Standard Model Particles from Split Octonions

2010

Starting with the usual definitions of octonions and split octonions in terms of Zorn vector matrix realization, we have made an attempt to write the consistent form of generalized Maxwell's equations in presence of electric and magnetic charges (dyons). We have thus written the generalized potential, generalized field, and generalized current of dyons in terms of split octonions and accordingly the split octonion forms of generalized Dirac Maxwell's equations are obtained in compact and consistent manner. This theory reproduces the dynamic of electric (magnetic) in the absence of magnetic (electric) charges.

2003

There is a growing interest in the logical possibility that exceptional mathematical structures (exceptional Lie and superLie algebras, the exceptional Jordan algebra, etc.) could be linked to an ultimate "exceptional" formulation for a Theory Of Everything (TOE). The maximal division algebra of the octonions can be held as the mathematical responsible for the existence of the exceptional structures mentioned above. In this context it is quite motivating to systematically investigate the properties of octonionic spinors and the octonionic realizations of supersymmetry. In particular the $M$-algebra can be consistently defined for two structures only, a real structure, leading to the standard $M$-algebra, and an octonionic structure. The octonionic version of the $M$-algebra admits striking properties induced by octonionic $p$-forms identities.

Journal of Physics: Conference Series, 2014

A special treatment based on the highest division algebra, that of octonions and their split algebraic formulation is developed for the description of diquark states made up of two quark pairs. We describe symmetry properties of mesons and baryons through such formulation and derive mass formulae relating π, ρ, N and ∆ trajectories showing an incredible agreement with experiments. We also comment on formation of diquark-antidiquark as well as pentaquark states and point the way toward applications into multiquark formulations expected to be seen at upcoming CERN experiments. A discussion on relationship of our work to flux bag models, string pictures and to string-like configurations in hadrons based on spectrum generating algebras will be given.

Physics of Particles and Nuclei Letters, 2017

⎯Octonions and their split versions are shown to be applicable to the solutions of a large number of problems in hadronic physics, from the foundations of exceptional groups that are used in grand unified theories, to heterotic strings, to the non-Desarguesian geometric property of space-time symmetries, twistors, harmonic superspace, conformal field theories, etc. Upon a brief review of these investigations we proceed to show how they are used in the unification of ancient and modern geometries, which in turn open new avenues for, and goes far beyond in providing, geometric foundations for the existence of internal symmetries such as color and flavor.

In this paper, we have made an attempt to discuss the role of division algebras (octonions) in gravity and dark matter where, we have described the octonion space as the combination of two quaternionic spaces namely gravitational G-space and electromagnetic EM-space. We have discussed the standard model and GUTs in terms of split octonion formulation. At last, we have formulated the theory of dark matter in terms of octonion variables i.e. hot dark matter and cold dark matter.

Advances in Applied Clifford Algebras, 2000

As is well-known, the real quaternion division algebra ℍ is algebraically isomorphic to a 4-by-4 real matrix algebra. But the real division octonion algebra can not be algebraically isomorphic to any matrix algebras over the real number field ℝ, because is a non-associative algebra over ℝ. However since is an extension of ℍ by the Cayley-Dickson process and is also finite-dimensional, some pseudo real matrix representations of octonions can still be introduced through real matrix representations of quaternions. In this paper we give a complete investigation to real matrix representations of octonions, and consider their various applications to octonions as well as matrices of octonions.

2021

The known equivalence of 8-dimensional chiral spinors and vectors is discussed for (4+4)space within the context of the algebra of the split octonions. It is shown that the complete algebra of hyper-complex octonionic basis units can be recovered from the Moufang and Malcev relations for the three vector-like elements of the split octonions. Trilinear form, which is invariant under SO(4,4) transformations for vectors and corresponding Spin(4,4) transformations for spinors, is explicitly written using both purely matrix and purely octonionic representations.

viXra, 2016

The vast majority of the dark matter in the universe is believed to be nonbaryonic, which means that it contains no atoms and does not interact with ordinary matter via electromagnetic forces. The nonbaryonic dark matter includes neutrinos, and possibly hypothetical entities such as axions, or supersymmetric particles. Unlike baryonic dark matter, nonbaryonic dark matter does not contribute to the formation of the elements in the early universe (big bang nucleosynthesis) and so its presence is revealed only via its gravitational attraction. The nonbaryonic dark matter is evident through its gravitational effect only. There are two type of nonbaryonic dark matter respectively defined as hot dark matter and cold dark matter. For the existence of nonbaryonic dark matter in this universe we can used the higher dimensional division algebra. There exists four normed division algebras: the real numbers, complex numbers, quaternions, and octonions. Since, the octonions are the last division...

Advances in Applied Clifford Algebras, 2005

We extend vector formalism by including it in the algebra of split octonions, which we treat as the universal algebra to describe physical signals. The new geometrical interpretation of the products of octonionic basis units is presented. Eight real parameters of octonions are interpreted as the space-time coordinates, momentum and energy. In our approach the two fundamental constants, c and , have the geometrical meaning and appear from the condition of positive definiteness of the octonion norm. We connect the property of non-associativity with the time irreversibility and fundamental probabilities in physics.

International Journal of Engineering Research and Technology, 2013

In this paper, we have made an attempt to discuss the role of octonions in superstring theory (i.e. a theory of everything to describe the unification of all four types of forces namely gravitational, electromagnetic, weak and strong) where, we have described the octonion representation of the super-string (SS) formulation as the combination of four complex (C) spaces namely associated with the gravitational (G-space), electromagnetic (EM-space), weak (W-space) and strong (S-space) interactions. We have discussed the octonionic differential operator, octonionic valued potential wave equation, octonionic field equation and other various quantum equations of superstring formulation in simpler, compact and consistent manner. Consequently, the generalized Dirac-Maxwell's equations are studied with the preview of superstring theory by means of octonions.