On the structure of the split octonion algebra

1990 ◽  
Vol 105 (1) ◽  
pp. 31-41 ◽  
Author(s):  
P. L. Nash
Keyword(s):  
Octonion Algebra

scholarly journals An octonion algebra originating in combinatorics

2010 ◽  
Vol 138 (12) ◽  
pp. 4187-4187
Author(s):  
Dragomir Ž. Đoković ◽  
Kaiming Zhao
Keyword(s):  
Octonion Algebra

scholarly journals Trace dynamics and division algebras: towards quantum gravity and unification

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Tejinder P. Singh
Keyword(s):  
Lie Group ◽  
Planck Scale ◽  
The Self ◽  
Division Algebras ◽  
Quantum Field ◽  
Octonion Algebra ◽  
Gauge Transformations ◽  
Yang Mills ◽  
Massless Spin ◽  
Correct Understanding

AbstractWe 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.

Quadratic functionals on modules over alternative *-rings

2005 ◽  
Vol 42 (4) ◽  
pp. 459-469 ◽  
Author(s):  
Dijana Ilišević
Keyword(s):  
Octonion Algebra ◽  
Sesquilinear Forms ◽  
Quadratic Functionals

The aim of this paper is to give a solution to the problem of the representability of quadratic functionals by symmetric weak sesquilinear forms on modules over alternative *-rings. The main result is applied on quadratic functionals acting on modules over the octonion algebra and an alternative H*-algebra.

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 Inflation Model and Riemann Tensor on Non-associative Algebra

2020 ◽  
Author(s):  
Vyacheslav Dorofeev
Keyword(s):  
Gravitational Constant ◽  
Dimensional Space ◽  
Riemann Tensor ◽  
Cross Linking ◽  
Octonion Algebra ◽  
Inflation Model ◽  
Space Reduction ◽  
Physical Fields ◽  
Definition Of ◽  
The Universe

In this article the reduction of a $n$-dimensional space to a $k$-dimensional space is considered as a reduction of $N^n$ states to $N^k$ states, where $N$ stands for the number of single-particle states per unit of spatial length. It turns out, this space reduction could be understood as another definition of inflation. It is shown that the introduction of the non-associativity of the algebra of physical fields in a homogeneous space leads to a nonlinear equation, the solutions of which can be considered as two-stage inflation. Using the example of reduction $T\times R^7$ to $T\times R^3$, it is shown that there is a continuous cross-linking of the Friedmann and inflationary stages of algebraic inflation at times $10^{-15}$ with the number of baryons $10^{80}$ in the Universe. In this paper, we construct a new gravitational constant based on a nonassociative octonion algebra.

On octonion quark confinement condition

2021 ◽  
Vol 36 (37) ◽  
Author(s):  
B. C. Chanyal
Keyword(s):  
Quark Confinement ◽  
Octonion Algebra ◽  
Color Confinement ◽  
Color Field ◽  
Confinement Condition

The octonion algebra is analyzed using a formalism that demonstrates its use in color quark confinement. In this study, we attempt to write a connection between octonion algebra and SU(3)[Formula: see text] group generators, as well as color quarks representation. We demonstrated the glueballs construction in the extended octonionic color field and also proposed the prerequisite for octonion color confinement of hadrons.

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