Generalised proper time as a higher-order polynomial yields octonionic exceptional-group structures whose symmetry breaking resembles Standard Model matter, with full unification deferred to a predicted E8 form.
Finite Lorentz Transformations, Automorphisms, and Division Algebras
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abstract
We give an explicit algebraic description of finite Lorentz transformations of vectors in 10-dimensional Minkowski space by means of a parameterization in terms of the octonions. The possible utility of these results for superstring theory is mentioned. Along the way we describe automorphisms of the two highest dimensional normed division algebras, namely the quaternions and the octonions, in terms of conjugation maps. We use similar techniques to define $SO(3)$ and $SO(7)$ via conjugation, $SO(4)$ via symmetric multiplication, and $SO(8)$ via both symmetric multiplication and one-sided multiplication. The non-commutativity and non-associativity of these division algebras plays a crucial role in our constructions.
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Octonions in Particle Physics through Structures of Generalised Proper Time
Generalised proper time as a higher-order polynomial yields octonionic exceptional-group structures whose symmetry breaking resembles Standard Model matter, with full unification deferred to a predicted E8 form.