Octonionic gauge theory from spontaneously broken SO(8)

Generalized non-associative structures on the 7-sphere

“…in [6] are not concerned and are very far beyond the scope of the present paper, since the main aim here was to introduce and investigate a more general class of non-associative products on S 7 . Also, the framework here introduced is a promising tool for considering its interplay with some applications, as introduced in [28,35,36,37,38,39,40,41,42].…”

Generalised Split Octonions and Their Transformation in SO(7) Symmetry

“…These are an algebraic structure defined on the 8-dimensional real vector space and they are non-associative and non-commutative extension of the algebra of quaternions. Günaydin and Gürsey discussed the Lie algebra of G 2 group and its embedding in SO (7) and SO(8) groups [4]. A dynamical scheme of quark and lepton family unification based on non associative algebra has also been discussed [5].…”

“…Generators of SO (8) are constructed by using Octonion structure tensors [6] and the representations of these generators are also given as the products of Octonions. Lassig and Joshi [7] introduced the bi-modular representation of octonions and formulated the SO (8) gauge theory equivalent to the octonionic construction, also the peculiarities of the eight dimensional space has been described by Gamba [8]. Octonionic description has been used to describe the various applications in quantum chromo dynamics [9], in the study of representations of Clifford algebras [10], non associative quantum mechanics [11], in the study of symmetry breaking [12], in the study of flavor symmetry [13], in proposals of unified field theory models [14], unitary symmetry [15], octonionic gravity [16].…”

“…Since SO(7) ⊂ SO (8), therefore the presented representation also gives the generators of SO(7) symmetry group. These generators of SO (7) group have been used to describe the rotational transformation in 7 dimensional space.…”

Glimpses of the Octonions and Quaternions History and Today’s Applications in Quantum Physics

“…This note being on infinite subject has also its references grouped into two parts. First part starts from [1] and runs up to [40] - [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40] -which might mark this note temporal end as these are sufficiently abundant and indicative. But life goes on and then follows an endless sequence, starting from [41].…”