scholarly journals The Characteristic Equation of the Exceptional Jordan Algebra: Its Eigenvalues, and their relation with the Mass Ratios of Quarks and Leptons

2021 ◽  
Author(s):  
Tejinder Pal Singh
Keyword(s):  
Standard Model ◽  
Jordan Algebra ◽  
Characteristic Equation ◽  
Structure Constant ◽  
Space Time ◽  
Inner Product ◽  
Fine Structure Constant ◽  
The Standard Model ◽  
Mass Ratios ◽  
Free Parameters

We have recently proposed a pre-quantum, pre-space-time theory as a matrix-valued Lagrangian dynamics on an octonionic space-time. This pre-theory offers the prospect of unifying the internal symmetries of the standard model with gravity. It can also predict the values of free parameters of the standard model, because these parameters arising in the Lagrangian are related to the algebra of the octonions which define the underlying non-commutative space-time on which the dynamical degrees of freedom evolve. These free parameters are related to the algebra $J_3(\mathbb O)$ [exceptional Jordan algebra] which in turn is related to the three fermion generations. 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 groups $F_4$ and $E_6$ 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 quarks and charged leptons. We also propose a diagrammatic representation of the standard model bosons, Higgs and three fermion generations, in terms of the octonions, exhibiting an $F_4$ and $E_6$ symmetry. 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.

scholarly journals The Characteristic Equation of The Exceptional Jordan Algebra: Its Eigenvalues, And Their Relation With Mass Ratios For Quarks And Leptons

2021 ◽  
Author(s):  
Tejinder P. Singh
Keyword(s):  
Standard Model ◽  
Jordan Algebra ◽  
Characteristic Equation ◽  
Structure Constant ◽  
Space Time ◽  
Inner Product ◽  
Fine Structure Constant ◽  
The Standard Model ◽  
Mass Ratios ◽  
Free Parameters

Abstract We have recently proposed a pre-quantum, pre-space-time theory as a matrix-valued La-grangian dynamics on an octonionic space-time. This pre-theory offers the prospect of unifying the internal symmetries of the standard model with gravity. It can also predict the values of free parameters of the standard model, because these parameters arising in the Lagrangian are related to the algebra of the octonions which define the underlying non-commutative space-time on which the dynamical degrees of freedom evolve. These free parameters are related to the algebra J3 (O) [exceptional Jordan algebra] which in turn is related to the three fermion generations. 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 F4. 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 groups F4 and E6 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 quarks and charged leptons. We also propose a diagrammatic representation of the standard model bosons, Higgs and three fermion generations, in terms of the octonions, exhibiting an F4 and E6 symmetry. 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.

scholarly journals The Characteristic Equation of the Exceptional Jordan Algebra: Its Eigenvalues, and Their Possible Connection with the Mass Ratios of Quarks and Leptons

2021 ◽  
Author(s):  
Tejinder Pal Singh
Keyword(s):  
Standard Model ◽  
Jordan Algebra ◽  
Characteristic Equation ◽  
Structure Constant ◽  
Space Time ◽  
Inner Product ◽  
Fine Structure Constant ◽  
The Standard Model ◽  
Mass Ratios ◽  
Free Parameters

We have recently proposed a pre-quantum, pre-space-time theory as a matrix-valued Lagrangian dynamics on an octonionic space-time. This pre-theory offers the prospect of unifying the internal symmetries of the standard model with gravity. It can also predict the values of free parameters of the standard model, because these parameters arising in the Lagrangian are related to the algebra of the octonions which define the underlying non-commutative space-time on which the dynamical degrees of freedom evolve. These free parameters are related to the algebra $J_3(\mathbb O)$ [exceptional Jordan algebra] which in turn is related to the three fermion generations. 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.

scholarly journals The Characteristic Equation of the Exceptional Jordan Algebra: Its Eigenvalues, and their relation with the Mass Ratios of Quarks and Leptons

2021 ◽  
Author(s):  
Tejinder Pal Singh
Keyword(s):  
Standard Model ◽  
Jordan Algebra ◽  
Characteristic Equation ◽  
Structure Constant ◽  
Space Time ◽  
Inner Product ◽  
Fine Structure Constant ◽  
The Standard Model ◽  
Mass Ratios ◽  
Free Parameters

We have recently proposed a pre-quantum, pre-space-time theory as a matrix-valued Lagrangian dynamics on an octonionic space-time. This pre-theory offers the prospect of unifying the internal symmetries of the standard model with gravity. It can also predict the values of free parameters of the standard model, because these parameters arising in the Lagrangian are related to the algebra of the octonions which define the underlying non-commutative space-time on which the dynamical degrees of freedom evolve. These free parameters are related to the algebra $J_3(\mathbb O)$ [exceptional Jordan algebra] which in turn is related to the three fermion generations. 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 quarks and charged leptons. We also propose a diagrammatic representation of the standard model bosons, Higgs and three fermion generations, in terms of the octonions, exhibiting an $F_4$ and $E_6$ symmetry. 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.

scholarly journals The Characteristic Equation of the Exceptional Jordan Algebra: Its Eigenvalues, and Their Possible Connection with the Mass Ratios of Quarks and Leptons

