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Trace dynamics and division algebras: towards quantum gravity and unification

  • Tejinder P. Singh EMAIL logo
Published/Copyright: November 4, 2020
Zeitschrift für Naturforschung A
From the journal Zeitschrift für Naturforschung A Volume 76 Issue 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.

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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

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https://doi.org/10.1515/zna-2020-0255
Keywords for this article
spontaneous localisation; standard model; trace dynamics; unification; quantum gravity

Articles in the same Issue

  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
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