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The Characteristic Equation of the Exceptional Jordan Algebra: Its Eigenvalues, and Their Possible Connection with the Mass Ratios of Quarks and Leptons
Version 1
: Received: 22 January 2021 / Approved: 25 January 2021 / Online: 25 January 2021 (10:35:06 CET)
Version 2
: Received: 17 March 2021 / Approved: 18 March 2021 / Online: 18 March 2021 (09:51:15 CET)
Version 3
: Received: 16 June 2021 / Approved: 17 June 2021 / Online: 17 June 2021 (09:33:35 CEST)
Version 4
: Received: 26 July 2021 / Approved: 27 July 2021 / Online: 27 July 2021 (11:26:14 CEST)
Version 5
: Received: 7 October 2021 / Approved: 8 October 2021 / Online: 8 October 2021 (10:58:00 CEST)
Singh, T. P. Quantum Theory without Classical Time: Octonions, and a Theoretical Derivation of the Fine Structure Constant 1/137. International Journal of Modern Physics D, 2021, 30. https://doi.org/10.1142/s0218271821420104.
Singh, T. P. Quantum Theory without Classical Time: Octonions, and a Theoretical Derivation of the Fine Structure Constant 1/137. International Journal of Modern Physics D, 2021, 30. https://doi.org/10.1142/s0218271821420104.
Singh, T. P. Quantum Theory without Classical Time: Octonions, and a Theoretical Derivation of the Fine Structure Constant 1/137. International Journal of Modern Physics D, 2021, 30. https://doi.org/10.1142/s0218271821420104.
Singh, T. P. Quantum Theory without Classical Time: Octonions, and a Theoretical Derivation of the Fine Structure Constant 1/137. International Journal of Modern Physics D, 2021, 30. https://doi.org/10.1142/s0218271821420104.
Abstract
The exceptional Jordan algebra [also known as the Albert algebra]is the finite dimensional algebra of 3x3 Hermitean matrices with octonionic entries. Its automorphism group is the exceptional Lie group $F_4$. These matrices admita cubic characteristic equation whose eigenvalues are real and depend on the invariant trace, determinant, and an inner product made from the Jordan matrix. Also, there is some evidence in the literature that the group $F_4$ could play a role in the unification of the standard model symmetries, including the Lorentz symmetry. The octonion algebra is known to correctly yield the electric charge values (0, 1/3, 2/3, 1) for standard model fermions, via the eigenvalues of a $U(1)$ number operator, identified with $U(1)_{em}$. In the present article, we use the same octonionic representation of the fermions to compute the eigenvalues of the characteristic equation ofthe Albert algebra, and compare the resulting eigenvalues with the known mass ratios for quarks and leptons. We find that the ratios of the eigenvalues correctly reproduce the [square root of the] knownmass ratios for up, charm and top quark. We also propose a diagrammatic representation of the standard model bosons, Higgs and three fermion generations, in terms of the octonions, exhibitingan $F_4$ symmetry. We motivate from our Lagrangian as to why the eigenvalues computed in this work could bear a relation with mass ratios of quarks and leptons. In conjunction with the trace dynamics Lagrangian, the Jordan eigenvalues also providea first principles theoretical derivation of the low energy value of thefine structure constant, yielding the value $1/137.04006$. The Karolyhazy correction to this value gives an exact match with the measured value of the constant, after assuming a specific value for the electro-weak symmetry breaking energy scale.
Keywords
Trace dynamics; Exceptional Jordan algebra; Octonions; Fine structure constant; Standard model
Subject
Physical Sciences, Acoustics
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
