Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

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)

How to cite: Singh, T.P. The Characteristic Equation of the Exceptional Jordan Algebra: Its Eigenvalues, and Their Possible Connection with the Mass Ratios of Quarks and Leptons. Preprints 2021, 2021010474 (doi: 10.20944/preprints202101.0474.v1). Singh, T.P. The Characteristic Equation of the Exceptional Jordan Algebra: Its Eigenvalues, and Their Possible Connection with the Mass Ratios of Quarks and Leptons. Preprints 2021, 2021010474 (doi: 10.20944/preprints202101.0474.v1).

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 admit a 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 of the 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] known mass 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, exhibiting an $F_4$ 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 provide a first principles theoretical derivation of the low energy value of the fine 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.

Subject Areas

Trace dynamics; Exceptional Jordan algebra; Octonions; Fine structure constant; Standard model

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