F. Karolyhazy, A. Frenkel, and B. Lukacs, in Physics as Natural Philosophy, A. Shimony, and H. Feshbach, Eds., Cambridge, MIT Press, 1982.Suche in Google Scholar

[64] F. Karolyhazy, and A. Miller, Eds., Sixty-Two Years of Uncertainty, New York, Plenum, 1990.Suche in Google Scholar

[65] F. Karolyhazy, M. Ferrero, and A. van der Merwe, Eds., Fundamental Problems of Quantum Physics, Netherlands, Kluwer Acad. Publ., 1995.Suche in Google Scholar

[66] Y. Jack Ng, “Entropy and gravitation: from black hole computers to dark energy and dark matter,” Entropy, vol. 21, p. 1035, 2019. https://doi.org/10.3390/e21111035.Suche in Google Scholar

[67] G. Amelino-Camelia, “Gravity-wave interferometers as quantum gravity detectors,” Nature, vol. 398, p. 216, 1999. https://doi.org/10.1038/18377.Suche in Google Scholar

[68] T. P. Singh, “Quantum gravity, minimum length and holography,” Pramana J. Phys., p. 06350, 2020, [to appear], arXiv:1910.10.1007/s12043-020-02052-2Suche in Google Scholar

[69] S. Vermeulen, L. Aiello, E. Aldo, et al.., “An experiment for observing quantum gravity phenomena using twin table-top 3d interferometers,” p. 2020, arXiv:2008.04957.10.1088/1361-6382/abe757Suche in Google Scholar

[70] M. Carlesso and M. Paternostro. “Opto-mechanical tests of collapse models,” arXiv:1906.11041, 2019. https://doi.org/10.1364/qim.2019.s1c.3.Suche in Google Scholar

[71] T. P. Singh, “Dark energy as a large scale quantum gravitational phenomenon,” Mod. Phys. Lett. A, vol. 35, p. 2050195, 2020, arXiv:1911.02955 https://doi.org/10.1142/S0217732320501953.Suche in Google Scholar

[72] T. P. Singh, “Nature does not play dice on the Planck scale,” Int. J. Mod. Phys., arXiv:2005.06427, 2020. https://doi.org/10.1142/S0218271820430129.Suche in Google Scholar

[73] V. Vanchurin, “The world as a neural network,” arXiv:2008.01540, 2020.10.3390/e22111210Suche in Google Scholar

[74] R. Gallego Torrome, “On the origin of the weak equivalence principle in a theory of emergent quantum mechanics,” arXiv:2005.12903, 2020.10.1142/S0219887820501571Suche in Google Scholar

[75] K. Shima, “Nonlinear SUSY general relativity and significances,” arXiv:1112.3098 [hep-th], 2011, https://doi.org/10.1088/1742–6596/343/1/012111.10.1088/1742-6596/343/1/012111Suche in Google Scholar

[76] Kazunari Shima, “New Einstein–Hilbert type action of space-time and matter-nonlinear-supersymmetric general relativity theory,” arXiv:2009.06266 [hep-th], 2020.10.22323/1.364.0067Suche in Google Scholar

[77] C. Castro Perelman, “R × C × H × O valued gravity as a grand unified field theory,” Adv. Appl. Clifford Algebras, vol. 27, no. 22, 2019. https://doi.org/10.1007/s00006-019-0937-4.Suche in Google Scholar

[78] S. Lee, “The exceptional Jordan algebra and the matrix string,” arXiv:hep-th/0104050, 2001.Suche in Google Scholar

[79] D. Choudhury, and B. Dasgupta, Private Communication, 2020.Suche in Google Scholar

Received: 2020-09-10
Accepted: 2020-10-04
Published Online: 2020-11-04
Published in Print: 2021-02-23

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. Dynamical Systems & Nonlinear Phenomena
  3. In search of hyperchaos in a high dimensional unmagnetized quantum plasma
  4. Multistability and chaotic scenario in a quantum pair-ion plasma
  5. Dust-acoustic Gardner solitons in cryogenic plasma with the effect of polarization in the presence of a quantizing magnetic field
  6. Quantum Theory
  7. Trace dynamics and division algebras: towards quantum gravity and unification
  8. Solid State Physics & Materials Science
  9. Birefringence and order parameter studies in ferroelectric liquid crystals using laser transmission technique
  10. Investigations on the g factors and local structure for the trigonal Ce3+ center in YAl3(BO3)4 crystal
  11. Fiber-optic Fabry–Perot temperature sensor based on the ultraviolet curable glue-filled cavity and two-beam interference principle
  12. Tuning the properties of RF sputtered tin sulphide thin films and enhanced performance in RF sputtered SnS thin films hetero-junction solar cell devices
https://doi.org/10.1515/zna-2020-0255
Schlagwörter für diesen Artikel
spontaneous localisation; standard model; trace dynamics; unification; quantum gravity

Artikel in diesem Heft

  1. Frontmatter
  2. Dynamical Systems & Nonlinear Phenomena
  3. In search of hyperchaos in a high dimensional unmagnetized quantum plasma
  4. Multistability and chaotic scenario in a quantum pair-ion plasma
  5. Dust-acoustic Gardner solitons in cryogenic plasma with the effect of polarization in the presence of a quantizing magnetic field
  6. Quantum Theory
  7. Trace dynamics and division algebras: towards quantum gravity and unification
  8. Solid State Physics & Materials Science
  9. Birefringence and order parameter studies in ferroelectric liquid crystals using laser transmission technique
  10. Investigations on the g factors and local structure for the trigonal Ce3+ center in YAl3(BO3)4 crystal
  11. Fiber-optic Fabry–Perot temperature sensor based on the ultraviolet curable glue-filled cavity and two-beam interference principle
  12. Tuning the properties of RF sputtered tin sulphide thin films and enhanced performance in RF sputtered SnS thin films hetero-junction solar cell devices
Jetzt anmelden und 20% sparen
Institutioneller Zugang
Wie funktioniert Authentifizierung?
Haben Sie eine Idee, wie wir unsere Website verbessern können?
Bitte schreiben Sie uns.
© 2025 De Gruyter Brill
Heruntergeladen am 2.12.2025 von https://www.degruyterbrill.com/document/doi/10.1515/zna-2020-0255/html
Button zum nach oben scrollen