2021 ◽  
Author(s):  
Tejinder Pal Singh
Keyword(s):  
Standard Model ◽  
Jordan Algebra ◽  
Characteristic Equation ◽  
Top Quark ◽  
Structure Constant ◽  
Inner Product ◽  
Fine Structure Constant ◽  
Octonion Algebra ◽  
The Standard Model ◽  
Mass Ratios

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.

scholarly journals The characteristic equation of the exceptional Jordan algebra: its eigenvalues, and their possible connection with the mass ratios of quarks and leptons

2021 ◽  
Author(s):  
Tejinder P. Singh
Keyword(s):  
Standard Model ◽  
Jordan Algebra ◽  
Characteristic Equation ◽  
Top Quark ◽  
Structure Constant ◽  
Inner Product ◽  
Fine Structure Constant ◽  
Octonion Algebra ◽  
The Standard Model ◽  
Mass Ratios

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 F4. 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 F4 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 F4 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.

scholarly journals The Characteristic Equation of the Exceptional Jordan Algebra: Its Eigenvalues, and Their Possible Connection with the Mass Ratios of Quarks and Leptons

2021 ◽  
Author(s):  
Tejinder Pal Singh
Keyword(s):  
Standard Model ◽  
Jordan Algebra ◽  
Characteristic Equation ◽  
Top Quark ◽  
Structure Constant ◽  
Inner Product ◽  
Fine Structure Constant ◽  
Octonion Algebra ◽  
The Standard Model ◽  
Mass Ratios

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.

scholarly journals Enhanced Derivation of the Electron Magnetic Moment Anomaly from the Electron Charge using Geometric Principles

2018 ◽  
Vol 10 (6) ◽  
pp. 24 ◽  
Author(s):  
Andrew Worsley ◽  
J.F. Peters
Keyword(s):  
Fine Structure ◽  
Standard Model ◽  
Magnetic Moment ◽  
Geometric Model ◽  
Structure Constant ◽  
Fine Structure Constant ◽  
Electron Charge ◽  
Mathematical Equation ◽  
The Standard Model ◽  
Electron Magnetic Moment

The electron magnetic moment anomaly is conventionally derived from the fine structure constant using a complex formula requiring over 13,000 evaluations. However, the charge of the electron is an important parameter of the Standard Model and could provide an enhanced basis for the derivation of the electron magnetic moment anomaly. This paper uses a geometric model to reformulate the equation for the electron’s charge, this is then used to determine a more accurate value for the electron magnetic moment anomaly from first geometric principles. This enhanced derivation uses a single evaluation, using a concise mathematical equation based on the natural log e^pi. This geometric model will lead to further work to theoretically improve the understanding of the electron.

scholarly journals Primordial nucleosynthesis with varying fundamental constants

2020 ◽  
Vol 633 ◽  
pp. L11 ◽  
Author(s):  
M. T. Clara ◽  
C. J. A. P. Martins
Keyword(s):  
Standard Model ◽  
Broad Class ◽  
Statistical Significance ◽  
Structure Constant ◽  
Big Bang ◽  
Fine Structure Constant ◽  
Fundamental Constants ◽  
Primordial Nucleosynthesis ◽  
The Standard Model ◽  
Theoretical Predictions

Primordial nucleosynthesis is an observational cornerstone of the Hot Big Bang model and a sensitive probe of physics beyond the standard model. Its success has been limited by the so-called lithium problem, for which many solutions have been proposed. We report on a self-consistent perturbative analysis of the effects of variations in nature’s fundamental constants, which are unavoidable in most extensions of the standard model, on primordial nucleosynthesis, focusing on a broad class of Grand Unified Theory models. A statistical comparison between theoretical predictions and observational measurements of 4He, D, 3He and, 7Li consistently yields a preferred value of the fine-structure constant α at the nucleosynthesis epoch that is larger than the current laboratory one. The level of statistical significance and the preferred extent of variation depend on model assumptions but the former can be more than four standard deviations, while the latter is always compatible with constraints at lower redshifts. If lithium is not included in the analysis, the preference for a variation of α is not statistically significant. The abundance of 3He is relatively insensitive to such variations. Our analysis highlights a viable and physically motivated solution to the lithium problem, which warrants further study.

scholarly journals Measurement of the fine-structure constant as a test of the Standard Model

Science ◽  
2018 ◽  
Vol 360 (6385) ◽  
pp. 191-195 ◽  
Author(s):  
Richard H. Parker ◽  
Chenghui Yu ◽  
Weicheng Zhong ◽  
Brian Estey ◽  
Holger Müller
Keyword(s):  
Fine Structure ◽  
Standard Model ◽  
Structure Constant ◽  
Fine Structure Constant ◽  
The Standard Model

scholarly journals STATUS OF STANDARD MODEL CALCULATION OF LEPTON g - 2

2014 ◽  
Vol 35 ◽  
pp. 1460405 ◽  
Author(s):  
MARC KNECHT
Keyword(s):  
Standard Model ◽  
Magnetic Moments ◽  
Structure Constant ◽  
Fine Structure Constant ◽  
Precision Determination ◽  
The Standard Model ◽  
Charged Leptons ◽  
Experimental Values ◽  
High Precision Determination

The contributions from the standard model interactions to the anomalous magnetic moments of the two lightest charged leptons, the electron and the muon, are reviewed. Comparison with the very accurately measured experimental values is made, using the most recent high-precision determination of the fine structure constant.

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